CORNELL UNIVERSITY LIBRARY PHYSICAL SCIENCES Cornell University Library QB 43.Y69 1895 A text-book of general astronomy for col 3 1924 004 196 923 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004196923 FRONTISPIECE. The Gbeat Telescope of the Lick Observatory, Mt. Hamilton, Cal. Object-Glass made by A. Clark & Sons: Aperture, 36 in. ; Focal Length, 56 ft. 2 in. Mounting by Warner & Swasey. A TEXT-BOOK GENERAL ASTRONOMY COLLEGES AND SCIENTIFIC SCHOOLS CHARLES A. YOUNG, Ph.D., LL.D. Pbofessok of Astbonomy in the College of New Jersey (Peiscetos) Boston, U.S.A., and London GESTN & COMPANY, PUBLISHERS 18?5 Entered at Stationers' Hall. Copyright, 1888, by CHARLES A. YOUNG. All Rights Reserved. Typography by J. S. Gushing & Co., Boston U.S.A. Presswork by Ginn & Co., Boston, U.S.A. PREFACE. The present work is designed as a text-book of Astronomy suited to the general course in our colleges and schools of science, and is meant to supply that amount of information upon the subject which may fairly be expected of every "liberally educated" person. While it assumes the previous discipline and mental maturity usually corresponding to the latter years of the college course, it does not demand the peculiar mathematical training and aptitude necessary as the basis of a special course in the science — only the most ele- mentary knowledge of Algebra, Geometry, and Trigonometry is required for its reading. Its aim is to give a clear, accu- rate, and justly proportioned presentation of astronomical facts, principles, and methods in such a form that they can be easily apprehended by the average college student with a reasonable amount of effort. The limitations of time are such in our college course that probably it will not be possible in most cases for a class to take thoroughly everything in the book. The fine print is to be regarded rather as collateral reading, important to any- thing like a complete view of the subject, but not essential to the course. Some of the chapters can even be omitted in cases where it is found necessary to abridge the course as much as possible; e.g., the chapters on Instruments and on Perturbations. While the work is no mere compilation, it makes no claims to special originality: information and help have been drawn VI PBBFACE. from all available sources. The author is under great obliga- tions to the astronomical histories of Grant and Wolf, and especially to Miss Clerke's admirable "History of Astronomy in the Nineteenth Century." Many data also have been drawn from Houzeau's valuable "Vade Mecum de l'Astronome." It has been intended to bring the book well down to date, and to indicate to the student the sources of information on subjects which are necessarily here treated inadequately on account of the limitations of time and space. Special acknowledgments are due to Professor Langley and to his publishers, Messrs. Ticknor & Co., for the use of a number of illustrations from his beautiful book, " The New Astronomy " : and also to D. Appleton & Co. for the use of several cuts from the author's little book on the Sun. Pro- fessor Trowbridge of Cambridge kindly provided the original negative from which was made the cut illustrating the com- parison of the spectrum of iron with that of the sun. Warner & Swasey of Cleveland and Fauth & Co. of Washington have also furnished the engravings of a number of astronomical instruments. Professors Todd, Emerson, Upton, and McNeill have given most valuable assistance and suggestions in the revision of the proof ; as indeed, in hardly a less degree, have several others. The author will consider it a great favor if those who may use the book will kindly communicate to him, either directly or through the publishers, any errata, in order that they may be promptly corrected. Princeton, N. J., August', 1888. Note. — In this issue of the book all the important errata known to exist in previous impressions have been corrected, and a number of notes have been pre- fixed, embodying recent important discoveries. Feb., 1893. TABLE OF CONTENTS. PAGE 8 INTRODUCTION 1-4 CHAPTER I. — The Doctrine of the Sphere; Definitions and Gen- eral Considerations 5-20 CHAPTER II. — Astronomical Instruments: the Telescope; Time- Keepers and Chronograph; the Transit Instrument and Accessories; the Meridian Circle and Reading Microscope ; the Altitude and Azi- muth Instrument; the Equatorial Instrument and Micrometer; the Sextant , 21-57 CHAPTER III. — Corrections to Astronomical Observations : the Dip of the Horizon ; Parallax ; Semi-Diameter ; Refraction ; and Twilight . 58-69 CHAPTER IV. — Problems of Practical Astronomy : the Determi- nation of Latitude, of Time, of Longitude, of a Ship's Place at Sea, of Azimuth, and of the Apparent Right Ascension and Declination - of a Heavenly Body ; the Time of Sunrise or Sunset .... 70-90 CHAPTER V. — The Earth : the Approximate Determination of its Dimensions and Eorm ; Proofs of its Rotation ; Accurate Determina- tion of its Dimensions by Geodetic Surveys and Pendulum Observa- tions; Determination of its Mass and Density .... 91-116 CHAPTER VI. — The Earth's Orbital Motion: the Motion of the Sun among the Stars ; the Equation of Time ; Precession ; Nutation ; Aberration ; the Calendar 117-143 CHAPTER VII. — The Moon: her Orbital Motion; Distance and Di- mensions; Mass, Density, and Superficial Gravity; Rotation and Librations ; Phases ; Light and Heat ; Physical Condition ; Influence exerted on the Earth ; Surface Structure ; and Possible Changes 144-171 CHAPTER VIII. — The Sun : Distance and Dimensions ; Mass and Den- sity; Rotation; Solar Eye-Pieces, and Study of the Sun's Surface; General Views as to Constitution; Sun Spots: their Appearance, Nature, Distribution, and Periodicity ; the Spectroscope and the Solar Spectrum ; Chemical Composition of the Sun ; the Chromosphere and Prominences ; the Corona 172-211 CHAPTER IX. — The Sun's Light and Heat : Comparison of Sunlight with Artificial Lights ; the Measurement of the Sun's Heat, and Deter- mination of the Solar Constant; the Pyrheliometer, Actinometer, and Bolometer ; the Sun's Temperature ; Maintenance, of the Sun's Radia- tion ; and Conclusions as to its Age and Future Endurance . . 212-227 yiii TABLE OF CONTENTS. PAGES CHAPTER X. — Eclipses; Form and Dimensions of Shadows; Lunar Eclipses; Solar Eclipses, Total, Annular, and Partial; Ecliptic Limits, and Number of Eclipses in a Year; the Saros; Occultations . 228-246 CHAPTER XI. — Central Forces; Equable Description of Areas; Areal, Linear, and Angular Velocities; Kepler's Laws and Inferences from them; Gravitation demonstrated by the Moon's Motion; Conic Sections as Orbits ; the Problem of Two Bodies ; the " Velocity from Infinity " and its Relation to the Species of Orbit described by a Body moving under Gravitation ; Intensity of Gravitation . . 247-267 CHAPTER XII. — The Problem op Three Bodies ; Disturbing Forces; Lunar Perturbations and the Tides 268-292 CHAPTER XIII. — The Planets: their Motions, Apparent and Real ; the Ptolemaic, Tychonic, and Copernican Systems ; the Orbits and their Elements ; Planetary Perturbations .... 293-315 CHAPTER XIV. — The Planets themselves; Methods of determining their Diameters, Masses, Densities, Times of Rotation, etc. ; the " Ter- restrial Planets," — Mercury, Venus, and Mars ; the Asteroids ; Intra- Mercurial Planets and the Zodiacal Light 316-347 CHAPTER XV. — The Major Planets, — Jupiter, Saturn, Uranus, and Neptune 348-374 CHAPTER XVI. — The Determination op the Sun's Horizontal Par- allax and Distance ; Oppositions of Mars and Transits of Venus ; Gravitational Methods ; Determination by Means of the Velocity of Light 375-393 CHAPTER XVII. — Comets : their Number, Motions, and Orbits ; their Constituent Parts and Appearance ; their Spectra, Physical Consti- tution, and Probable Origin 394-429 CHAPTER XVIII. — Meteors : Aerolites, their Fall and Physical Char acteristics; Shooting Stars and Meteoric Showers; Connection be- tween Meteors and Comets 430-446 CHAPTER XIX. — The Stars: their Nature and Number; the Constella- tions ; Star-Catalogues ; Designation of Stars ; their Proper Motions, and the Motion of the Sun in Space ; Stellar Parallax . . . 447-466 CHAPTER XX. — The Light of the Stars ; Star Magnitudes and Pho- tometry ; Variable Stars ; Stellar Spectra ; Scintillation of the Stars 467-492 CHAPTER XXI. — Aggregations of Stars : Double and Multiple Stars; Clusters ; Nebula? ; the Milky Way, and Distribution of Stars in Space; Constitution of the Stellar Universe; Cosmogony and the Nebular Hypothesis .... 493-525 APPENDIX. — Tables of Astronomical Data .... 527-535 INDEX 537-551 INTRODUCTION. 1. Astronomy (aorpov vo/uos) is the science which treats of the heavenly bodies. As such bodies we reckon the sun and moon, the planets (of which the earth is one) and their satellites, comets and meteors, and finally the stars and nebulas. "We have to consider in Astronomy : — (a) The motions of these bodies, both real and apparent, and the laws which govern these motions. (6) Their forms, dimensions, and masses. (c) Their nature and constitution. (d) The effects they produce upon each other by their attractions, radiations, or by any other ascertainable influence. It was an early, and has been a most persistent, belief that the heavenly bodies have a powerful influence upon human affairs, so that from a knowledge of their positions and "aspects" at critical moments (as for instance at the time of a person's birth) one could draw up a "horoscope" which would indicate the probable future. The pseudo-science which was founded on this belief was named Astrology, — the elder sister of Alchemy, — and for centuries As- tronomy was its handmaid; i.e., astronomical observations and cal- culations were made mainly in order to supply astrological data. At present the end and object of astronomical study is chiefly knowledge pure and simple ; so far as now appears, its development has less direct bearing upon the material interests of mankind than that of any other of the natural sciences. It is not likely that great inventions and new arts will grow out of its laws and principles, such as are continually arising from physical, chemical, and biological discoveries, though of course it would be rash to say that such out- growths are impossible. But the student of Astronomy must expect his chief profit to be intellectual, in the widening of the range of thought and conception, in the pleasure attending the discovery of simple law working out the most complicated results, in the delight 2 INTRODUCTION. over the beauty and order revealed by the telescope in systems other- wise invisible, in the recognition of the essential unity of the material universe, and of the kinship between his own mind and the infinite Reason that formed all things and is immanent in them. At the same time it should be said at once that, even from the lowest point of view, Astronomy is far from a useless science. The art of navigation depends for its very possibility upon astronomical prediction. Take away from mankind their almanacs, sextants, and chronometers, and commerce by sea would practically stop. The science also has important applications in the survey of extended regions of country, and the establishment of boundaries, to say nothing of the accurate determination of time and the arrangement of the calendar. It need hardly be said that Astronomy is not separated from kin- dred sciences by sharp boundaries. It would be impossible, for in- stance, to draw a line between Astronomy on one side and Geology and Physical Geography on the other. Many problems relating to the formation and constitution of the earth belong alike to all three. 2. Astronomy is divided into many branches, some of which, as ordinarily recognized, are the following : — 1. Descriptive Astronomy. — This, as its name implies, is merely an orderly statement of astronomical facts and principles. 2. Practical Astronomy. — This is quite as much an art as a science, and treats of the instruments, the methods of observation, and the processes of calculation by which astronomical facts are ascertained. 3. Theoretical Astronomy, which treats of the calculations of orbits and ephemerides, including the effects of so-called "perturbations." 4. Mechanical Astronomy, which is simply the application of me- chanical principles to explain astronomical facts (chiefly the planetary and lunar motions) . It is sometimes called Gravitational Astronomy, because, with few exceptions, gravitation is the only force sensibly concerned in the motions of the heavenly bodies. Until within thirty years this branch of the science was generally designated as Physical Astronomy, but the term is now objectionable because of late it has been used by many writers to denote a very different and compara- tively new branch of the science ; viz., — INTRODUCTION. 3 5. Astronomical Physics, or Astro-physics. — This treats of the physical characteristics of the heavenly bodies, their brightness and spectroscopic peculiarities, their temperature and radiation, the nature and condition of their atmospheres and surfaces, and all phenomena which indicate or depend on their physical condition. 6. Spherical Astronomy. — This, discarding all consideration of absolute dimensions and distances, treats the heavenly bodies simply as objects moving on the " surface of the celestial sphere " : it has to do only with angles and directions, and, strictly regarded, is in fact merely Spherical Trigonometry applied to Astronomy. 3. The above-named branches are not distinct and separate, but they overlap in all directions. Spherical Astronomy, for instance, finds the demonstration of many of its formulae in Gravitational Astronomy, and their application appears in Theoretical and Prac- tical Astronomy. But valuable works exist bearing all the different titles indicated above, and it is important for the student to know what subjects he may expect to find discussed in each ; for this reason it has seemed worth while to name and define the several branches, although they do not distribute the science between them in any strictly logical and mutually exclusive manner. In the present text-book little regard will be paid to these sub- divisions, since the object of the work is not to present a complete and profound discussion of the subject such as would be demanded by a professional astronomer, but only to give so much knowledge of the facts and such an understanding of the principles of the science as may fairly be considered essential to a liberal education. If this result is gained in the reader's case, it may easily happen that he will wish for more than he can find in these pages, and then he must have recourse to works of a higher order and far more difficult, dealing with the subject more in detail and more thoroughly. To master the present book no further preparation is necessary than a very elementary knowledge of Algebra, Geometry, and Trigo- nometry, and a similar acquaintance with Mechanics and Physics, especially Optics. While nothing short of high mathematical attain- ments will enable one to become eminent in the science, yet a perfect Comprehension of all its fundamental methods and principles, and a very satisfactory acquaintance with its main results, is quite within the reach of every person of ordinary intelligence, without any more extensive training than may be had in our common schools. At the 4 INTRODUCTION. same time the necessary statements and demonstrations are so much facilitated by the use of trigonometrical terms and processes that it would be unwise to dispense with them entirely in a work to be used by pupils who have already become acquainted with them. In discussing the different subjects which present themselves, the writer will adopt whatever plan appears best fitted to conve}- to the student clear and definite ideas, and to impress them upon the mind. Usually it will be best to proceed in the Euclidean order, by first stating the fact or principle in question, and then explaining its demonstration. But in some cases the inverse process is preferable, and the conclusion to be reached will appear gradually unfolding itself as the result of the observations upon which it depends, just as Its discovery came about. The frequent references to " Physics " refer to the " Elementary Text-Book of Physics," by Anthony & Brackett; 3d edition, 1887. Wiley & Sons, N.Y. THE "DOCTRINE OF THE SPHERE." CHAPTER I. THE " DOCTRINE OP THE SPHERE," DEFINITIONS, AND GENERAL CONSIDERATIONS. Astronomy, like all the other sciences, has a terminology of its own, and uses technical terms in the description of its facts and phenomena.. In a popular essay it would of course be proper to avoid such terms as far as possible, even at the expense of circum- locutions and occasional ambiguity ; but in a text-book it is desirable that the reader should be introduced to the most important of them at the very outset, and made sufficiently familiar with them to use them intelligently and accurately. 4. The Celestial Sphere. — To an observer looking up to the heavens at night it seems as if the stars were glittering points attached to the inner surface of a dome ; since we have no direct perception of their distance there is no reason to imagine some nearer than others, and so we involuntarily think of the surface as spherical with our- selves in its centre. Or if we sometimes feel that the stars and other objects in the sky really differ in distance, we still instinctively imagine an immense sphere stirrou tiding and enclosing all. Upon this SDhere we imagine lines and circles traced, resembling more or less the meridians and parallels upon the surface of the earth, and by reference to these circles weVre able to describe intelligently the apparent positions and motions of the heavenly bodies. This celestial sphere may be regarded in either of two different ways, both of which are correct and lead to identical results. (a) We may imagine it, in the first place, as, transparent, and of merely finite (though undetermined) dimensions, but in some way so attached to, and connected with, the observer that his eye always remains dt its centre wherever he goes. Each observer, in this way of viewing it, carries his own sky with him, and is the centre of his own heavens. (&) Or, in the second place, — and this is generally the more con- venient way of regarding the matter, — we may consider the celestial 6 THE "DOCTRINE OF THE SPHERE. sphere as mathematically infinite in its dimensions: then, let the observer go where he will, he cannot sensibly get away from its centre. Its radius being "greater than any assignable quantity," the size of continents, the diameter of the earth, the distance of the sun, the orbits of planets and comets, even the spaces between the stars, are all insignificant, and the whole visible universe shrinks relatively to a mere point at its centre. In what follows we shall use this conception of the celestial sphere. 1 The apparent place of any celestial body will then be the point on the celestial sphere where the line drawn from the eye of the observer in the direction in which he sees the object, and produced indefinitely, pierces the sphere. Thus, in Figure 1, A, B, C are the apparent places of a, b, and c, .^1 — j _ the observer being at 0. The appar- y^ '' j ;*? en * P^ce of a heavenly body evidently ■*^ i ! / \ depends solely upon its direction, and / \^ \l c /' \ is wholly independent of its distance I e \ \| y' \ from the observer. \> minus angle XPQ. for ordinary purposes, is about 4 minutes longer than the sidereal day. The exact difference is 3 m 56 s .556 (sidereal), or just one day in a year ; there being 366^- sidereal days in the year, as against 365J solar days. 27. Observatory Definition of Right Ascension. —It is evident from the above definition of sidereal time, that we may also define the Right Ascension of a star as the sidereal time when the star crosses the meridian. The Star and the Vernal Equinox are both of them 16 DEFINITIONS AND GENERAL CONSIDERATIONS. fixed points in the sky, and do not change their relative position dur- ing the sky's apparent daily revolution ; a given star, therefore, always comes to the meridian of any observer the same number of hours after the vernal equinox has passed ; and this number of hours is the sidereal time at the moment of the star's transit, and measures its right ascension. In the observatory, this definition of right ascen- sion is the most natural and convenient. It is obvious that the right ascension of a star corresponds in the sky exactly with the longitude of a place on the earth ; terrestrial longitude being reckoned from Greenwich, just as right ascension is reckoned from the vernal equinox. N.B. We shall find, hereafter that the stars have latitudes and longitudes of their own; but unfortunately these celestial latitudes and longitudes do not correspond to the terrestrial, and great care is neces- sary to prevent confusion. (See Art. 179.) 28. An armillary sphere, or some equivalent apparatus, is almost essential to enable a beginner to get correct ideas of the points, circles, and co-ordinates defined above, but the figures will perhaps be of assistance. The first of them (Fig. 4) represents the horizon, meridian, and prime vertical, and shows how the position of a star is indicated by its altitude and azimuth. This framework of circles, depending upon the direction of gravity, of course always remains apparently unchanged in position, as if attached directly to the earth, while the sky apparently turns around outside it. The other figure (Fig. 5) represents the system of points and circles which depend upon the earth's rotation, and are independent of the direction of gravity. The vernal equinox and the hour-circles apparently revolve with the stars while the pole remains fixed upon the meridian, and the equator and parallels of declination, revolving truly in their own planes, also appear to be at rest in the sky. But the whole system of lines and points represented in the figure (hori- zon and meridian alone excepted) may be considered as attached to, or marked out upon, the inner surface of the celestial vault and whirling with it. It need hardly be said that the "appearances are deceitful" — that which is really carried around by the earth's rotation is the observer, with his plumb-line and zenith, his horizon and meridian ; while the stars stand still — at least, their motions in a day are in- sensible as seen from .the earth. DEFINITIONS AND GENERAL CONSIDERATIONS. 17 At the poles of the earth, which are, mathematically speaking, " singular " points, the definitions of the Meridian, of North and South, etc., break down. There the pole (celestial) and zenith coincide, and any number of circles may be drawn through the two points, which have now become one. The horizon and equator coalesce, and the only direction on the earth's surface is due south (or north) — east and west have vanished. A single step of the observer will, however, remedy the confusion : zenith and pole will separate, and his meridian will again become determinate. 29. To recapitulate : The direction of gravity at the point where the observer stands determines the Zenith and Nadir, the Horizon, and the Almucantars (parallel to the Horizon), and all the vertical circles. One of the verticals, the Meridian, is singled out from the rest by the circumstance that it passes through the pole of the sky, marking the North and South Points where it cuts the horizon. Altitude and Azimuth (or their complements, Zenith Distance and Amplitude) are the co-ordinates which designate the position of a body by reference to the Zenith and the Meridian. Similarly, the direction of the earth's aotis (which is independent of the observer's place on the earth) determines the Poles, the Equator, the Parallels of Declination, and the Hour-Circles. Two of these Hour-Circles are singled out as reference lines ; one of them, the Meridian, which passes through the Zenith, and is a purely local reference line ; the other, the Equinoctial Colure, which passes through the Vernal Equinox, a point chosen from its relation to the sun's annual motion. Declination and Hour-Angle are the co-ordi- nates which refer the place of a star to the Pole and the Meridian ; while Declination and Bight Ascension refer it to the Pole and Equi- noctial Colure. The latter are the co-ordinates usually employed in star-catalogues and ephemerides to define the positions of stars and planets, and. correspond exactly to Latitude and Longitude on the earth, by means of which geographical positions are designated. 30. Relation of the Apparent Diurnal Motion of the Sky to the Observer's Latitude. — "Evidently the apparent motions of the stars will be considerably influenced by the station of the observer, since the place of the pole in the sky will depend upon it. The Altitude of the pole, or its height in degrees above the horizon, is always equal to the Latitude of the observer. Indeed, the German word for lati- tude (astronomical) is'Polhohe; i.e., simply "Pole-height." 18 DEFINITIONS AND GENERAL CONSIDERATIONS. This will be clear from Fig. 6. The latitude of a place is the angle between its plumb-line and the plane of the equator; the angle ONQ in the figure. [If the earth were truly spherical, N would coincide with C, the centre of the earth. The ordinary definition of latitude given in the geographies disregards the slight difference.] Now the angle H'OP" is equal to ONQ, because their sides are mutually perpendicular ; and it is also the altitude of the pole, be- cause the line HH' is horizontal at 0, and OP" is parallel to PP, the earth's axis, and therefore points to the celestial pole. This fundamental relation, that the altitude of the celestial pole is the Latitude of the observer, cannot be too strongly impressed on the student's mind. The usual symbol for the latitude of a place is <£. Fig. 8. — Relation of Latitude to the Elevation of the Pole. 31. The Right Sphere. — If the observer is situated at the earth's equator, i.e., in latitude zero ( = o), the pole will be in the horizon, and the equator will pass vertically overhead through the zenith. The stars will rise and set vertically, and their diurnal circles will ill be- bisected by the horizon, so that they will be 12 hours above it and 12 below. This aspect of the heavens is called the Bight Sphere. 32. The Parallel Sphere. —If the observer is at the pole of the earth (0=90°), then the celestial pole will be in the zenith, and the equator will coincide with the horizon. If he is at the North Pole, all stars north of the celestial equator will remain permanently DEFINITIONS AND GENERAL CONSIDERATIONS. 19 above the horizon, never rising or falling at all, but sailing around on circles of altitude (or Almucantars) parallel to the horizon. Stars in the Southern Hemisphere, on the other hand, would never rise to view. As the sun and the moon move in such a way that during half the time they are alternately north and south of the equator, they will be half the time above the horizon and half the time below it. The moon would be visible for about a fortnight at a time, and the sun for six months. 33. The Oblique Sphere (Fig. 7). — At any station between the pole and equator the stars will move in circles oblique to the horizon, SENW in the figure. Those whose distance from the elevated pole is less than the latitude of the place will, of course, never sink below the horizon, — the radius of the "Circle of Perpetual Apparition," T' Fig. 7. — The Oblique Sphere and Diurnal Circles. as it is called (the shaded cap around P in the figure), being just equal to the height of the pole, and becoming larger as the latitude increases. On the other hand, stars within the same distance of the depressed pole will lie within the " Circle of Perpetual Occupation" and will never rise above the horizon. A star exactly on the celestial equator will have its diurnal circle EQ WQ' bisected by the horizon, and will be above the horizon just as long as below it. A star north of the equator (if the North Pole is the elevated one) will have more than half of its diurnal circle above the horizon, and will be visible more than half the time ; as, for instance, a star at A : and of course the reverse will be true of stars 20 DEFINITIONS AND .GENERAL CONSIDERATIONS. on the other side of the equator. 1 Whenever the sun is north of the equator, the day will therefore be longer than the night for all stations in northern latitude : how much longer will depend both on the latitude of the place and the sun's distance from the celestial equator. 1 A Celestial Globe will be of great assistance in studying these diurnal circles. The north pole of the globe must be elevated to an angle equal to the latitude of the observer, which can be done by means of the degrees marked on the brass meridian. It will then at once be easily seen what stars never set, which ones never rise, and during what part of the 24 hours any heavenly body at a known distance from the equator is above or below the horizon. ASTRONOMICAL INSTRUMENTS. 21 CHAPTER II. ASTRONOMICAL INSTRUMENTS. 84. Astronomical observation are of various kinds : sometimes we desire to ascertain the apparent distance between two bodies at a given time ; sometimes the position which a body occupies at a given time, or the moment it arrives at a given circle of the sky, usually the meridian. Sometimes we wish merely to examine its surface, to measure its light, or to investigate its spectrum ; and for all these purposes special instruments have been devised. We propose in this chapter to describe very briefly a few of the most important 35. Telescopes in General. — Telescopes are of two kinds, refract- ing and reflecting. The former were first invented, and are much more used, but the largest instruments ever made are reflectors. In both the fundamental principle is the same. The large lens, or mir- ror, of the instrument forms at its focus a real image of the object looked at, and this image is then examined and magnified by the eye- piece, which in principle is only a magnifying-glass. In the form of telescope, however, introduced by Galileo, 1 and still used as the " opera-glass," the rays from the object-glass are intercepted by a con- cave lens which performs the office of an eye-piece before they meet at the focus to form the "real image." But on account of the smallness of the field of view, and other objections, this form of telescope is never used when any considerable power is needed. 1 In strictness, Galileo did not invent the telescope. Its Jirst invention seems to have been in 1608, by Lipperhey, a spectacle-maker of Middleburg, in Holland; though the honor has also been claimed for two or three other Dutch opticians. Galileo, in his "Nuncius Sydereus," published in March, 1610, himself says that he had heard of the Dutch instruments in 1609, and by so hearing was led to construct his own, which, however, far excelled in power any that had been made previously ; and he was the first to apply the telescope to Astronomy. See Grant's "History of Astronomy," pp. 614 and seqq. 22 ASTRONOMICAL INSTRUMENTS. 36. Simple Refracting Telescope.— This consists essentially as shown in the figure (Fig. 8), of a tube containing two lenses : a single convex lens, A, called the object-glass ; and another, of smaller size and short focus, B, called the eye-piece. Recalling the principles of lenses the student will see that if the instrument be directed at a dis- tant object, the moon, for instance, all the rays, ^0^, which fall upon the object-glass from a point at the top of the moon, will be collected at a in the focal plane, at the bottom of the image. Simi- larly rays from the bottom of the moon will go to b at the top of the image ; moreover, since the rays that pass through the optical centre of the lens, o, are undeviated, 1 the angle a ob will equal boa ; or, in other words, if the focal length of the lens be five feet, for instance, then the image of the moon, seen from a distance of five feet, will appear just as large as the moon itself does in the sky, — it will subtend the same angle. If we look at it from a smaller distance, «! A ao Mo j ' :^bjc=i _^L Ao _ ■ = ps"^^ ^ /„•--"•"'" Fig. 8. — Path of the Rays in the Astronomical Telescope. say from a distance of one foot, the image will look larger than the moon ; and in fact, without using an eye-piece at all, a person with normal eyes can obtain considerable magnifying power from the object-glass of a large telescope. With a lens of ten feet focal length, such as is ordinarily used in an 8-inch telescope, one can easily see the mountains on the moon and the satellites of Jupiter, by taking out the eye-piece, and putting the eye in the line of vision some eight or ten inches back of the eye-piece hole. The image is a real one; i.e., the rays that come from different points in the object actually meet at corresponding points in the im- age, so that if a photographic plate were inserted at ab, and prop- erly exposed, a picture would be obtained. If we look at the image with the naked eye, we cannot come nearer 1 In this explanation, we use the approximate theory of lenses (in which their thickness is neglected), as given in the elementary text-books. The more exact theory of Gauss and later writers would require some slight modifications in our statements, but none of any material importance. For a thorough discussion, see Jamin, " Traits de Physique," or Encyc. Britannica, — Optics. ASTRONOMICAL INSTRUMENTS. 23 to the image (unless near-sighted) than eight or ten inches, and so cannot get any great magnifying power ; but if we use a magnify- ing-glass, we can approach much closer. 37. Magnifying Power. — If the eye-piece B is set at a distance from the image equal to its principal focal distance, then any pencil of rays from any point of the image will, after passing the lens, be con- verted into a parallel beam, and will appear to the eye to come from a point at an infinite distance, as if from an object in the sky. The rays which came from the top of the moon, for instance, and are col- lected at a in the image, will reach the eye as a beam parallel to the line ac, which connects a with the optical .centre of the eye-piece. Simi- larly with the rays which meet at b. The observer, therefore, will see the top of the moon's disc in the direction ck, and the bottom in the direction cl. It will appear to him inverted, and greatly magni- fied ; its apparent diameter, as seen by the naked eye and measured by the angle aob (or its equal b oa ) , having been increased to acb. Since both these angles are subtended by the same line ab, and are small (the figure, of course, is much out of proportion), they must be inversely proportional to the distance ob and cb ; i.e., boa : bca = cb:ob; or, putting this into words : The ratio between the natural apparent diameter of the object, and its diameter as seen through the telescope, is equal to the ratio between the focal lengths of the eye- lens and object-glass. This ratio is called the magnifying power of the telescope, and is therefore given by the simple formula F M= — , where F is the focal length of the object-glass and / that of eye-piece, 1 while M is the magnifying power, If, for example, the object-glass have a focal length of thirty feet, and the eye-piece of one inch, the magnifying power will be 360 ; the power may be changed at pleasure by substituting different eye- pieces, of which every large telescope has an extensive stock. 38. Brightness of Image. — Since all the rays from a star which fall upon the large object-glass are transmitted to the observer's eye (neglecting the losses by absorption and reflection) , he obviously re- 1 A magnifying power of 1 is no magnifying power at all. Object and image subtend equal angles. A magnifying power denoted by a fraction, say \, would be a minifying power, making the object look smaller, as when we look at an ob- ject through the wrong end of a spy-glass. 24 ASTRONOMICAL INSTRUMENTS. ceives a quantity of light much greater than he would naturally get : as many times greater as the area of the object-lens is greater than that of the pupil of the eye. If we estimate this latter as having a diameter of one-fifth of an inch, then a 1-inch telescope would in- crease the light twenty-five times, a 10-inch instrument 2500 times, and the great Lick telescope, of thirty-six inches' aperture, 32,400 times, the amount being proportional to the square of the diameter of the lens. It must not be supposed, however, that the apparent brightness of an object like the moon, or a planet which shows a disc, is increased in any such ratio, since the eye-piece spreads out the light to cover a vastly more extensive angular area, according to its magnifying power ; in fact, it can be shown that no optical arrangement can show an extended surface brighter than it appears to the naked eye. But the total quantity of light utilized is greatly increased by the telescope, and in consequence, multitudes of stars, far too faint to be visible to the unassisted eye, are revealed ; and, what is practically very important, the brighter stars are easily seen by day with the telescope. 39. Distinctness of Image. — This depends upon the exactness with which the lens gathers to a single point in the focal image all the rays which emanate from the corresponding point in the object. A single lens, with spherical surfaces, cannot do this very perfectly, the " aberrations " being of two kinds, the spherical aberration and the chromatic. The former could be corrected, if it were worth while, by slightly modifying the form of the lens-surfaces ; but the latter, which is far more troublesome, cannot be cured in any such way. The violet rays are more refrangible than the red, and come to a focus nearer the lens; so that the image of a star formed by such- a lens can never be a luminous point, but is a round patch of light of different color at centre and °dge. 40. Long Telescopes. — By making the diameter of the lens very small as compared with its focal length, the aberration becomes less con- spicuous ; and refractors were used, about 1660, having a length of more than 100 feet and a diameter of five or six inches. The object-glass' was mounted at the top of a high pole and the eye-piece was on a separate stand below. With such an " aerial telescope," of six inches aperture and 120 feet focus, Huyghens discovered the rings of Saturn. His object-glass still exists, and i» preserved in the collection of the Koyal Society in London. ASTRONOMICAL INSTRUMENTS. 25 41. The Achromatic Telescope.— The chromatic aberration of a lens, as has been said, cannot be cured by any modification of the lens itself; but it was discovered in England about 1760 that it can be nearly. corrected by making the object-glass of two (or more) lenses, of different kinds of glass, one of the lenses being convex and the other concave. The convex lens is usually made of crown glass, the concave of flint glass. At' the same time, by properly choosing the curves, the spherical aberration can also be destroyed, so that such a compound object-glass comes reasonably near to fulfilling the coiv dition, that it should gather to a mathematical point in the image all the rays that reach the object-glass from a single point in the object. These object-glasses admit of a considerable variety of forms. Formerly they were generally made, as in Fig. 9, No. 3, having the two lenses close together, and the adjacent surfaces of the same, or nearly the same, curva- ture. In small object-glasses the lenses are often cemented together with Canada balsam or some other transparent medium. At present some of the best makers separate the two lenses by a considerable distance, so as to admit a free circulation of air between them ; in the Pulkowa and Prince- ton object-glasses, constructed by Clark, the lenses are seven inches apart, and in the Lick telescope six and a half inches ; as in No. 1. In a form devised by Gauss (No. 2), Claris which has some advantages, but is *■"»•»• -Different Forms of the Achromatic •,.„, ,, . ° ' Object-glass, difficult or construction, the curves are very deep, and both the lenses are of watch-glass form — concave on one side and convex on the other. In all these forms the crown glass is outside ; Steinheil, Hastings, and others have constructed lenses with the flint-glass lens outside. Object-glasses are sometimes made with three lenses instead of two ; a slightly better correction of aberrations can be obtained in this way, but the gain is too small to pay for the extra expense and loss of light. 42. Secondary Spectrum. — It is not, however, possible with the kinds of glass at present available to secure a perfect correction of the color. Our best achromatic lenses bring the yellowish green rays to a focus nearer the lens than they do the red and violet. In conse- quence, the image of a bright star is surrounded by a purple halo, which is not very noticeable in a good telescope of small size, but is very conspicuous and troublesome in a large instrument. This imperfection of achromatism makes it unsatisfactory to use an ordi- nary lens {visually corrected) for astronomical photography. To fit it to make good photographs, it must either be specially corrected for the rays Gaim Littrow 26 ASTBONOMICAL INSTRUMENTS. that are most effective in photography, the blue and violet (in which case it will be almost worthless visually), or else a subsidiary lens, known as a "pho- tographic corrector," may be provided, which can be put on in front of the object-glass when needed. A new form of object-glass, devised independ- ently by Pickering in this country and Stokes in England, avoids the necessity of a third lens by making the crown-glass lens of such a form that when put close to the flint lens, with the flatter side out, it makes a perfect object- glass for visual purposes ; but by simply reversing the crown lens, with the more convex side outward, and separating the lenses an inch or two, it be- comes a photographic object-glass. A 13-inch object-glass of this construc- tion at Cambridge performs admirably. Much is hoped from the new kind of glass now being made at Jena. In combination with crown glass it produces lenses almost free from chromatic aberration, and if it can be produced in homogeneous pieces of sufficient size, it will revolutionize the art of telescope making. 43. Diffraction and Spurious Disc. — Even if a lens were perfect as regards the correction of aberrations, the "wave" nature of light prevents the image of a luminous point from being also a point ; the image must necessarily consist of a central disc, brightest in the cen- tre and fading to darkness at the edge, and this is surrounded by a series of bright rings, of which, however, only the smallest one is generally easily seen. The size of this disc-and-ring system can be calculated from the known wave-lengths of light and the dimensions of the lens, and the results agree very precisely with observation. The diameter of the " spurious disc" varies inversely with the aper- ture of the telescope. According to Dawes, it is about 4". 5 for a 1-inch telescope ; and consequently 1" for a 4^-inch instrument, 0".5 for a 9-inch, and so on. This circumstance has much to do with the superiority of large instru- ments in showing minute details. No increase of magnifying power on a small telescope can exhibit things as sharply as the same power on the larger one ; provided, of course, that the larger object-glass is equally perfect in workmanship, and the air in good optical condition. If the telescope is a good one, and if the air is perfectly steady, — which unfortunately is seldom the case, — the apparent disc of a star should be perfectly round and well defined, without wings or tails of any kind, having around it from one to three bright rings, separated by distances somewhat greater than the diameter of the disc. If, however, the magnifying power is more than about 50 to the inch of aperture, the edge of the disc will begin to appear hazy. There is seldom any advantage in the use of a magnifying power exceeding 75 to the inch, and for most purposes powers ranging from 20 to 40 to the inch are most satisfactory. ASTRONOMICAL INSTRUMENTS. 27 44. Eye-Pieces. — For many purposes, as for instance the examina- tion of close double stars, there is no better eye-piece than the simple convex lens ; but it performs well only when the object is exactly in the centre of the field. Usually it is best to employ for the eye-piece a combination of two or more lenses which will give a more exten- sive field of view. Eye-pieces belong to two classes, the positive and the negative. The former, which are much more generally useful, act as simple magnify- ing-glasses, and can be used as hand magnifiers if desired. The focal image formed by the object-glass lies outside of the eye-piece. In the negative eye-pieces, on the other hand, the rays from the object-glass are intercepted before they come to the focus, and the image is formed between the lenses of the eye-piece. Such an eye- piece cannot be used as a hand magnifier. Steinhcil 'Monocentric* {Positive) Samsden (Positive) Kuyghenian (Negative) Fig. 10. — Various Forms of Telescope Eye-piece. 45. The simplest and most common forms of these eye-pieces are the Ramsden (positive) and Huyghenian (negative). Each is composed of two plano-convex lenses, but the arrangement and curves differ, as shown in Fig. 10. The former gives a very flat field of view, but is. not achro- matic ; the latter is more nearly achromatic, and possibly defines a little better just at the centre of the field ; but the fact that it is a negative eye-piece greatly restricts its usefulness. In the Rams- den eye-piece the focal lengths of the two component lenses, both of which have their flat sides out, are about equal to each other, and their distance is about one-third of the sum of the focal lengths. In the Huyghenian the curved sides of the lenses are both turned towards the object-glass; the focal distance of the field lens should be exactly three times that of the lens next the eye, and the distance between the lenses one-half the sum of the focal lengths. The peculiarity of the Steinheil " monocentric " eye-piece which is a triple achromatic positive lens, consisting of a central convex lens of crown glass, with a concave meniscus of flint glass cemented to each side, is that the curves are all struck from the same centre, the thickness of the lenses being so computed as to produce the needed corrections. It is free from all internal reflections, which in other eye-pieces often produce " ghosts," as they are called. There are numerous other forms of eye-piece, each with its own advan- tages and disadvantages. The erecting eye-piece, used in spy-glasses, is 28 ASTRONOMICAL INSTRUMENTS. essentially a compound microscope^ and gives erect vision by again invert- ing the already inverted image formed by the object-glass. It is obvious that in a telescope of any size the object-glass is the most important and expensive part of the instrument. Its cost varies from a few hundred dollars to many thousands, while the eye-pieces generally cost only from $5 to $20 apiece. 46. Reticle. — When a telescope is used for pointing, as in most astronomical instruments, it must be provided with a reticle of some sort. This is usually a metallic frame with spider lines stretched across it, placed, not near the object-glass itself (as is often sup- posed), but at the focus of the object-glass, where the image is formed, as at a b in Fig. 8. It is usually so arranged that it can be moved in or out a little to get it exactly into the focal plane, and then, when the eye-piece (positive) is ad- justed for the observer's eye to give distinct vision of the object, the " wires," as they are called, will also be equally distinct. As spider-threads are very fragile, and likely to get broken or displaced, it is often better to substitute a thin plate of glass with lines ruled upon it and blackened. Of course, provision must be made for illuminating either the field of view or the threads themselves, in order to make them visible in darkness. 47. The Reflecting Telescope. — When the chromatic aberration of lenses came to be understood through the optical discovery of the dispersion of light by Newton, the reflecting telescope was invented, and held its place as the instrument for star-gazing until well into the present century, when large achromatics began to be made. There are several varieties of reflecting telescope, all agree- ing in the substitution of a large concave mirror in place of the object- glass of the refractor, but differing in the way in which they get at the image formed by this mirror at its focus in order to examine it with the eye-piece. 48. In the Herschelian form, which is the simplest, but only suited to very large instruments, the mirror is tipped a little, so as to throw the image to the side of the tube, and the observer stands with his back to the object and looks down into the tube. If the telescope is as much as two or three feet in diameter, his head will not intercept enough light to do much harm, — not nearly so much .is would be lost by the second reflection necessary in the other forms of the instrument. But the inclination of the mirror, and the heat from the observer's person, are fatal to any very accurate definition, and unfit this form of h strument for anything but the observation of neb-las and objects which mainly require light-gathering power. ASTRONOMICAL INSTRUMENTS. 29 In the Newtonian telescope, a small plane reflector standing at an angle of 45° is placed in the centre of the tube, so as to intercept the rays reflected by the large mirror a little before they come to their focus, and throw them to the side of the tube, where the eye-piece is placed. In the Gregorian form (which was the first invented), the large mirror is pierced through its centre, and the rays from it are reflected through the hole by a small concave mirror, placed a little outside of the principal focus at the mouth of the tube. With this instrument one looks directly at the stars as with a refractor, and the image is erect. The Cassegrainian form is very similar, except that the small concave mirror of the Gregorian is replaced by a convex mirror, placed a little inside the focus of the large mirror, which makes the instrument a little shorter, and gives a flatter field A B —IT an A 'B 40 Qa> °M v A ~B Fig. 11. — Different Forms of Reflecting Telescope. 1. The Herschelian ; 2. The Newtonian ; 3. The Gregorian. of view. Formerly the great mirror was always made of a composition of cop- per and tin (two parts of copper to one of tin) known as " speculum metal." At present it is usually made of glass silvered on the front sur- face, by a chemical pro- cess which deposits the metal in a thin, brilliant film. These silver-on- glass reflectors, when new, reflect much more light than the old specula, but the film does not retain its polish so long. It is, however, a comparatively simple matter to renew the film when necessary . The largest telescopes ever made have been reflectors. At the head of the list stands the enormous instrument of Lord Rosse,constructed in 1842, with a mirror six feet in diameter and sixty feet focal .length. Next in order are a number of instruments of four feet aperture, first among which is the great telescope of the elder Herschel, built in 1789, followed by the telescope erected by Lassell at Malta in 1860, the Melbourne reflector by Grubb in 1870, and the still more recent silver-on-glass reflector of the Paris observatory, which, however, has proved a failure, owing to defective support of the mirror. 49. Relative Advantages of Refractors and Reflectors. — There has been a good deal of discussion on this point, and each construction has its partisans. In favor of the reflectors we may mention, — First. Ease of construction and consequent cheapness. The concave mirror 30 ASTRONOMICAL INSTRUMENTS. has but one surface to figure and polish, while an object-glass has four. Moreover, as the light goes through an object-glass, it is evident that the glass employed must be perfectly clear and of uniform density through and through ; while in the case of the mirror, the light does not penetrate the material at all. This makes it vastly easier to get the material for a large mirror than for a large lens. Second (and immediately connected with the preceding). The possibility of making reflectors much larger than refractors. Lord Rosse's great reflector is six feet in diameter, while the Lick telescope, the largest of all refractors, is only three. Third. Perfect achromatism. This is unquestionably a very great advan- tage, especially in photographic and spectroscopic work. But, on the whole, the advantages are generally considered to lie with the refractors. In their favor we mention : — First. Great superiority in light. No mirror (unless, perhaps, a freshly polished silver-on-glass film) reflects much more than three-quarters of the incident light; while a good (single) lens trans- mits over 95 per cent. In a good re- fractor about 82 per cent of the light reaches the eye, after passing through the four lenses of the object-glass and eye-piece. In a Newtonian reflector, in average condition, the percentage sel- dom exceeds 50 per cent, and more frequently is lower than higher. Second. Better definition. — Any slight error at a point in the surface of a glass lens, whether caused by faulty workman- ship or by distortion, affects the direc- tion of the ray passing through it only one-third as much as the same error on the surface of a mirror would do. If, for instance, in Fig. 12, an element of the surface at P is turned out of its proper direction, aa', by a small angle, so as to take the direction bb\ then the reflected ray will be sent to/, and its deviation will be twice the angle aPb. But since the index of refraction of glass is about 1.5 the change in the direction of the refracted ray from R to r will only be about two-thirds of aPb. Moreover, so far as distortions are concerned, when a lens bends a little by its own weight, both sides are affected in a nearly compensatory manner, while in a mirror there is no such compensation. As a consequence, mirrors very seldom indeed give any such definition as lenses do. The least fault of workmanship, the least distortion by their own weight, the slightest difference of temperature, between front and back, will absolutely ruin the image, while a lens would be but slightly affected in its performance by the same circumstances. 12. —Effect of Surface Errors in a Mirror and in a Lena. ASTRONOMICAL INSTRUMENTS. 31 Third. Permanence. The lens, once made, and fairly taken care of, suffers no deterioration from age ; but the metallic speculum or the silver film soon tarnishes, and must be repolished every few years. This alone is decisive in most cases, and relegates the reflector mainly to the use of those who are themselves able to construct their own instruments. To these considerations we may add that a refractor, though more expen- sive than a reflector of similar power, is not only more permanent, and less likely to have its performance affected by accidental circumstances, but is lighter and more convenient to use. 50. Time-Keepers and Time-Recorders. — Tlie Clock, Chronometer, and Chronograph. — Modern practical astronomy owes its develop- ment as much to the cloQ and chronometer as to the telescope. The ancients possessed no accurate instruments for the measurement of time, and until within 200 years, the only reasonably precise method of fixing the time of an important observation, as, for instance, of an eclipse, was by noting the altitude of the sun, or of some known star at or very near the moment. It is true that the Arabian astronomer Ibn Jounis had made some use of the pendulum about the year 1000 a.d., more than 500 years before Galileo introduced it to Europeans. But it was not until nearly a century after Galileo's discovery that Huyghens applied it to the construction of clocks (in 1657). So far as the principles of construction are concerned, there is no difference between an astronomical clock and any other. As a matter of convenience, however, the astronomical clock is almost invariably made to beat seconds (rarely half-seconds), and has a conspicuous second-hand, while the hour-hand makes bat one revolution a day, instead of two, as usual, and the face is marked for twenty-four hours instead of twelve. Of course it is constructed with extreme care in all respects. The Escapement, or "Scapement," is often of the form known as the " Graham Dead-beat"; but it is also frequently one of the numerous " gravity " escape- ments which have been invented by ingenious mechanicians. The office of the escapement is to be " unlocked " by the pendulum at each vibration, so as to permit the wheel-work to advance one step, marking a second (or some- times two seconds), upon the clock-face ; while, at the same time, the escape- ment gives the pendulum a slight impulse, just equal to the resistance it has suffered in performing the unlocking. The work done by the pendulum in "unlocking" the train, and the corresponding impulse, ought to he perfectly constant, in spite of all changes in the condition of the train of wheels ; and it is desirable, though not essential, that this work should be as small as 32 ASTRONOMICAL INSTRUMENTS. 51. The pendulum itself is usually suspended by a flat spring, aud great pains should be taken to have the support extremely firm : this is often neglected, and the clock then cannot perform well. Compensation for Temperature. — In order to keep perfect time, the pendulum must be a " compensation pendulum" ; i.e., constructed in such a way that changes of temperature will not change its length. An uncompensated pendulum, with steel rod, changes its daily rate about one-third of a second for each degree of temperature (centigrade). A wooden pendulum rod is much less affected by temperature, but is very apt to be disturbed by changes of moisture. Graham's mercurial pendulum (Fig. 13) is the one most commonly used. It consists simply of a jar (usually steel), three or four inches in diameter, and about eight inches high, containing forty or fifty pounds of mercury, and suspended at the end of a steel rod. When the temperature rises, the rod lengthens (which would make the clock go slower) ; but, at the same time, the mercury expands, from the bottom upwards, just enough to compensate. This pendulum will perform well only when not exposed to rapid changes of temperature. Under rapid changes the compensation lags. If, for in- stance, it grows warm quickly, the rod will expand before the mercmy does ; so that, while the mercury is growing warmer, the clock will run slow, though after it has become warm the rate may. be all right. A compensation pendulum, constructed on the principle of the old gridiron pendulum of Harrison, but of zinc and steel instead of brass and steel, is now much used. The compensation is not so easily adjusted as in the mercurial pendulum, but when properly made the mechan- ism acts well, and bears rapid alterations of temperature much better than the mercurial pendulum. The heavy pendulum-bob, a lead cylinder, is hung at the end of a steel rod, which is suspended from the top of a zinc tube, and hangs through the centre of it. This tube is itself supported at the bottom by three or four steel rods which hang from a piece attached to the pendu- lum spring. The standard clock at Greenwich has a pendulum of this kind. Fia. 13. Compensation Pendulums. 1. Graham's Pendulum. 2. Zinc-Steel Pendulum. 52. Effect of Atmospheric Pressure. — In consequence of the buoyancy of the air, and its resistance to motion, a pendulum swings ASTRONOMICAL INSTRUMENTS. 33 a little more slowly than it would in vacuo, and every change in the density of the air affects its rate more or less. With mercurial pendulums, of ordinary construction, the " barometric coefficient," as it is called, is about one-third of a second for an inch of the barometer; i.e., an increase of atmospheric density which would raise the barometer one inch would make the clock lose about one- third of a second daily. It varies considerably, however, with differ ent pendulums. It is not very usual to take any notice of this slight disturbance ; but when the extremest accuracy of time-keeping is aimed at, the clock is either sealed in an air-tight case from which the air is partially exhausted (as at Berlin), or else some special mechanism, controlled by a barometer, is de- vised to compensate for the barometric changes, as at Greenwich. In the Greenwich clock a magnet is raised or lowered by the rise or fall of the mercury in a barometer attached to the clock-case. When the magnet rises, it approaches a bit of iron two or three inches above it, fixed to the bottom of the pendulum, and the increase of attraction accelerates the rate just enough to balance the retardation due to the air's increased density and viscosity. There are several other contrivances for the same purpose. 53. Error and Rate. — The "error," or "correction" of a clock is the amount that must be added to the indication of the clock-face at any moment in order to give the true time; it is, therefore, plus (+) when the clock is slow, and minus (— ) when it is fast. The rate of a clock is the amount of its daily gain or loss; plus ( + ) when the clock is losing. Sometimes the hourly rate is used, but " hourly " is then always specified. A perfect clock is one that has a constant rate, whether that rate be large or small. It is desirable, for convenience' sake, that both error and rate should be small ; but this is a mere matter of adjust- ment by the user of the clock, who adjusts the error by setting the hands, and the rate by raising or lowering the pendulum-bob. The final adjustment of rate is often obtained by first setting the pendu- lum-bob so that the clock will run slow a second or two daily, and then putting on the top of the bob little weights of a gramme or two, which will accelerate the motion. They can be dropped into place or knocked off with- out stopping the clock or perceptibly disturbing it. The very best clocks will run three or four years without being stopped for cleaning, and will retain their rate without a change of more than one- fifth of a second, one way or the other, during the whole time. But this is 34 ASTRONOMICAL INSTRUMENTS. exceptional performance. In a run as long as that, most clocks would be liable to change their rate as much as half a second or more, and to do it somewhat irregularly. 54. The Chronometer. — The pendulum-clock not being portable, it is necessary to provide time-keepers that are. The chronometer is merely a carefully made watch, with a balance wheel compensated to run, as nearly as possible, at the same rate in different tempera- tures, and with a peculiar escapement, which, though unsuited to watches exposed to ordinary rough usage, gives better results than any other when treated carefully. Fig. 14. — A Chronograph by Warner and Swasey. The box-chronometer used on ship-board is usually about twice the diameter of a common pocket watch, and is mounted on gimbals, so as to keep hori- zontal at all times, notwithstanding the motion of the vessel. It usually beats half-seconds. It is not possible to secure in the chronometer-balance as perfect a temperature correction as in the pendulum. For this and other ASTRONOMICAL INSTRUMENTS. 35 reasons the best chronometers cannot quite compete with the best clocks in precision of time-keeping; but they are sufficiently accurate for most pur- poses, and of course are vastly more convenient for field operations. They are simply indispensable at sea. Never turn the hands of a chronometer backward. 55. Before the invention of the telegraph it was customary to note time merely " by eye and ear." The observer, keeping his time- piece near him, listened to the clock-beats, and estimated as closely as he could, in seconds and tenths of seconds, the moment when the phenomenon he was watching occurred — the moment, for instance, when a star passed across a wire in the reticle of his telescope. At present the record is usually made by simply pressing a "key" in the hand of the observer, and this, by a telegraphic connection, makes a mark upon a strip or sheet of paper, which is moved at a uniform rate by clock-work, and graduated by seconds-signals from the clock or chronometer. 56. The Chronograph. —This is the instrument which carries the marking-pen and moves the paper on which the time-record is made. 50 s a H ^ * S b 25 m OO'.O n i A A A . A A A Flo. 15. — Part of a Chronograph Record. The paper is wrapped upon a cylinder, six or seven inches in diameter, and fifteen or sixteen inches long. This cylinder is made to revolve once a minute, by clock-work, while the pen rests lightly upon the paper and is slowly drawn along by a screw-motion, so that it marks a continuous spiral. The pen is carried on the armature of an electro-magnet, which every other second (or sometimes every second) receives a momentary current from the clock, causing it to make a mark like those which break the lines in the figure annexed. The beginning of a new minute (the 60th sec.) is indicated either by a double mark as shown, or by the omission of a mark. When the observer touches his key he also sends a current through the magnet, and thus interpolates a mark of his own on the record, as at X in the figure : the beginning of the mark is the instant noted — in this case 54.9 s . Of course the minutes when the chronograph was started and stopped are noted by the observer on the sheet, and so enable him to identify the minutes and seconds all through the record. 36 ASTRONOMICAL INSTRUMENTS. Many European observatories use chronographs in which the record is made upon a long fillet of paper, instead of a sheet on a cylinder. The instrument is lighter and cheaper than the -American form, but much less convenient. The regulator of the clock-work must be a "continuous " regulator, work- ing continuously, and not by beats like a clock-escapement. There are various forms, most of which are centrifugal governors, acting either by friction (like the one in the figure) or by the resistance of the air ; or else "spring-governors," in which the motion of a train, with a pretty heavy fly-wheel, is slightly checked at regular intervals by a pendulum. 57. Clock-Breaks. — The arrangements by which the clock is made to send regular electric signals are also various. One of the earliest and simplest is a fine platinum wire attached to the pendulum, which swings through a drop of mercury at each vibration. All of the arrangements, however, in which the pendulum itself has to make the electric contact are objectionable, and for clocks using the Graham dead-beat scapement no absolutely satis- factory means of giving the signals has yet been devised. Clocks with the gravity escapements have a decided advantage in this respect. Their wheel- work has no direct action in driving the pendulum, and so may be made to do any reasonable amount of outside work in the way of " key-manipula- tion " without affecting the clock-rate in the least. Usually a wheel on the axis of the scape-wheel is made to give the electric signals by touching a light spring with one of its teeth every other second. Chronometers are now also fitted up in the same way, to be used with the chronograph. The signals sent are sometimes " breaks " in a continuous current, and sometimes " makes " in an open circuit. Usage varies in this respect, and each method has its advantages. The break-circuit system is a little simpler in its connections, and possibly the signals are a little more sharp, but it involves a much greater consumption of battery material, as the current is always circulating, except during the momentary breaks. 58. Meridian Observations. — A large proportion of all astronomi- cal observations are made at the time when the heavenly body observed is crossing the meridian, or very near it. At that moment the effects of refraction and parallax (to be discussed hereafter) are a minimum, and as they act only in a vertical plane, they do not have any influence on the time at which the body crosses the meridian. 59. The Transit Instrument is the instrument used, in connection with a clock or chronometer, and often with a chronograph also, to observe the time of a star's " transit" across the meridian. If the error of the (sidereal) clock is known at the moment, this observation will determine the right ascension of the body, which, it ASTRONOMICAL INSTRUMENTS. 37 will be remembered, is simply the sidereal time at which it crosses the meridian; i.e., the number of hours, minutes, and seconds by which it follows the vernal equinox. Vice versa, if the right ascension is known, the error or correction of the clock will be determined. The instrument (Fig. 16) consists essentially of a telescope mounted upon a stiff axis perpendicular to the telescope tube. This axis is placed horizontal, east and west, and turns on pivots at its extremi- ties, in Y-bearings upon the top of two fixed piers or pillars. A Fig. 16. — The Transit Instrument (Schematic). small graduated circle is attached, to facilitate "setting" the telescope at any designated altitude or declination. The telescope carries at the eye-end, in the focal plane of the object-glass, a reticle of some odd number of vertical wires, — five or more, — one of which is always in the centre, and the others are usually placed at equal distances on each side of it. One or two wires also cross the field horizontally. If the pivots are true, and the instrument accurately adjusted, it is evident that the central vertical wire ivill always follow the meridian as the instrument is turned; and the instant when a star crosses this wire will be the true moment of the star's meridian transit. The 38 ASTRONOMICAL INSTRUMENTS. object in having a number of wires is, of course, simply to gain accuracy by taking the mean of a number of observations instead of depending upon a single one. In order to "level" the axis properly, a delicate spirit-level is an essential adjunct ; it is usual, also, (and important) to provide a con- venient "reversing apparatus," by which the instrument can be turned half round, making the eastern and western pivots change places. The instrument must be thoroughly stiff and rigid, without loose joints or shaky screws ; and the two pivots must be accu- rately round, precisely in line with each other, free from taper, and precisely of the same size ; all of which conditions may be summed up by saying that they must be portions of one and the same geometrical cylinder. Fig. 17. — Reticle of the Transit Instrument. The proper construction and grinding of these pivots, which are usually of hard bell metal (sometimes of steel), taxes the art of the most skilful mechanician. The level, also, is a delicate instrument, and difficult to construct. Provision is made, of course, for illuminating the field of view at night so as to make the reticle wires visible. Usually one (or both) of the pivots is pierced, and a lamp throws light through the opening upon a small mirror in the centre of the tube, which reflects- it down upon the reticle. The Y's are used instead of round bearings, in order to prevent any rolling or shake of the pivots as the instrument turns. Fig. 18 shows a modern transit instrument (portable) as actually con- structed by Fauth & Co. Another form of the instrument is much used, which is often designated as the " Broken Transit." A reflector in the central cube throws the rays coming from the object-glass, out at right angles through one end of the axis, where the eye-piece is placed ; so that the observer does not have to change his position at all for differ- ent stars, but simply looks straight forward horizontally. It is very convenient and rapid in actual work, but the observations require a considerable correction for flexure of the axis. 60. Adjustments. — (1) Focus and verticality of wires. (2) Collimation. (3) Level. (4) Azimuth. First. The first thing to do after the instrument is set on its supports and the axis roughly levelled, is to adjust the reticle. The eye-piece is drawn out ASTRONOMICAL INSTRUMENTS. 39 iMMmm ■MHIRffl Fig. 18. — A 3-inch Transit, with reversing apparatus. Fauth & Co. or pushed in until the wires appear perfectly sharp, and then the instrument is directed to a star or to some distant object (not less than a mile away), and without disturbing the eye-piece, the sliding-tube, which carries the reticle, is drawn out or pushed in until the object is also distinct at the 40 ASTRONOMICAL INSTRUMENTS. same time with the wires. If this adjustment is correctly made, motion of the eye in front of the eye-piece will not produce any apparent displacement of the object in the field, with reference to the wires. To test the verticality of the wires, the telescope is moved up and down a little, while looking at the object ; if the axis is level and the wires vertical, the wire will not move off from the object sideways. There are screws provided to turn the reticle a little, so as to effect this adjustment. When the wires have been thus adjusted for focus and verticality, the reticle-slide should be tightly clamped ^,nd never disturbed again. The eye- piece can be moved in and out at pleasure, to secure distinct vision for differ- ent eyes, but it is essential that the distance between the object-glass and the reticle remain constant. Second. Collimation. The line joining the optical centre of the object- glass with the middle wire of the reticle is called the " line of collimation," and this line must be made exactly perpendicular to the axis of rotation by moving the reticle slightly to one side or the other by means of the adjusting screws provided for the purpose. The simplest way of effect- ing the adjustment is to point the instrument on some well-defined dis- tant object, like a nail-head or a joint in brickwork, and then carefully to " reverse " the instrument without disturbing the stand. If the middle wire, after reversal, points just as it did before, the " collimation " is correct ; if not, the middle wire must be moved half way towards the object by the screws. Collimator. — It is not always easy to find a distant object on which to make this adjustment, and a "collimator" may be substituted with advantage. This is simply a telescope mounted horizontally on a pier in front of the transit instrument, so that when the transit telescope is horizontal, it can look straight into the collimator, which ought to be of about the same size as the transit itself. In the focus of the collimator object-glass are placed two wires forming an X, and thus placed they can be seen by a telescope looking into the colli- mator just as distinctly as if they were at an infinite distance and really celes- tial objects. The instrument furnishes us a mark optically celestial, but mechanically within reach of our finger-ends for illumination, adjustment, etc. If the pier on which it is mounted is firm, the collimator cross is in all respects as good as a star, and much more convenient. Third. Level. The adjustment for level is made by setting a striding level on the pivots of the axis, reading the level, then reversing the level (not the transit) and reading it again. If the pivots are round and of the same size, the difference between the level-readings direct and reversed will indicate the amount by which one pivot is higher than the other. One of the Y's is made so that it can be raised and lowered slightly by means of a screw, and this gives the means of making the axis horizontal. If the pivots are not of the same size (and they never are absolutely), the astronomer must determine and allow for the difference. ASTRONOMICAL INSTRUMENTS. 41 fourth. Azimuth. In order that the instrument may indicate the meridian truly, its axis must lie exactly east and west ; i.e., its azimuth must be 90°. This adjustment must be made by means of observations upon the stars, and is an excellent example of the method of successive approximations, which is so characteristic of astronomical investigation, (a) After adjusting care- fully the focus and collimation of the instrument, we set it north and south by guess, and level it as precisely as possible. By looking at the pole star, and remembering how the pole itself lies with reference to it, one can easily set the instrument pretty nearly; i.e., within half a degree or so. The middle wire will now describe in the sky a vertical circle, which crosses the meridian at the zenith, and lies very near the meridian for a considerable distance each side of the zenith. (fi) We must next get an "approximate" time; i.e., set our clock or chronometer nearly right. To do this, we select from the list of several hundred stars in the Nautical Almanac (which is to be regarded in about the same light with the clock and the spirit level, as an indispensable accessory to the transit) a star which is about to cross the meridian near the zenith. The difference between the right ascension of the star as given in the Almanac, and the time shown by the clock-face, will be very nearly the error of the clock at the time of the observation : not exactly, unless the dec- lination of the star is such that it passes exactly through the zenith, but very nearly, since the star crosses the meridian near the zenith. We now have the time within a second or two. (c) Next turn down the telescope upon some Almanac star, which is soon to cross the meridian within 10° of the pole. It will appear to move very slowly. A little before the time it should reach the meridian, move the whole frame of the instrument until the middle wire points upon it, and then, by means of the " Azimuth Screw," which gives a slight horizontal motion to one of the Y*s, follow the star until the indicated moment of its tran- sit; i.e., until the clock (corrected for clock error) shows on its face the star's right ascension. If the clock correction had been known with absolute exact- ness, the instrument would now be truly in the meridian : as the clock error, however, is only approximate, the instrument will only be approximately in the meridian; but — and this is the essential point — it will be very much more nearly so than at the beginning of the operation. The supposed incor- rectness, amounting perhaps to one or two seconds, in the time at which the instrument was set on the circumpolar star will, on account of the slew mo- tion of the star, make almost no perceptible difference in the direction given to the axis. A repetition of the operation may possibly be needed to secure all the desired precision. The accuracy of this azimuth adjustment can then be verified by three successive " culminations " or transits of the pole star, or any other circumpolar. The interval occupied in passing from the upper to the lower culmination on the west side of the meridian ought, of course, to be exactly equal to the time on the eastern side ; i.e., twelve sidereal hours. 42 ASTRONOMICAL INSTRUMENTS. 61. The flual test of all the adjustments, and of the accurate going of the clock, is obtained by observing a number of Almanac stars of widely different declination. If they all indicate identically the same clock correction, the instrument is in adjustment ; if not, and if the differences are not very great, it is possible to deduce from the observations themselves the true clock error, and the adjustment errors of the instrument. It is to be added, in this connection, that the astronomer can never assume that adjustments are perfect : even if once perfect, they would not stay so, on account of changes of temperature and other causes. Nor are observations ever absolutely accurate. The problem is, from observations more or less inaccurate but honest, with instruments more or less maladjusted but firm, to find the result that would have been obtained by a perfect observation with a perfect and perfectly adjusted instrument. It can be more nearly done than one might suppose. But the discussion of the subject belongs to Practical Astronomy, and cannot be entered into here. 62. Prime Vertical Instrument. — For certain purposes, a Transit Instrument, provided with an apparatus for rapid reversal, is turned quarter-way round and mounted with the axis north and south, so that the plane of rotation lies east and west, instead of in the meri- dian. It is then called a Prime Vertical Transit. 63. The Meridian Circle. — In order to determine the Declina- tion or Polar Distance of an object, it is necessary to have some instrument for measuring angles ; mere time-observations will not suffice. The instrument most used for this purpose is the Meridian Circle, or Transit Cir- cle, which is simply a transit in- strument, with a graduated circle attached to its axis, and revolv- ing with the telescope. Some- times there are two circles, one "at each end of the axis. Fig. 19 represents the instru- ment " schematically," showing merely the essential parts. Fig. 20 is a meridian circle, with a 4-inch telescope, constructed by Fautli & Co. Km. 19. — The Meridian Circle (Schematic). ASTKONOMICAL INSTRUMENTS. 43 Fig. 20. — Meridian Circle. B, C, D, the Reading MicroBoopes. the Graduated Circle. . the Roughly Graduated Setting Circle, the Index Microscope. This is usually, however placed half way between A and D. F, the Clamp. G, the Tangent Screw. LL, the Level, only placed in position occasionally AT, the Right Ascension Micrometer. WW, Counterpoises, which take part of the weigh of the instrument off from the Y 's. 44 ASTRONOMICAL INSTRUMENTS. In observatory instruments the circle is usually from two to four feet in diameter ; larger circles were once used, but it is found that their weight, and the consequent strains and flexures, render them actually less accurate than the smaller ones. The utmost resources of mechanical art are ex- hausted in making the graduation as precise as possible and in providing for its accurate reading, as well as in securing the maximum firmness and sta- bility of every part of the instrument. The actual divisions are usually 5' apart (in very large instruments sometimes only 2'), but the circle is ■'read" to seconds and tenths of seconds of arc by means of reading micro- scopes, from two to six in number, fixed to the pier of the instrument. In a circle of forty inches diameter, 1" is a little less than -^^ of an inch, GtfI?5J inch), so that the necessity of fine workmanship is obvious. 64. The Reading Microscope (Fig. 21). — This consists essen- tially of a compound microscope, which forms a magnified image of the graduation at the focus of its object-glass, where this image is viewed by a positive eye-piece. At the place where the image is formed a pair of parallel spider-lines or a cross is placed, movable in the plane of the image by a ''micrometer screw"; i.e., a fine screw with a graduated head, usually divided into sixty parts. One revolution of the screw carries the wire l r of arc, which makes one division of the screw-head 1", the tenths of seconds being estimated. n oaa» r Limb of Circle 1 The adjustment of the microscope for " runs," as it is called (that is, to make one revolution of the micrometer screw exactly equal to 1'), is effected as follows. By setting the wires first on one of the graduation marks visible in the field of view, and then on the Fia. 21. —The Reading Microscope, next mark, it is immediately evident whether five revolutions of the screw "run" over or fall short of 5' of the graduation. If they overrun, it shows that the image of the graduation formed by the microscope objective is too small to fit the screw, and vice-versa. Now, by simply increasing or decreasing the distance AB between the objective and the micrometer box, the size of the image can be altered at will, and the objective is therefore so mounted that this can be done. Of course, every change in the length of the microscope tube will also require a readjustment of the distance between the "limb," or graduated surface, of the circle and the microscope, in order to secure distinct vision ; but by a few trials the adjustment is easily made sufficiently precise. ASTRONOMICAL INSTRUMENTS. 45 The reading of the circle is as follows : An extra index-microscope, with low power and large field of view, shows by inspection the de- grees and minutes. The reading-microscopes are only used to give the odd seconds, which is done by turning the screw until the parallel spider-lines are made to include one of the graduation lines half-way between themselves ; the head of the screw then shows directly the seconds and tenths, to be added to the degrees and minutes shown by the index. Thus in Fig. 22, the reading of the microscope is 3' 22".l, the 3' being given by the scale in the field, the 22".l by the screw-head. 65. Method of observing a Star. — A minute or two before the star reaches the meridian the instrument is approximately pointed, so that the star will come into the field of view. As soon as it makes its appearance, the instrument is moved by the slow-mo- tion tangent- screw until the star is ' ' bisected " by the fixed horizontal wire of the reticle, and the star is kept bisected until it reaches the middle ver- tical wire which marks the meridian. The microscopes are then read, and their mean result is the star's " circle-reading." Fig. 22. — Field of View of Reading Microscope. Frequently the star is bisected, not by moving the whole instrument, but by means of a " micrometer wire," which moves up and down in the field of view. The micrometer reading then has to be combined with the reading of the microscope, to get the true circle-reading. 66. Zero Points. — -In determining the declination or meridian altitude of a star by means of its circle-reading, it is necessary to know the '■'zero -point" of the circle. For declinations, the "zero point" is either the polar or the equatorial reading of the circle ; i.e., the reading of the circle when the telescope is pointed at the pole or at the equator. The " polar point" may be found by observing some circumpolar star above the pole, and again, twelve hours later, below it. "When the two circle-readings have been duty corrected for refraction and instrumental errors, their mean will be the polar point. 46 ASTRONOMICAL INSTRUMENTS. Suppose, for instance, that 5 Ursse Minoris, at the " upper culmination," gives a corrected reading of 52° 18' 25".3, while at the lower culmination the reading is 45° 31' 35".7, then the mean of these, 48° 55' 00".5, is the polar point, and of course the equatorial reading is 138° 55' 00".5, — just 90 c greater. The polar distance of the star would be the half-difference of the two readings, or 3° 23' 24".8. 67. Nadir Point. • — The determination of the polar point requires two observations of the same star at an interval of twelve hours. It is often difficult to obtain such a pair ; moreover, the refraction compli- cates the matter, and renders the result less trustworthy. Accord- ingly it is now usual to use the nadir or the horizontal reading as the zero, rather than the polar point. The nadir point is determined by pointing the telescope down- wards to a basin of mercury, moving the telescope until the image of the horizontal wire of the reticle, as seen by reflection, coincides with the wire itself. Since the reticle is exactly in the principal focus of the object-glass, rays of light emitted by any point in the reticle will become a parallel beam after passing the lens, and if this , beam strikes a plane mirror perpendicularly and k ^.i—p is returned, the rays will come just as if from a ! / \ real object in the sky, and will form an image J?——\ f *» at the focal plane. "When, therefore, the image of the central wire of the reticle, seen in the — ; ■"< mercury basin by reflection, coincides with the ^Reticle wire itself, we know that the line of collimation Fig. 23. must be exactly perpendicular to the surface of The Collimating Eye-Piece. the mercui .y . i\ e ., Vertical. To make the image visible it is necessary to illuminate the reticle by light thrown towards the object-glass from behind the wires, instead of light coming from the object-glass towards the eye as usual. This peculiar illu- mination is commonly effected by means of Bohnenberger's "collimating eye-piece," shown in Fig. 23. In the simplest form it is merely a common Kamsden eye-piece, with a hole in one side, and a thin glass plate inserted at an angle of 45°. A light from one side, entering through the hole, will be (partially) reflected towards the wires, and will illuminate them sufficiently. The horizontal point of course differs just 90° from the nadir point. It may also be found independently by noting the circle-readings of some star observed one night directly, and the next night by reflection in mercury; or, if the star is a close circumpolar, both observations may be made the same evening, one a few minutes before its meridian passage, the other just as long after. But the method of the collimating eye-piece is fully as accurate and vastly more convenient. ASTRONOMICAL INSTRUMENTS. 47 68. Differential Use of the Instrument. — We now know the places of several hundred stars with so much precision that in many cases it is quite sufficient to observe one or two of these " standard stars " in connection with the bodies whose places we wish to determine. The difference between the declination of the known star and that of any star whose place is to be determined, will, of course, be simply the difference of their circle-readings, corrected for refraction, etc. The meridian circle is said to be used "differ- entially " when thus treated. 69. Errors of Graduation, etc. — If the circle is from a reputable maker, and has four or six microscopes, and if the observations are carefully made and all the microscopes read each time, results of sufficient precision for most purposes may be obtained by merely correcting the observations for "runs" and refraction. The out- standing errors ought not to exceed a second or two. But when the tenths of a second are in question, the case is different. It will not then do for the astronomer to assume the accuracy of the graduation of his circle, but he must investigate the errors of its divisions, the errors of the micrometer screws in the microscopes, the flexure of the telescope, and the effect of differences of temperature in shifting the zero points of the circle, by slightly disturbing the position or direction of the microscopes. Of course this is not the place to enter into such details, but it is an opportunity to impress again upon the student the fact that truth and accuracy are only attainable by immense painstaking and labor. 70. Ulural Circle. — This instrument is in principle the same as the meridian circle, which has superseded it. It consists of a circle, carrying a telescope mounted on the face of a wall of masonry (as its name implies) and free to revolve in the plane of the meridian. The wall furnishes a con- venient support for the microscopes. 71. Altitude and Azimuth Instrument. — Since the transit instru- ment and meridian circle are confined to the plane of the meridian, their usefulness is obviously limited. Meridian observations, when they are to be had, are better and more easily used than any others, but are not always attainable. We must therefore have instruments which will follow an object to any part of the heavens. The altitude and azimuth instrument is simply a surveyor's theodo- lite on a large scale. It has a horizontal circle turning upon a verti- cal axis, and read by verniers or microscopes. Upon this circle, and turning with it, are supports which carry the horizontal axis of the telescope with its vertical circle, also read by microscopes. Obvi- 48 ASTRONOMICAL INSTRUMENTS. ously the readings of these two circles, when the instrument is prop- erly adjusted and the zero points determined, will give the altitude Fxa. 24. — Altitude and Azimuth Instrument. and azimuth of the body pointed on. Fig. 24 represents a small in- strument of this kind. ASTRONOMICAL INSTRUMENTS. 49 72. The Equatorial.— The essential characteristic of this instrument is that its principal axis, i.e., the axis which rests in fixed bearings, instead of being either horizontal or vertical, is inclined at an angle equal to the latitude of the place, and directed towards the pole, thus placing it parallel to the earth's axis of rotation. This axis of the instrument is called its polar axis; and the graduated circle which it carries, and which is parallel to the celes- tial equator, is called the hour-circle, be- cause its reading gives the how-angle of the body upon which the telescope hap- pens to be pointed. Sometimes, also, it is called the Right Ascension Circle. Upon this polar-axis are secured the bearings of the declination axis, which is perpen- dicular to the polar axis, and carries the telescope itself and the declination circle. In the instruments before described, the telescope is a mere pointer, and wholly subsidiary to the circles ; in the equatorial the telescope is usually the main thing, and the circles are subordinate, serving only to aid the observer in finding or identifying the body upon which the telescope is directed. Fig. 25 exhibits schematically the ordinary form of equatorial mounting, of which there are numerous modifications. Fig 26 is the 23-inch Clark telescope at Princeton, and Fig. 27 is the 4-foot Melbourne reflector. The frontispiece is the great Lick telescope of thirtj'-six inches diameter. The advantages of the equatorial mounting for a large telescope are very great as regards convenience. In the first place, when the telescope is once pointed upon a star or planet, it is only necessary to turn the polar axis with a uniform motion in order to " follow " the star, which otherwise would be carried out of the field of view in a few moments by the diurnal motion. This motion, since it is uni- form, can be, and in all large instruments usually is, given by clock- work, with a continuous regulator of some kind, similar to that used in the chronograph. The instrument once directed and clamped, and the clock-work started, the object will continue apparently im- movable in the .field of view as long as may be desired. Tn the next place, it is very easy to find an object, even if invisible to the naked eye, like a faint comet or nebula, or a star in the day- FlO. 25. The Equatorial (Schematic). 50 ASTRONOMICAL INSTRUMENTS. time, provided we know its declination and right ascension, and have the sidereal time ; for which reason a sidereal clock or chro- nometer is an indispensable adjunct of the equatorial. Fig. 26. — The 23-inch Princeton Telescope. To find an object, the telescope is turned in declination until the reading of the declination circle corresponds to the declination of the object, and then the polar axis is turned until the hour-circle of the instrument (not to be confounded with an hour-circle in the sky) reads the hour-angle of the object. This hour-angle, it will be remembered, is simply the difference be- ASTRONOMICAL INSTRUMENTS. 51 tween the sidereal time and the right ascension of the object. The horn- angle is east if the right ascension exceeds the time; west, if it is less When the telescope is thus set, the object will be found (with a low mag- nifying- power) in the field of- view, unless it is near the horizon, in which case refraction must be taken into account. Fig. 27. — The Melbourne Reflector. While the instrument cannot give very accurate determinations o the positions of bodies by the direct readings of its circles, on account of the irregular flexures of its axes, it may do so indirectly ; that is, it may be used to determine very accurately the difference between the right ascension and declination of a comet or planet, for instance, and that of some neighboring star, whose place has been already determined by the meridian circle ; and this is one of the most im portant uses of the instrument. 52 ASTRONOMICAL INSTRUMENTS. 73. The Micrometer. —Micrometers of various sorts are employed for the purpose. The most common and most generally useful is the so-called "■filar position-micrometer," Fig. 28, which is an indispen- sable auxiliary of every good telescope. It is a small instrument, much like the upper part of the read- ing microscope, but more com- plicated. It usually contains a reticle of fixed wires, two or three parallel to each other, and crossed at right angles by a second set. Then there are two or three wires parallel to the first set, and movable by an accu- rately made screw with a gradu- ated head and a counter, or scale, for indicating the number of entire revolutions made by the screw. The box containing these wires, and carrying the eye-piece and screw, can itself be turned around in a plane perpendicular to the optical axis of the telescope, and set in any desired position ; for example, so that the movable wires shall be parallel to the celestial equator, while the Fig. 28. — The Filar Position-Micrometer. Fig. 29. — Construction of the Micrometer. other set run north and south. This "position angle" is read on a graduated circle, which forms part of the instrument. Means of illumination are provided, giving at pleasure either dark wires in a bright field, or vice versa. With this instrument one can measure the distance (in seconds of arc), and the direction between any two stars which are near enough to be seen at once in the same field of view. This range in small ASTRONOMICAL INSTRUMENTS. 53 telescopes may reach 30' of arc; while in the larger instruments, which, with the same eye-pieces have much higher magnifying powers, it is necessarily less, — not more than from 5' to 10'. 74. A new form of equatorial, known as the Equatorial Coude, or Elbowed Equatorial, has been recently introduced at the Paris Observatory. With large instruments of the ordinary form a great deal of inconvenience is en- countered by the observer, in moving about to follow the eye-piece into the various positions into which it is forced by the inconsiderateness of the Fig. 30. — The Equatorial Coude\ heavenly bodies. Moreover, the revolving dome, which is usually erected to shelter a great telescope, is an exceedingly cumbrous and expensive affair. In the Equatorial Coude - , Fig. 30, these difficulties are overcome by the use of mirrors. The observer sits always in one fixed position, looking obliquely down through the polar axis, which is also the telescope tube. The Paris instrument has an object-glass about ten inches in diameter, and performs very satisfactorily. The two reflections, however, cause a considerable loss of light, and some injury to the definition. The mirrors, and the consequent complications, also add heavily to the cost of the in strument. Fig. 30 is from a photograph of this instrument. 75. All the instruments so far described, except the chronometer, are fixed instruments ; of use only when they can be set up firmly and carefully adjusted to established positions. Not one of them would be of the slightest use on shipboard. 54 ASTRONOMICAL INSTRUMENTS. We have now to describe the instrument which, with the help of the chronometer, is the main dependence of the mariner. It is an instrument with which the observer measures the angular distance between two objects ; us, for instance, the sun and the visible horizon, not by pointing first on one and then afterwards on the other, but by sighting them both, simultaneously and in apparent coincidence; which can be done even when he has no fixed position or stable footing. 76. The Sextant. — The graduated limb of the sextant is carried by a light framework, usually of metal, provided with a suitable handle X. The arc is about one-sixth of a circle, as the name implies, and Fig. 31. — The Sextant. is usually from five to eight inches radius. It bears a graduation of half-degrees, numbered as whole degrees, so that it can measure any angle less than 120°. An " index-arm," MN in the figure, is pivoted at the centre of the arc, and carries a vernier which slides along the limb, and can be fixed at any point by a clamp and delicately moved by the attached tangent screw, T. The reading of this vernier gives the angle measured by the instrument. The best instruments read to 10"- Just over the centre of motion, the "index-mirror" M, about two inches by one and one-half in size, is fastened securely to the index-arm, so as to be perpendicular to the plane of the limb. At ASTRONOMICAL INSTRUMENTS. 55 H, the " horizon-glass," about an inch wide and of the same height as the index-glass, is secured firmly to the frame of the instrument, in such position that, when the vernier of the index-arm reads zero, the index-mirror and horizon-glass will be parallel to each other. Only half of the horizon-glass is silvered, the upper half being left transparent. E is a small telescope. If the vernier stands near, but not at zero, the observer look- ing into the telescope will see together in the field of view two sepa- rate images of the object; and if, while still looking, he slides the vernier a little, lie will see that one of the images remains fixed, while the other moves. The fixed image is due to the rays which reach the object-glass of the telescope directly, coming through the unsilvered half of the horizon-glass : the movable image, on the other hand, is produced by rays which have suffered two reflections, — first, from the index-mirror to the horizon-glass ; and second, at the lower half of the horizon-glass. "When the two mirrors are parallel, and the vernier reads zero, the two images coincide, pro- vided the object is at a considerable distance. If now the vernier does not stand at or near zero, the observer, looking at any object directly through the horizon-glass, will see, not only that object, but also whatever other object is so situated as to send its rays to the telescope by reflection upon the mir- rors ; and the reading of the vernier will give the angle at the instru- ment between the two objects whose images thus coincide; the angle between the planes of the two mirrors being just half that between the objects, and the half-degrees on the limb being numbered as whole ones. 77. The principal use of the instrument is in measuring the altitude of the sun. At sea the observer, holding the instrument with his right hand and keeping the plane of the arc vertical, looks directly towards the visible horizon at the point under the sun, through the horizon- glass (whence its name) ; then by moving the vernier with his left hand, ha inclines the index-glass upwards until one edge of the reflected image of the sun is brought just to touch the horizon-line, noting the exact time by the chronometer, if necessary. The reading of the vernier, after correcting for the semi-diameter of the sun, the dip of the horizon, the refraction, and the parallax (and for the " index -error " of the sextant, if the vernier does not read strictly zero when the mirrors are parallel) gives the sun's true altitude at the moment. 56 ASTRONOMICAL INSTRUMENTS. 78. On laud the visible horizon is of no use, and we have recourse to an "artificial horizon," as it is called. This is merely a shallow basin of mercury, covered, when necessary to protect it from the wind, with a roof made of glass plates having their sides plane and parallel. In this case we measure the angle between the sun's image reflected in the mercury and the sun itself. The reading of the instrument, corrected for index-error, gives twice the sun's apparent altitude ; which apparent altitude, corrected as before for refraction and ,S\ parallax, but not for dip of the horizon, \> gives the true altitude. The skilful use n. of the sextant requires steadiness of p ____— dP^ '^ p 7>~ nan d and considerable dexterity, and /iX^^^ from the small size of the telescope the /^C*^ angles measured are of course less pre- Q_ \r^ i fi'- -. pj _ cise than if determined by large fixed \ I E " instruments. But its portability and \ j applicability at sea render it absolutely \ invaluable. \ 1 \| _, 79. The principle that the true angle K between the objects whose images coin- Kig. 32. -Principle of the Sextant. cide ^ twice the an S le between the mir- rors (or between their normals) is easily demonstrated as follows (Fig. 32) : — The ray SM coming from an object, after reflection first at M (the index- mirror), and then at H (the horizon-glass), is made to coincide with the ray OH coming from the horizon. We must prove that the angle SEO, be- tween the object and the horizon, as seen from the point E in the instrument, is double the angle Q, between MQ and HQ, which are normals to the mir- rors, and therefore double Q', which is the angle between the planes of the mirrors. First, from the law of reflection, we have, SMP=HMP, or SMH=2xPMH. Similarly, MHE = 2x MHQ. From the geometric principle that the exterior angle SMH of the triangle HME is equal to the sum of the opposite interior angles at H and E, we get HEM = SMH -MHE = 2 PMH - 2 MHQ = 2{PMH- MHQ ) . Similarly, from the triangle HMQ, we have HQM=PMH-MHQ, which is half the value just found for HEM, and proves the proposition. ASTRONOMICAL INSTRUMENTS. 57 Of course with the sextant, as with all other instruments, it is necessary for the observer who aims at the utmost precision to inves- tigate, and take into account its errors of graduation, construction and adjustment ; but their discussion does not belong here. 80. Besides the instruments we have described, there are many others designed for special work, some of which, as the zenith tele- scope, and heliometer, will be mentioned hereafter as it becomes necessary. There is also a whole class of physical instruments, photometers, spectroscopes, heat-measuring appliances, and photo- graphic apparatus, which will have to be considered in due time. But with clock, meridian circle, and equatorial and their usual accessories, all the fundamental observations of theoretical and spheri- cal astronomy can be supplied. The chronometer and sextant are practically the only astronomical instruments of any use at sea. 58 DIP OF THE HORIZON. CHAPTER III. CORRECTIONS TO ASTRONOMICAL OBSERVATIONS, DIP OF THE HORIZON, PARALLAX, SEMI-DIAMETER, REFRACTION, AND TWILIGHT. 81. Dip of the Horizon. — In observations of the altitude of a heavenly body at sea, where the measurement is made from the sea-line, a correction is needed on account of the fact that this visible horizon does not coincide with the true astronomical horizon (which is 90° from the zenith), but -jl falls sensibly below it by an amount known as the Dip of the Horizon. The amount of this dip depends upon the size of the earth and the height of the observer's eye above the sea-level. In Fig. 33, is the centre of the earth, AB a portion of its level surface, and the observer, at an elevation h above A. The line OH is truly horizontal, while the tangent line, OB, corresponds to the line drawn from the eye to the visible horizon. The angle HOB is the dip. This is obviously equal to the angle OCB at the centre of the earth, if we regard the earth as spherical, as we may do with quite sufficient accuracy for the purpose in hand. From the right-angled triangle OBO we have directly cos OCB = BQ. CO Putting R for the radius of the earth, and A for the dip, this becomes cos A = — — . R + h This formula is exact, but inconvenient, because it gives the small angle A by means of its cosine. Since, however, 1 — cosA = 2sin 2 JA, we easily obtain the following : — Fig. 33. — Dip of the Horizon. sin£A = y 2(R + h) DIP OF THE HORIZON. 59 This gives the true depression of the sea horizon, as it would be if the line of sight, drawn from the eye to the horizon line, were straight. On account of refraction it is not straight, however, and the amount of this "terrestrial refraction" is very variable and uncertain. It is usual to diminish the dip computed from the formula by one-eighth its whole amount. An approximate formula 1 for the dip is A (in minutes of arc) = V/i (feet) ; or, in words, the square root of the elevation of the eye {in feet) gives the dip in minutes. This gives a value about ^\ part too large. < Since the dip is applicable 01113- to sextant observations made at sea, where, from the nature of the instrument, and the rising and falling of the observer with the vessel's motion, it is not possible to measure altitudes more closely than within about 15", there is no need of any extreme precision in its calculation. 1 This approximate formula may be obtained thus : — " ' R + h \Rj \ Rj But since - is a very small fraction, it may be neglected in the divisor [ 1 + - ), R \ R) and the expression becomes simply, 2sin 2 jA = --; whence sin J A = -» l-i-. R \2 R Since A is a very small angle, A = sin A = 2 sin \ A, so that ~h A (in radians) = 2 ^= ^ To reduce radians to minutes, we must multiply by 3438, the number of minutes in a radian. Accordingly, A' (in minutes of are) = 3438 .« fr-p- If we express h in feet, we must also use the same units for R. The mean radius of the earth is about 20,884,000 feet, one-half of which is 10,442,000, and the square root of this is 3231 ; so that the formula becomes A . 3438 . . A= 3231 V ^ feet )' which is near enough to that given in the text. In fact, the refraction makes so much difference that even after taking the numerical factor, — 8 , as unity, the formula still gives A' about 2 V part too large. The formula A' = V 3 h {metres) is yet more nearly correct 60 CORRECTIONS TO OBSERVATIONS. 82. Parallax. — In the most general sense, "parallax " is the change of a body's direction resulting from the observer's displacement. In the restricted and technical sense in which we are to employ it now, it may be defined as the difference between the direction of a body as actually observed and the direction it would have if seen from the earth's centre. Thus in the figure, Fig. 34, where the observer is supposed to be at 0, the position of P in the^JK' (as seen from 0) would be marked by the point where OP pifli^red would pierce the celestial sphere. Its position as seen from C would be determined in the same way by producing . OP to which OX is drawn parallel. The angle POX, therefore, or its equal, OPG, is the parallax of P for an observer at 0. Obviously, from the figure, we may also give the following defini- tion of the parallax. It is the angu- lar distance (number of seconds of arc) between the observer's station and the centre of the earth's disc, as seen from the body observed. The moon's parallax at any moment for me is my angular distance from the earth's cen- tre, as seen by ' ' the man in the moon." "When a body is in the zenith its parallax is zero, and it is a maxi- mum at the horizon. In all cases it depresses a body, diminishing the altitude without changing tlie azimuth. The ' ' law " of the parallax is , that it varies as the sine of the zenith dis- tance directly, and inversely as the linear distance (in miles) of the body. This follows easily from the triangle COP, where we have PC : 00 = sin COP : sin CPO. Put D for PC, the distance of the body from the earth ; R for the earth's radius, CO ; p for CPO, the parallax ; £ for ZOP, the appar- ent zenith distance, and remember that the sine of - £ is equal to the sine of its supplement, COP: we then have as the translation of the above proportion, D : R = sin £ : sin p. R Fig. 34. — Diurnal Parallax. This gives us or, since p is always a small angle, '» P = -p siu £ ; R p" = 206265"-. sin £. D b PARALLAX. 61 83. Horizontal Parallax. — When a body is at the horizon ^P h in the figure), then £ becomes 90°, and sin £ =1. In this case the par- allax reaches its maximum value, which is called the horizontal paral- lax of the body. Takings as the symbol for this, we have sinft=- ; or, nearly enough, p h = 206265"-. Comparing this with the fonmm above, we see that the parallax of a body at any zenith distance equals the horizontal parallax multiplied by the sine of the zenith distance; i.e.,^}=p 4 sin£. N.B. A glance at the figure will show that we may define the horizontal parallax, OP h C, of any body, as the angular semi-diameter of the earth seen from that body. To say, for instance, that the sun's horizontal parallax is 8".8, amounts to saying that, seen from the sun, the earth's apparent diameter is twice 8".8, or 17".6. 84. Relation between Horizontal Parallax and Distance. — Since we have B sm p h = - , it follows of course that D = R -=- sin p h ; / i \ r> 206265" „ or, (nearly) D = „ X R- p" If the sun's parallax equals 8"-8, its distance = 2 ° Q 62 Q 65 X R = 23439 R. 8.8 85. Equatorial Parallax. — Owing to the " ellipticity " or "ob- lateuess " of the earth the horizontal parallax of a body varies slightly at different places, being a maximum at the equator, where the distance of an observer from the earth's centre is greatest. It is agreed to take as the standard the equatorial horizontal parallax ; i.e. , the earth's equatorial semi-diameter as seen from the body. 86. Diurnal Parallax. — The parallax we have been discussing is sometimes called the diurnal parallax, because it runs through all its possible changes in one day. When the sun, for instance, is rising, its parallax is a maximum, and by throwing it down towards the east, increases its apparent right ascension. At noon, when the sun is on the meridian, its parallax is a minimum, and 62 CORRECTIONS TO OBSERVATIONS. affects only the declination. At sunset it is again a maximum, but now throws the sun's apparent place down towards the west. Although the sun is invisible while below the horizon, yet the parallax, geometrically considered, again becomes a minimum at midnight, regaining its original value at the next sunrise. The qualifier, '" diurnal," is seldom used except when it is neces- sary to distinguish between this kind of parallax and the annual parallax of the fixed stars, which is due to the earth's orbital motion. The stars are so far away that they have no sensible diurnal parallax (the earth is an infinitesimal point as seen from them) ; but some of them do have a slight and measurable annual parallax, by means of which we can roughly determine their distances. (Chap. XIX.) 87. Smalhiess of Parallax. — The horizontal parallax of even the nearest of the heavenly bodies is always small. In the case of the moon the average value is about 57', varying with her continually changing distance. Excepting now and then a stray comet, no other heavenly body ever comes within a distance a hundred times as great as hers. Venus and Mars approach nearest, but the parallax of neither of them ever reaches 40". 88. Semi-Diameter. — In order to obtain the true altitude of an object it is necessary, if the edge, or "limb," as it is called, has been observed, to add or deduct the apparent semi-diameter of the object. In most cases this will be sensibly the same in all parts of the sky, but the moon is so near that there is quite a perceptible difference between her diameter when in the zenith and in the horizon. A glance at Fig. 34 shows that in the zenith the moon's distance is less than at the horizon, by almost exactly the earth's radius — the difference between the lines OZ and OP h . Now this is very nearly one-sixtieth part of the moon's distance, and consequently the moon, on a night when its apparent diameter at rising is 30*, will be 30" larger when near the zenith. Since the semi-diameter given in the almanac is what would be seen from the centre of the earth, every measure of the moon's distance from stars or from the horizon will require us to take into account this " augmentation of the semi-diameter," as it is technically called. The formula, easily deduced from the figure by remembering that the angle PCO = £-p (zenith distance -parallax), and that the apparent and " almanac " diameters will be inversely proportional to the two distances UP and CP, is apparent semi-diameter = almanac s. d. x — sln ^ ■ sin (£,-p) REFRACTION. 63 This measurable increase of the moon's angular diameter at high altitudes has nothing- to do with the purely subjective illusion which makes the disc look larger to us when near the horizon. That it is a mere illusion may be made evident by simply looking through a dark glass just dense enough to hide the horizon and intervening land- scape. The moon or stra then seems to shrink at once to normal dimensions. 89. Refraction. — Rays of light have their direction changed by refraction in passing through the air, and as the direction in which we see a body is that in which its light reaches the eye, it follows that thb refraction apparently dis- places the stars and all bodies seen through the atmosphere. So far as the action is regular, the effect is to bend the rays directly downwards, and thus to make the objects appear higher in the sky. Refraction increases the altitude of a celestial ob- ject without altering the azimuth. Like parallax, it is zero at the zenith and a maximum at the horizon ; but it follows a different law. It is entirely independent of the distance of the object, and its amount varies (nearly) as the tangent of the zenith distance — not as the sine, as in the case of parallax. 90. This approximate law of the refraction is easily proved. Suppose in Fig. 35 that the observer at sees a star in the direction OS, at the zenith distance ZOS or £. The light has reached him from S> by a path which was straight until the ray met the upper surface of the air at A, but afterwards curved continually downwards as it passed from rarer to denser regions. We know that the atmosphere is very shallow as compared with the size of the earth, and it is exceedingly rare in the upper portions, so that, as far as concerns refraction, we may assume that the point A, where the first perceptible bending of the ray occurs, is not more than fifty miles high, and that the vertical AZ> is sensibly parallel to OZ ; consequently also, Fig. 35. — Atmospheric Refraction. 64 CORRECTIONS TO OBSERVATIONS. that all the successive "strata of equal density" are parallel to each other and to the upper surface of the air. [This amounts to neglecting the earth's curvature between O and B.] The true zenith distance (as it would be if there were no refraction) is ZDS', which equals Z'AS'; and since the refraction, r, may be defined as the difference between the true and apparent zenith distances, this true zenith distance will = £ + r. Now from optical principles, when a ray of light passes through a medium composed of parallel strata, the final direction of the ray is the same as if the medium had throughout the density of the last stratum, and therefore the final direction, SO, will be the same as if all the air, from A down, had the same density as at 0, with the same index of refraction, n. We may therefore apply the law of refraction directly at A, and write sin Z'A S<= n sin BA C ( = ZOS), or sin (£ + r) = n sin £ ; AC being drawn parallel to OS. Developing the first member, we have sin £ cos r + cos £ sin r = n sin £. But r is always a small angle, never exceeding 40' ; we may therefore take cos r= 1. Doing this and transposing the first term, we get cos £ sin r= n sin £ — sin £ = (n — 1) sin £. Whence, sin r = (n — 1) tan £ ; or, r» =(»»-!) 206265 tan £ (nearly) . The index of refraction for air, at zero centigrade and a barometric pressure of 760 mm , is 1.000294 ; whence, r" = .000294 x 206265 X tan£ = 60". 6 tan £. This equation holds very nearly indeed down to a zenith distance of 70°, but fails as we approach the horizon. For rays coming nearly horizontal, the points A and B are so far from O that the normal AZ' is no longer practically parallel to OZ ; and many of the other fundamental assumptions on which the formula is based also break down. At the horizon, where £ = 90° and tan £ = infinity, the formula would give sin r = infinity also ; an absurdity, since no sine can exceed unity. The refraction there is really about 37', under the circumstances of temperature and pressure above indicated. REFRACTION. 65 91. Effect of Temperature and Barometric Pressure. — The index of refraction of air depends of course upon its temperature and pressure. As the air grows warmer, its refractive power decreases; as it grows denser, the refraction increases. Hence, in all precise observations of the altitude (or zenith distance), it is necessary to note both the thermometer and the barometer, in order to compute the refraction with accuracy. For rough work, like ordinary sextant observations, it will answer to use the " mean refraction," corresponding to an average state of things. Tables of Refraction. — The computation of the refraction is best effected by special tables made for the purpose ; of these, Bessel's tables are the most convenient, best known, and probably even yet the most accurate. It must be always borne in mind, however, that from the action of wind and other causes the condition of the air along the path of the ray is seldom per- fectly normal; in consequence, the actual refraction in any given case is lia- ble to differ from the computed by as much as one or even two per cent. No amount of care in observation can evade this difficulty ; the only remedy is a sufficient repetition of observations under varying atmospheric condi- tions. Observations at an altitude below 10° or 15° are never much to be trusted. Lateral Refraction. — When the air is much disturbed, sometimes ob- jects are displaced horizontally as well as vertically. Indeed, as a general rule, when one looks at a star with a large telescope and high power, it will seem to "dance" more or less — the effect of the varying refraction which continually displaces the image. 92. Effect on the Time of Sunrise and Sunset. —The horizontal re- fraction, ranging as it does from 34' to 39', according to temperature, is always somewhat greater than the diameter of either the sun or the moon. At the moment, therefore, when the sun's lower limb appears to be just rising, the whole disc is really below the plane of the horizon; and the time of sunrise in our latitudes is thus accelerated from two to four minutes, according to the inclination of the sun's diurnal circle to the horizon, which inclination varies with the time of the year. Of course, sunset is delayed by the same amount, and thus the day is lengthened by refraction from four to eight minutes, at the expense of the night. 93. Effect on the Form and Size of the Discs of the Sun and Moon. — Near the horizon the refraction changes very rapidly. While un- der ordinary summer temperature it is about 35' at the horizon, it is 66 DETERMINATION OF THE REFRACTION. only 29' at an elevation of half a degree ; so that, as the sun or moon rises, the bottom of the disc is lifted 6' more than the top, and the vertical diameter is thus made apparently about one-fifth part shorter than the horizontal. This distorts the disc into the form of an oval, flattened on the under side. In cold weather the effect is much more marked. As the horizontal diameter is not at all increased by the refraction, the apparent area of the disc is notably diminished by it ; so that it is evident that refraction cannot be held in any way re- sponsible for the apparent enlargement of the rising luminary. 94. Determination of the Refraction. — 1. Physical Method. Theory furnishes the law of astronomical refraction, though the mathematical expression becomes rather complicated when we attempt to make it exact. In order, therefore, to determine the astronomical refraction under all possible circumstances, it is only necessary to determine the index of refraction of air, and its variations with tem- perature and pressure, by laboratory experiments, and to introduce the constants thus obtained into the formulae. It is difficult, however, to make these determinations with the necessary precision . In fact, at present our knowledge of the constauts of air rests mainly on astro- nomical work. 2. By Observations of Circumpolar Stars. At an observatory whose latitude exceeds 45° select some star which passes through the zenith at the upper culmination. (Its declination must equal the latitude of the observatory.) It will not be affected by refraction at the zenith, while at the lower culmination, twelve hours later, it will. With the meridian circle observe its polar distance in both positions, determin- ing the " polar point " of the circle as described on pp. 46—47. If the polar point were not itself affected by refraction, the simple differ- ence between the two results for the star's polar distance, obtained from the upper and lower observations, would be the refraction at the lower point. As a first approximation, however, we may neglect the refraction at the pole, and thus obtain a first approximate lower refraction. By means of this we may compute an approximate polar refraction, and so get a first "corrected polar point." With this compute a second approximate lower refraction, which will be much more nearly right than the first ; this will give a second " corrected polar point" ; this will in turn give us a third approximation to the refraction ; and so on. But it would never be necessary to go beyond the third, as the approximation is very rapid. If the star does not go exactly TWILIGHT. 67 through the zenith, it is only necessary to compute each time approxi- mate refractions for its upper observation, as well as for the polar point. At present, however, the refraction is so well known that the method actually used is to form "equations of condition" from the observations of the altitude of known stars under varying circum- stances, and from these to deduce such corrections to the star places and refraction constants as will best harmonize the whole mass of material. 95. 3. By Observations of the Altitudes of Equatorial Stars made at an Ob- servatory near the Equator. For an observer so situated, stars that are on the celestial equator (8 = 0) will come to the meridian at the zenith, and will rise and fall vertically, with a motion strictly proportional to the time; the true zenith distance of the star at any moment being just equal to its hour-angle in degrees. We have only, then, to observe the apparent zenith distance of a star with the corresponding time, and the refraction comes out directly. If the station is not exactly on the equator, and if the star's declination is not exactly zero, it is only necessary to know the latitude and declination approximately in order to get the refraction very accurately : a considerable error in either latitude or declination will affect the result but slightly. 4. The French astronomer Loewy has recently proposed a method which promises well. He puts a pair of reflectors, inclined to each other at a con- venient angle of from 45° to 50° (a glass wedge with silvered sides), in front of the object-glass of an equatorial. This will bring to the eye two rays which make a strictly constant angle with each other, and there is no diffi- culty in finding pairs of stars so situated that their images will come into the field of view together. Now, were it not for refraction, these images would always keep their relative position unchanged, notwithstanding the diurnal motion; but on account of the changes in the refraction, as one star rises and the other falls, they will shift in the field, and micrometric meas- ures will determine the shifting, and so the refraction, with great precision. 96. Twilight. — (Although this subject is outside the main purpose of this chapter, which deals with corrections to be applied to astronomical observations, we treat it here because, like refraction, it is a purely atmospheric phenomenon, and finds no other more convenient place.) Twilight, the illumination of the sky which begins before sunrise, and continues after sunset, is caused by the reflection of light to the observer from the upper regions of the earth's atmosphere. It is not yet certain whether this is due to reflection from foreign matter in the air, such as minute crystals of ice and salt, particles of dust of various kinds, and infinitesimal drops of water, or whether the pure gases themselves have some power of reflecting light. There is no 68 TWILIGHT. doubt, however, that air, under the ordinary conditions, possesses considerable power of reflection ; so that, as long as any air upon which the sun is shining is visible to the observer, it will send him more or less light, and appear illuminated. Suppose the atmosphere to have the depth indicated in the figure. Then, if the sun is at S, Fig. 36, it will just have set to an observer at 1 , but all the air within his range of vision will still be illuminated. When, by the earth's rotation, he has been transported to 2, he will see the "twilight bow" rising in the east, a faintly reddish arc separating the illuminated part of the sky from the darkened part below, which lies in the shadow of the earth. When he reaches 3, the western half of the sky alone remains blight, but the arc of separation be- Fig. 36. — Twilight. tween the light and darkness has become vague and indefinite ; when he reaches 4, only a glow remains in the west ; and when he comes to 5, night closes in on him. Nothing remains in sight on which the sun is shining. 97. Duration of Twilight. — This depends upon the height of the atmos- phere, and the angle at which the sun's diurnal circle cuts the horizon. It is found as a matter of observation, not admitting, however, of much precision, that twilight lasts until the sun has sunk about 18° below the horizon ; that is to say, the angle 1 C 5 in the figure is about 18°. The time required to reach this point in latitude 40° varies from two hours at the longest days in summer, to one hour thirty minutes about Oct. 12 and March 1, when it is least. At the winter solstice it is about one hour and thirty-five minutes. In higher latitudes the twilight lasts longer, and the variation is more considerable : the date of the minimum also shifts. Near the equator the duration is shorter, hardly exceeding an hour at the HEIGHT OP THE ATMOSPHERE. 69 sea-level; while at high elevations (where the amount of air above the observer's level is less) it becomes very brief. At Quito and Lima it is said not to last more than twenty minutes. Probably, also, in mountain regions the clearness of the air, and its purity contribute to the effect. 98. Height of the Atmosphere. — It is evident from the figure that at the moment twilight ceases, the last visible portion of illuminated air is at the top of the atmosphere, and just half-way between the ob- server and the nearest point where the sun is setting. If the whole arc 1, 5 is 18°, 1, 3 is 9° : then calling the height of the atmosphere H and the earth's radius R, and neglecting refraction (i.e., supposing the lines 1 m and 5 m to be straight) , we have from the right-angled triangle 1 Cm, Cm = 1 C X sec 9°, or R + H = R x sec 9° ; whence H= R (sec 9°— 1) = 0.0125 R, or almost exactly fifty miles. This must be diminished about one-fifth part on account of the curvature of the lines \m and 5m by refraction, making the height of the atmos- phere about forty miles. The result must not, however, be accepted too confidently. It only proves that we get no sensible twilight illumination from air at sc greater height : above that elevation the air is either too rare, or too pure from foreign particles, to send us any perceptible reflection. There is abundant evidence from the phenomena of meteors that the atmosphere extends to a height of 100 miles at least, and it cannot be asserted positively that it has any definite upper limit. 99. Aberration. — There is yet one more correction which has to be applied in order to get the true direction of the line which at the instant of observation joins the eye of the observer to the star he is pointing at. The aberration of light is an apparent displacement of the object observed, due to the combination of the earth's orbital motion with the progressive motion of light. It can be better discussed, however, in a different connection (see Chap. VI.), and we content ourselves with merely mentioning it here. 70 PEOBLBMS OF PRACTICAL ASTRONOMY. CHAPTER IV. PROBLEMS OF PRACTICAL ASTRONOMY, LATITUDE, TIME, LONGITUDE, AZIMUTH, AND THE RIGHT ASCENSION AND DECLINATION OF A HEAVENLY BODY. 100. There are certain problems of Practical Astronomy which have to be solved in obtaining the fundamental facts from which we deduce our knowledge of the earth's form and dimensions, and other astronomical data. The first of these problems is that of the LATITUDE. The latitude (astronomical) of a place (Art. 30) is simply the altitude of the celestial pole (Polhohe), or, what comes to the same thing, as is evident from Fig. 7 (Art. 33 ) , it is the declination of the zenith. It may also be defined, from the mechanical point of view, as the angle between the plane of the earth's equator and the observer's plumb-line or vertical. Neither of these definitions assumes anything as to the form of the earth, and we shall find farther on that this astronomical latitude is seldom identical with the geocentric, nor even with the geodetic latitude of a place. It is, however, the only kind of latitude which can be directly determined from astronomical observations, and its determination is one of the most impor- tant operations of what may be called Economic Astronomy. ( 101. Determination of Latitude. — First: By Circumpolars. The most obvious method of determining the latitude is to observe, with the meridian circle or some analogous instrument, the altitude of a circum- polar star at its upper culmination, and again, twelve hours later, at its lower. Each of the observations must be corrected for refraction, and then the mean of the two corrected altitudes ivill be the latitude. This method has the advantage of being an independent one; i.e., it does not require any data (such as the declination of the stars used) to be ac- cepted on the authority of previous observers. But to obtain much accuracy it requires considerable time and a large fixed instrument. In low latitudes the refraction is also very troublesome. LATITUDE. 1\ 102. Second : By the meridian altitude or zenith distance of a body of known declination. In Fig. 37 the semicircle AQPB is the meridian, Q and P being respectively the equator and the pole, and Z the zenith. QZ is the declination of the zenith, or the observer's latitude (=PB=). Suppose now that we observe Zs (=£,), the zenith distance of a star s (south of the zenith), as it crosses the meridian, and that its declination Qs (= 8„) is known ; then evidently = 8, + £,. In the same way, if the star were at n, between zenith and pole, If we use the meridian circle, we can always select stars that pass near the zenith where the refraction will be small; moreover, we can select them in such a way that some will be as much north of the zenith as others are south, and thus eliminate the refraction errors. But we have to get our star declinations out of catalogues made by previous observers, and so the method is not an independent one. 103. At Sea the latitude is usually obtained by observing with the sex- tant the sun's maximum altitude, which of course occurs at noon. Since at sea it is seldom that one knows beforehand precisely the moment of local noon, the observer takes care to begin to ob- serve the sun's altitude some ten or fifteen minutes earlier, repeating his ° . . Fig. 37. — Determination of Latitude. observations every minute or two. At first the altitude will keep increasing, but immediately after noon it will begin to decrease. The observer uses therefore the maximum' altitude obtained, which, corrected for refraction, parallax, seiui- diameter, and dip of the horizon, will give him the true latitude of his ship, by the formula <£ = S ± £. 104. Third : By Circum-meridian Altitudes. — If the observer knows his time with reasonable accuracy, he can obtain his latitude from observa- tions made when the body is near the meridian, with practically the same precision as at the moment of meridian passage. It would take us a little 1 On account of the sun's motion in declination, and the northward or south- ward motion of the ship itself, the sun's maximum altitude is usually attained not precisely on the meridian, but a few seconds earlier or later. This requires a slight correction to the deduced latitude, explained in books on Navigation or Practical Astronomy. 72 PROBLEMS OP PRACTICAL ASTBONOMY. too far to explain the method of reduction, which is given with the necessary tables in all works on Practical Astronomy. The great advantage of this method is that the observer is not restricted to a single observation at each meridian-passage of the sun or of the selected star, but can utilize the half- hours preceding and following that moment. The meridian-circle cannot be used, as the instrument must be such as to make extra-meridian observations possible. Usually the sextant or universal instrument is employed. This method is much used in the French and German geodetic surveys. 105. Fourth : The Zenith Telescope Method. — (Sometimes known as the American method, because first practically introduced by Captain Talcott of the United States Engineers, in a boundary survey in 1845.) The essential characteristic of the method is the micrometric meas- urement of the difference between the nearby equal zenith distances of two stars which culminate within a few minutes of each other, one north and the other south of the zenith, and not very far from it : such pairs of stars can always be found. When the method was first introduced, a special instrument, known as the zenith telescope, was generally employed, but at present a simple transit instrument, with declination micrometer, aud a delicate level attached to the tele- scope tube, is ordinarily used. The telescope is set at the proper alti- tude for the star which first comes to the meridian, and the " latitude level," as it is called, is set horizontal ; as the star passes through the field of view its distance north or south of the central wire is measured by the micrometer. The instrument is then reversed, and so set by turning the telescope up or down {without, however, disturbing the angle $ (Fig. 38) between the level and telescope), that the level is again horizontal. After this reversal and adjustment, the telescope tube is then evidently elevated at exactly the same angle, £, as before, but on the opposite side of the zenith. As the second star passes through the field, we measure with the micrometer its distance north or south of the centre of the field ; the comparison of the two micrometer measures gives the difference of the two zenith distances. Fig. 38. Principle of the Zenith Telescope. LATITUDE. 73 From Fig. 37 we have for star south of zenith, = 8, + £, ; for star north of zenith, = tan S sec t. 74 PROBLEMS OF PRACTICAL ASTRONOMY. if 8 nearly equals , t will be small, and a considerable error in the obser- vation of l will then produce very little change in its secant or m the com- puted latitude. The observations are not so convenient and easy as in the case of the zenith telescope, and the number of stars available is less; but the method presents the great advantage of requiring nothing but an ordinary transit instrument, without any special outfit of micrometer and latitude level. It also entirely evades the difficulties caused by refraction. 107. Sixth: By the Gnomon. — The ancients had no instruments such as we have hitherto described, and of course could not use auy of the preceding methods of finding the latitude. They were, how- ever, able to make a very respectable approximation by means of the simplest of all astronomical instruments, the gnomon. This is merely a vertical shaft or column of known height erected on a perfectly horizontal plane ; and the observation consists in not- ing the length of the shadow cast at noon at certain times of the year. Suppose, for instance, that on the day of the summer solstice, at noon, the length of the shadow is AC, Fig. 40. The height AB being given, we can easily compute in the right-angled triangle the angle ABO, which equals SBZ, the sun's zenith dis- tance when farthest north. Again observe the length AD of the shadow at noon of the shortest day in winter, and compute the angle ABD, which is the sun's corresponding zenith distance when farthest south. Now, since the sun travels equal distances north and south of the celestial equator, the mean of the two results will give the angular distance between the equator and the zenith; i.e., the decli- nation of the zenith, which (Art. 100) is the latitude of the place. The method is an independent one, like that by the observation of cir-» cumpolar stars, requiring no data except those which the observer determines for himself. Evidently, however, it does not admit of much accuracy, since the penumbra at the end of the shadow makes it impossible to measure its length very precisely. It should be noted that the ancients, instead of designating the position A. C E Fig. 40. — Latitude by the G-nomon. DETERMINATION OP TIME. 75 of a place by means of its latitude, used its climate instead; the climate (from kXI/ul) being the slope of the plane of the celestial equator, the angle A EB, which is the complement of the latitude. Tt is supposed, indeed known, .that many of the Egyptian obelisks were erected primarily to serve as gnomons, and were used for that purpose. t 108. 1 Possible Variations of the Latitude, -it is an interesting ques- tion whether the position of the earth's axis is fixed with reference to its mass and surface. Theoretically it is hardly possible that it should be, because any change in the arrangement of the matter of the earth, by denudation, subsidence, or elevation, would almost necessarily disturb it. If so disturbed, the latitudes of places toward which the pole approached would be increased, and those on the opposite side would be decreased. At present we can only say that if such disturbance has occurred, it must have been extremely slight for the last 200 years, not exceeding 40 or 50 feet at most; but there are suspicions of a minute and progressive change of the latitude of some of the observatories (notably PuLkowa), which have drawn attention to the matter, and the subject is under investigation. TIME AND ITS DETERMINATION. 109. One of the most important problems presented to the astron- omer is the determination of Time. By universal consent the appar- ent rotation of the heavens is made to furnish the standard, and the determination of time is effected by ascertaining by observation the hour-angle of the object selected to mark the beginning of the day by its transit across the meridian. In practice, three kinds of time are now recognized, viz., sidereal time, apparent solar time, and mean solar time. 110. Sidereal Time. — As has already been explained (Art. 26), the sidereal time at any moment is the hour-angle of the vernal equinox at that moment; or, what comes to the same thing, though it sounds dif- ferently, it is the time marked by ajilock which is so set and adjusted as to show noon, or h 00 m 00 s , at each transit of the vernal equinox. The sidereal day, thus defined, is the time intervening between two successive transits of the same star; at least, it is so within the hundredth part of a second, though on account of the precession of the equinoxes (and the proper motions of the stars) the agreement is not absolute, the difference amounting to about one day in twenty- six thousand years. 111. Apparent Solar Time. — Just as sidereal time is the hour-angle of the vernal equinox, so at any moment the apparent solar time is the hour-angle of the sun. It is the time shown by the sun-dial, and i See p. 90\ 76 PEOBLEMS OP PRACTICAL ASTRONOMY. its noon is when the sun crosses the meridian. On account of the annual eastward motion of the sun among the stars (due to the earth's orbital motion) , this day is about four minutes longer than the sidereal ; and moreover, because the sun's motion in right ascen- sion is not uniform, the apparent solar days are not all of the same length, nor, consequently, its hours, minutes, or seconds. December 23d is fifty-one seconds longer from (apparent) noon to noon than Sept. 16th. For this reason, apparent solar time is not satisfactory for scientific use, and has long been discarded in favor of mean solar time. Until within about a hundred years, however, it was the only kind of time commonly employed, and its use in the city of Paris was not discontinued until the year 1816. 112 . Mean Solar Time. — A "fictitious sun " is therefore imagined, which moves uniformly and in the celestial equator, completing its annual course in exactly the same time as that in which the actual sun makes the circuit of the ecliptic. It is mean noon when this " ficti- tious sun " crosses the meridian, and at any moment the hour-angle of this "fictitious sun " is the mean time for that moment. Sidereal time will not answer for business purposes, because its noon (the transit of the vernal equinox) occurs at all hours of night and daylight in different seasons of the year. Apparent solar time is scientifically unsatis- factory, because of the variation in the length of its days and hours. And yet we have to live by the sun ; its rising and setting, daylight and night, control our actions. In mean solar time we find a satisfactory compromise, an invariable time unit, and still an agreement with sun-dial time close enough for convenience. It is the time now used for all purposes except in certain astronomical work. The difference between apparent time and mean time, never amounting to more than about a quarter of an hour, is called the equa- tion of time, and will be discussed hereafter in connection with the earth's orbital motion, Chap. VI. The nautical almanac furnishes data by means of which the sidereal time may be deduced from the corresponding solar, or vice versa, by a very brief and simple calculation. 113. In practice the problem of determining the time always takes the form of ascertaining the error of a time-piece; that is, the amount by which a clock or watch is fast or slow of the time it ought to show. The methods most in use by astronomers are the following : — First. By means of the transit instrument. Since the right ascen- sion of a star is the sidereal time of its passage across the meridian DETERMINATION OF TIME. 77 (Art. 26), it is obvious that the difference between the right ascension of a known star and the time shown by a sidereal clock at the instant when the star crosses the middle wire of an accurately adjusted transit instrument, is the error of the clock at that moment. Prac- tically, it is usual to observe a number of stars (from eight to ten), reversing the instrument once at least, so as to eliminate the collima- tion error (Art. 60) . With a good instrument a skilled observer can determine this clock error or "correction" within about one-thirtieth of a second of time, provided proper means are taken to ascertain and allow for his " personal equation." 114. Personal Equation. — It is found that every observer has his own peculiarities of time observation with a transit, and his "personal equation" is the amount to be added (algebraically) to the time observed by him, in order to get the actual moment of transit as it would be recorded by some supposable arrangement, which should automatically register the moment when the star's image was bisected by the wire. This personal equation differs for different observers, but is reasonably (though never strictly) constant for one who has had much practice. In the case of observations with the chronograph, it is usually less than ± B .2. It can be determined by an apparatus in which an artificial star, resembling the real stars as much as possible in appearance, is made to traverse the field of view and to telegraph its arrival at certain wires, while the observer notes the moments for himself. One of the most important problems of practical astronomy now awaiting solution is the contrivance of some practical method of time observation free from this annoying human element, the personal equation, which is always more or less uncertain and variable. If mean time is wanted, it can be deduced from the sidereal time by the data of the almanac. The sun can also be observed instead of the stars, the moment of the sun's .transit being that of apparent noon ; but this observation, for many reasons, is far less accurate and satisfactory than observa- tions of the stars. 115. Second. The method of equal altitudes. — If we observe with a sextant in the forenoon the time shown by the chronometer when the sun attains the height indicated by a certain reading of the sextant, and then in the afternoon, the time when the sun again reaches the same 78 PROBLEMS OF PRACTICAL ASTRONOMY. altitude, the moment of apparent noon will be half-way between the two observed times ; provided, of course, that the chronometer runs uniformly during the interval, and also provided that proper correc- tion is made for the sun's slight motion in declination — a correction easily computed. The advantage of this method is that the errors of graduation of the sextant h:ive no effect, nor is it necessary for the observer to know his latitude except approximately. Per contra, there is, of course, danger that the afternoon observa- tions may be interfered with by clouds ; and, moreover, both obser- vations must be made at the same place. A modification of this method is now coming into extensive use, in which two different stars of known right ascension and of nearly the same declination are used, at equal altitudes east and west of the meridian. 116. Third. By a single altitude of the sun, the observer's latitude being known. — This is the method usual at sea. The altitude of the sun having been measured with the sextant, and the corresponding time shown by the chronometer having been accurately noted, we compute the hour-angle of the sun, P, from the triangle ZPS (Fig. 41), and this hour-angle cor- rected for the equation of time, gives the true mean time at the observed moment. The difference between this hi fnd that shown by the chro- Fig. 41.- Determination of Time by a Single Altitude. n0meter IS the error of the chronometer. In the triangle ZPS all three of the sides are given, viz. : PZ is the complement of the latitude cj>, which is supposed to be known ; PS is the com- plement of the declination S, which is found in the almanac, as is also the equation of time ; while ZS or £, is the complement of the sun's altitude, as measured by the sextant, and corrected for semi- diameter, refraction, and parallax. The formula is sin t j. p = / sin j K + (*-«)] sin i K - (» - ») J \*. \ cos cos S J In order to accuracy, it is desirable that the sun should be on the prime vertical, or as near it as practicable. It should not be near the DETERMINATION OP TIME. 79 meridian. Any slight error in the assumed latitude produces no sensible effect upon the result, if the sun is exactly east or west at the time the observation is taken. The disadvantage of the methqd is that any error of graduation of the sextant enters into the result with its full effect. In some cases a person is so situated that it is necessary to determine his time roughly, without instruments ; and this can be done within about a half a minute by establishing a noon-mark, which is nothing but a line drawn exactly north and south, with a plumb-line, or some vertical edge, like the edge of a door-frame or window-sash, at its southern extremity. The shadow will then always fall upon the meridian line at apparent noon. 117. The Civil and the Astronomical Day. — The astronomical day begins at mean noon. The civil day begins at midnight, twelve hours earlier. Astronomical mean time is reckoned round through the whole twenty-four hours, instead of being counted in two series of twelve hours each. Thus, 10 a.m. of Wednesday, May 2, civil reckoning, is Tuesday, May 1, 22 h , by astronomical reckoning. Beginners need to bear this in mind in using the almanac. LONGITUDE. 118. Having now methods of obtaining the true local time, we can attack tlie problem of longitude, which is perhaps the most important of all the economic problems of astronomy. The great observatories at Greenwich and at Paris were established simply for the purpose of furnishing the observations which could be made the basis of the accurate determination of longitude at sea. The longitude of a place on the earth is the angle at the pole between the meridian of Greenwich and the meridian passing through the ob- server's place; or it is the arc of the equator intercepted between these meridians ; or, what comes to the same thing, since this arc is measured by the time required for the earth to turn sufficiently to bring the second meridian into the same position held by the first, it is simply the difference of their local times, — the amount by which the noon at Greenwich is earlier or later than at the observer's place. It is now usually reckoned in hours, minutes, and seconds, instead of degrees. Since it is easy for the observer to find his own local time by the methods which have been given, the knot of the problem is really this : being at any place, to find the corresponding local time at Green- wich without going there. 80 PROBLEMS OF PRACTICAL ASTRONOMY. The methods of finding the longitude may be classed under three different heads : First, By means of signals simultaneously observable at the places between which the difference of longitude is to be found. Second, By making use of the moon as a clock-hand in the sky. Third, By purely mechanical means, such as chronometers and the telegraph. 119. Under the first head we may make use of [A] A Lunar Eclipse. — When the moon enters the shadow of the earth, the phenomenon is seen at the same moment, no matter where the observer may be. By noting, therefore, his own local time at the moment, and afterwards comparing it with the time at which the phe- nomenon was observed at Greenwich, he will obtain his longitude from Greenwich. Unfortunately, the edge of the earth's shadow is so indistinct that the progress of events is very gradual, so that sharp observations are impossible. [B] Eclipses of the satellites of Jupiter may be used in the same way, with the advantage that they occur very frequently, — almost every night, in fact ; but the objection to them is the same as to the lunar eclipses, — they are not sudden. [C] The appearance and disappearance of meteors may be and has been used to determine the difference of longitude between places not more than two or three hundred miles apart, and gives very accu- rate results. (Now superseded by the telegraph.) [D] Artificial signals, such as flashes of powder and rockets, can be used between two stations not too far distant. Early in the. cen- tury the difference of longitude between the Black Sea and the Atlan- tic was determined by means of a chain of signal stations on thi mountain tops ; so also, later, the difference of longitude between the eastern and western extremities of the northern boundary of Mexico. This method is now superseded by the telegraph. 120. Second, the moon regarded as a clock. Since the moon revolves around the earth once a month, it is, of course, continually changing its place among the stars ; and as the laws of its motion are now well known, and as the place which it will occupy is predicted for every hour of every Greenwich day three years in advance in the nautical almanac, it is possible to deduce the corresponding Greenwich time by any observation which will determine the place of the moon among the stars. The almanac LONGITUDE. 81 place, however, is the place at which the moou would be seen by an observer at the centre of the earth, and consequently the actual ob- servations are in most cases complicated with very disagreeable reductions for parallax before they can be made available. The simplest lunar method is, [A] That of Moon Culminations. — AVe merely observe with a transit instrument the time when the moon's bright limb crosses the meridian of the place ; and immediately after the moon we observe one or more stars with the same instrument, to give us the error of our clock. As the moon is observed on the meridian, its paral- lax does not affect its right ascension, and accordingly, by a simple reference to the almanac, we can ascertain the Greenwich time at which the moon had the particular right ascension determined by the observation. The method has been very extensively used, and would be an admirable one were it not for the effects of personal equation. It seldom happens that the personal equation of an observer is the same for such an object as the limb of the moon as it is for a star ; and since the moon's motion among the stars is very slow, the effect of such a difference is multiplied by about 30 (roughly the number of days in a month) in its effect upon the longitude deduced. [B] Lunar-Distances. — At sea it is, of course, impossible to observe the moon with a transit instrument, but we can observe its distance from the stars near its path by means of a sextant. The distance observed will not be the same that it would be if the observer were at the centre of the earth, but by a mathematical process called "clearing a lunar" the distance as seen from the centre of the earth can be easily deduced, and compared with the distance given in the almanac. From this the longitude can be determined. Any error, however, in measuring a lunar-distance entails an error about thirty times as great in the resulting longitude, and the method is at present very little used, the moon having been superseded by the chronometer for such purposes. [C] Occultations. — Occasionally, in its passage through the sky, the moon over-runs a star, or " occults" it. The star vanishes instan- taneously, and, of course, at the moment of its disappearance the distance from the centre of the moon to the star is precisely equal to the apparent semi-diameter of the moon ; we thus have a " lunar- distance " self-measured. Observations of this kind furnish one of the most accurate methods 82 PROBLEMS OF PRACTICAL ASTRONOMY. of determining the difference of longitude between widely separated places, the only difficulty arising from the fact that the edge of the moon is not smooth, but more or less mountainous, so that the dis- tance of a star from the moon's centre is not always the same at the moment of its disappearance, [D] In the same way a solar eclipse may be employed by observing the moment when the moon's limb touches that of the sun. It -will be noticed that these two last methods (the methods of occultation and solar eclipse) do not belong in the same class with the method of lunar eclipse, because the phenomena are not seen at the same instant at different places, but the calculation of longitude depends upon the determination of the moon's place in the sky at the given time, as seen from the earth's centre. There are still other methods, depending upon measurements of the moon's position by observations of its altitude or azimuth. In all such cases, however, every error of observation entails a vastly greater error in the final results. Lunar methods (excepting oecul- tations) are only used when better ones are unavailable. 121. Finally we have what may be called the mechanical methods of determining the longitude. [A] By the chronometer ; which is simply an accurate watch that has been set to indicate Greenwich time before the ship leaves port. In order to find the longitude by the chronometer, the sailor has to determine its "error" upon local time by an observation of the alti- tude of the sun when near the prime vertical, as indicated on page 78. If the chronometer indicates true Greenwich time, the error deduced from the observation will be the longitude. Usually, however, the indi- cation of the chronometer face requires correction for the rate and run of the chronometer since leaving port. Chronometers are only imperfect instruments, and it is important, there- fore, that several of them should be used to check each other. It requires three at least, because if only two chronometers are carried and they disagree, jhere is nothing to indicate which one is the delinquent. On very long voyages the errors of chronometers are cumulative, and the uncertainty accumulates, not merely in proportion to the time, but more nearly in proportion to the square of the time ; i.e., if the error to be feared in the use of a chronometer in longitude determinations at the end of a week is about two seconds of time, at the end of the month it would be, not eight seconds, but about thirty-two seconds. If, therefore, a ship is to be at sea, without making port, more than three LONGITUDE. 83 or four months at a time, the method becomes untrustworthy, and it may be necessary to recur to lunar distances ; for voyages of less than a month the method is now, practically, all that could be desired. [B] But the method which, wherever it is applicable, has super- seded all others, is that of The Telegraph. "When we wish to find the longitude between two stations connected by telegraph, the process is usually as follows : The observers at both stations, after ascer- taining that they both have clear weather, proceed to determine their own local time by extensive series of star observations with the transit instrument. Then, at an agreed-upon time, the observer at Station A " switches his clock" into the telegraphic circuit, so that its beats are communicated along the line and received upon the chron- ograph of the other, say the western station. After the eastern clock has thus sent its signals, say for two minutes, it is switched out of the circuit, and the western observer now switches his clock into the circuit, and its beats are received upon the eastern chronograph. The operation is closed by another series of star observations. We have now upon each chronograph sheet an accurate comparison of the two clocks, showing the amount by which the western clock is •slow of the eastern. If the transmission of electric signals were instantaneous, the difference shown upon the two chronograph sheets would agree precisely. Practically, however, there will always be a small discrepancy amounting to twice the time occupied in the trans- mission of the signals ; but the mean of the two differences will be the true difference of longitude of the places after the proper correc- tions have been applied. Especial care must be taken to determine with accuracy, or to eliminate, the personal equations of the observers. It is customary to make observations of this kind on not less than five or six evenings in cases where it is necessary to determine the difference of longitude with the highest accuracy. The astronomical difference of longi- tude between two places can thus be telegraphically determined within about the one-hundredth part of a second of time ; i.e., within about ten feet or so, in the latitude of the United States. It may be noted here that the time occupied by the transmission of elec- tric signals in longitude operations is not to be taken as the real measure of " the velocity of the electric fluid " upon the wires, as was once supposed. The time apparently consumed in the transmission is simply the time re- quired for the current at the receiving station (which current probably begins at the very instant the key is touched at the other end of the line) to become strong enough to do its work in making the signal j and this time depends upon a multitude of circumstances. 84 PROBLEMS OF PRACTICAL ASTRONOMY. 122. Local and Standard Time. — In connection with time and longitude determinations, a few words on this subject will be in place. Un- til recently it has always been customary to use only local time, each observer determining his own time by his own observations. Before the days of the telegraph, and while travel was comparatively slow and infrequent, this was best ; but the telegraph and railway have made such changes that, for many reasons, it is better to give up the old system of local times in favor of a system of standard time. It facilitates all railway and telegraphic busi- ness in a remarkable degree, and makes it practically easy for every one to keep accurate time, since it can be daily wired from some observatory to every telegraph office. According to the system that is now established in this country, there are five such standard times in use, — the colonial, the eastern, the central, the mountain, and the Pacific, — which differ from Greenwich time by exactly four, five, six, seven, and eight hours respectively, the minutes and seconds being identical everywhere. At most places only one of these times is employed ; but in cities where different systems join each other, there are two standard times in use, differing from each other by exactly one hour, and from the local time by about half an hour. In some such places the local time also maintains its place. In order to determine the standard time by observation, it is only nec- essary to determine the local time by one of the methods given, and correct it according to the observer's longitude from Greenwich. 123. Where the Day Begins. — If we imagine a traveller starting from Greenwich on Monday noon, and journeying westward as swiftly as the earth turns to the east under his feet, he would, of course, keep the sun exactly on the meridian all day long, and have continual noon. But what noon ? It was Monday when he started, and when he gets back to London, twenty- four hours later, it is Tuesday noon there, and there has been no intervening sunset. When does Monday noon become Tuesday noon? The conven- tion is that the change of date occurs at the 180th meridian from Greenwich. Ships crossing this line from the east skip one day in so doing. If it is Monday forenoon when the ship reaches the line, it becomes Tuesday fore- noon the moment it passes it, the intervening twenty-four hours being dropped from the reckoning on the log-book. Vice versa, when a vessel crosses the line from the western side, it counts the same day twice, passing from Tuesday forenoon back to Monday, and having to do its Tuesday over again. This 180th meridian passes mainly over the ocean, hardly touching land anywhere. There is a little irregularity in the date upon the different islands near this line. Those which received their earliest European inhabi- tants via the Cape of Good Hope have, for the most part, the Asiatic date, belonging to the west side of the 1 80th meridian ; while those that were ap- proached via Cape Horn have the American date. POSITIOK AT SEA. 85 When Alaska was transferred from Russia to the United States, it was necessary to drop one day of the week from the official dates. THE PLACE OF A SHIP AT SEA. 124. The determination of the place of a ship at sea is commer- cially of such importance that, at the risk of a little repetition, we collect together here the different methods available for its determi- nation. The methods employed are necessarily such that observa- tions can be made with the sextant and chronometer, the only instruments available under the circumstances. The Latitude is usually obtained by observations of the sun's altitude at noon, according to the method explained in Art. 103. The Longitude is usually found by determining the error upon local time of the chronometer, which carries Greenwich time. The nec- essary observations of the sun's altitude should be made when the sun is near the prime vertical, as explained in Art. 116. In the case of long voyages, or when the chronometer has for any reason failed, the longitude may also be obtained by measuring a lunar-distance and comparing it with the data of the nautical almanac. By these methods separate observations are necessary for the lati- tude and for the longitude. 125. Sumner's Method. — Recently a new method, first proposed by Captain Sumner, of Boston, in 1843, has been coming largely into use. In this method, each observation of the sun's altitude, with the corresponding chronometer time, is made to define the position of the ship upon a certain line, called the circle of position. Two such ob- servations will, of course, determine the exact place of the vessel at one of the intersections of the two circles. At any moment the sun is vertically over some point upon the earth's surface, which maybe called the sub-solar point. An observer there would have the sun directly overhead. Moreover, if at any point on the earth an observer measures the altitude of the sun with his sextant, the zenith distance of the sun (which is the complement of this altitude) will be Ms distance from the sub-solar point at the moment of observation, reckoned in degrees of a great circle. If, then, I take a terrestrial globe, and, opening the dividers so as to cover an arc equal to this observed zenith distance of the sun, put one foot of the dividers upon the sub-solar point, and sweep a 86 PEOELEMS OF PRACTICAL ASTRONOMY. circle on the surface of the globe around that point, the observer must be somewhere on the circumference of that circle; and moreover, if to the observer the sun is in the southwest, he himself must be in the opposite direction from this sub-solar point ; i.e. , northeast of it. In other words, the azimuth of the sun at the time of observation informs him upon what part of the circle he is situated. Suppose a similar observation made at the same place a few hours later. The sub-solar point, and the zenith distance of the sun, will have changed ; and we shall obtain a new circle of position, with its centre at the new sub-solar point. The observer must be at one of its two intersections with the first circle — which of the two inter- sections is easily determined from the roughly observed azimuth of the sun. If the ship moves between the two observations, the proper allow- ance must be made for the motion. This is easily done by shift- ing upon the chart that part of the first circle of position where the ship was situated, carrying the line forward parallel to itself, by an amount just equal to the ship's run between the two observations, as shown by the log. The intersection with the second circle then gives the ship's place at the time of the second observation. The only problem remaining is to find the position of the " sub-solar point " at any given moment. Now, the latitude of this point is ob- viously the declination of the sun (which is found in the almanac) . If the sun's declination is zero, the sun is vertically over some point upon the equator. If its declination is + 20°, it is vertically over some point on the twentieth parallel of north latitude, etc. In the next place, its longitude is equal to the Greenwich apparent solar time at the moment of observation ; and this is given by the chronometer (which keeps Greenwich mean solar time) , by simply add- ing or subtracting the equation of time ; so that, by looking in his almanac and at his chronometer, the observer has the position of the sub-solar point immediately given him. (See note, page 90.) Suppose, for example, that on May 20 (the sun's declination being + 20°), at 11 a.m., Greenwich apparent time (i.e., May 19, 23 h by astronomical reck- oning), according to the chronometer, the sun is observed to have an altitude of 40° by a ship in the North Atlantic. The sub-solar point will then be (Fig. 42) at a point in Africa having a latitude of + 20°, and an east longi- tude of 15°— at A in the figure. And the radius of the " circle of position," i.e., the distance from A to C — will be 50°. Again, a second observation is made three hours later, when the sun's altitude is found to be 65°. The sub-solar point will then be at B, latitude POSITION AT SEA. 87 20°, longitude 30° W., and the radius of the circle of position BC will be 25°, C being the ship's place. Of course it would be impracticable to carry on a vessel a terrestrial globe large enough for the accurate working out of the graphical operation indicated, but tables are provided, by which the necessary portions of the position circles can be easily drawn upon the ordinary charts. W- FlG. 42. — Sumner's Method . 126. The peculiar advantage of the method is, that a single obser- vation is used for all it is worth, giving accurately the position of a line upon which the ship is somewhere situated, and approximately (by the rough observation of the sun's azimuth) the part of that line upon which its place will be found. In approaching the American coast, for instance, if an observation be taken in the forenoon when the sub-solar point is over the continent of Africa, the ship's position circle will lie nearly parallel to the coast, and then a single observa- tion will give approximately the distance of the ship from land, which may be all the sailor wishes to know. The observations need not be taken at any particular time. We are not limited to observations at noon, or to the time when the sun is on the prime vertical. It is to be noted, however, that everything depends upon the chronometer, as much as in the ordinary cbronometric determination of longitude. 127. Determination of Azimuth. — A problem, important, though not so often encountered as that of latitude and longitude determina- 88 PROBLEMS OP PRACTICAL ASTRONOMY. tions, is that of determining the azimuth, or true bearing, of aline upon the earth's surface. The process is this : With a theodolite having an accurately graduated horizontal circle the observer points alter- nately upon the pole star and upon a dis- tant signal erected for the purpose; the signal being an artificial star consisting of a small hole in a plate of metal, with a bull's-eye lantern or other light behind it. It is desirable that it should be at least a mile away from the observer, so that auy small displacement of the instrument will be harmless. The theodolite must be carefully adjusted for collimation, and especial pains must be taken to have the „ „ _. ... ... ,, axis of the telescope perfectly level. Fig. 43.— Determination of Azimuth. r r J The next morning by daylight the ob- server measures the angle or angles between the night-signal and the objects whose azimuth is required. If the pole star were exactly at the pole, the mere difference between the two readings of the circle, obtained when the telescope is pointed on the star and on the signal, would directly give the azimuth of the signal. As this is not the case, however, the time at which each observation of the pole star is made must be noted, and the azimuth of the star must be computed for that moment. This can easily be done, as the right ascension and declination of this star are given in the almanac for every day of the year. Recurring to the Z.P.S. [zenith-pole-star] triangle, N (Fig. 43) being the north point of the horizon, P the pole, and NZ the meridian, we at once see that the side PS is the complement of the star's declination ; the side PZ is the complement of the observer's latitude (which must be known) ; and the angle at P is the difference between the right ascension of the pole star and the sidereal time of the observation; [(t — a) if the star is west of the meridian at the time, and (a — t) if it is east.] This will come out in hours, of course, and must be reduced to degrees before making the com- putation. We thus have two sides of the triangle, viz., PS and PZ, with the included angle at P, from which to compute the angle Z at the zenith. This is the star's azimuth. The pole star is used because, being so near the pole, any sbght error in the assumed latitude of the place or in the sidereal time of the observation will hardly produce any effect upon the result, especially if the star be caught between five and six hours before or after its upper culmination, at POSITION OP A HEAVENLY BODY. 89 a time when it changes its azimuth very slowly (near S' or S" in the figure). The sun, or any other heavenly body whose position is given in the almanac, can also be used as a reference point in the same way, provided sufficient pains is taken to secure an accurate observation of the time at the instant when the pointing is made. The altitude should not exceed thirty degrees or so. But the results are usually rough compared with those obtained by means of the pole star. DETERMINATION OF THE POSITION OF A HEAVENLY BODY. 128. The position of a heavenly body is defined by its right ascension and declination. These quantities may be determined — (1) By the meridian circle, provided the body is bright enough to be seen by the instrument and comes to the meridian in the night- time. If the instrument is in exact adjustment, the sidereal time when the object crosses the middle wire of the reticle of the instrument is directly (according to Art. 27) the right ascension of the object. The reading of the circle of the instrument, corrected for refraction and parallax if necessary, gives the polar distance of the object, if the polar point of the circle has been determined (Art. 66) ; or it gives the zenith distance of the object if the nadir point has been deter- mined (Art. 67) . In either case the declination can be immediately deduced, beiug the complement of the polar distance, and equal to the latitude of the observer, minus the distance of the star south of the zenith. One complete observation, then, with the meridian circle, determines both the right ascension and declination of the object. If a body (a comet, for instance) is too faint to be observed by the telescope of the meridian circle, which is seldom very powerful, or if it does not come to the meridian during the night, we usually accomplish our object — 129. (2) By the Equatorial, determining the position of the body by measuring the difference of right ascension and declination be- tween it and some neighboring star, whose place is given in a star catalogue, and of course has been determined by the meridian circle of some observatory. In measuring this difference of right ascension and declination, we usually employ a filar micrometer fitted lite the reticle of a meridian circle. It car- ries a number of wires which lie north and south in the field of view, and these are crossed at right angles by one or more wires which can be moved 90 PROBLEMS OP PRACTICAL ASTRONOMY. by the micrometer screw. The difference of right ascension between the star and the object to be determined is measured by simply observing with the chronograph the transits of the two objects across the north and south wives ; the difference of declination, by bisecting each object with one of the micrometer wires as it crosses the middle of the field of view. The ob- served difference must be corrected for refraction and for the motion of the body, if it is appreciable. Other less complicated micrometers are also in use. One of them, called the ring micrometer, consists merely of an opaque ring supported in the field of view either by being cemented to a glass plate or by slender arms of metal. The observations are made by noting the transits of the comparison star and of the object to be determined across the outer and inner edges of the ring. If the radius of the ring is known in seconds of arc, we can from these observations deduce the differences both of right ascension and declination. The results are less accurate than those given by the wire micrometer, but the ring micrometer has the advantage that it can be used with any telescope, whether equatorially mounted or not, and requires no adjustment. There are also many other methods of effecting the same object. 130. To Compute the Time of Sunrise or Sunset. — To solve this prob- lem, it is only necessary to work out the Z.P.S. triangle and find the hour-angle P, having given precisely the same data as in finding the time by a single altitude of the sun (Art. 116). PZ is the observer's co-latitude, PS is the complement of the sun's declination (given by the almanac); and the true distance from the zenith to the centre of the sun at the moment when its upper edge is at the horizon is 90° 50', which is made up of 90°,+ 16' (the mean semi-diameter of the sun), plus 34' (the mean refraction at the horizon). The resulting hour-angle P, corrected for the equation of time, gives the mean time (local) at which the sun's upper limb touches the horizon, under the average circumstances of temperature and barometric pressure. If it is very cold, with the barometer standing high, sunrise will be accelerated, or sunset retarded, by a considerable fraction of a minute. If the sun rises or sets over the sea-horizon, and the observer's eye is at any considerable elevation above the sea-level, the dip of the horizon must also be added to the 90° 50' before making the computation. The beginning and end of twilight may be computed in the same way by merely substituting 108° for 90° 60'. 131. Note to Art. 125. — In the explanation of Sumner's method it is assumed that the earth is a perfect sphere. In the actual application of the method certain corrections are therefore necessary to take into account the earth's ellipticity. ADDENDUM TO CHAPTER IV. 90 A In the text all articles to -which Addenda have been prepared are distinguished by a dagger (t) prefixed to their designating number. Art. 108. Observations made during the last few years at various European stations, combined with observations in this country and at Honolulu, have established beyond question the fact that during a year the latitudes of all places really oscillate to the extent of nearly half a second of arc. The axis of the earth changes its position within the globe, so that the pole of the earth (at present) describes approximately a circle 50 or 60 feet in diameter in a period of about 14 months. But the investiga- tions of Chandler show that the range and period of this motion of the pole are probably both of them variable to a considerable extent, — as might be expected. Indeed his latest results show that the matter is quite complicated, involving the combination of at least two separate motions, one annual, and the other running through its course in 14 months, as just stated. While the periodical variation of latitude has thus been established as an observed fact, the question of a progressive, secular, variation remains unsettled. THE EABTH AS AN ASTBONOMICAL BODY. 91 CHAPTER V. THE EABTH AS AN ASTBONOMICAL BODY. Approximate Dimensions — Proofs of its Rotation — Accurate Determination of its Form and Size by Geodetic Operations and Pendulum Observations — Astronomical, Geodetic and Geo- centric Latitude — Determination of the Earth's Mass and Density. 132. Having discussed the methods of making astronomical ob- servations, we are now prepared to consider the earth in its astro- nomical relations ; i.e., those facts relating to the earth which are ascertained by astronomical methods, and are similar to the facts which we shall have to consider in the case of the other planets. The facts are broadly these : — 1. Tlie earth is a great ball, about 7918 miles in diameter. 2. It rotates on its axis once in twenty-four sidereal hours. 3. It is flattened at the poles, the polar diameter being nearly twenty-seven miles, or one two hundred and ninety-fifth part less than the equatorial. 4. It has a mean density of about Jive and six-tenths times that of water, and a mass represented in tons by six with twenty-one ciphers after it (or six sextillions of tons, according to the French numeration). 5. It is flying through space in its orbital motion around the sun, with a velocity of about nineteen miles a second; i.e., about seventy- live times as swiftly as any cannon-ball. I. 133. The Earth's Approximate Form and Size. — It is not necessary to dwell upon the ordinary proofs of its globularity. We merely men- tion them. 1. It can be circumnavigated. 2. The appearance of vessels coming in from sea indicates that the surface is everywhere convex. 3. The fact that the sea-horizon, as seen from an emi- nence, is everywhere depressed to the same extent below the level line, shows that the surface is approximately spherical. 4. The fact that as one goes from the equator toward the north, the elevation of 92 THE EARTH AS AN ASTRONOMICAL BODY. the pole increases proportionally to the distance from the equator, proves the same thing. 5. The shadow of the earth, as seen upon the moon at the time of a lunar eclipse, is that which only a sphere could cast. We may add as to the smoothness and globularity of the earth, that if the earth be represented by an 18-inch globe, the difference between the polar and equatorial diameter would only be about one- sixteenth of an inch, the highest mountains upon the earth's surface would be represented by about one-eightieth of an inch, and the aver- age elevation of the continents would be hardly greater than that of a film of varnish. The earth is really relatively smoother and rounder than most of the balls in a bowling-alley. 134. An approximate measure of the diameter is easily obtained. Erect upon a level plain three A -? n rods in line, a mile apart, and cut off their tops at the same level, carefully determined with a sur- veyor's levelling instru- ment. It will then be found that the line AC, Fig. 44, joining the ex- tremities of the two terminal rods, passes about eight inches below B, the top of the middle rod. Suppose the circle ABC completed, and that E is the point on the cir- cumference opposite B, so that BE equals the diameter of the earth (= 2 R). By geometry, BD : BA = BA : BE, Fia. 44. — Curvature of the Earth's Surface. whence 2*E = ^-*, or fl=*^. BD 2BD Now BA is one mile, and BD= § of a foot, or y-^ of a mile. I 2 Hence 2R = , or 7920 miles : a very fair approximation. 7920 On account of refraction, however, the result cannot be made exact by any care in observation. The line of sight, AC, is not strictly straight, but curves slightly towards the earth, and differently as the weather changes. 135. The best method of ascertaining the size of the earth — in fact the only one of real value — is by measuring arcs of the meridian in order to ascertain the number of miles or "kilometres in one degree, from which we immediately get the circumference of the earth. SIZE OF THE EARTH REGARDED AS A SPHERE. 93 This measure involves two distinct operations. One — the measure of the number of miles — is purely geodetic; the other— the deter- mination of the number of degrees, minutes, and seconds between ,the two stations — is purely astronomical. {jf We have to find by astronomical observation the angle between two /radii drawn from the centre of the earth to the two stations (regarding ithe earth as spherical) ; or, what is the same thing, the. angular dis- | tance in the sky between their respective zeniths. The two stations being ;!onthe same meridian, all that is necessary is to measure their latitudes by any of the methods which have been given in Chapter IV. and take the difference. This will be the angle wanted. If, for instance, the distance between the two stations was found by measurement to be 120 miles, and the difference of latitude was found by astronomical observations to be 1° 44'.2, we should get 69.27 miles for one degree. Three hundred and sixty times this would be the circumference of the earth, a little less than 25,000 miles, and the diameter would be found by dividing this by w, which would give 7920 miles. » 136. Eratosthenes of Alexandria seems to have understood the mattei as early as 250 b.c. His two stations were Alexandria and Syene in Upper Egypt. At Syene he observed that at noon of the longest day in summer there was no shadow at the bottom of a well, the sun being then vertically overhead. On the other hand, the" gnomon at Alexandria, on the same day, by the length of the shadow, gave him ^ of a circumference, or 7° 12' as the distance of the sun from the zenith at that place, which, therefore, is the difference of latitude between Alexandria and Syene. The weak place in his work was in the measurement of the distance be- tween the two places. He states it as 5000 stadia, thus making the circum- ference of the earth 250,000 stadia; but we do not know the length of his stadium, nor does he give any account of the means by which he measured the distance, if he measured it at all. There seem to have been as many different stadia among the ancient nations as there were kinds of " feet " in Europe at the beginning of this century. The first really valuable measure of the arc of a meridian was that made by Picard in Northern France in 1671 — the measure which served Newton so well in his verification of the idea of gravitation. H- 137. The Rotation of the Earth, — At the time of Copernicus the only argument in favor of the earth's rotation 1 was that the hypoth- 1 The word rotate denotes a spinning motion like that of a wheel on its axis. The word revolve is more general in its application, and may be applied either to 94 THE EARTH AS AN ASTRONOMICAL BODY. esis was more probable than that the heavens themselves revolved. All phenomena then known would be sensibly the same on either supposition. A little later, analogy could be adduced, for when the telescope was invented, we could see that the sun, moon, and several of the planets are rotating globes. At present we are able to adduce experimental proofs which abso- lutely demonstrate the earth's rotation, and some of them even make it visible. 138. 1 . The Eastward Deviation of Bodies falling from a Great Height. — The idea that such a deviation ought to occur was first suggested by Newton. Evidently, since the top of a tower, situated anywhere but at the pole of the earth, describes every day a larger circle than its base, it must move faster. A body which is dropped from the top, retaining its ex- cess of eastward motion as it descends, must therefore strike to the east of the point which is vertically under its starting-point, provided it is not deflected in its fall by the resistance of the air or by air-currents. Fig. 45 illustrates the principle. A body starting from A, the top of the tower, reaches the earth at D (BD be- ing equal very approximately to AA'), while during O its fall the bottom of the tower has only moved from fig. 45. B to jB'. The experiments are delicate, since the devi- Eastward Devia- ation is very small, and it is not easy to avoid the «on of a Failing fftf a ; r _ currents . It fe a i s0 extremelv difficult to Body. get balls so perfectly spherical that they will not sheer off to one side or the other in falling. The best experiments of this kind so far have been those of Benzenberg, performed at Hamburg in 1802, and those of Reich, performed in 1831, in an abandoned mine shaft near Freiberg, in Saxony. The latter obtained a free fall of 520 feet, and from the mean of 106 trials, the eastern deviation observed was 1.12 inches, while theory would make it 1.08. The experiment also gave a southern deviation of 0.17 of an inch, unexplained by theory. It seems to indicate the probable error of observation. The balls in falling sometimes deviated two or three inches one side or the other from the average. describe such a spinning motion, or (and this is the more usual use in astronomy) to describe the motion of one body around another, as that of the earth around the sun. PROOFS OF THE EARTH'S ROTATION. 95 The formula given by Worms in his treatise on " The Earth and its Mechanism," is 4 irf (if— £ a) cos

is the latitude of the place of observation. In latitude 45° a fall of 576 feet should give, neglecting the resistance of the air, a deviation of 1.47 inches. The resistance would increase it a little. It will be noted that at the pole, where the cosine of the latitude equals zero, the experiment fails. The largest deviation is obtained at the equator. 139. 2. Foucault's Pendulum Experiment. — In 1851 Foucault, that most ingenious of French physicists, devised and first exe- cuted an experiment which actually shows the earth's rotation to the eye. From the dome of the Pan- theon in Paris he suspended a heavy iron ball about a foot in diameter by a wire more than 200 feet long (Fig. 46). A circular rail some "twelve feet across, with a little ridge of sand built upon it, was placed under the pendulum in such a way that a pin attached to the swinging ball would just scrape the sand and leave a mark at each vibration. The ball was drawn aside by a cotton cord and allowed to come absolutely to rest; then the cord was burned, and the pen- dulum set to Swinging in a true Fio. 46. -Foucault's Pendulum Experiment plane ; but this plane seemed to deviate slowly towards the right, cutting the sand in a new place at each swing and shifting at a rate which would carry it completely around in about thirty-two hours if the pendulum did not first come to rest. In fact, the floor of the Pantheon was seen turning under the plane of the pendulum's vibration. The experiment created 96 THE EAKTH AS AN ASTRONOMICAL BODY. great enthusiasm at the time, and has since been very frequently performed, and always with substantially the same results. 140. The approximate theory of the experiment is very simple. Such a pendulum, consisting of a round ball hung by a round wire or else suspended on a, point, so as to be equally free to swing in any plane (unlike the common clock pendulum in this freedom) , being set up at the pole of the earth, would appear to shift around in twenty-four hours. Really, the plane of vibration remains invariable and the earth turns under it, the plane of vibration in this case being un- affected by the motion of the earth. This can be easily shown by setting up a similar apparatus, consisting of a ball hung by a thread, upon a table, and then turning the table around with as little jar as possible. The plane of the swing will remain unchanged by the motion of the table. It is easy to see, further, that at the equator there would be no such tendency to shift. In any other latitude the effect will be intermedi- ate, and the time required for the pendulum to complete the revolu- tion of its plane will be twenty-four hours divided by the nine of the latitude. The north- ern edge of the floor of a room (in the northern hemisphere) is nearer the axis of the earth than its southern edge, and therefore is car- ried more slowly eastward by the earth's rota- tion. Hence it must skew around continually )m like a postage stamp gummed upon a whirling globe anywhere except at the globe's equator. The southern extremity of every north and south line on the floor continually works to- ward the east faster than the northern ex- tremity, causing the line itself to shift its direc- tion accordingly, compared with the direction it had a few minutes before. A free pendu- lum, set at first to swing along such a line, must therefore apparently deviate continually at the same rate in the opposite direction. In the northern hemisphere its plane moves dextrorsum; i.e., with the hands of a watch : in the southern, its motion is sinistrorsum. 141. Suppose a parallel of latitude drawn through the place in question, and a series of tangent lines drawn toward the north at points an inch or so apart on this parallel. All these tangents would meet at some point, V, Fig. 47, which is on the earth's axis produced ; and. taken together these tangents Fia. 47. Explanation of the Foucault Pendulum Experiment. PROOFS OP THE EARTH'S ROTATION. 97 would form a cone with its point at V. Now if we suppose this cone cut down upon one side and opened up (technically, -developed"), it would give us a sector of a circle, as in Fig. 48, and the angle of the sector would be the sum total of the angles between all the adjacent meridians tangent to the earth on that parallel. ! Now it is easy to prove that the angle of this sec- 1 tor 'equals 360° X sin $> ($> being the latitude). (1) The circumference of the parallel AB (Fig. 47) = 2irXiJcoS(/>, since Rxcos

. (2) The side 4 7 of the cone (which will be the radius of the sector when the cone is developed) = R cot

. Fio. 48. - Developed Cone. Hence (ABA' heiag the circumference of the parallel forming the edge of the developed sector ABA' Fin Fig. 48), the angle of the sector AVA' (greater than 180° in the figure) : 360° = arc ABA': whole circumference ABA'm; or angle V : 360° = R cos

; whence F=360°5°ii = 360 o sm<*. COt cf> V is the total angle described by the plane of the pendulum in a day. At the pole the cone produced by the tangent lines becomes a little "button," a complete circle. At the equator it becomes a cylinder, and the angle is zero. In order to make the experiment successfully, many precautions must be taken. It is specially important that the pendulum should vibrate in a true plane, without any lateral motion. To secure this end, it must be carefully guarded against all jarring motion and air-currents. To diminish the effect of all such disturbances, which will always occur to a certain extent, the pendulum should be very heavy and very long, and of course the suspended ball must be truly round and smooth. Ordinary clock-work cannot be used to keep the pendulum in vibration, since it must be free to swing in every plane. Usually, the apparatus once started is left to itself until the vibra- tions cease of their own accord ; but Foucault contrived a most ingenious electrical apparatus, which we have not space to describe, by means of which the vibration could be kept up for days at a time without producing any hurtful disturbance whatever. It will be noticed that this experiment is most effective precisely where the experiment of falling bodies fails. This is best near the pole, the other at the equator. 142. 3. By the Gyroscope, an experiment also due to Foucault, and proposed and executed soon after the pendulum experiment. 98 THE EARTH AS AN ASTRONOMICAL BODY. The instrument shown in Fig. 49 consists of a wheel so mounted in gimbals that it is free to turn in every direction, and so delicately balanced that it will stay in any position if undisturbed. If the wheel be set to rotating rapidly, it will maintain the direction of its axis invariable, unless acted upon by extraneous force. If, then, we set the axis horizontal and arrange a microscope to watch a mark upon one of the gim- bals, it will appear slowly to shift its position as the earth revolves, in the same way as the plane of the pendulum behaves. 143. 4. There are many other phenomena which depend upon and really demonstrate the earth's rotation. We merely mention them : — a. The Deviation of Projectiles In the northern hemisphere a projectile always deviates towards the right ; in the southern hemi- sphere toward the left. b. The Trade Winds. c. The Vorticose Revolution of the Wind in Cyclones. In the northern hemisphere the wind in a cyclone moves spirally towards the centre of the storm, whirling counter clock-wise, while in the southern the spiral motion is with the hands of a watch. The motion is explained in either case by the fact that currents of air, setting out for the centre of disturbance where the cyclone is formed, deviate like projectiles, to the right in the northern hemisphere, and towards the left in the southern hemisphere, so that they do not meet squarely in the centre of disturbance. d. The Ordinary Law of Wind-change ; that is, in the northern hemisphere the north wind, under ordinary circumstances, changes to a northeast, a northeast wind to au east, east to southeast, etc. When the wind changes in the opposite direction, it is said to " back '' around. In the southern hemisphere it of course usually backs around, much to the disconcertment of the early Australian settlers. Fig. 49. — Foucault'e Gyroscope. It might seem at first that the rotation of the earth, which occupies twenty-four hours, is not a very rapid motion. A point on the equa- SPHEROID FORM OF THE EARTH. 99 tor, however, has to move nearly one thousand miles an hour, which is about fifteen hundred feet per second, and very nearly the speed of a cannon-ball. 144. Invariability of the Earth's Rotation. — It is a question of great importance whether the day changes its length. Theoretically it must almost necessarily do so. The friction of the tides, and the deposits of meteoric matter upon the earth both tend to lengthen it ; while on the other hand, the earth's loss of heat by radiation and consequent shrinkage must tend to shorten it. Then geological changes, the elevation and subsidence of continents, and the trans- portation of matter by rivers, act, some one waj r , some the other. At present it can only be said that the change, if any has occurred since astronomy became accurate, has been too small to be detected. The day is certainly not longer or shorter by jfo of a second than in the days of Ptolemy, and probably has not changed by 1 1 00 of a second. The criterion is found in comparing the times at which celestial phenotnena, such as eclipses, transits of Mercury, etc., occur. III. 145. The Earth's Form, more accurately stated, in that of a spheroid of revolution, having an equatorial radius of 6,377,377 metres, and a polar radius of 6,355,270 metres, according to Listing (1873); or of 6,378,206.4 and 6,356,583.8 respectively, according to Clarke. 1 It must be understood, also, that this statement is only a second approximation (the first being that the earth is a globe). Owing to mountains and valleys, etc., the earth's surface does not strictly correspond to that of any geometrical solid whatever. The flattening at the poles is the necessary consequence of the earth's rotation, and might have been cited in the preceding section as proving it. 146. There are two ways of determining the form of the earth : one, by measurement of distances upon its surface in connection with the latitudes and longitudes of the points of observation. This gives not only the form, but the dimensions. The other method is by the observation of the varying force of gravity at various points, — obser- vations which are made by means of a pendulum apparatus of some kind, and determine only the form, but not the size of the earth. 1 This is Clarke's spheroid of 1866, and is adopted by the United States Coast and Geodetic Survey. See Appendix for his spheroid of 1878. 100 THE EAKTH AS AN ASTKONOMICAL BODY. 147. 1 . Measurements of Arcs of Meridian in Different Latitudes. — To determine the size of the earth regarded as a sphere, a single are of meridian in any latitude is sufficient. Assuming, however, that the earth is not a sphere, but a spheroid with elliptical meridians, we must measure at least two such arcs, one of which should be near the equator, the other near the pole. The astronomical work consists simply in finding with the greatest possible accuracy the difference of latitude between the terminal sta- tions of the meridian arc. The geodetic work consists in measuring their distance from each other in miles, feet ? or metres, and it is this part of the work which consumes the most time and labor. The process is generally that known as triangulation. Two stations are selected for the extremities of a base line six or seven miles long, and the ground between them is levelled as if for a railroad. The A distance between these stations (A and B in Fig. 50) "i is then carefully measured by an apparatus especially designed for the purpose and with an error not to exceed half an inch or so in the whole distance. A third station, 1, is then chosen, so situated that it will be visible from both A and B, and all the angles of the triangle AB 1 are measured with great care by a theodolite. A fourth station, 2, is then selected, such that it will be visible from A and 1 (and if possible from B also), and the angles of the triangle A 1 2 are Fio. 50.— A Triangulation. measured in the same way. In this manner the whole ground between the two terminal stations is covered with a network of triangulation, the two terminal stations themselves being made two of the triangulation points. Knowing one distance and all the angles in this system, it is possible to compute with great accuracy the exact length of the line 1 5 and its direction. The sides of the triangles are usually from twenty-five to thirty miles in length, though in a mountainous country not infrequently much longer ones are available. Generally speaking, the fewer the stations necessary to connect the extremities of the arc, and the longer the lines, the greater will be the ultimate accuracy. In this way it is possible to measure distances of 200 or 300 miles with a probable error not exceeding two or three feet. Many arcs of meridians have been measured in this way, — not less than twenty- or thirty in different parts of the earth, the most extensive being the DIMENSIONS OF THE EARTH FROM GEODETIC MEASURES. 101 so-called Anglo-French arc, extending more than twelve degrees in length; the Indian arc, nearly eighteen degrees long ; and the great Russo-Scandi- navian arc, more than twenty-five degrees in length, and reaching from Hammerfest to the mouth of the Danube. One short arc has been measured in South America and one in South Africa. In a general way, it appears that the higher the latitude the longer the arc. Thus, near the equator the length of a degree has been found to be 362,800 feet in round numbers, while in northern Sweden, in latitude 66°, it is 365,800 feet ; in other words, the earth's surface is flatter near the poles. It is necessary to travel 3000 feet further in Sweden than in India to increase the latitude one degree, as measured by the elevation of the celestial pole. 148. The deduction of the exact form of the earth from such measurements is an abstruse problem. Owing to errors of observation and local deviations in the direction of gravity, the different arcs do not give strictly accordant results, and the best that can be done is to find the result which most nearly satisfies all the observations. If we assume that the form is that of an exact spheroid of revolution, with all the meridians true ellipses and all exactly alike, the problem is simplified somewhat, though still too complicated for discussion here. Theory indicates that the form of a revolving mass, fluid enough to yield to the forces acting in such a case, might, and prob- ably would, be such a spheroid; but other forms are also theoreti- cally possible, and some of the measurements rather indicate that the equator of the earth is not a true circle, but an oval flattened by nearly half a mile. On the whole, however, astronomers are dis- posed to take the ground that since no regular geometrical solid whatsoever can absolutely represent the form of the earth, we may as well assume a regular spheroid for the standard surface, and consider all variations from it as local phenomena, like hills and valleys. 149. Each measurement of a degree of latitude gives the "radius of cur- vature,'' as it is called, of the meridian at the degree measured. The length of a degree from 44° 30' to 45° 30', multiplied by 57.29 (the number of de- grees in a radian), gives the radius of the " osculatory circle," which would just fit the curve of the meridian at that point. Having a table giving the actual length of each degree of latitude, we could construct the earth's meridian graphically as follows : — Draw the line AX, Fig. 51. On it lay off Aa, equal to the radius of curva- ture of the first measured degree (that is, 57.3 times the length of the degree), 102 THE EAKTH AS AN ASTRONOMICAL BODY. and with a as centre, describe an arc AB, making the angle AaB just one Fig. 51. Radii of Curvature of the Meridian. degree. Next produce the line Ba to b, making Bb the radius of curvature of the second degree, and draw this second degree-arc; and so proceed until the whole ninety have been drawn. This will give one quarter of the meridian, and of course the three other quarters are all just like it. a, b, c, etc., are called the " centres of curvature " of the different degrees. If we assume the curve to be an ellipse, then the equatorial semidiameter AO, and the polar, PO, are given respec- tively by the two formulas, AO= y/qp* and PO = y/fp, q and p being the radii of curvature (Aa and Pe in the figure) at the equator and pole. 150. The li ellipticity" or " oblateness " of an ellipse is the fraction found by dividing the difference of the polar and equatorial diameters by the equatorial, and is expressed by the equation d = A-B In the case of the earth this is -^, according to Clarke's spheroid, of 1866. Until within the last few years Bessel's smaller value, viz., ^j, was generally adopted. Listing's larger value, ^^-j, is now preferred by some. The ellipticity of an ellipse must not be confounded with its eccentricity. The latter is y/A 2 - B 2 e = —A—' and is always a much larger numerical quantity than the ellipticity. In the case of the earth's meridian, it is j-^ as against ^-g-. Its symbol is usually e. 151. Arcs of longitude are also available for determining the earth's form and size. On a spherical earth a degree of longitude measured along any parallel of latitude would be equal to one degree of the equator multiplied by the cosine of the latitude. On an oblate or orange-shaped spheroid (the sur- face of which lies wholly within the sphere having the same equator) the de- PENDULUM EXPERIMENTS. 103 grees of longitude are evidently everywhere shorter than on the sphere, the difference being greatest at a latitude of 45°. In fact, arcs in any direction between stations of which both the latitude and longitude are known can be utilized for the purpose ; and thus the extensive surveys that have been made in different countries have given us a pretty accurate knowledge of the earth's dimensions. It is very desirable that in some way the chain of actual measurements should be extended from the eastern continent to the western, but the immense difficulties of so doing are obvious. At present the distance from a point on the earth's surface (say the ob- servatory at Washington) to any other point in the opposite hemisphere (say the observatory at the Cape of Good Hope) is uncertain to perhaps the extent of a quarter of a mile. 152. 2. Pendulum Experiments. — Since J=T-\[ { (Physics, p. 72), g = ^; we can therefore measure the variations of the force of gravity, g, at different parts of the earth, either by taking a pendulum of in- variable length and determining t, the time of its vibration ; or by measuring the length, I, of a pendulum which will vibrate seconds. Extensive surveys of this sort have been made, and are still in prog- ress, and it is found that the/o?-ce of gravity at the pole exceeds that at the equator by about j^ part. In other words, a person who weighs 190 pounds at the equator (by a spring balance) would, if carried to the pole, show 191 pounds by the same balance. The apparatus most used at present for the purpose of measuring the force of gravity is a modification of the so-called Kater's pendulum. The pendulum itself usually consists of a brass tube about an inch in diameter and about four feet long, carrying a ball three or four inches in diameter at each end, both balls being exactly of the same size, but one solid while the other is hollow. Two knife edges are inserted through the rod at right angles, one near the heavy ball and the other at just the same distance from the lighter one, and the weights and dimensions of the apparatus are so adjusted that the time of vibration will be very approximately the same whether the pendu- lum is swung heavy end up or light end up, and will be not far from one second. The distance between the knife edges will then, according to the theory of the pendulum, be very nearly equal to the length of a simple pendulum • vibrating in the same time; and the small difference can be accurately cal- culated when we know the exact time of vibration, each end up. The knife edges swing on agate planes which are fastened upon a firm support : 104 THE EARTH AS AN ASTRONOMICAL BODY. and great pains must be taken to have the support really firm. Professor Peiree of our Coast Survey a few years ago detected important errors in a majority of the earlier pendulum observations, due to insufficient care in this respect. 153. The observations consist in comparing the pendulum with a clock, either by noting the "coincidences," or by an electrical record automatically made on a chronograph. A pin attached to the end of the pendulum touches a globule of mercury (which is momentarily raised for the purpose once in eight or ten minutes), and so records the swing upon the chrono- graph sheet. The observations need to be carefully corrected for temperature (which, of course, affects the distance between the knife edges), for the length of arc through which the pendulum is swinging, and for the resistance of the air. The observations determine the "force of gravity" (French "pesanteur") at the station. This "force of gravity," however, thus deter- mined, is not simply the earth's attraction, but includes also the effects of the centrifugal force, due to the earth's rotation, which we must consider and allow for. 154. At the equator the centrifugal force acts vertically in direct opposition to gravity, and is given by the well-known formula (see Physics, p. 62), in which V is the velocity of the earth's sur- face at the equator, and R the earth's radius. Since V is equal to the earth's circumference divided by the number of seconds in a sidereal day, we have „ 2ttR , n i-rSR V= » and (7= — - — c c Now R, the radius of the earth, equals 20,926,000 feet ; and t equals 86,164 mean-time seconds. C, therefore, comes out 0.111 feet, which is rh oi 9^9 bein g 32 l feet - We may remark in passing that if the rate of rotation were seventeen times as great, C would be 17 2 , or 289 times greater than now, and would equal gravity; so that on that supposition bodies at the equator would weigh absolutely nothing, and any greater velocity of rotation would send them flying. At any other latitude, since MN= OQ cos MOQ, 1 the centrifugal 1 This is not exact, since MN in an oblate spheroid is less than OQ X cos MOQ ; but the difference is unimportant in the case of the earth. the earth's centrifugal force. 105 force, c, eq^ls q,cos<£, acting at right angles to the axis of the earth and parallel to the plane of the equator. Now, this centrifugal force c is not wholly effective in dimin- ishing the weight of a body, but only that portion of c (MB in Fig. 52) which is directed vertically, c is MT in the figure, and MB is equal to c multiplied by the cosine of <£, which finally gives usOxcos s <£ for the amount by which the centrifugal force diminishes gravity at a station whose latitude is <&. FlG ' 52- _, , , , The Earth's Centrifugal Force. iiVery determination, therefore, of the " force of gravity," obtained by the pendulum, needs to be increased by the quantity -2- X cos 2 d., 289 ^ in order to get the real value of the earth's gravitational attraction at the point of observation. The other component of c (viz. MS) acts at right angles to gravity and parallel to the earth's surface, and is given by the formula C cos sin

ce of the earth's attraction. This is greater the nearer each station is to the centre of the earth ; but unfortunately there is no simple relation connecting the force with the distance. The attraction depends not only on the distance from the centre of the earth, but also upon the form of the earth and the constitution of its interior, and the arrangement of its strata of different density. We may safely assume, however, that the earth is made up concentrically, so to speak ; the strata of equal density being arranged like the coats of an onion. On this hypothesis Clairaut, in 1742, demonstrated the relation giyen below, which is always referred to as Clairaut's equation. 106 THE EARTH AS AN ASTRONOMICAL BODY. Let w be the loss of weight between the equator and the pole, and C the centrifugal force at the planet's equator, both being expressed as fractions of the equatorial force of gravity, and let d be the ellipticity of the planet. Then, as Clairaut proved, d + w=2h X C; whence d=2bC — w. In the case of the earth we have d = 2Jx — -, 289 190 which gives d = 292.8 Considering all the data, the most that can safely be said as to d is that it lies between the fractions -^^ and ^^-j. (Clarke's later values for d are larger than that adopted by the Coast Survey.) 156. Astronomical, Geographical, and Geocentric latitudes. — The astronomical latitude of a place has been defined as the elevation of the pole, or, what comes to the same thing, it is the angle between the plane of the equator and the direction of gravity at that place, how- ever that direction may be affected by local causes. The geocentric latitude, on the other hand, is the angle made at the centre of the earth (as the word implies) between the plane of the equator and a line drawn from the observer to the centre of the earth, which line of course does not coincide with the direction of gravity, since the earth is not spherical. The geographical or geodetic latitude of -j^-* — — q a station is the angle formed with the plane FlG . 53, of the equator by a line drawn from the Astronomical and Geocentric station perpendicular to the surface of the Latitude. standard spheroid. If the earth's surface were strictly spheroidal, and there were no local varia- tions of gravity, the astronomical latitude and the geographical latitude would coincide — and they never differ greatly ; but the geocentric latitude differs from them by a very considerable quantity— as much as 11' in lati- tude 45°. The geocentric latitude is but little used except in certain astro- nomical calculations where parallax is involved. DIFFERENT KINDS OF LATITUDE. 107 In Fig. 53, the angle MOQ is the geocentric latitude of M, while MNQ is the geographical latitude. MNQ is also the astronomical latitude, unless there is some local disturbance of the direction of gravity. The angle OMN, which is the difference between the geocentric and astronomical latitudes, is called " the angle of the vertical." 157. It will be noticed that the astronomical latitude of a place is the only one of these three latitudes which is determined directly by observation. In order to know the geocentric and geographical lati- tudes of a place, we must know the form and dimensions of the earth, which are ascertained only by the help of observations made elsewhere. The geocentric degrees are longer near the equator than near the poles, and it is worth noticing that if we form a table giving the length of each degree of geographical latitude from the equator to the pole, the same table, read backwards, gives the length of geocentric degrees. Since the earth is ellipsoidal instead of spherical, it is evident that lines of " level " on the earth's surface are affected by the earth's ro- tation. If this rotation were to cease, the direction of gravity would be so much changed that the Gulf of Mexico would run up the Mis- sissippi River, because the distance from the centre of the earth to the head of the river is less by some thousands of feet than the distance from the mouth of the river to the centre of the earth. 158. Station Errors. — The irregularities in the direction of gravity are by no means insensible as compared with the accuracy of modern astro- nomical observation, and the difference between the astronomical latitude and longitude of a place and the geographical latitude and longitude of the same place constitute what is called the "station error." In the eastern part of the United States these station errors, according to the Coast Survey observations, average about 1J"- Errors of from 4" to 6" are not uncom- mon, and in mountainous countries, as for instance in the Caucasus and in Northern India, these errors occasionally amount to 30" or 40". They are not " errors " in the sense that the astronomical latitude of the place has not been determined correctly, but are merely the effects of the irregular distri- bution of matter in the crust of the earth in altering the direction of gravity. Pendulum observations show local variations in the force of gravity quite proportional to the deviations which the station-errors show in its direction. IV. 159. The Earth's Mass and Density. — The ' mass ' of a body is the quantity of matter that it contains, the unit of mass being the quantity of matter contained in a certain arbitrary body which is taken as a standard. For instance, a "kilogram" is the quantity of matter 108 THE EARTH AS AN ASTRONOMICAL BODY. contained in the block of platinum preserved at Paris as the standard of mass. 1 A pound is similarly denned by reference to the proto- types at "Washington and London. Two masses of matter are defined as equal which require the same expenditure of energy to give them the same velocity; or vice versa, those are equal which, when they have the same velocity ; possess the same energy, and, in giving up their motion and coming to rest, do the same amount of work. Masses can therefore be compared by placing them in the same field of force and comparing the energies developed in them when they have moved equal distances under the action of the force. This method, however, is seldom convenient. 160. Proportionality of Mass to Weight. — Newton showed by his experiments with pendulums of different substances, that at any given point the attraction of the earth for a body of any kind of maflfer is proportional to the mass of that body ; the attraction being measured as a pull or "stress" in this case, and called '■'■the weight" of the body. In other and more common language, the mass of a body is proportional to its weight (we must not say it is its weight), provided the weighing of the bodies thus compared is done, in cases where scientific accuracy is essential, at the same place on the earth's surface. Practically, therefore, we usually measure the masses of bodies by simply weighing them. It is to be carefully observed, how- ever, that the words "kilogram," "pound,"' "ton," etc., have also a secondary meaning, as denoting units of pull and push, — of "stress," speaking strictly and technically, — or of "force," as that much abused word is very generally used. It is, from a literary point of view, just as proper to speak of a stress or a pull of "a hundred pounds 2 as of a mass of a hundred pounds, but the word " pound " means an entirely different thing in the two cases. At the sur- face of the earth the relation between the ideas, however, is so close that the way in which the ambiguity came about is perfectly obvious, and it is hardly probable that language will ever change so as to remove it. To a certain extent it is admittedly unfortunate, and the student must always be on his guard against it. At the earth's surface a mass of 100 pounds always 1 This was meant to be just equal to the mass contained in a cubic decimeter of water at its maximum density, and is so very nearly indeed: 2 Of course the student will remember that we have a unit of stress, — the dyne, — which is wholly free from this objection of ambiguity. GRAVITATION. 109 "weighs" very nearly 100 pounds; but, to anticipate slightly, at an eleva- tion of 4000 miles above the surface, the same mass would "weigh" only 25 pounds; at the distance of the moon about half an ounce; while on the surface of the sun it would " weigh " nearly 2800 pounds. 161. Gravity. — The law of gravitation discovered by Newton de- clares that any particle of matter attracts any other particle with a force ("stress," if the bodies are prevented from moving) proportional inversely to the square of the distance between them, and directly to the product of their masses; or, as a formula, we may write, F = 1c M k xMz d 2 ' in which M x and M 2 are the two masses, and d the distance between them, while h is a constant numerical factor depending upon the units employed. 1 We must not imagine the word "attract" to mean too much. It merely states the fact that there is a tendency for the bodies to move toward each other, without including or implying any explanation of the fact. So far, no explanation has appeared which is less difficult to comprehend than the fact itself. Whether bodies are drawn together by some outside action, or pushed together ; or whether they themselves can act across space with mathematical intelligence, — in what way it is that "attraction" comes about, is still unknown, — apparently as inscrutable as the very nature and constitution of an atom of matter itself; it is simply a fundamental fact. 162. When the distance between attracting bodies is large as com- pared with their own magnitude, then reckoning the distance between their centres of mass as their true distance, the formula is sensibly true for them as it would be for mere particles. When, however, the distance is not thus great, the calculation of the attraction becomes a very serious problem, involving what is known as a " double integra- 1 It will not do to write the formula F = — *-= — - a' (omitting the k), unless the units are so chosen that the unit of force shall be equal to the attraction between two masses each of one unit, at A distance of one unit. It is not true that the attraction between two particles, each having a mass of one pound, at a distance of one foot, is equal to a stress of one pound (of force), as would rather naturally be inferred if we should write the equation without the constant factor, k is the so-called "Newtonian constant," and on the centimetre-gramme-second system of units equals 0.000 000 0666 dynes, according to the determination of Boys in 1894. 110 THE EARTH AS AN ASTRONOMICAL BODY. tion." "We must find the attraction of each particle of the first body upon each particle of the other body, and take the sum of all these infinitesimal stresses. Newton, however, showed that if the bodies are spheres, either homogeneous or of concentric structure, then they attract and are attracted precisely as if the matter in them were wholly collected at their centres. The earth, for instance, attracts a body at its sur- face very nearly as if it were all collected at its own centre, 4000 miles distant; not exactly so, because the earth is not strictly spherical ; but in what follows we shall neglect this slight inaccuracy. 163. In order, then, to find the mass of the earth in kilograms, pounds, or tons, we must find some means of accurately comparing its attraction for some object on its own surface with the attraction of the same object by some body of known mass, at a measured distance. The difficulty lies in the fact that the attraction produced by any body, not too large to be handled conveniently, is so excessively small that only the most delicate operations serve to detect and measure it. The first successful attack upon the problem was made in 1774 by Maskelyne, the Astronomer Eoyal, by means of what is now usually ref erred to as, — 164. 1. " The Mountain Method," because, in fact, the earth in this operation is weighed against a mountain. Two stations were chosen on the same meridian, one north and one south of the mountain Schehallien, in Scotland. In the first place, a careful topographical survey was made of the whole region, giving the precise distance between the sta- tions, as well as the exact dimen- yks.m. sious of the mountain, which is a The Mountain Method of Determining the ' ' ho g" Da C k " of very regular Contour. Earth's Density. From the known dimensions of the earth and the measured distance, the difference of the geographical latitudes of the two places M and N (Fig. 54) can be accurately computed ; i.e., the angle which the plumb lines at M and N would have made if there were no mountain there. In this case it was 41". The next operation was to observe the astronomical latitude at each station. This astronomical difference of latitude, i.e., the angle which the plumb lines actually do make, was found to be 53", the plumb lines at M and N being drawn inward out of their normal position by the attraction of the mountain to the MASS AND DENSITY OF THE EARTH. Ill extent of 6" on each side ; so that the astronomical difference of latitude was increased by 12" over the geographical. Now, in such a case the ratio of gravity to the deflecting force, according to the laws of the composition of forces, is that of aM to aA' in the figure (Fig. 55), or the ratio of 1 to the tan- gent of the deflection, 8; that is, calling the deflecting force f, we have 2.= cot 8, = cot 6" in this case. By the law of gravitation, the earth's attracting force at its surface is given by the formula where E is the mass of the earth (the unknown quantity of our problem), and R its radius, 4000 miles. Similarly, if C in the figure is the centre of attraction of the mountain, we have m being the mass of the mountain, and d the distance from C to the station. Combining this with the preceding, we get E or, in this case, -m M. = cot I R\* "(f) We thus get the ratio of the earth's mass to that of the moun- tain; and provided we can find the mass of the mountain in tons or any other known unit of mass, the problem will be completely solved. By a careful geological survey of the mountain, with deep borings into its strata, the mass of the mountain was determined as accurately as it could be (though here is the weakest point of the method), and thus the mass of the earth was finally computed. Now, knowing the diameter of the earth, its volume in cubic feet is easily found, and from the volume and the known number of mass- pounds, (62| nearly) in a cubic foot of water, the weight the earth would have, if composed of water, follows. Comparing this with the mass actually found, we get the density, which in this experiment came out 4.71. A repetition of the work in 1832 at Arthur's Seat, near Edinburgh, gave 5.32. 112 THE EARTH AS AN ASTRONOMICAL BODY. 165. 2. Much more trustworthy results, however, are obtained by the method of the toesion balance, first devised by Michell, but first employed by Cavendish in 1798. A light rod, carrying two small balls at its extremities, is suspended horizontally at its centre by a long fine metallic wire. If it be al- lowed to come to rest, and then a very slight deflecting force be applied, the rod will be pulled out of position by an amount depending on the stiffness and length of the wire, as well as the force itself. When the deflecting force is removed, the rod will vibrate back and forth until brought to rest by the resistance of the air. The ' ' torsional coefficient," as it is called (i.e., the stress corresponding to a„tor-' sion of one revolution), can be accu- rately determined by observing the time of vibration when the dimensions and weight of the rod and balls are known. If, now, two large balls A and B are brought near the smaller ones, as in Fig. 56, a deflection will be produced by their attraction, and the small balls will move from a and b to a' and V. By shifting the large balls to the other side at A' and B', we get an equal deflection in the opposite direction, i.e., to a" and b", and the difference be- tween the two positions assumed by the small balls, i.e., a'a" and b'b", will be twice the deflection. Fig. 56. — Plan of the Torsion Balance. It is not necessary, nor even best, to wait for the balls to come to rest. We note the extremities of their swing. The middle point of the swing gives the point of rest, and the time occupied by the swing is the time of- vibration, which we need in determining the coefficient of torsion. We must also measure accurately the distance, A a' -and BV between the centre of each of the large balls and the point of rest of the small ball when deflected. The earth's attraction on each of the small balls of course equals the ball's weight. The attractive force of the large ball on the small one near it is found directly from the experiment. If the deflection, for instance, is 1° and the coefficient of torsion is such that it takes one grain to twist the wire around one whole revolution, then the MASS AND DENSITY OF THE EARTH. 113 deflecting force, which we will call / as before, will be ^ of a grain. Call the mass of the large ball B, and let d be the measured distance from its centre to that of the deflected ball. We shall then have also, w being the weight of the small ball, 7 E w = /<;—, whence we get, very much as in the preceding case, E_ = w(E B f\d 1 which gives the mass of the earth in terms of B. The method differs from the preceding in that we use a large ball of metal instead of a mountain, and measure its deflecting force by a laboratory experiment instead of comparing astronomical observations with geodetic measurements. 166. In the earlier experiments by this method the small balls were of lead, about two inches in diameter, at the extremities of a light wooden rod, five or six feet long, enclosed in a case with glass ends, and their posi- tion and vibration was observed by a telescope looking directly at them from a distance of several feet. The attracting masses, B, were balls also of lead, about one foot in diameter, mounted on a frame pivoted in such a way that they could be easily brought to the required positions. Great difficulty was caused by air currents in the case, and it was neces- sary to enclose the whole apparatus in a small room of its own which was covered with tin-foil on the outside, and to avoid going near the room or allowing any radiant heat to strike it for hours before the observations. Baily, in England, and Eeich, in Germany, between 1838 and 1842, made very extensive series of observations of this kind. Baily obtained 5 66 for the earth's density, and Reich 5.48. The experiment was repeated in 1872 by Cornu, in Paris, with a modified apparatus. The horizontal bar was in this case only half a metre long, of aluminium, with small platinum balls at the end. For the large balls, glass globes were used, which could be pumped full of mercury or emptied at pleasure. The whole was enclosed in an air-tight case, and the air exhausted by an air- pump. The deflections and vibrations were observed by means of a tele- scope watching the image of a scale reflected in a small mirror attached to the aluminium beam near its centre, according to the method first devised 114 THE EARTH AS AN ASTRONOMICAL BODY. by Gauss and now so generally used in galvanometers and similar apparatus. Cornu obtained 5.56 as the result, and showed that Baily's figure required a correction which, when applied, would reduce it to 5.55. 167. 3. Potsdam Observations. — During 1886 and 1887 another series of observations was made by Wilsing, at Potsdam, with apparatus similar in principle to the torsion balance, except that the bar carry- ing the balls to be attracted was vertical, and turned on knife edges very near its centre of gravity. The knife edges, like those of an ordinary balance, rested upon agate planes, and the centre of gravity of the apparatus was so adjusted that one vibration of the pendulum, under the influence of gravity alone, would occupy from two to four minutes. The deflecting weights in this case were large cylinders of cast iron, suspended in such a way that they could be brought oppo- site the small balls, first on one side and then on the other. The whole was set up in a basement, and carefully and very effectually guarded against all changes of temperature, the arrangements being such that all manipulations and observations could be effected from the outside without entering the room. The deflections and vibra- tions were observed by a reflected scale, as in Cornu's observations. The result obtained was 5.59. Several other methods have been used ; of less scientific value, however. 168. a. The mass of the earth can be deduced by ascertaining the force of gravity at the top of a mountain and at its base, by means of pendulum experi- ments. The mass of the mountain must be determined by a survey, just as in the Schehallien method, which makes the method unsatisfactory. At the top of a mountain the height of which is h, and the distance of its centre of attraction from the top is d, gravity will be made up of two parts, one the attraction of the earth at a distance from its centre equal to R + h, and the other the attraction of the mountain alone considered. Calling the mass of the mountain m, and gravity at its summit cf (g being the force of gravity at the earth's surface), we shall have the proportion , E p E , m 9-9 ,_EV E ml 1 iJ 2 l(R + hY d^J' (R+hy the second fraction in the last term of the proportion being the attraction of the mountain. When g and g' are ascertained by the pendulum experi- ments, E remains as the only unknown quantity, and can be readily found- Observations of this kind were made by Carlini, in 1821, at Mt. Cenis, and the result was 4.95. MASS AND DENSITY OP THE EARTH. 115 169. b. By means of pendulum observations at the earth's surface compared with those at the bottom of a mine of known depth. This method was employed by Airy in 1843, at Harton Colliery, 1200 feet deep ; result, 6.56. In this case the principle involved is somewhat different. At any point within a hollow, homogeneous, spherical shell, gravity is zero, as Newton has shown. The attraction balances in all directions. If, then, we go down into a mine, the effect on gravity is the same as if a shell composed of all that part of the earth above our level had been removed. At the same time our distance from the earth's centre has been decreased by d, the depth of the mine. V At the surf ace g—h — , as before. At the bottom of the mine q'=fc g ~" she11 " . y (R-dy Comparing the two equations, we find E in the terms of the shell, since the ratio of g to g 1 is given by pendulum observations. Obviously, however, the mass of the " shell " is difficult to determine with accuracy. And it is by no means homogeneous, so that there is no great reason for surprise at the discordant result, g 1 was found to be actually greater than g, showing that although at the centre of the earth the attraction necessarily becomes zero, yet as we descend below ike surface, gravity increases for a time down to some unknown but probably not very great depth, where it becomes a maximum. 170. c. By experiments with a common balance. If a body be hung from one of the scale-pans of a balance, its apparent weight will obviously be increased when a large body is brought very near it underneath; and this increase can be measured. Poynting in England and Jolly in Germany have recently used this method, and have obtained results agreeing very fairly with those got from the torsion balance. The experiment, with some modifications, is soon to be tried again on a very large scale in Germany. 171. Constitution of the Earth's Interior. — Since the average den- sity of the earth's crust does not exceed three times that of water, while the mean density of the whole earth is about 5.58 (taking the average of- all the most trustworthy results) , it is obvious that at the centre the density must be very much greater than at the surface, — very likely as high as eight or ten times that of water, and equal to the density of the heavier metals. There is nothing in this that might not have been expected. If the earth were ever fluid, it is natural to suppose that in the solidification the densest materials would settle towards the interior. Whether the interior of the earth is solid or fluid it is difficult to say with certainty. Certain tidal phenomena, to be discussed hereafter, have led Sir 116 THE EARTH AS AN ASTRONOMICAL BODY. William Thomson and the younger Darwin to conclude that the earth as a whole is solid throughout, and "more rigid than glass," volcanic centres being mere pustules iu the general mass. To this many geologists demur. As regards the temperature at the earth's centre, it is hardly an astronomical question, though it has very important astronomical relations. We can only take space to say that the temperature appears to increase from the surface downward at the rate of about one degree Fahrenheit for every fifty or sixty feet, so that at the depth of a few miles the temperature must .be very high. THE ANNUAL MOTION OF THE SUN. 117 CHAPTER VI. THE APPARENT MOTION OP THE SUN AMONG THE STARS, AND THE EARTH'S ORBITAL MOTION. — THE EQUATION OP TIME, PRECESSION, NUTATION, AND ABERRATION. — VARIOUS KINDS OP "YEAR." — THE CALENDAR. 172. The Annual Motion of the Sun. — The apparent annual mo- tion of the sun must have been one of the earliest noticed of all astro- nomical phenomena. Its discovery antedates history. As seen by the people in Europe and Asia, the sun, starting in the spring, mounts higher in the sky each day at noon for three months, appears to stand still for a few days at the summer sol- stice, and then descends towards the south, reaching in the autumn the same noonday elevation it had in the spring. It keeps on its southward course to a winter solstice in December, and then returns to its original height at the end of a year, marking and causing the seasons by its course. A year, the interval between the successive returns of the sun to the same position, was very early found to consist of a little more than three hundred and sixty days. Nor is this all. The sun's motion is not merely a north-and-south motion, but it also moves eastward among the stars.; for in the spring the stars which are rising in the eastern horizon at sunset are different from those which are found there in the summer or winter. In the spring, the most conspicuous of the eastern constellations at sunset are Leo and Bootes ; a little later, Virgo appears ; in the sum- mer, Ophiuchus and Libra ; still later, Scorpio ; and in mid-winter, Orion and Taurus are in the eastern sky. 173. So far as mere appearances go, everything would be ex- plained by assuming that the earth is at rest and the sun moving around it ; but equally by the converse supposition., — for if the earth as seen from the sun appears at any point in the heavens, the sun as seen from the earth must appear in exactly the opposite point, and must keep opposite, moving through the same path in the sky (but 118 THE ECLIPTIC AND ZODIAC. six months behind), and always in the same " angular direction," if we may use the expression. (Just as two opposite teeth on a gear- wheel move in the same angular direction, though at any moment they are moving in opposite linear directions.) 174. That it is really the earth which moves, and not the sun, is absolutely demonstrated by two phenomena, too minute and delicate for pre-telescopic observations, but accessible enough to modern methods. We can only mention them here, leaving their fuller dis- cussion for the present. One of them is the aberration of light, the other the annual parallax of the fixed stars. These can be explained only by the actual motion of the earth. 175. The Ecliptic. — By observing with a meridian circle daily the declination of the sun, and the difference between its right ascension and that of some- star (Flamsteed used a Aquilae for the purpose), we shall obtain a series of positions of the sun's centre which can be plotted on a celestial globe ; and we can thus make out the path of the sun among the stars, and find the place where it cuts the celestial equator, and the angle it makes. This path turns out to be a great circle, as is shown by its cutting the equator at two points just 180° apart (the so-called equinoctial points or equinoxes), and makes an angle with it of approximately 23^°. This great circle is called the Ecliptic, because, as was early discovered, eclipses happen onty when the moon is crossing it. It may be defined as the trace of the plane of the earth's orbit upon the celestial sphere, just as the celestial equator is the trace of the plane of the terrestrial equa- tor on the same sphere. 176. Definitions. — The angle which the ecliptic makes with the equator is called the Obliquity of the ecliptic, and the points midway between the equinoxes are called the Solstices (sol-stitium) , because at these points the sun " stands," or stops moving in declination for a short time. Two circles parallel to the equator, drawn through the solstices, are called the Tropics (Greek rpiww), or "turning-lines," because there the sun turns from its northward motion to a southward, or vice versa. The obliquity is, of course, simply equal to the sun's maximum decli- vxtion, or greatest distance from the equator, which is reached in ane and December. CELESTIAL LATITUDE AND LONGITUDE. 119 The ancients were accustomed to determine it by means of the gnomon 1 (Art. 107). The length of the shadow at noon on the solstitial days deter- mines the zenith distance of the sun on those days, and the difference of the zenith distances at the two solstices is twice the angle desired. The gnomon also determined for the ancients the length of the year, it being only neces- sary to observe the interval between days in the spring or autumn, when the shadow had the same length at noon. 177. The Zodiac and its Signs. — A belt 16° wide, 8° on each side of the ecliptic, is called the Zodiac. The name is said to be derived from £$ for Tp (Tpayos). CELESTIAL LATITUDE AND LONGITUDE. 178. Since the moon and all the principal planets always keep within the zodiac, the ecliptic is a very convenient circle of reference, and was used as such by the ancients. Indeed, until the invention of pendulum clocks, it was on the whole more convenient than the equator, and more used. The two points in the heavens 90° distant from the ecliptic are called the Poles of the ecliptic The northern one is in the constellation 1 The Chinese claim to have made an observation of this kind about 4000 b.c, and the result given is very nearly what it should have been at that time. (The obliquity changes slightly in centuries.) If their observation is genuine, it is by far the oldest of all astronomical records. 120 CELESTIAL LATITUDE AND LONGITUDE. of Draco, about, half-way between the stars 8 and £ Dracouis. Now, suppose a set of great circles drawn, like meridians, through these poles of the ecliptic, and hence perpendicular to that circle ; these are Circles of latitude or secondaries to the ecliptic. The Longitude of a star or any other heavenly body is, then, the angle made at the pole of the ecliptic, between the circle of latitude, which passes through the vernal equinox, and the circle of latitude passing through the tody; or, what comes to the same thing, it is the arc of the ecliptic included between the vernal equinox and the foot of the circle of latitude pass- ing through the body. Celestial longitude is always reckoned east- ward from the vernal equinox, completely around the ecliptic, so that the longitude of the sun when 10° west of the vernal equinox would be written as 350°, and not as ^-10°. The Latitude of a star is simply its distance north or south of the ecliptic measured on the star's circle of latitude. 179. It will be seen that longitude differs from right ascension in being reckoned on the ecliptic instead of on the equator, nor can it be reckoned in time, but only in degrees, minutes, and seconds. Latitude differs from de- clination in that it is reckoned from the ecliptic instead of from the equator. The relation between right ascension and declination on the one hand, and longitude and latitude on the other, may be made clearer by. the accom- panying diagram (Fig;. 57), in which Relation between Celestial Latitude and -rnn • it, t a- % nf\ , i Longitude, and Eight Asceneion and EG 1S the ecliptic and EQ the equa- Deciination. tor, E being the vernal equinox. S being a star, its right ascension (a) is ER and its declination (8) is SB ; its longitude (A.) is EL, and its lati- tude (/3) is SL. Pand K are the poles of the equator and ecliptic respectively, and the circle KPCQ is the Solstitial Colure, so called. The student can hardly take too great care to avoid confusion of celestial latitude and longitude with right ascension and declination or -with terres- trial latitude and longitude. It is, of course, unfortunate that latitude in the sky should not be analogous to latitude upon the earth, or celestial longi- tude to terrestrial. The terms right ascension and declination are, however, of comparatively recent introduction, and found the ground preoccupied, celestial latitude and longitude being much older. CELESTIAL LATITUDE AWD LONGITUDE. 121 180. Conversion of \ and /3 into a and 8, or Vice Versa. — Right ascension and declination can, of course, always be converted into longitude and latitude by a trigonometrical calculation. We proceed as follows : In the triangle ERS, right-angled at R, we have given ER and RS (o and 8), from which we find the hypothenuse ES and the angle RES. Next in the triangle ELS, right-angled at L, we have the hypothenuse ES and the angle LES, which is equal to RES—LEQ (LEQ being a, the obliquity of the ecliptic). Hence we easily find EL and LS. 181. The Earth's Orbit in Space. — The ecliptic is not the earth's orbit, and must not be confounded with it. It is a great circle of the infinite celestial sphere, the trace made upon the sphere by the plane of the earth's orbit, as was stated in its definition. The fact that it is a great circle gives us no information about the earth's orbit, except that the orbit all lies in one plane passing through the sun. It tells us nothing as to its real form and size. By reducing the observations of the sun's right ascension and declination through the year to longitude and latitude (the latitude will always be zero, of course, except for some slight perturbations) and combining them with observations of the sun's apparent diameter, we can, however, ascertain the real form of the earth's orbit and the law of its motion in this orbit. But the size of the orbit — the scale of miles — cannot be fixed until we can find the sun's distance. 182. To find the Form of the Orbit, we may proceed thus : Take a point S for the sun and draw from it a line SO, Fig. 58, directed towards the vernal equinox as the origin of longi- tudes. Lay off from S indefi- nite lines, making angles with SO equal to the earth's longi- tude on each of the days ob- served through the year; i.e., the angle OS 10, is the longi- tude at the time of the 10th observation ; and so on. We shall thus get a sort of " spi- der," showing the directions as seen from the earth on those days. Next, as to relative distances. While the apparent diameter of the sun does not tell us its real distance from the earth, unless we first ¥ie. 58. Determination of the Form of the Earth's Orbit. 122 CELESTIAL LATITUDE AND LONGITUDE. know the sun's real diameter in miles, the changes in the apparent diameter do inform us as to the relative distance of the earth at different times, since the nearer we are, the larger the sun appears, — the distance being inversely proportional to the apparent diameter (Art. 6). If, then, we lay off on the arms of our "spider" dis- tances inversely proportional to the number of seconds of arc in the sun's measured diameter at each date, these distances will be pro- portional to the true distance of the earth from the sun, and the curve joining the points thus obtained will be a true map of the earth's orbit, though without any scale of miles upon it. When the operation is performed, we find that the orbit is an ellipse of small eccentricity (about one- sixtieth), with the sun, not in the centre, but at one focus. Fiq. 59. — The Ellipse. 183. For the benefit of any who may not have studied conic sections we define the ellipse. It is a curve such that the sum of the two distances from any point on its circumference to two points within, called the foci, is always constant, and equal to what is called the major-axis of the ellipse. SP + PF= A A', in Fig. 59. AC is called the semi-major-axig, and is usually denoted by A or a. BC is the semi-minor-axis, denoted by B or b. The eccentricity, denoted by e, is the SC fraction . AC Since BS is equal to A, SC= VA^-B 2 ; and VA*-B* The points where the earth is nearest to and most remote from the sun are called respectively perihelion and aphelion, and the line that joins them is, of course, the major-axis of the orbit. This line, con- sidered as indefinitely produced in both directions, is called the line of apsides, — the major-axis being a limited piece or " sect " of the line of apsides. 184. The variations of the sun's diameter are too small to be detected without a telescope (amounting only to about three per cent), so that the ancients were unable to perceive them. Hipparchus, however, about LAW OF THE EARTH'S MOTION. 123 150 B.C., discovered that the earth is not in the centre of the circular orbit which he supposed the sun to describe around it. Everybody assumed, on a priori grounds, never disputed until the time of Kepler, that the sun's orbit must be a circle and described with a uniform motion, because a circle is the only " perfect " curve, and uniform motion the only perfect motion. Obviously, however, the sun's apparent motion is not uniform, because it takes 186 days for the sun to pass from the vernal equinox to the autumnal through the summer months, and only 179 days to return during the win- ter. Hipparchus explained this difference by the hypothesis that the earth is out of the centre of the sun's path. 185. To find the Eccentricity of the Orbit. — Having the greatest and least apparent diameters of the sun, the eccentricity, e, is easily found. In Fig. 59, since, by definition, e = 08 -f- GA, we have OS = GA x e, or Ae. The perihelion distance AS is therefore equal to A X (1 — e), and the aphelion distance SA' to A (1 + e). Suppose now that the greatest and least measured diameters of the sun are p and q. This gives us the proportion p:q=A (1+e) : A (1 — e), since the diameters are inversely proportional to the distances. From this we get _ p + q The actual values of p and q are 32' 36" .4 and 31' 31".8, which give e = 0.01678 : this is about T ] T , as has been stated. 186. To find the Law of the Earth's Motion. — By comparing the measured apparent diameter with the differences of longitude from day to day, we can also deduce the law of the earth's motion. On making a table of daily motions and apparent diameters, we find that these daily motions vary directly as the squares of the diameters; from which it directly follows that the earth moves in such a way that its radius-vector describes areas proportional to the times (a law which Kepler first brought to light in 1609). The radius-vector is the line which joins the earth to the sun at any moment. 187. Consider a small elliptical sec- fib. wi. tor, dSc (Fig. 60), described by the earth BquabIe DeBC1 . iption f Areae. - in a unit of time. Regarding it as a trian- gle, its area is given by the formula ^ScXSd sin cSd ; and calling this angle 6 (which will be very small), and considering that in so short a time 124 KEPLER'S 1'EOBLEM. Sd and Sc would remain sensibly equal, each being equal to R (the radius- vector at the middle point of the arc), this formula becomes, Area of sector = \ R 2 $. Now, calling the sun's apparent diameter D, we have : R =i' (k being a constant, and depending on the sun's diameter in miles) ; whence ii 2 =— — . But our measurements show that 0= k^D 2 , k 1 being another constant. Substitute these values of R 2 and in the formula for the area, and we have Z.2 Area of sector = J — Xk l D 2 ={ k' 2 k„ a constant ; that is, the area described by the radius-vector in a unit of time is always the same. The planet near perihelion moves so much faster, that the areas aSb, cSd, and eSf are all equal to each other, if the arcs are de- scribed in the same time. 188. Kepler's Problem. — As the case stands so far, this is a mere fact of observation ; but as we shall see hereafter, and as was demon- strated by Newton, the fact shows that the earth moves under the ac- tion of a force always directed in line with the sun. In such a case the "equable description of areas" is a necessary mechanical consequence. It is true in every case of elliptical motion, and enables us to find the fig. 6i. -Kepler's Problem. position of the earth or any planet in its orbit at any time, when we once know the time of its orbital revolution (technically the period) , and the time when it was at perihelion. Thus, the angle ASP (Fig. 61), which is called the Anomaly of the planet, must be such that the area of the elliptical sector ASP will be that portion of the whole ellipse which is represented by the fraction ~, t being the number of days since the planet last passed the perihelion, and T the number of days in the whole period. For instance, if the earth last passed perihelion on Dec. 31 (which it did), its place on May 1 must "be such that the ANOMALY AND EQUATION OP THE CENTRE. 125 sector ASP will be ^££ } of the whole of the earth's orbit ; since from Dec. 31 to May 1 is 121 days. The solution of this problem, known as " Kepler's problem," leads to transcendental equations, and lies beyond our scope. See Watson's "Theoretical Astronomy," pp. 53 and 54, or any other.simi- lar worki 189. Anomaly and Equation of the Centre. — The angle ASP, which has been termed simply the "Anomaly," is strictly the true Anomaly, as distinguished from the mean Anomaly. The former may be defined as the angle actually made at any time by the radius- vector of a planet with the line of apsides, the angle being reckoned from the perihelion point completely around in the direction of the planet's motion. The mean Anomaly is what the Anomaly would be at the given moment if the planet had moved with uniform angular velocity, completing the orbit in the same period, and passing perihelion at the same time, as it actually does. The difference between the two anomalies is called the Equation of the Centre. This is zero at perihelion and aphelion, and a maximum midway between them. In the case of the sun, its greatest value is nearly 2°, the sun getting alternately that amount ahead of, and behind, the position it would occupy if its apparent daily motion were uniform. 190. The Seasons. — The earth in its motion around the sun always keeps its axis parallel to itself, for the mechanical reason that . a revolving body necessarily maintains the direction of its axis inva- riable, unless disturbed by extraneous force, as is very prettily illus- trated by the gyroscope. About March 20 the earth is so situated that the plane of its equator passes through the sun, the sun's decli- nation being zero on that day. At that time, the line which separates the illuminated portions of the earth passes through the two poles, and day and night are every- where equal. The same is again true of the 22d of September, when the sun is at the autumnal equinox on the opposite side of the orbit. About the 21st of June the earth is so situated that its north pole is inclined towards the sun by about 23£°, which is the sun's northern declination on that date. The south pole is then in the obscure half of the earth's globe, while the north pole receives sunlight all day long ; and in all portions of the northern hemisphere the day is longer than the night, the difference between the day and night depending upon the latitude of the place, while in the southern hemisphere the 126 DIURNAL PHENOMENA NEAR THE POLE. days are shorter than the nights. At the time of the winter solstice these conditions are reversed. At the equator (of the earth) the day and night are equal at all times of the year. The sun when in north- ern declination of course always rises at a point on the horizon north of east, and sets at a point north of west, so that for a portion of the time each day it shines on the north side of a house. 191. Diurnal Phenomena near the Pole. — At the north pole, where the celestial pole is in the zenith, and the diurnal circles are parallel with the horizon, the sun will maintain the same elevation all day long, except for the slight change caused by the variation of its declination in twenty- four hours. The sun will appear on the horizon at the date of the vernal equinox (in fact, about three days before, on account of refraction), and slowly wind upward in the sky until it reaches its maximum elevation of 23|° on June 21. Then it will retrace its course until a day or two after the autumnal equinox, when it sinks out of sight. At points between the north pole and the polar circle the sun will appear above the horizon earlier in the year than March 20, and will rise and set daily until its declination becomes equal to the observer's distance from the pole, when it will make a complete circuit of the heavens, touching the hori- zon at midnight at the northern point ; and after that never setting again until it reaches the same declination in its southward course after passing the solstice. From that time it will again rise and set daily until it reaches a southern declination just equal to the observer's polar distance, when the long night begins ; to continue until the sun, having passed the southern solstice, returns again to the same declination at which it made its appear- ance iu the spring. At the polar circle itself (or, more strictly speaking, owing to refraction, about one-half a degree south of it) the "midnight sun" will be seen on just one day in the year, the day of the summer solstice ; and there will also be one absolutely sunless day, viz., the day of the winter solstice. The same remarks apply in the southern hemisphere, by making the obvious changes. 192. Effects on Temperature. — The changes in the duration of 11 insolation" (exposure to sunshine) at any place involve changes of temperature and other climatic conditions, thus producing the sea- sons. Taking as a standard the amount of heat received in twenty-four hours on the day of the equinox, it is clear that the surface of the soil at any place in the northern hemisphere will receive more than this average amount of heat whenever the sun is north of the celestial equator, for two reasons. 1. Sunshine lasts more than half the day. 2. The mean elevation of the sun during the day is greater than TIME OP HIGHEST TEMPERATURE. 127 when it is at the equinoxes, since it is higher at noon, and in any case reaches the horizon at rising and setting. Now, the more obliquely the rays strike, the less heat they bring to each square inch of surface, as is obvious from Fig. 62. A beam of sunshine having the cross- section ABOD, when it strikes the surface at an angle h (equal to the sun's altitude) is spread over a much larger surface, Ac, and of course the amount of heat per square inch is proportionately reduced. If Q is the amount of heat per square inch brought by the ray when fall- ing perpendicularly, as on the sur- Bio. 62. face AC, then on Ac the amount Efleot it 8 ™,' 8 Ele ™ t ' on ° n H A , monnt ' of Heat imparted to the Soil. per square inch will be Q x sin li, since AB = Ab X sin h. This difference in favor of the more nearly vertical rays is exaggerated by the absorption of heat in the atmo- sphere, because rays that are nearly horizontal have to traverse a much greater thickness of air before reaching the ground. For these two reasons, at a place in the northern hemisphere, the temperature rises rapidly as the sun comes north of the equator, thus giving us our summer. 193. Time of Highest Temperature. — We, of course, receive the most heat per diem at the time of the summer solstice ; but this is not the hottest time of the summer, for the obvious reason that the weather is then all the time getting hotter, and the maximum will not be reached until the increase ceases; that is, not until the amount oj heat lost in the night equals that stored up by day. H the earth's surface threw off the same amount of heat hourly whether it were hot or cold, then this maximum would not come until the autumnal equinox. This, however, is not the case. The soil loses heat faster when warm than it does when cold, the loss being nearly proportional to the dif- ference between the temperature of the soil and that of surrounding space ; (Newton's law of cooling) ; and so the time of the maximum is made to come not far from the end of July, or the first of August, in our latitude. For similar reasons the minimum temperature of winter occurs about Feb. 1, about half-way between the solstice and the vernal equinox. Since, how- ever, our weather is not entirely " made on the spot where it is used," but is affected by winds and currents that come from great distances, the actual time of the maximum temperature cannot be determined by any mere astro- nomical considerations, but varies considerably from year to year. 128 DIFFERENCE BETWEEN SEASONS. 194. Difference between Seasons in Northern and Southern Hemi- spheres. — Since in December the distance of the earth from the sun is about three per cent less than it is in June, the earth (as a whole) receives hourly about six per cent more heat in December than in June, the heat varying inversely as the square of the distance. Foi this reason the southern summer, which occurs in December and January, is hotter than the northern. It is, however, seven days shorter, because the earth moves more rapidly in that part of its orbit. The total amount of heat per acre, therefore, received during the summer is sensibly the same in each hemisphere, the shortness of the southern summer making up for its increased warmth. 195. The southern winter, however, is both longer and colder than the northern ; and it has been vigorously maintained by certain geologists, that, on the whole, the mean annual temperature of the hemisphere which has its winter at the time when the earth is in aphelion is lower than that of the opposite one. It has been attempted to account for the glacial epochs in this way. It is certain that at present, at any place in the southern hem- isphere, the difference between the maximum temperature of summer and the minimum of winter must be greater than in the case of a station in the - northern hemisphere, similarly situated as to elevation, etc. We say " at present " because, on account of certain slow changes in the earth's orbit, to be spoken of immediately, the state of things will be reversed in about ten thousand years, the northern summer being then the hotter and shorter one. 196. Secular Changes in the Orbit of the Earth. — The orbit of the earth is not absolutely unchangeable in form or position, though it is so in the long run as regards the length of its major axis and the duration of the year. 197. 1. Change in Obliquity of the Ecliptic. — The ecliptic slightly and very slowly shifts its position among the stars, thus altering the latitudes of the stars and the angle between the ecliptic and equator, i.e., the obliquity of the ecliptic. This obliquity is at present about 24' less than it was 2000 years ago, and is still decreasing about half a second a year. It is computed that this diminution will continue for about 15,000 years, reducing the obli- quity to 22|°, when it will begin to increase. The whole change, according to J. Herschel, can never exceed about 1° 20' on each side of the mean. 198. 2. Change of Eccentricity.— At present the eccentricity of the earth's orbit (which is now 0.0168) is also slowly diminishing. EQUATION OF TIME. 129 According to Leverrier, it will continue to decrease for about 24,000 years, until it becomes 0.003, and the orbit will be almost circu- lar. TJhen it will increase again for 40,000 years, until it becomes 0.02. x In this way the eccentricity will oscillate backwards and forwards, always, however, remaining between zero and 0.07 ; but the oscillations are not equal either in amount or time, and so cannot properly be compared to the "vibrations of a mighty pendulum,'' which is rather a favorite figure of speech. 199. 3. Revolution of the Apsides of tlie Earth's Orbit. — The line of apsides of the orbit, which now stretches in both directions towards the constellations of Sagittarius and Gemini, is also slowly and steadily moving eastward, and at a rate which will carry it around the circle in about 108,000 years. 200. These so-called '■'■secular" changes are due to the action of the other planets upon the earth. Were it not for their attraction, the earth would keep her orbit with reference to the sun strictly unaltered from age to age, except that possibly in the course of millions of years the effects of falling meteoric matter and the attraction of the nearer fixed stars might make themselves felt. Besides these secular perturbations of the earth's orbit, the earth itself is continually being slightly disturbed in its orbit. On account of its con- nection with the moon, it oscillates each month a few hundred miles above and below the true plane of the ecliptic, and by the action of the other planets it is sometimes set forwards or backwards to the extent of a few thousand miles. Of course every such change produces a corresponding slight change in the apparent position of the sun. 201. Equation of Time. — We have stated a few pages back (Art. Ill), that the interval between the successive passages of the sun across the meridian is somewhat variable, and that for this reason apparent solar, or sun-dial, days are unequal. On this ac- count mean time has been adopted, which is kept by a " fictitious" or il mean," sun moving uniformly in the equator at the same average rate as that of the real sun in the ecliptic. The hour- angle of this mean sun is, as has been already explained, the local mean time (or clock time) ; while the hour-angle of the real sun is the apparent or sun-dial time. The Equation of Time is the difference between these two times, reckoned as plus when 130 EQUATION OF TIME. the sun-dial is shiver than the clock, and minus when it is faster. It is the correction which must be added (algebraically) to apparent time in order to get mean time. As it is the difference between the two hour-angles, it may also be defined as the difference between the right ascensions of the mean sun and the true sun; or as a formula we may write : E=a t —a m , in which a m is the right ascension of the mean sun, and a, of the true sun. The principal causes of this difference are two : — • 202. 1. The Variable Motion of the Sun in the Ecliptic, due to the Eccentricity of the Earth's Orbit. ■ — Near perihelion, which occurs about Dec. 31, the sun's motion in longitude is. most rapid. Ac- cordingly, at this time the apparent solar days exceed the sidereal by more than the average amount, making the sun-dial days longer than the mean. (The average solar day, it will be remembered, is 3 m 56" longer than the sidereal.) The sun-dial will therefore lose time at this season, and will continue to do so for about three months, until the angular motion of the sun falls to its mean value. Then it will gain until aphelion, when, if the clock and the sun were started together at perihelion, they will once more be together. During the next half of the year the action will be reversed. Thus, twice a year, so far as the eccentricity of the earth's orbit is concerned, the clock and sun would be together at perihelion and aphelion, while half-way between they would differ by about eight minutes; the equation of time (so far as due to this cause only) being+ 8 minutes in the spring, and — 8 minutes in the autumn. 203. 2. The Inclination of the Ecliptic to the Equator. — Even if the sun's (apparent) motion inlongU tude (i.e., along the ecliptic) were uniform, its motion in right ascen- sion would be variable . If the true and fictitious suns started together at the equinox, they would indeed be together at the solstices and at the other equinox, because it is iust Effect of Obliquity of Ecliptic in producing , o „ „ . ' . , Equation of Time. 18(J from equinox to equinox, and the solstices are exactly half-way between them. But at intermediate points, between the equinoxes and solstices, they would not be together on the same hour-circle. EQUATION OF TIME. 131 This is best seen by taking a celestial globe and marking on the eclip- tic a point, m, half-way between the vernal equinox and the solstice, and also marking a point n cm the equator, 45° from the equinox. It will at once be seen that the former point, m in Fig. 63, 1 is west of n, so that m in the daily westward motion of the sky will come to the meridian first; in other words, when the sun is half-way between the vernal equinox and the summer solstice, the sun-dial is faster than the clock, and the equation of time is minus. The difference, measured by the arc m'n, amounts to nearly ten minutes ; and of course the same thing holds, mutatis mutandis, for the other quad- rants. 204. Combination of the Effects of the Two Causes. — We can rep- resent graphically these two components of the equation of time and the result of their combination as follows (Fig. 64) : — The central horizontal line is a scale of dates one year long, the letters denoting the beginning of each month. The dotted curve Fig. 64. — The Equation of Time. shows the equation of time due to the eccentricity of the earth's orbit, above considered. Starting at perihelion on Dec. 31, this com- ponent is then zero, rising from there to a value of about + 8 ra on April 2, falling to zero on June 30, and reaching a second maximum of — 8 m on Oct. 1 . In the same way the broken-line curve denotes the effect of the obliquity of the ecliptic, which, by itself alone considered, would produce an equation of time having four maxima of, approximately, 10 m each, about the 6th of February, May, 1 Fig. 63 represents a celestial globe viewed from the west side, the axis being vertical, and K, the pole of the ecliptic, on the meridian, while E is the vernal equinox. 132 EQUATION OF TIME. August, and November (alternately + and — ),and reducing to zero at the equinoxes and solstices. The full-lined curve represents their combined effect, and is con- structed by making its ordinate at each point equal to the sum (algebraic) of the ordinates of the two other curves. At the 1st of February, for instance, the distance F, 3, in the figure = F, 1 +F,2. So, also, M , 6 = M , 4 + M. , 5 ; the components, however, in this case have opposite signs, so that the difference is actually taken. The equation of time is zero four times a year, viz. : on April 15, June 14, Sept. 1, and Dec. 24. The maxima are February 11,+ 14 m 32" ; May 14, - 3 ra 55' ; July 26, + 6 m 12", and Nov. 2,- 16 m 18'. But the dates and amounts vary slightly from year to year. The two causes above discussed are only the principal ones effective in producing the equation of time. Every perturbation suffered by the earth comes in with its own effect ; but all other causes combined never alter the equation by more than a few seconds. 205- Precession of the Equinoxes. — The length of year was found in two ways by the ancients : — 1. By the gnomon, which gives the time of the equinox and sol- stice ; and 2. By observing the position of the sun with reference to the stars, — their rising and setting at sunrise or sunset. Comparing the results of observations made by these two methods at long intervals, Hipparchus (120 B.C.) found that the two do not agree ; the former year (from equinox to equinox) being 20 m 23" shorter than the other (according to modern data). The equinox is plainly moving westward on the ecliptic, as if it advanced to meet the sun on each annual return. He therefore called the motion the " precession" of the equinoxes. On comparing the latitudes of the stars in the time of the ancient astronomers with the present latitudes, we find that tbey have changed very slightly indeed ; and we know therefore that the ecliptic and the plane of the earth's orbit maintains its position sensibly unaltered. On the other hand, the longitudes of the stars have been found to increase regularly at the rate of about 50".2 annually, — fully 30° in the last 2000 years. Since longitudes are reckoned from the equinox (the intersection between the ecliptic and .equa- tor), and since the ecliptic does not move, it is evident that the motion must be in the celestial equator; and accordingly we find that both the right ascension and the declination of the stars are constantly changing. PRECESSION. 133 206. Motion of the Pole of the Equator around the Pole of the Ecliptic. — The obliquity of the ecliptic, which equals the distance in the sky between the pole of the equator and the pole of the ecliptic (Art. 178), has remained nearly constant. Hence the pole of the equator must be describing a circle around the pole of the ecliptic in a period of about 25,800 years (360° divided by 50".2). The pole of the ecliptic has remained practically fixed among the stars, but the other pole has changed its position materially. At present the pole star is about 1^° from the pole. At the time of the star catalogue of Hipparchus it was 12° distant from it, and during the next century it will approach to within about 30', after which it will recede. 207. If upon a celestial globe we take the pole of the ecliptic as a cen- tre, and describe about it a circle with a radius of 23J°, we shall get the track of the celestial pole among the stars, and shall find that the circle passes very near the star a Lyrae, on the opposite side of the pole of the ecliptic from the present pole star. About 12,000 years hence a Lyrse will be the pole star. Reckoning backwards, we find that some 4000 years ago a Draconis was the pole star; and it is a curious circumstance that certain of the tunnels in the pyramids of Egypt face exactly to the north, and slope at such an inclination that this star at its lower culmination would have been visible from their lower end at the date when the pyramids are supposed to have been built. It is probable that these passages were arranged to be used for the purpose of observing the transits of their then pole star. 208. Effect of Precession upon the Signs of the Zodiac. — Another effect of precession is that the signs of the zodiac do not now agree with the constellations which bear the same name. The sign of Aries is now in the constellation of Pisces ; and so on, each sign having " backed," so to speak, into the constellation west of it. 209. Physical Cause of Precession. — The physical cause of this slow conical rotation of the earth's axis around the pole of the eclip- tic lies in the two facts that the earth is not exactly spherical, and that the attractions of the sun and moon 1 act upon the equatorial ring of mat- ter which projects above the true sphere, tending to draw the plane of the equator into coincidence with the plane of the ecliptic by their greater attraction on the nearer portions of the ring. The action is just what it would be if a spheroidal ball of iron of the shape of the earth had 1 The planets, by their action upon the plane of the earth's orbit (Art. 197), slightly disturb the equinox in the opposite direction. This effect amounts to about 0'M6 annually. / 134 PRECESSION. a magnet brought near it. The magnet, as illustrated in Fig. 65, would tend to draw the plane of the equator into the line CM joining its pole with the centre of the globe, because it attracts the nearer portion of the equatorial protuberance at E more strongly than the remoter at Q. If it were not for the earth's rotation, this attraction would bring the two planes of the ecliptic and equator together ; but since the earth is spinning on its axis, we get the same result that we do with the whirl- ing wheel of a gyroscope by hanging a weight at one end of the axis. We then have the result of the combination of two rotations at right angles with each other, one the whirl of the wheel, the other the "tip" which the weight tends to give the axis. (See Brackett's Phys- ics, pp. 53-56.) Fig. 65. Effect of Attraction on a Spheroid. Fig. 66. Precession illustrated by the Gyroscope. 210. In this case, if the wheel of the gyroscope is turning swiftly dock-wise, as seen from above (Fig. 66) , the weight at the (lower) end of the axis will make the axis move slowly around, counter-clock- wise, without at all changing its inclination. If we regard the hori- zontal plane passing through the gyroscope as representing the eclip- tic, and the point in the ceiling vertically above the gyroscope as the pole of the ecliptic, the line of the axis of the wheel produced up- ward would describe on the ceiling a circle around this imaginary ecliptic pole, acting precisely as does the pole of the earth's axis in the sky. The swifter the wheel's rotation, the slower would be this " precessional " motion of its axis ; and of course, the rate of motion also depends upon the magnitude of the suspended weight. 211. A full treatment of the subject would te too complicated for our pages. An elementary notion of the way the action takes place, correct as PRECESSION. 135 far as it goes, is easily obtained by reference to Fig. 67. Let X Y be the axis of the gyroscope, the wheel being seen in section edge-wise, and the eye being \>n the same level as the centre of the wheel; the wheel turning so that the point B is coming towards the observer. Now, suppose a weight hung on the lower end of the axis. If the wheel were not turning, the point B would come to some point F in the sarie time it now takes to reach C (that is, after a quarter of a revolution). By combina- tion of the two motions it will come to a point K at the end of the same time, having crossed the horizontal plane AD at L; and this can be effected only by a backward "skewing around" of the whole wheel, axis and all. This does not, FlQ * 67 - of course, explain why the inclination of Regression of the Gyroscope Wheel, the axis does not change under the action of the weight, but is only a very partial illustration, showing merely why the plane of the wheel regresses. A complete discussion would require the consideration of the motion of every point on the wheel by a thorough and difficult analytical treatment, in order to give the complete explanation of the reason why the depressing weight, however heavy, does not cause the end if the axis to fall perceptibly. (See article, "Gyroscope," in Johnson's Cyclopaedia.) 212= Why Precession is so Slow. — The slowness of the preces- sion depends on three things: (a) the enormous " moment of rota- tion "of the earth — a point on the equator moves with the speed of a cannon ball ; (b) the smallness of the mass (compared with that of the whole earth) of the protuberant ring to which precession is due ; and (c) the minuteness of the force which tends to bring this ring into coincidence with the ecliptic, a force which is not constant and per- sistent, like the weight hung on the gyroscope axis, but very variable. 213. The Equation of the Equinox. — Whenever the sun is in the plane of the equator (whi ;h is twice a year, at' the time of the equinoxes), the sun's precessional force disappears entirely, its attraction then having no tendency to draw the equator out of its position. The moon's action, on ac- count of her proximity, is still more powerful than that of the sun ; on the average two and a half times as great. Now, the moon crosses the celestial equator twice every month, and at those times her action ceases. There -is still another cause for variation in the effectiveness of the moon's attraction. As we shall see hereafter, she does not move in the ecliptic, but in a path which cuts the ecliptic at an angle of about 5°, at two points called the Nodes; the ascending node being the point where she crosses the ecliptic 136 NUTATION. from south to north. These nodes move westward on the ecliptic (Art. 455), making the circuit once in about nineteen years. Now, when the ascending node of the moon's orbit is at B (Fig. 68), near the autumnal equinox F, its inclination to the equator will J3 be, as the figure shows, less than the obliquity of the eclip- tic by about 5°; i.e., it will Fig. 68. be only about 18°. On the Variation in the Inclination of Moon'B Orbit to Equator, other hand, nine and a half years later, when the node has backed around to a point A, near the vernal equinox, the inclination of the moon's orbit to the equator will be nearly 28°. When the node is in this position, the moon will produce nearly twice as much precession al move- ment each month as when the node was at B. The precession, therefore, is not uniform, but variable, almost ceasing at some times and at others becoming rapid. The average amount, as has been stated, is 50". 2 a year; and the variation is taken account of in what is called the equation of the equinox, which is the difference between the actual position of the equinox at any time and the position it would have at that moment if the precession had been all the time going on uniformly. 214. Nutation. — Not only does the precessional force vary in amount at different times, but in most positions of the disturbing body with respect to the earth's equator there is a slight thwartwise component of the force, tending directly to accelerate or retard the pre- cessional movement of the pole — just as if one should gently draw the weight W (Fig. 66) horizontally. The consequence is what is willed Ntittition or " nodding." The axis of the earth, instead of moving smoothly in a circle, nods in and out a little with respect to the pole of the ecliptic, describing a wavy curve resembling that shown in Fig. 69, but with nearly 1400 indentations in the entire cir- cumference traversed in 26,000 years. 215. Vi T e distinguish three of these nutations, (a) The Lunar Nutation, depending upon the motion of the moon's nodes. This has a period of a little less than nineteen years, and amounts to 9".2. (b) The Solar Nuta- tion, due to the changing declination of the sun. Its period is a year, and its amount 1".2. (c) The Monthly Nutation, precisely like the solar nutation, except that it is due to the moon's changes of dec- lination during the month. It is, however, too small to be certainly measured, not exceeding one- tenth of a second. iuaUon THE CALENDAR. 137 Nutation was detected by Bradley in 1728, but not fully explained until 1748. Neither precession nor nutation affects the latitudes of the stars, since they are not due to any change in the position of the ecliptic, but only to displacements of the earth's axis. The longitudes alone are changed by them. The right ascension and declination of a star are both affected. 216. The Three Kinds of Year. — In consequence of the motion of the equinoxes caused by precession, the sidereal year and the equinoctial or " tropical" year do not agree in length. Although the sidereal year is the one which represents the earth's true orbital revo- lution around the sun, it is not used as the year of chronology and the calendar, because the seasons depend on the sun's place in rela- tion to the equinoxes. The tropical year is the year usually employed, unless it is expressly stated to the contrary. The length of the Sidereal year is 365 d 6 h 9 m 9 s ; that of the Tropical year is about 20 m less, 365 d 5 h 48 m 46 8 . The third kind of year is the anomalistic year, which is the time from perihelion to perihelion again. As the line of apsides of the earth's orbit moves always slowly towards the east, this year is a littl*. longer than the sidereal. Its length is 365 d 6 h 13 m 48 s . 217. The Calendar. — The natural units of time are the day, the month, and the year. The clay, however, is too short for convenient use in designating extended periods of time, as for instance in expressing the age of a man. The month meets with the same objection, and for all chronological purposes, therefore, the year is the unit practically employed. In ancient times, however, so much regard was paid to the month, and so many of the religious beliefs and observances connected themselves with the times of the new and full moon, that the early history of the calendar is largely made up of attempts to fit the month to the year in some convenient way. Since the two are incommensurable, the problem is a very difficult, and indeed strictly speaking, an impossible, one. In the earliest times matters seem to have been wholly in the hands of the priesthood, and the calendar then was predominantly lunar, With months and days intercalated from time to time to keep the seasons in place. The Mohammedans still use a purely lunar calen- dar, having a year of twelve lunar months, and containing alternately 354 and 355 days. In their reckoning the seasons fall, of course, continually in different months, and their calendar gains about one year in thirty-three upon the reckoning of Christian nations. 138 THE CALENDAR. 218. The Metonic Cycle. — Among the Greeks the discovery of the so-called lunar or Metonic cycle by Meton, about 433 B.C., con- siderably simplified matters. This cycle consists of 235 synodic months (from new moon to new again) , which is very approximately equal to 19 common years of 365£ clays. 235 months equal 6939 d 16 h 31 m ; 19 tropical years equal 6939 d 14 h 27 m : so that at the end of the 19 years, the new and full moon recur again on the same days of the year, and at the same time of day within about two hours. The calendar of the phases of the moon, for instance, for 1889 is the same as for 1870 and 1908 (except that intervening leap-years may change the dates by one day). The " Golden number " of a year is its number in this Metonic cycle, and is found by adding 1 to the " date-number " of the year and dividing by 19. The remainder, unless zero, is the "golden number" (if it comes out zero, 19 is taken instead). Thus the golden number for 1888 is found by dividing 1889 by 19, and the remainder 8, is the golden number of the year. This cycle is still employed in the ecclesiastical calendar in finding the time of Easter. For further information on the subject, consult Johnson's Encyclopaedia, or Sir Edmund Becket's "Astronomy without Mathematics." 219. Julian Calendar. — Until the time of Julius Caesar the Roman calendar seems to have been based upon the lunar year of twelve months, or 355 days, and was substantially like the modern Mohamme- dan calendar, with arbitrary intercalations of months and days made by the priesthood and magistrates from time to time in order to bring it into accordance with the seasons. In the later days of the Eepublic, the confusion had become intolerable. Csesar, with the help of the as- tronomer Sosigenes, whom he called from Alexandria for the purpose, reformed the system in the year 45 B.C., introducing the so-called "Julian calendar," which is still used either in its original shape or with a very slight modification . He gave up entirely the attempt to co- ordinate the month with the year, and adopting 365J days as the true length of the tropical year, he ordained that every fourth year should contain an extra clay, the sixth day before the Kalends of March on that year being counted twice, whence the year was called " bissextile." Before his time the year had begun in March (as indicated by the Roman names of the months, — September, seventh month ; October, eighth month, etc.), but he ordered it to begin on the 1st of January, which in that year (45 b.c.) was on the day of the new moon next fol- lowing the winter solstice. In introducing the change it was necessary to make the preceding year 445 days long, and it is still known in THE CALENDAR. 139 the annals as " the year of confusion." He also altered the name of the month Quintilis, calling it " July " after himself. There was some irregularity in the bissextile years for a few years after Caesar's death, from a misunderstanding of his rule for the intercalary day ; but his successor Augustus remedied that, and to put himself on the same level with his predecessor, he took possession of the month Sextilis, calling it " August " ; and to make its length as great as that of July, he robbed February of a day. From that time on, the Julian calendar continued unbrokenly in use until 1582 ; and it is still the calendar of Russia and of the Greek Church. 220. The Gregorian Calendar. — The Julian calendar is not quite correct. The true length of the tropical year is 365 days 5 hours 48 minutes and 46 seconds, and this leaves a difference of 11 minutes and 14 seconds by which the Julian calendar year is the longer, being exactly 365J days. As a consequence, the date of the equinox comes gradually earlier and earlier by about three days in 400 years. (400 X ll£ m = 4467 minutes = 3 d 2 h 27 m .) In the year 1582, the date of the vernal equinox had fallen back 10 days to the 11th of March, instead of occurring on the 21st of March, as at the time of the Council at Nice, 325 a.d. Pope Gregory, therefore, acting under the advice of the Jesuit astronomer, Clavius, ordered that the day following Oct. 4 in the year 1582 should be called not the 5th, but the loth, and that the rule for leap-year should be slightly changed so as to prevent any such future displacement of the equinox. The rule now stands : All years whose date-number is divisible by four without a remainder are leap-years, unless they are century years (1700, 1800, etc.). The' century years are not leap-years unless their date- number is divisible by 400, in which case they are: that is, 1700, 1800, and 1900 are not leap years ; but 1600, 2000, and 2400 are. 221. Adoption of the New Calendar. — The change was imme- diately adopted by all Catholic nations ; but the Greek Church and most of the Protestant nations, rejecting the Pope's authority, de- clined to accept the correction. In England it was at last adopted in the year 1752, at which time there was a difference of eleven days between the two calendars. (The year 1600 was a leap-year accord- ing to the Gregorian system as well as the Julian, but 1700 was not.) Parliament in 1751 enacted that the clay following the 2d of Septem- ber, in the year 1752, should be called the 14th instead of the 3d; and also that this year (1752), and all subsequent years, should be- gin on the first of January. 140 BEGINNING OF THE YEAR. The change was made under very great opposition, and there were violent riots in consequence in different parts of the country, especially at Bristol, where several persons were killed. The cry of the populace was, " Give us back our. fortnight," for they supposed they had been robbed of eleven days, although the act of Parliament was carefully framed to prevent any injus- tice in the collection of interest, payment of rents, etc. At present, since the year 1800 was not a leap-year according to the Gregorian calendar, while it was so according to the Julian, the difference between the two calendars amounts to twelve days ; thus in Russia the 19th of August would be reckoned as the 7th. In Russia, however, for scientific and commercial purposes both dates are very generally used, so that the date mentioned would bs written Aug. -£ Xl . When Alaska was annexed to the United States, its calendar had to be altered by eleven days. (See Art. 123.) 222. The Beginning of the Year. — The beginning of the year has been at several different dates in the different countries of Europe. Some have regarded it as beginning at Christmas, the 25th of December ; others, on the 1st of January ; others still, on the 1st of March ; others, on the 25th ; and others still, at Easter, which may fall on any day between the 22d of March and the 25th of April. In England previous to the year 1752 the legal year commenced on the 25th of March, so that when the change was made, the year 1751 necessarily lost its months of January and February, and the first twenty-four days of March. Many were slow to adopt this change, and it becomes necessary, therefore, to use considerable care with respect to English dates which occur in the months of January, February, or March about that period. The» month of February, 1755, for instance, would by some writers be reckoned as occurring in 1754. Confusion is best avoided by writing, Feb. yfff. 223. First and Last Days of the Year. — Since the ordinary civil year consists of 365 days, which is 52 weeks and one day, the last day of each common year falls on the same day as the first ; so that any given date will fall one day later in the week than it did on the preceding year, unless a 29th of February has intervened, in which case it will be two days later ; that is, if the 3d of January, 1889, falls on Thursday, the same date in 1890 will fall on Friday. 224. Aberration. — Although in strictness the discussion of aberration does not belong to a chapter describing the earth and its motions, yet since it is a phe- nomenon due to the earth's motion, and affects the right ascension and declination of the stars in much the same way as do precession and nutation, it may properly enough be considered here. Aberration is the apparent displacement of a star, due to the combi- nation of the motion of light with the motion of the observer. The direction in which we have to point our telescope in observing a star is not the same that it w^ould be if the earth were at rest. It ABERRATION. 141 Fig. 70. — Aberration of a Raindrop. lies beyond our scope to show that according to the wave theory of light the apparent direction of a ray will be affected by the observer's motion precisely in the same way (within very narrow limits) as it would be if light consisted of corpuscles shot off from a luminous body, as Newton supposed. This is the case, however, as Doppler and others have shown ; and as- suming it, the explanation of aberration is easy : — Suppose an observer standing at rest with a tube in his hand in a shower of- rain where the drops are falling vertically. If he wishes to have the drops descend axially through the tube without touching the sides, he must of course keep it vertical ; but if he advances in any direction, he must draw back the bottom of the tube by an amount which equals the advance he makes in the time while the drop is falling through the tube, so that when the drop falling from B reaches A', the bottom of the tube will be there also ; i.e., he must incline the' tube forward by an angle a, such that tan a = u -=- V, where V is the velocity of the raindrop and u that of his own motion. In Fig. 70 BA'= V and AA'=u. 225. Now take the more general case. Suppose a star sending us light with a velocity V in the direction SP, Fig. 71, which makes the angle with the line of the observer's motion. He himself is carried by the earth's orbital velocity in the direction QP- In pointing the tele- scope so that the light may pass exactly along its optical axis, he will have to draw back the eye-end by an amount QP, which just equals the distance he is carried, by the earth's motion during the time that the light moves from to P. The star will thus appar- ently be displaced towards the point towards which he is moving, the angle of displacement POQ, or a, being determined by the rela- tive length and direction of the two sides OP and QP of tjtie triangle YQ u JJp Fig. 71. — Aberration of Light. 142 ABERRATION. whence and OPQ. These sides are respectively proportional to the velocity of light, V, and the orbital velocity of the earth, u. The angle at P being 0, the angle OQP will be (0-a), and we shall have from trigonometry the proportion sin a : sin (0 — a) = « : V. To find a from this, develop the second term of the proportion and divide the first two terms by sin a, which gives us l:sin0cota — cos0=u: V, u sin 6 cot a = V+u cos 6, V+ucos$ COta = — ; — yr- ' usintf Taking the reciprocal of this, we have U « A tan a = — t sin 0. V + u cos The second term in the denominator is insensible, since u is only about one ten-thousandth of V, so that we may neglect it. 1 This gives the formula in the shape in which it ordinarily appears, viz., tana=-sin(/. The value of «. (denoted by a<>) which obtains when = 90° and sin 6 = unity, is called the Constant of Aberration. The latest, and probably the most accurate, determination of this constant (derived from the Pulkowa Ob- servations by Nyren in 1882) is 20".492. Aberration was discovered and explained by Bradley, the English Astronomer Royal, in 1726. 226. The Effect of Aberration upon the Apparent Places of the Stars. — As the earth moves in an orbit nearly circular, and with a velocity so nearly uniform that we may for our present purpose disregard its varia- tions, it is clear that a star at the pole of the ecliptic will be always displaced by the same amount of 20". 5, but in a direction ■fnr* B Fig. 72. Aberrational Orbit of a Star. 1 The velocity of light, according to the latest determinations of Newcomb and Michelson, is 299860 kilometers ± 30 kilometers (which equals 186,330 miles ± 20 miles). The mean velocity of the earth in its orbit, if we assume the solar parallax to be 8".8, is 29.77 kilometers, or 18.50 miles : this makes the constant of aberration 20".478, a little smaller than that given in the text. ABERRATION. I43 continually changing. It must, therefore, appear to describe a little circle 41" in diameter during the year, as shown in Fig. 72. Now the direction of the earth's orbital motion is always in the plane of the ecliptic, and towards the right hand as we stand facing the sun. At the vernal equinox, therefore, we are moving toward the point of the ecliptic, which is 90° west of the sun, i.e., towards the winter solstitial point, and the star is then displaced in that direction. Three months later the star will be displaced in a line directed towards the vernal equinox, and so on. The earth, there- fore, so to speak, drives the star before it in the aberrational orbit, keeping it just a quarter of a revolution ahead of itself. A star on the ecliptic simply appears to oscillate back and forth in a Straight line 41" long. Generally, in any latitude whatever, the aberrational orbit is an ellipse, having its major axis parallel to the ecliptic, and always 41" long, while its minor axis is 41" x sin /J, /} being the star's lati- tude, or distance from the ecliptic. 226*. Diurnal Aberration. — The motion of an observer due to the earth's rotation also produces a slight effect known as the diurnal aberration. Its " constant " is only 0".31 for an observer situated at the equator ; any- where else it is 0".31 cos cf>, being the latitude of the observer. For any given star it is a maximum when the star is crossing the merid- ian, and then its whole effect is slightly to increase the right ascension by an amount given by the formula Aa = 0".31 cos sec 8, S being the star's declination. See' Chauvenet, "Practical Astronomy," I. p. 638. 144 THE MOON. CHAPTER VII. THE MOON: HER ORBITAL MOTION AND VARIOUS KINDS OF MONTH. — DISTANCE AND DIMENSIONS, MASS, DENSITY AND GRAVITY. — ROTATION AND LIBRATIONS. — PHASES. LIGHT AND HEAT. — PHYSICAL CONDITION . AND INFLUENCES EX- ERTED ON THE EARTH. — TELESCOPIC ASPECT. — SURFACE AND POSSIBLE CHANGES UPON IT. 227. "We pass next to a consideration of our nearest neighbor in the celestial spaces, the moon, which is a satellite of the earth and accompanies us in our annual motion around the sun. She is much smaller than the earth, and compared with most of the other heav- enly bodies, a very insignificant affair ; but her proximity makes her far more, important to us than any of them except the sun. The very beginnings of Astronomy seem to have. originated in the study of her motions and iu the different phenomena which she causes, such as the eclipses and tides ; and in the development of modern theoretical astronomy the lunar theory with the problems it raises has been perhaps the most fertile field of invention and discovery. 228. Apparent Motion of the Moon. — Even superficial observa- tion shows that the moon moves eastward among the stars' every night, completing her revolution from star to star again in about 27^ days. In other words, she revolves around the earth in that time ; or, more strictly speaking, they both revolve about their common centre of gravity. But the moon is so much smaller than the earth that this centre of gravity is situated within the ball of the earth on the line joining the centres of the two bodies at a point about 1100 miles below its surface. As the moon moves eastward so much faster than the sun, which takes a year to complete its circuit, she every now and then, at the time of the new moon, overtakes and passes the sun; and as the phases of the moon depend upon her position with reference to the sun, this interval from new moon to new moon is what we ordinarily understand as the month. MONTHS. 145 229. Sidereal and Synodic Revolutions. — The Sidereal revo- lution of the moon is the time occupied in passing from a star to the same star again, as the name implies. It is equal to 27 d 7 h 43 m 11 8 .545±0 S .01, or 27 a .32166. The moon's mean daily motion among the stars equals 360° divided by this, which is 13° 11' (nearly). The Synodic revolution is the interval from new moon to new moon again, or from full to full. It varies somewhat on account of the eccentricity of the moon's orbit and of that of the earth around the sun, but its mean value is 29 d 12 h 44 m 2 8 .684±0 8 .01, or 29 d . 53059 ; and this is the ordinary month. (The word syn- odic is derived from the Greek \nji> and 6S05;, and has nothing to do with the nodes of the moon's orbit. The word is syn-odic, not sy-nodic) . A synodical revolution is longer than the sidereal, because during each sidereal month of 27.3 days the sun has advanced among the stars, and must be caught up with. 230. Elongation, Syzygy, etc. — The angular distance of the moon from the sun is called its Elongation. At new moon it is zero, and the moon is then said to be in " Conjunction." At full moon it is 180°, and the moon is then in " Opposition." In either case; the moon is said to be in " Syzygy" (ow £vy6v). ^-When the elongation is 90°, as at the half-moon, the moon is in " Quadrature." 231. Determination of the Moon's Sidereal Period. — This is effected directly by observations of the moon's right ascension and declination (with the meridian circle) , kept up systematically for n sufficient time. If it were not for the so-called "secular acceleration" of the moon's motion (Arts. 459-461), an exceedingly accurate determina- tion of the moon's synodic period could be obtained by comparing ancient eclipses with modern. The earliest authentically recorded eclipse is one that was observed at Nineveh in the year 763 B.C. between 9 and 10 o'clock on the morning of June 15th. By comparing this eclipse with (say) the eclipse of August, 1887, we have an interval of more than 35,000 months, and so an error of ten hours even, in the observed time of the Nineveh eclipse, would make only about one second in the length of the month. But the month is a little shorter now than it was 2000 years ago. 146 THE MOON. 232. Relation of Sidereal and Synodic Periods. — The fraction of a revolution described by the moon in one day equals — , M being the length of the sidereal month. In the same way — represents Hi the earth's daily motion in its orbit, E being the length of the year. The difference of these two, equals the fraction of a revolution which the moon gains on the sun during one day. In a synodic month, S, it gains one whole revolution, and therefore must gain each day -ofa revolution ; so that we have the equation 1 }__1 . M E S' or, substituting the numerical values of E and 8, 1_ _ 1 _1 M ' 365.25635 29.53059' whence we derive the value of M. Another way of looking at it is this : In a year there must be exactly one more sidereal revolution than there are synodic revolutions, because the sun completes one entire circuit in that time. Now the number of synodic revo- lutions in a year is given by the fraction 55M= 12.369+. S There will therefore be 13.369 sidereal revolutions in the year, and the length of one sidereal revolution equals S65\ days divided by this number 13.369, which will be found to give the length of the sidereal revolution as before. 233. Moon's Path among the Stars. — By observing with the me- ridian circle the right ascension and declination of the moon daily during the month, just as in the case of the sun, we obtain the posi- tion of the moon for each day, and joining the points thus found, we can draw the path of the moon in the sky. It is found to be a great circle inclined at a mean angle of 5° 8' to the ecliptic, which it cuts in two points called the nodes (from nodus, a " knot "). We say the path is found to be a great circle. This must be taken with some reservation, since at the end of the month the moon never returns precisely to the position it occupied at the beginning, owing HARVEST AND HUNTER'S MOONS. 147 to the regression of the nodes and other so-called " perturbations," which will be discussed hereafter. 234. Moon's Meridian Altitude. — Since the moon's orbit is inclined to the ecliptic 5° 8', its inclination to the equator varies from 28° 36' (23° 28' -f 5° 8'), when the moon's ascending node is the vernal equinox, to 18° 20', when, 9J years later, the same node is at the autumnal equinox. In the first case the moon's declination will change dui'ing the month by 57° 12', from - 28° 36' to + 28° 36'. In the other case it will change only by 36° 40', so that at different times the difference in the behavior of the moon in this respect is very striking. 235. Interval between Moon's Transits. — On the average the moon gains 12° 11'.4 on the sun daily, so that it comes to the meridian about 51 minutes of solar time later each day. To find the mean interval between the successive transits of the moon we may use the proportion (360° -12° 11'.4) : 360° = 24 h : x ; whence x = 24 h 50 m .6. The variations of the moon's motion in right ascension, which are very considerable (mucb greater than in the case of the sun) , cause this interval to vary from 24 h 38 m to 25 h 06 m . 236. The Daily Retardation of the Moon's Rising and Setting. — The average daily retardation of the moon's rising and setting is, of course, the same as that of her passage across the meridian, viz., 51 m ; but the actual retardation of rising is subject to very much greater variations than those of the meridian passage, being affected by the moon's changes in declination as well as by the inequalities of her motion in right ascension. When the moon ia very far north, having her maximum declination of 28° 36', she will rise in our latitudes much earlier than when she is farther south. In the latitude of New York the least possible daily retardation of moon-rise is 23 minutes, and the greatest is 1 hour and 17 minutes. In higher latitudes the variation is greater yet. 237. Harvest and Hunter's Moons. — The variations in the retarda- tion of the moon's rising attract most attention when they occur at the time of the full moon. When the retardation is at its minimum, the moon rises soon after sunset at nearly the same time for several successive evenings ; 'whereas, when the retardation is greatest, the moon appears to plunge nearly 148 THE MOON. vertically below the horizon by her daily motion. When the full moon occurs at the time of the autumnal equinox, the moon itself will be near the first of Aries. Now, as will be seen by reference -to Fig. 73, the portion of the ecliptic near the first of Aries makes a much smaller angle with the eastern horizon , than the equator. [The line HN is the horizon, E being the east point — the figure being drawn to represent a celestial globe, as if the observer were looking at the eastern side of the celestial sphere from the outside."] EQ is the equator. Now, when the autumnal equinoctial point or first of Libra is on the horizon at E, the position of the ecliptic will be that repre- sented by ED; more steeply inclined to the horizon than EQ is, by the angle QED, 2Z\°. But when the first of Aries is at E, the ecliptic will be in Fig. 73. — Explanation of the Harvest Moon. the position JJ'. And if the ascending node of the moon's orbit happens then to be near the first of Aries, the moon's path will be MM' . Accordingly, when the moon is in Aries, it, so to speak, coasts along the eastern horizon from night to night, its time of rising not varying very much ; and this, when it occurs near the full of the moon, gives rise to the phenomenon known as the harvest moon, the harvest moon being the full moon nearest to the autumnal equinox. The full moon next following is called the hunter's moon. In Norway and Sweden, under these circumstances, the moon's orbit may actually coincide with the horizon, so that she will rise at absolutely the same time for a considerable number of successive evenings. 238. The Moon's Orbit. — As in the case of the sun, the observa- tion of the moon's path in the sky gives no information as to the real size of its orbit ; but its form may be found by measuring the appar- ent diameter of the moon, which ranges from 33' 30" to 29' 21" at different points. The orbit turns out to be an ellipse like the orbit of the earth, but with an eccentricity more than three times as great THE MOON'S PARALLAX AND DISTANCE. 149 — about T \ on the average, but varying from T \ to ^ ou account of perturbations. The extremities of the major axis of the moon's orbit are called the jwrigee and apogee (from irepl yyj and A.™ yrj) . The line of a psides, which passes through these two points, moves around towards the east once in about nine years, also on account of pert urbat ions. 239. Parallax and Distance of the Moon. — These can be found in several ways, of which the simplest is the following : At two observa- tories B and (Fig. 74) on, or very nearly on, the same merid- ian and very far apart (in the northern and southern hemi- spheres if possible ; Greenwich and the Cape of Good Hope, for instance) let the moon's zenith Q distance ZBM and Z'OMbe ob- served simultaneously with the meridian circle. This gives in the quadrilateral BOCM the two angles at B and C, each of which is the supplement of the geocentric zenith distance. The angle at the centre of the earth, BOC, is the difference of the geo- centric latitudes and is known from the geographical positions of the two observatories. Knowing the three angles in the quadrilat- eral, the fourth at M is of course known, since the sum of the four must be four right angles. The sides BO and GO are known, being radii of the earth ; so that we can solve the whole quadrilateral by a simple trigonometrical process. First find from the triangle BOC the partial angles OCB and OBC, and the side BC. Then in the triangle BCMwe have BC and the two angles CBM and MCB, from which we can find the two sides BM and CM. Finally, in the triangle OBM, we now know the sides OB and BM and the included angle OBM, so that the side OM can be computed, which is the distance of the moon from the earth's centre. Knowing this, the horizontal parallax KM 0, or the semi-diameter of the earth as seen from the moon, follows at once. The moon's parallax can also be deduced from observations at a single station on the earth, but not so simply. If she did not move among the stars, it would be very easy, as all we should have to do would be to compare her apparent right ascension and declination at different points in her diur- Fig. 74. — Determination of the Moon's Parallax. 150 THE MOON. nal circle. Near the eastern horizon the parallax (always depressing an ob ject) increases her right ascension ; at setting, vice versa. On the meridian the declination only is affected. But the motion of the moon must be al- lowed for, as the observations to be compared are necessarily separated by considerable intervals of time, and this complicates the calculation. A third, and a very accurate, method is by means of occultations of stars, observed at widely separated points on the earth. These occultations furnish the moon's place with great accuracy, and so determine the paral- lax very precisely; but the calculation is not very simple, as the moon's motion in this case also enters into it, since the observations cannot be simultaneous. 240. The Distance of the Moon is continually changing on account of the eccentricity of its orbit, varying all the way, according to Nei- son, between 252,972 and 221,614 miles; the mean distance being 238,840 miles, or 60.27 times the equatorial radius of the earth. The mean parallax of the moon is 57' 2", subject to a similar per- centage of change. This value of the parallax, it will be noted, indicates that the earth, as seen from the moon, has a diameter of nearly 2°. Knowing the size of the moon's orbit and the length of the month, the velocity of her motion around the earth is easily calculated. It comes out 2288 miles per hour, or about 3350 feet a second. Fig. 75. — Moon's Path with Reference to the Sun. Fig. 76. False Representations of Moon's Motions. Fig. 77. 241. Form of the Moon's Orhit with Reference to the Sun.— While the moon moves in a small elliptical orbit around the earth, it also moves around the sun in company with the earth. This common THE MOON'S MASS. 151 motion of the moon and earth, of course, does not affect their relative motion ; bnt to an observer outside the system the moon's motion around the earth would only be a very small component of the moon's movement as seen by him. The distance of the moon from the earth, 239,000 miles, is very small compared with that of the earth from the sun, 93,Q00 000 miles — being only about ¥ ^ part. The speed Of the earth in its orbit around the sun is also more' than thirty times faster than that of the moon in its orbit around the earth, so that for the moon the result- ing path in space is one which is always concave towards ihe sun, as shown in Fig. 75. It is not like Figs. 76 and 77, as often rep- resented. If we represent the orbit of the earth by a circle with a radius of 100 inches (8 feet 4 inches) , the moon would only move out and in a quarter of an inch, crossing the circumference twenty- five times in going once around it. 242. Diameter of the Moon. — The mean apparent diameter of the moon is 31' 7". This gives it a real diameter of 2163 miles (plus or minus one mile), which equals 0.273 of the earth's diameter. Since the surfaces of globes are as the squares of their diameters, and their volumes as their cubes, this makes the surface of the moon 0.0747 of the earth's (between fa and fa) ; and the volume 0.0204 of the earth's volume (almost exactly fa) ; that is, it would take 49 balls each as large as the moon in bulk to make a ball of the size of the earth. 243. Mass of the Moon. — This is about fa of the earth's mass, different authorities giving the value from fa to -fa. It is not easy to determine it with accuracy. In fact, though the moon is the nearest of all the heavenly bodies, it is more difficult to " weigh" her than to weigh Neptune, although he is the most remote of the planets. There are four ways of approaching the problem : (1) (perhaps easiest to understand) by finding the position of the common centre of gravity of the earth and moon with reference to the centre of the earth. Since it is this common centre of gravity of the two bodies which describes around the sun the ellipse which we have called the earth's orbit, and since the earth and moon revolve around this common centre of gravity once a month, it follows that this monthly motion of the earth causes an alternate eastward and westward displacement of the sun in the sky, which can be measured. At the time of the new and full moon this displacement is zero, the Centre of gravity being on the line which joins the earth and sun ; 152 THE MOON. M 3 (A) but when the moon is at quadrature (that is, 90° from the sun, as at the time of half-moon), the sun is apparently displaced in the sky Mjowards the moon, as i^ evident from Fig. 78. It will be about 6"-3 east of its mean place at the first quarter of the moon, Fig. 78 (B) , a"nd as much west at the time of the last quarter, Fig. 78 (A) ; (i.e., when the angle MGS is 90°, the angle MCS is always less than 90° by 6".3, which is therefore the value of the angle CSG). Now since the parallax of the sun (which is the earth's semi-diam- eter seen from the sun — the angle CSK) is about 8".8, it fol- lows that the distance of the cen- tre of gravity of the earth and moon from the centre of the earth is the fraction f s of the earth radius, or, about 2830 miles. This is just about ■£$ of the dis- tance-fiom the earth to the moon, whence we conclude that the mass of the earth is 80 times that of the moon. Fig. 78. Apparent Displacement of Sun at First and Third Quarters of the Month. 244. (2) A second method is by comparing the moon's actual period ~mth the computed period which a single particle at the moon's distance from the earth iught to have, according to the known force of gravity of the earth, as deter- mined by pendulum experiments. The explanation of this method cannot be given until we have further studied the motion of bodies under the law of gravitation. It is not susceptible of great accuracy. (3) Still another method is by comparing the tides produced by the moon with those produced by the sun. This gives us the mass of the moon as com- pared with that of the sun ; and the mass of the sun compared with that of the earth being known, it gives us ultimately the mass of the moon com- pared with that of the earth. (4) The ratio of the moon's mass to the sun's can also be computed from the nutation of the earth's axis. (See Chap. XIII.) 245. No other satellite is nearly as large as 1 the moon, in comparison with its primary planet. The earth and moon together, as seen from a the moon's density. 153 distant star, are really in many respects more like a double planet than like a planet and satellite, as ordinarily proportioned to each other. At a time, for instance, when Venus happens to be near the earth, at a distance of about twenty-five millions of miles, the earth to her would appear just about as bright as Venus at her best does to us ; and the moon would be about as bright as Sirius, at a distance of about half a degree from the earth. 246. Density and Superficial Gravity of the Moon. — Since the density of a body is equal to , the density of the moon as volume compared with that of the earth is found from the fraction tAt 0.0125 M, or — ■fa' 0.0204 This makes the moon's density 0.613 of the earth's density, or about 3-j*^ the density of water — somewhat above the average den- sity of the rocks which compose the crust of the earth. This small density of the moon is not surprising, nor at all inconsistent with the belief that it once formed part of the same mass with the earth, since if such were the case, the moon was probably formed by the separation of the outer portions of that mass, which would be likely to have a smaller specific gravity than the rest. 247. The superficial gravity, or the attraction of the moon for bod- ies at its own surface, may be found by the equation i v, in in which g' signifies the superficial gravity of the moon , g is the force of gravity of the earth, while m and r are the mass and radius of the moon as compared with those of the earth. This gives us ,_ 0.0125 9 - 9 *oau7' or (very approximately) g' equals one-sixth of g ; that is, a body which weighs six pounds on the earth's surface would at the surface of the moon weigh only one (in a spring balance). A man on the moon could jump six times as high as he could on the earth and could throw 154 TH E MOON. a stone six times as far. This is a fact to be remembered in connec- tion with the enormous scale of the surface-structure of the moon. Volcanic forces, for instance, upon the moon would throw the rejected materials to a vastly greater distance there than on the earth. 248. Rotation of the Moon. — The moon rotates on its axis once a month, in precisely the same time as that occu- Pl f~^\ pied by its revolution around the earth. In the Z] \.J long run it therefore keeps the same side always qJ towards the earth : we see to-day precisely the same face and aspect of the moon as Galileo did when he first looked at it with his telescope, and the same will continue to be the case for thousands of years more, if not forever. XT It is difficult for some to see why a motion of this Fiq. 79. sor (; should be considered a rotation of the moon, since it is essentially like the motion of a ball carried on a revolving crank. See Fig. 79. Such a ball, they say, " revolves around the shaft, but does not rotate on its own axis." It does rotate, however. The shaft being vertical and the crank horizontal, suppose that a compass needle be substituted for the ball, as in Fig. 80. The pivot turns under- neath it as the crank whirls, but the compass needle does not rotate, maintaining always ^J}. its own direction with the marked end north. On the other hand, if we mark one side of the ball (in the preceding figure), we shall find the marked side presented successively to every point of the compass as the crank re- volves, so that the ball as really turns on its |J own axis as if it were whirling upon a pin Fig. 80. fastened to a table. The ball has two dis- tinct motions by virtue of its connection with the crank : first, the motion of translation, which carries its centre of gravity, like that of the compass needle, in a circle around the axis of the shaft; secondly, an additional motion of rotation around a line drawn through its centre of gravity parallel to the shaft. 248*. Definition of Rotation. — A body "rotates" whenever a line drawn from its centre of gravity outward, through any point selected at random in its mass, describes a circle in the heavens'. In every rotating body, one such line can be so drawn that the circle described by it in the sky becomes in- finitely small. This is the axis of the body. Another set of points can be found such that lines drawn from the centre of gravity outward through L1BRATI0NS. 155 them describe a great circle in the sky 90° distant from the point pierced by the axis, and these points constitute the equator of the body. 249. Librations of the Moon. — 1. Libration in Latitude. The axis of revolution of the moon is not perpendicular to its orbit. It makes a constant angle of about 88£° with the ecliptic, and the moon's equator is so placed that it is always edge-wise to the earth when the moon is at her node, being maintained in that position by an action of the earth, which produces a precessional motion of the moon's axis. The angle between the moon's equator and the plane of her orbit, therefore, is 1^° + the inclination of the moon's orbit, which together make up an angle of a little more than 6£° ; but, as the inclination of the moon's orbit to the ecliptic is constantly varying slightly, this inclination of the moon's axis to her orbit also changes correspondingly. This inclina- tion of the moon's axis produces changes in the aspect of the moon towards the earth similar to those produced by the inclination of the earth's axis towards the ecliptic. At one time, just as the north pole of the earth is turned towards the sun, so also the north pole of the moon is tipped towards the earth at an angle of 6-|-°, and in the oppo- site half of the moon's orbit the south pole is similarly presented to us. In consequence we alternately look over the northern and southern portions of the moon's disc. The period of this libration is the time of the moon's revolution from node to node, called a nodical revolution. This is 27.21 days — about 2 hours and 38 minutes shorter than the sidereal revo- lution of the moon, since the nodes always move westward, completing the circuit in about 19 years. 250. 2. Libration in Lon- gitude. The moon's orbit being eccentric, she moves faster when near perigee, and slower when near apo- gee ; half-way between peri- gee and apogee she is more than 6° ahead of the position she would have if she had moved with the mean angular velocity. Now the rotation is uniform. A point, therefore, on the moon's Fig. 81. — The Libration in Longitude. 156 THE MOON. surface which is directed toward the earth at perigee will Dot have revolved far enough to keep it directed toward the earth when she is half-way (in time) between perigee and apogee, as is evident from Fig. 81. For in the quarter-month next following the perigee, the moon will travel to a point M, considerably more than half-way to apogee. But the point a will have made only one quarter-turn, which is not enough to bring it to the line ME. We shall therefore see a little around the western edge. Similarly on the other side of the orbit, half-way between apogee and perigee, we shall look around the eastern edge to the same extent. At perigee and apogee both, the libration is, of course, zero. The amount of this libration is evidently at any moment just the same as that of the so-called "equation of the centre," which, it will be remembered, is the difference between the mean and true anomalies of the moon at any moment. Its maximum possible value is 7°45'. The period of this libration is the time it takes the moou to go around from perigee to perigee — the so-called anomalistic revolution, which is 27.555 days, about 5 hours and 3G minutes longer than the sidereal month, and 8 hours 14 minutes longer than the moon's nodical revolution, which deter- mines the libration in latitude. The cause of the increased length of the anomalistic revolution is of course the fact that the line of apsides continually advances eastward, mak- ing one revolution every nine years. 251. 3. Diurnal Libration. This is strictly a libration not of the moon, but of the observer ; still, as far as the aspect of the moon goes, the effect is precisely the same as if it were a true lunar libra- tion. The moon's motions have reference to the earth's centre. We, on the surface of the earth, look down over the western edge of the moon when it is rising, and over the eastern when it is setting, by an amount which is equal to the semi-diameter of the earth as seen from the moon ; that is, about one degree (the moon's parallax). On the whole, taking all three librations into account, we see con- siderably more than half the moon, the portion which never disappears being about forty-one per cent of the moon's surface, that never visi- ble also forty-one per cent, while that which is alternately visible and invisible is eighteen per cent. 252. The agreement between the moon's time of rotation and of her orbital revolution cannot be accidental. It is probably due to the action of the earth on some slight protuberance on the moon's surface, THE MOON'S PHASES. 157 analogous to a tidal wave. If the moon were ever plastic, such a bulge must have been formed on the side of the moon next the earth, and would serve as the handle by which the earth always keeps the same face of the moon towards herself. This subject will be re- sumed later. 253. The Phases of the Moon. — Since the moon is an opaque globe, shining entirely by reflected light, we can see only that hemi- sphere of her surface which happens to be illuminated, and of course Fig. 82. — Explanation of the Phases of the Moon. only that part of the illuminated hemisphere which is at the time turned towards the earth. At new moon, when the moon is between the earth and the sun, the dark side is towards us. A week later, at the end of the first quarter, half of the illuminated hemisphere is seen, and we have the half moon, just as we do a week after the full. Be- tween the new moon and the half moon, during tne first and last quarters of the lunation, we see less than half of the illuminated por- tion, and then have the "crescent" phase. See Fig. 82 (in which the 158 THE MOON. light is supposed to come from a point far above the moon's orbit) . Between the half moon and the full, during the second and third quarters of the luuation, we see more than half of the moon's illumi- nated side, and have what is called the "gibbous" phase. Since the terminator or line which separates the dark portion of the disc from the bright is always a semi-ellipse (being a semi-circle viewed obliquely) , the illuminated surface is always a figure made up of a semi-circle plus or minus a semi-ellipse, as shown in Fig. 83, A. It is sometimes incorrectly attempted to represent the crescent form by a construction like Fig. 83, B (where a smaller circle is cut by a larger one). It is to be noticed that ab, the line which joints the cusps, is always perpendicular to the line directed to the sun, and the horns are always turned away from the sun; so. that the precise position in which they will stand at any time is always predictable, and has nothing whatsoever to do with the weather. Artists are sometimes careless in the manner in which they introduce the moon into landscapes. One occasionally sees the moon near the horizon with the horns turned downwards, a piece of drawing fit to go with Hogarth's barrel which shows both its heads at once. 254. Earth-Shine on the Moon. — Near the time of new moon the whole disc of the satellite is easily visible, the portion on which sunlight does not fall being illuminated by a pale ruddy light. This light is earth-shine, the earth as seen from the moon being then nearly full ; for seen from the moon the earth shows all the phases that the moon does, the earth's phase in every case being exactly supplementary to that of the moon as seen by us. As the earth has a diameter nearly four times that of the moon, the earth- shine at any phase would be about thirteen times as strong as moonlight, if the reflective power of the earth's surface were the same. Probably, taking the clouds and snow into account, the earth's surface on the whole is rather more brilliant than the moon's, so that near new moon the earth-shine, by which the dark side of the moon is then illuminated, is- from fifteen to twenty times as strong as full moonlight. The ruddy color is due to the fact that light sent to the moon from the earth has twice penetrated our atmosphere and so has acquired the sunset tinge. 255. Physical Characteristics of the Moon. — 1. Its Atmosphere. The moon's atmosphere, if it has any at all, is extremely rare, prob- ably not producing a barometric pressure to exceed ^_ of an inch of mercury, or -fa of the pressure at the earth's surface. The evidence on this point is twofold. ATMOSPHERE OF THE MOON. 159 (a) The telescopic appearance. The parts of the moon near the edge of the disc, which, if there were any atmosphere, would be seen through its greatest possible depth, are seen without the least distortion : there is no haze, and all shadows are perfectly black. There is no sensible twilight at the cusps of the moon ; no evidences of clouds or storms, or anything like atmospheric phenomena. (b) The absence of refraction when the moon intervenes between us and any more distant object. For instance, at an eclipse of the sun there is no distortion of the sun's limb where the moon cuts it, nor any ring of light running out on the edge of the moon like that which encircles the disc of Venus at the time of a transit. The most striking evidence of this sort comes, however, from occultations of the stars. When the moon hides a star from sight, the phenome- non, if it occurs at the moon's dark edge, is an exceedingly striking one. The star retains its full brightness in the field of the tele- scope until all at once, without the least warning, it simply is not there, the disappearance generally being absolutely instantaneous. Its reap- pearance is of the same sort, and still more startling. Now if the moon had any perceptible atmosphere (or the star any sensible diam- eter) the disappearance would be gradual. The star would change color, become distorted, and fade away more or less gradually. The spectroscope adds its evidence in the same direction. There is no modification of the spectrum of the star in any respect at the time of its disappearance ; and we may add that the spectrum of moonlight is identical with that of sunlight pure and simple, there being no traces of any effect whatever produced upon the sunlight by its re- flection from the moon, nor any signs of its having passed through an atmosphere. 256. The time during which a star would be hidden behind the moon would also be decreased by the refraction of any sensible atmosphere, making the observed duration of an occultation less than that computed from the known diameter of the moon and its rate of motion. Certain Greenwich observations apparently show a differ- ence, amounting to about two seconds of time. This may possibly be due in some part to the action of a real, but exceedingly rare, lunar atmosphere ; for if the whole phenomenon were due simply to atmos- pheric action, it would indicate an atmosphere having a density about yvw part of our own, — far within the limits which were stated above. But the difference may be, and very probably is, attributable, in part at least, to a slight error in the measured diameter of the moon, due to 160 THE MOON. irradiation : the diameter of a bright object always appears a little larger than it really is. An error of about 2" of this sort would explain the whole discrepancy, without any need of help from an atmosphere. 257. What has become of the Moon's Atmosphere. — If the moon ever formed a part of the same mass as the earth, she must once have had an atmosphere. There are a number of possible and more or less probable hypotheses to account for its disappearance. It has been surmised (1) that there may be great cavities left within the moon's mass by volcanic erup- tions, and that the rocks themselves have been transformed into a sort of pumice-stone structure, and that the air has retired into these internal cavities. (2) That the air has been absorbed by the inner lunar rocks in cooling. A heated rock expels any gases that it may have absorbed ; but if it afterwards cools slowly, it reabsorbs them, and can take up a very great quantity. The earth's core is supposed to be now too intensely heated to absorb much gas ; but if it goes on cooling, it will absorb more and more, and in time it may rob the surface of the earth of all its air. There are still other hypotheses, which we can not take space even to mention. 258. Water on the Moon's Surface. — Of course without an atmos- phere there can be no water, since the water would immediately evap- orate and form an atmosphere of water vapor if there were no air present. It is not impossible, however, or even improbable, that solid water, that is, ice and snow, may exist on the moon's surface at a temperature too low for any sensible evaporation. There are manj' things in the moon's appearance that seem to indicate the for- mer existence of seas and oceans on her surface, and the same hypotheses have been suggested to account for their disappearance that were suggested in -the case of the moon's atmosphere. It may be added also that many kinds of molten rock in crystallizing would take up large quantities of water of crystallization, not merely ab- sorbed as a sponge absorbs water, but chemically united with the other constituents of the rock. In whatever way, however, it may have come about, it is certain that now no substances that are gaseous, or that can be evaporated at low temperatures, exist in any quantity on the moon's surface — at least, not on our side of the moon. There have been speculations that on the other side — that celestial coun- try so near us and so absolutely concealed from us — there may be air and water and abundant life ; the idea being that our side of the moon is a great table land many miles in elevation, while the other side is a corresponding THE MOON'S LIGHT. 161 depression, like the valley of the Caspian Sea, only vastly deeper. An in- sufficiently grounded conclusion of Hansen's, that the centre of gravity of the moon is some thirty miles farther from us than its centre of figure, for a time gave color to the idea, but it is now practically abandoned, Hansen's conclusion having been shown to be unwarranted by the facts. 259. The Moon's Light. — As to quality it is simple sunlight, show ing a spectrum which, as has been said, is identical in every detail with that of light coming directly from the sun. Its brightness as compared with that of sunlight is difficult to measure accurately, and different experimenters have found results for the ratio between full moonlight and sunlight ranging all the way from ^TroW (Bouguer) *° Tinnnnr (Wollaston) . The value now usually accepted is that determined by Zollner, viz., ^y^^. According to this, if the whole visible hemisphere were packed with full moons, we should receive from it about one-eighth part of the light of the sun. It is found, also, that the half moon does not give even nearly half as much light as the full moon. The law which connects the phase of the moon with the amount of light given at the time, is rather complicated, but the gist of the matter is that at any time, except at the full, the visible sur- face is more or less darkened by the shadows cast by the irregularities of the surface. Zollner has calculated that an average angle of 52° for these eleva- tions and depressions would account for the law of illumination actually observed. The average '■'■albedo" or reflecting power of the moon's surface, Zollner states as 0.174; that is, the moon's surface reflects a little more than one-sixth part of the light that falls upon it. This is about the albedo of a rather light-colored sandstone, and agrees well with the estimate of Sir John Herschel, who found the moon to be very exactly of the same brightness as the rock of Table Mountain when it was setting behind it, illuminated as were the rocks themselves by the light of the rising sun. There are, however, great variations in the brightness of different portions of the moon's surface. Some spots are nearly as white as snow or salt, and others as dark as slate. 260. Heat of the Moon. — For a long time it was impossible to detect the moon's heat. It is too feeble to be detected by the most delicate mercurial thermometer even when concentrated by a large lens. The first sensible effect was obtained by Melloni, in 1846, 162 THE MOOK. with the then newly invented thermopile, by a series of observations from the summit of Vesuvius. Since then several physicists have worked upon the subject with more or less success, but the most re- cent and reliable investigations are those of Lord Rosse and Profes- sor Langley. With modern apparatus there is no difficulty in detect- ing the heat in the lunar radiations, but measurements are extremely difficult and liable to error. A considerable percentage of the lunar heat seems to be heat simply reflected (like light), while the rest, perhaps three-fourths of the whole, is "obscure heat" ; that is, heat which has been first absorbed by the moon's surface and then radiated, like the heat from a brick surface that has been warmed by sunshine. This is shown by the fact that a comparatively thin plate of glass cuts off some 86 per cent of the heat received from the moon in the same way that it does the heat of a stove, while the heat of direct sunlight, or of an electric arc, would pass through the same plate with very little diminution. The same thing appears also from direct measurements upon the tient-spectrum of the moon made by Langley with his bolometer, described further on. (Art. 343.) 261. As to the temperature of the moon's surface, it is difficult to affirm much with certainty. On one hand, the lunar rocks are exposed to the sun's rays in a cloudless sky for fourteen days at a time, so that if they were blanketed by air like our own rocks they would cer- tainly become intensely heated. A few years ago, Lord Rosse in- ferred from his observations that the temperature of the lunar surface rose at its maximum (about three days after full moon) far above that of boiling water. But his own later investigations and those of Langley throw great doubt on this conclusion. There is no air-blanket at the moon's surface to prevent it from losing heat as fast as it receives it ; and it now seems rather more probable that the temperature never rises above the freezing-point of water, as is the ease on the highest of our mountains, where there is perpetual ice, and the temperature is always low even at noon. So far as we can judge, the condition of things on the moon's surface must correspond to an elevation many times higher than any mountain on the earth ; for no terrestrial moun- tain is so high that the density of the air at its summit is even nearly as low as that of the densest supposable lunar atmosphere. This idea, that the temperature is low, is borne out, also, by the fact that the bolometer shows the presence, in the lunar radiations, of a considerable quantity of heat having a wave-length greater than that of the heat radiated from a block of ice. LUNAR INFLUENCES. 163 4 At the end of the long lunar night of fourteen days the tempera- ture must fall appallingly low, certainly 200° below zero. The whole amount of heat sent by the full moon to the earth is estimated by Kosse as about one eighty-thousandth part of that sent by the sun. 262. Lunar Influences on the Earth. — The moon's attraction co- operates with that of the sun in producing tides, of which we shall speak hereafter. There are also certain distinctly ascertained dis- turbances of terrestial magnetism connected with the approach and recession of the moon at perigee and apogee ; and this ends the chapter of ascertained lunar influences. The multitude of current beliefs as to the controlling influence of the moon's phases and changes over the weather and the various conditions of life are mostly unfounded, and in the strict sense of the word "supersti- tions," — mere survivors from a past credulity. It is quite certain that if there is any influence at all of the sort it is ex- tremely slight — so slight that it cannot be demonstrated with certainty, although numerous investigations have been made expressly for the purpose of detecting it. We have never been able to ascertain, for instance, with certainty, whether it is warmer or not, or less cloudy or not, at the time of the full moon. Different investigations have led to contradictory results. The frequency of the moon's changes is so great that it is always easy to find instances by which to verify a belief that changes of the moon control conditions on the earth. A change of the moon necessarily occurs about once a week, the interval from quarter to quarter being between seven and eight days. All changes, of the weather for instance, must therefore occur within three and three-fourths days of a change of the moon, and fifty per cent of them ought to occur within forty-six hours of a change, even if there were no causal connection whatever. Now it requires only a very slight prepossession in favor of a belief in the effectiveness of the moon's changes to make one forget a few of -the weather changes that occur too far from the proper time. Coincidences enough can easily be found to justify a preexisting belief. THE MOON'S SURFACE. 263. Even to the naked eye the moon is a beautiful object, diversified with darker and lighter markings which have given rise to numerous popular superstitions. With a powerful telescope these Daked-eye markings mostly vanish, and are replaced by a countless multitude of smaller details, which are interesting in the highest degree. The moon on the whole, on account of this diversity of 164 THE MOON. detail, is the finest of all telescopic objects ; especially to moderate- sized instruments, say from six to ten inches in diameter, which generally give a more pleasing view of our satellite than instruments either larger or smaller. 264. How near the Telescope brings the Moon. — An instrument of this size, with magnifying powers between 250 and 500, brings up the moon virtually to a distance ranging from 1000 miles to 500 ; and since an object a mile in diameter on the moon subtends an angle of about 0".86, with the higher powers of such an instrument objects less than a mile in diameter become visible under favorable atmospheric conditions. A long Hue or streak, even less than a quarter of a mile across, could probably be seen. With larger tele- scopes the power can now and then be carried at least twice as high, and correspondingly smaller details made out. "When everything is at its best, the great Lick telescope of 36 inches aperture, with a power of 2500 or so, may possibly reduce the virtual distance of our satellite to about 100 miles for visual purposes. It is evident that while with our telescopes we should be able to see such objects as lakes, rivers, forests, and great cities, if they exist on the moon, it will be hopeless to expect to distinguish single buildings, or any of the ordinary operations and indications of life, if such there are. There are a few mountains on the earth from which a range of 100 miles is obtained in the landscape. Those who have seen such a landscape know how little is to be made out with the naked eye at that distance. Still, the comparison is not quite fair, because in looking at a terrestrial object a hun- dred miles away the line of vision passes through a dense atmosphere, while in looking upward towards the moon it penetrates a much less thickness of air. 265. The Moon's Surface Structure. — The moon's surface for the most part is extremely uneven and broken, far more so than that of the earth. The structure, however, is not like that of the earth's surface. On the earth the mountains are mostly in long ranges, such as the Alps, the Andes, and Himalayas. On the moon such moun- tain ranges are few in number, though they exist ; but the surface is pitted all over with great craters, resembling very closely the vol- canic craters on the earth's surface, though on an immensely greater scale. One of the largest craters upon the earth, if not the largest, is the Aso San in Japan, about seven miles across. Many of those on the moon are fifty and sixty miles in diameter, and some are THE MOON'S SURFACE. 165 over 100 miles across, while smaller ones from a half-mile to eight or teu miles in diameter are counted by the thousand. The normal lunar crater is nearly circular, surrounded by an ele- vated ring of mountains which rise anywhere from 1000 to 20,000 feet above the surrounding country. Within the floor of the crater the surface may be either above or below the outside level. Some cra- ters are deep, some filled nearly to the brim. In some Cases the surround- Fig. 84. - A Normal Lunar Crater (Nasmyth). ing mountain ring is en- tirely absent, and the crater is a mere hole in the plain. In the centre of the crater there usually rises a group of peaks, which attain about the same elevation as the encircling ring, and these central peaks often show little holes or craterlets in their summits. In most cases the resemblance of these formations to terrestrial volcanic structures, like those exemplified by Vesuvius and others in the surround- ing region, makes it natural to assume that they had a similar origin. This, however, is not absolutely certain, for there are considerable difficul- ties in the way, especially in the case of the great " Bulwark Plains," so called, which are so extensive that a person standing in the centre could not see the summit of the surrounding ring at any point ; and yet no line of de- marcation can be drawn between them and the smaller craters. ^The series is continuous. Moreover, on the earth, volcanoes necessarily require the action of air and water, which do not now exist on the moon. It is obvious, therefore, that if these lunar craters are the result of true volcanic eruptions, they must be fossil formations ; for it is quite certain that no evidence of existing volcanic activity has ever been found. The moon's surface appears to be absolutely quiescent — still in death. On some portions of the moon these craters stand very thickly. Older craters have been encroached upon, or more or less completely obliterated by the newer, and the whole surface is a chaos, of which the counterpart is hardly to be found on the earth, even in the rough- est portions of the Alps. This is especially the case near the moon's south pole. It is noticeable that, as on the earth the newest moun- tains are generally the highest, so on the moon the more newly formed craters are generally deeper and more precipitous than the older ones. 166 THE MOON. 266. lunar Nomenclature. — The great plains were called by Galileo oceans or seas (Maria), and some of the smaller ones marshes (Paludes) and lakes, for he supposed that the grayish sur- faces visible to the naked eye, and conspicuous in a small telescope, were covered with water. Thus we have the ' ' Oceanus Procellarum," the " Mare Imbrium," and a number of other "seas," of which Wo Fia. 85.— Map of the Moon. "Mare Fecunditatis," "Mare Serenitatis," and "Mare Tranquilitatis," are the most conspicuous. There are twelve of them in all, and eight or nine Paludes, Lacus, and Sinus. The ten mountain ranges on the moon are mostly named after terrestrial mountains, as Caucasus, Alps, Apennines, though two or three bear the names of astronomers, like Leibnitz, Dorfel. etc. LUNAR MOUNTAINS. 167 The conspicuous craters bear the names of the more eminent ancient and medise.val astronomers and philosophers, as Plato, Archimedes, Tycho, Copernicus, Kepler, and Gassendi ; while hundreds of smaller and less conspicuous formations bear the names of more modern or less noted astronomers. The system seems to have originated with Riccioli in 1650, but most of the names have been more recently assigned by the later map-makers, the most eminent of whom have been the German astronomers Beer and Maed- ler (who published their map in 1837), and Schmidt of Athens, whose great map of the moon, on a scale seven feet in diameter, was published by the Prussian government a few years ago. It is not at all too much to say that our maps of the earth's surface do not, on the whole, compare in fulness and accuracy with our maps of the moon. Of course this" is not true of such countries as France and England, or others that have been trigonometrically surveyed ; but there are no such lacunas in our maps of the moon as exist in our maps of Asia and Africa, for instance. 267. Other Lunar Formations. — The craters and mountains are not the only interesting for- mations on the moon's sur- face. There are many deep, narrow, crooked valleys that goby the name of ''rills" (German Rillen), some of which may once have been watercourses. Then there are numerous " clefts," half a mile or so wide and of un- known depth, running in some cases several hundred miles, straight through moun- tain and valley, without any apparent regard for the ac- cidents of the surface. They seem to be deep cracks in the crust of our satellite. Several of them are shown in Fig. 86 . Most curious and interesting of all are the light-colored streaks or "rays" which radiate from certain of the craters, extending in some cases a distance of several hundred miles. They are usually from Ave to ten miles wide, and neither elevated nor depressed to any extent with WP=°mmn or, since - = r sin p (p being the sun's horizontal parallax), we have finally It will be noticed that in this expression the cube of the parallax appears, and this is the reason why an uncertainty of one per cent in p involves an uncertainty of three per cent in S. 176 THE SUN. In obtaining the mass of the sun it will be seen that we require as data, T, the length of the sidereal year in seconds ; the value of gravity, g (which is derived from pendulum experiments) ; the radius of the earth, r (deduced from geodetic surveys) ; and finally (and most difficult to get), the sun's parallax, p, or else, what comes to v the same thing, the ratio — • H 279. The Sun's Density. — This density 1 as compared with that of the earth is found by simply dividing its mass by its volume (both as compared with the earth) ; that is, it equals the fraction 332000 = 0.255, 1300 000 a little more than a quarter of the earth's density. To get its " specific gravity" (i.e., density as compared with water), we must multiply this by 5.58, the earth's mean specific gravity. This gives 1.41 ; that is, the sun's mean density is not 1^ times that of water, — a most significant result as bearing on its physical condition. 280. Superficial Gravity. — This is found by dividing its mass by the square of its radius ; that is, 332 000 (109£) 2 ' which equals 27.6. A body weighing one pound on the earth's sur- face would there weigh 27.6 lbs. A body would fall 444 feet in a second, instead of sixteen feet, as here. 281. The Sun's Rotation. — The sun's surface often shows spots upon it, which pass across the disc from east to west. These are evidently attached to its surface, and not bodies circling around the sun at a distance above it, as was imagined by some early astrono- mers, because, as Galileo early demonstrated, they continue in sight just as long as the time during which they are invisible ; which would not be the case if they were at any considerable elevation. x The determination of the sun's density does not necessarily involve its parallax. Put p for sun's radius, and Ds for its density : let De be earth's mean density. Substitute in equation (c), and we have $ Tp*Ds = f mflDe \-^(- )(— ) 1, whence Ds = Def-^-^^-) 1 • But (7) = sin s > 2 bein S the sun ' s angular serai-diameter. Hence, finally, Ds = Del -=7-1 -H I - PERIOD OF ROTATION. 177 Period of Rotation. — The average time occupied by a spot in pass- ing around the sun and returning to the same position again is 27.25 days,— average because different spots show considerable differences in this respect. This interval, however, is not the true time of solar rotation, but the synodic, since the earth advances in the interval of a revolution so that the sun has to turn on its axis a little farther each time to bring the spot again into conjunction with the earth. The equation by which the true period is deduced from the synodic is the same as in the case of the moon (Art. 232) , viz. : 1-1 = 1 T E £' T being the true period of the sun's rotation, E the length of the year, and S the observed synodic rotation ; whence — = (- T 27.25 365.25' which gives !F=25 d .35. Different observers get slightly different results. Carrington finds 25 d . 38 ; Spoerer, 25 d .23. 282. Position of the Sun's Axis. — On watching the spots with care as they cross the disc, it appears that they usually describe pathr more or less oval, showing that the sun's axis is inclined to the ecliptic. Twice a year, however, the paths become straight, at the times when the earth is in the plane of the sun's rotation. These dates are about June 3 and Dec. 5. The ascending node of the sun's equator is in celestial longitude 73° 40' (Carrington), and the inclination of its equator to the plane of the ecliptic is 7° 15'. Its inclination to the plane of the terrestrial equator is 26° 25'. The position of the point in the sky towards which the sun's pole is directed is in right ascension 18 h 44 m , declination + 64°, almost exactly half-way between the bright star a Lyrse and the Pole Star. 283. Peculiar Law of the Sun's Rotation. — Equatorial Accelera- tion. The earth rotates as a whole, every point on its surface making its diurnal revolution in the same time ; so also with the moon and with the planet Mars. Of course it is necessarily so with any solid globe. But this is not the case with the sun. It was noticed quite early that the different spots give different results for the rotation period, but the researches of Carrington about thirty years ago first 178 THE SUN. brought out the fact that the differences follow a regular law, showing that at the solar equator the time of rotation is less than on either side of it. Thus spots near the sun's equator give T — 25 days ; at solar latitude 20°, T— 25.75 days; at solar latitude 30°, T= 26.5 days ; at solar latitude 40°, T = 27 days. The time of rotation in latitude 40° is fully two days longer than at the solar equator ; but we are unable to follow the law further towards the poles, because the spots are rarely found beyond the parallels of 45° on each side of the equator, and there are no well-defined markings between this point and the poles by which we can accurately determine the motion. 284. Various formulae have been proposed to represent this law of rota- tion. Carrington gives for the daily motion of a spot X = 865' — 165' X sin * I, I being the solar latitude of the spot. Faye, from the same observations, con- sidering that the exponent J could have no physical justification, deduced X= 862' — 186' x sin 2 1, which agrees almost as well with the observations. Still other formulae have been deduced by Spoerer, Zollner, and Tisserand, all giving substantially the same results. The law, in any case, is simply empirical; that is, it is deduced from the observations, without being based upon any satisfactory physical explanation, for no such explanation of this strange equatorial accel- eration has yet been found. Probably it has its origin somehow in the effects produced by the outpour of heat from the sun's surface ; still, just how such a result should follow in the case of a cooling globe, of which the particles are free to move among each other, is not yet evident. f 285. It is possible that the spots move on the surface of the sun, chang- ing their places just as do clouds or railroad trains upon the surface of the earth, so that their motion does not represent the sun's true rotation. This however, as we shall see later, is hardly probable, and if it were the case, it would still be no less difficult to account for this systematic difference in their behavior at the solar equator and in the higher latitudes. It has been suggested that the spots may be due to the fall of matter from a considerable elevation above the sun's surface, matter which has remained at that elevation for some time, and acquired a corresponding velocity of rotation due to that elevation. It can be shown that if the matter forming the spots had thus fallen from an elevation of about 20,000 miles, it would account for their apparent acceleration. Matter so falling would have an apparent eastward motion, just as do bodies on the earth when falling from the summit of a tower (Art. 138). From this point of view it is very interest- ing to inquire whether the minuter markings upon the sun's surface, such asthe "granules," to be spoken of very soon, do, or do not, possess the same THE PHENOMENA OF THE SUN'S SURFACE. 179 rate of motion as the spots. There is no decisive observational * evidence at present that they do not. But the subject is an extremely difficult one ; and yet important, because the solution of the problem of the sun's equatorial acceleration will probably throw much light upon its real constitution. 286. The Phenomena of the Sun's Surface.— In order to study the sun with the telescope it is necessary to be provided with some special forms of apparatus. Its heat and light are so intense that it is impossible to look directly at it, as we do at the moon. A very con- venient method of exhibiting the sun to a number of persons at once is sim- ply to attach to the telescope a frame carrying a screen of white paper at a distance of a foot or more from the eye-piece, as Shown in Fig. 91. On Fig. 91. -Telescope and Screen. pointing the instrument to the sun and properly adjusting the focus, a distinct image is formed on the screen, which shows the main features very fairly. It is, however, much more satisfactory to look at it directly, with a proper eye- piece. "With a small telescope, not more than two and a half or three inches in diameter, a mere dark glass between the eye-piece and the eye can be used, but this dark glass soon becomes very hot, and is apt to crack. With larger instruments, it is necessary to use eye-pieces especially designed for the purpose and known as solar eye-pieces or helioscopes. The simplest of them, and a very good one for ordinary purposes, is one known as Herschel's, in which the sun's rays are reflected at right angles by a plane of unsilvered glass ( Fig. 92) . This reflector is made either of a prismatic form or concave, in order that the reflection from the back surface may not 1 Mr. Crew has recently made at Baltimore, under the direction of Professor Rowland, an extensive series of observations upon the displacement of the lines of the spectrum at the eastern and western limbs of the sun. This displacement, which is very slight, is due, according to Doppler's principle (Art. 321), to the rotation of the sun; and Mr. Crew's results, so far as they can be considered decisive, go to show that the absorbing layer of gases by which the lYaunhofer lines are formed does not behave like the sun spots, but is slightly retarded at the sun's equator. The observations are so delicate, however, that the conclusion, though made very probable, can hardly be considered to be absolutely proved beyond question. 180 THE SUN. interfere with that from the front. About nine-tenths of the light passes through this reflector, and is allowed to pass out uselessly through the open end of the tube. The remaining tenth is sent through the eye-piece, and though still too intense for the eye to endure, it requires only a comparatively thin shade of neutral-tinted glass to reduce it sufficiently, and in this case the shade does not become uncomfort- ably heated. It is well to have the shade- glass made wedge-shaped, — thinner at one end than at the other — so that one can choose the particular thickness which is best adapted to the magnifying power Fig. 92. — Herecbel Eye-piece. employed. 287. The polarizing eye-pieces are still better when -well made. In these the light is reflected twice #t plane surfaces of glass at the " angle of polarization" (Physics, p. 480), and is then received on a second pair of reflectors of black glass. When the upper pair of reflectors is in either of the two positions shown in Fig. 93, a strong beam of light is received at C, — too strong for the eye to bear, although more than ninety per cent of it has already been rejected ; but by simply turning the box which carries the upper reflectors one-quar- ter of a revolution around the line BB! as an axis, the light may be wholly extinguished ; and any desired rtHfe, gradation may be obtained by setting L rr- 1 it at the proper angle, without the use of a shade-glass. 288. It may be asked why it will not answer merely to " cap " the object-glass, and so cut off part of the light, instead of re- jecting it after it has once been allowed to enter the telescope. It is because of the fact, meutioned in Article 43, that the smaller the object-lens of the telescope, the larger the image it makes of a lumi- nous point, or the wider its image of a sharp line. To cut down the Fig. 93. — Polarizing Helioscope. PHOTOGRAPHY. * 181 aperture, therefore, is to sacrifice the definition of delicate details. With a low power there is no objection to reducing the amount of heat admitted into the telescope tube in that way, but with the higher powers the whole aperture should always be used. 289. Photography. — In the study of the sun's surface photog- raphy is for some purposes very advantageous and much used. The instrumeut must have a special object-glass (Article 42) , with an appa- ratus for the quick exposure of plates. Such instruments are called photo-heliographs, and with them photographs of the sun are made daily at numerous observatories. The necessary exposure varies from -j^ T to -jJj of a second, in different cases. The pictures made by these instruments are usually from two inches up to eight or ten inches in diameter, and some of Janssen's, made at Meudon, bear* enlarging up to forty inches in diameter. Photographs have the advantage of freedom from prejudice and prepossession on the part of the ob- server ; but they take no advantage of the instants of fine seeing. They represent the surface as it happened to be at the moment when the plate was uncovered. 290. The study of the sun has become so important from a scientific point of view that several observatories have recently been established mainly for that purpose, though most of them connect with it that of other topics in astronomical physics. The two most important of these solar or astro-physical observatories, are the observatory at Meudon and the so-called " Sonnenwarte " at Potsdam. There ought to be one in this country. 291. General Views. — JJefoi'e passing to a discussion of the details of the different solar phenomena, it will be well to give a very brief summary of the objects and topics to be considered. 1. The pJwtospJiere ; i.e., the luminous surface of the sun directly visible to our telescopes. It is probably a sheet of luminous clouds formed by condensation into little drops and crystals (like the water- drops and ice-crystals in our terrestrial clouds) of certain substances which within the central mass of the sun exist in a gaseous form, but are cooled at its surface below the temperature necessary for their condensation ; perhaps such substances as carbon, boron, and silicon. The granules, faculee, and spots are all phenomena in this photosphere. 2. The so-called " reversing layer " is a stratum of unknown thick- ness, but probably shallow, just above the photosphere, containing the vapors of many of the familiar terrestrial elements ; of which 182 THE SUN. the presence, and to some extent their physical condition, can be in- vestigated by means of the spectroscope. 3. Above the photosphere, interpenetrating the atmosphere of, vapors just spoken of, and perhaps indistinguish- able from it, is an envelope of permanent gases ; that is, gases which, under the solar conditions, cannot be condensed into clouds - of solid or liquid parti- cles. Among them hydro- gen is most conspicuous. This envelope is the so- called Chromosphere; and from it the prominences of various kinds rise, sometimes to the height of hundreds of thousands of miles. These beauti? ful objects are best seen at total eclipses of the sun, but to a certain ex- tent they can also be stud- ied at any time by the help of a spectroscope. 4. Higher yet rises the mysterious Corona, of ma- terial still less dense,and so far observable only during total eclipses of the sun. Fig. 94 shows the relative positions of these different elements of the solar constitution. 5. A fifth subject deals with the measurement of the sun's light and the relative brightness of different parts of the solar surface. 6. Another most interesting and important topic relates to the amount of heat radiated by the sun, — the sun's probable tempera- ture and the mechanism by which its heat-supply is maintained. Fie. 94. Constitution of the Sun. From " The Sun," by permission of the Publishers. 292. The Photosphere. — The sun's visible surface is called the photosphere, and when studied under favorable atmospheric condi- THE PHOTOSPHERE. 183 tions, with rather a low magnifying power, it looks like rough draw- ing-paper. With higher powers it is seen to be, as shown in Fig. 95, made up of a comparatively darkish background sprinkled over with grains, or " nodules," as Herschel called them, of some- thing much more brilliant, — like snowflakes on gray cloth, according to Langley. These are from 400 to 600 miles across, and in the' finest seeing are themselves resolved into more minute " granules." For the most part, these nodules are about as broad as they are long, Fig. 95. The Great Sun Spot of September, 1870, and the Structure of the Photosphere. From a Drawing by Professor Langley. From " The New Astronomy," by permission of the Publishers. though of irregular form ; but here and there, especially in the neighborhood of the spots, they are drawn out into long streaks. Nasmyth seems first to have observed this structure, and called the filaments " willow leaves." Secchi called them " rice grains." According to Huggins they were "dots" ; and there was for a long time a pretty lively controversy as to their true form. Their shape, however, unquestionably varies very much in different parts of the 184 THE SUN. surface and under different circumstances. They are probably lumi- nous clouds floating in a less luminous atmosphere. Near the edge the photosphere appears generally much less brill- iant ; but certain bright streaks called " faculae " (from fax, a torch), which though visible are not very obvious at points further from the limb, become there conspicuous. These faculae are eleva- tions, — masses of the same material as the rest of the photosphere, but elevated above the general level and intensified in brightness. Fig. 96. — Faculao at Edge of the Sun. (De La Rue.) When one of them passes off the edge of the sun, it is sometimes seen as a little projection. They are most abundant near the sun- spots, and they are more conspicuous near the edge of the disc, as shown in Fig. 96, because the sun's surface is overlaid by a gaseous atmosphere which absorbs more of the light there than it does near the centre, and these faculae push up through it like mountains. 293. The Sun Spots. — Whenever these are present upon the sun's surface, they are the most conspicuous objects to be seen upon it. The appearance of a normal sun spot, Fig. 97, fully formed and not yet beginning to break up, is that of a dark central '■'■umbra" more or less nearly circular, with a fringing "penumbra," composed of filaments directed radially. The umbra itself is not uniformly dark throughout, but is overlaid with filmy clouds which require a good telescope and helioscope to make them visible. Usually, also, in the THE SUN SPOTS. 185 umbra there are several round and very black spots, which are some- times called u nucleoli" but are often referred to as "Dawes* holes," after the name of their first discoverer. But while this is the appearance of what may be taken as a normal spot, very .few are strictly normal. Most of them are more or less irregular in form. They are often gathered in groups with a common penumbra, and partly covered by brilliant " bridges" extending across from the out- side photosphere. Often the umbra is out of the centre of the pe- numbra, or has a penumbra only on one side, and the penumbral Fig. 97. — A Normal Sun Spot. (Secchi ; modified.) filaments, instead of being strictly radial, are frequently distorted in every conceivable way. In fact, the normal spots form a very small proportion of the whole number. The darkest portions of the umbra are dark only by contrast. Photometric observations (by Langley) show that even the nucleus gives at least one per cent as much light as a corresponding area of the photosphere ; that is to say, as we shall see hereafter, the darkest portion of a sun spot is brighter than a calcium light. 294. The spots are unquestionably cavities or depressions in the photosphere, filled with gases and vapors which are cooler than the surrounding portions, and therefore absorb a considerable propor- tion of light. The fact that they are cavities is shown by the change 186 THE STJN. in the appearance of a spot as it approaches the edge of the disc. When a normal spot is near the centre of the disc, the nucleus is nearly central. As it approaches the edge, the penumbra becomes wider on the outer edge and narrower on the inner, and just before the spot disappears around the limb of the sun, the penumbra on the inner edge entirely disappears, — the appearance being precisely such as would be shown by a saucer-shaped cavity in the surface of a globe, the bottom of the cavity being painted black to repre- sent the umbra, and the sloping sides gray for the penumbra. Fig. 98 represents the phenomena in a schematic way. Observations upon a single spot would hardly be sufficient to substantiate this, Fig. 98. — Sun Spots as Cavities. because the spots are so irregular in their form ; but by observing- the behavior of several hundred of them the truth comes out quite- clearly. Occasionally, when a very large spot passes off the sun's limb, the depression can be seen with the telescope. The fact was first discovered by Wilson of Glasgow somethings more than a hundred years ago. Previously it had very commonly- been supposed that the spots were elevated above the general surface - of the sun, and the idea still survives in certain quarters, though cer- tainly incorrect. 295. The penumbra is usually composed of "thatch-straws," or long drawn-out granules of photospheric matter, which, as has been said, converge in a general way towards the centre of the spot. At the inner edge the penumbra, from the convergence of these filaments, is usually brighter than the outer. The inner ends of the filaments are generally club-formed ; but sometimes they are drawn out into fine points, which seem to curve downward into the umbra like the rushes over a pool of water. The outer edge of the penumbra is usually pretty definite, and the penumbra there is darker. Around DIMENSIONS OF SUN SPOTS. 187 the spot the photosphere is much disturbed and elevated into faculae, which sometimes radiate outward from the spot like streams of lava from a crater, though, of course, they are really nothing of the sort. 296. Dimensions of Sun Spots. —The diameter of the umbra of a sun spot ranges all the way from 500 to 1000 miles in the case of a very small one, to 50,000 or 60,000 miles in the case of the lr.rger ones. The penumbra surrounding a group of spots is sometimes 150,000 miles across, though that would be rather an exceptional size. Not infrequently sun spots are large enough to be seen by the naked eye, and they have been often so seen at sunset or through a fog. The depth by which the umbra is depressed below the general surface of the photosphere is very difficult to determine, but accord- ing to Fay e, Carrington, and others, it seldom exceeds 2500 miles, and more often is between 500 and 1500. 297. Development and Changes of Form. — Generally the origin of a sun spot fails to be observed. It begins from an insensible point, and rapidly grows larger, the penumbra usually appearing only after the nucleus is fairly developed. If the disturbance which causes the spot is violent, the spot usually breaks up into several fragments, and these again into others which tend to separate from each other. At each new disturbance the for- ward portions of the group show a tendency to advance eastward on the sun's surface, leaving behind them a trail of smaller spots. • 298. The " segmentation " of a' spot, as Faye calls it, is usually effected by the formation of a " bridge," or streak of brilliant light, which projects itself across the penumbra and umbra from the outside photosphere. These bridges are mere extensions of the surrounding faculae, and are often in- tensely bright. Occasionally a spot shows a distinct cyclonic motion, the filaments being drawn inward spirally ; and in different members of the same group of spots the cyclonic motions are not seldom in opposite directions. When a spot at last vanishes it is usually by the rapid encroachment of the photospheric matter, which, as Secchi expresses it, appears to "fall pell- mell into the cavity," completely burying it and leaving its place covered by a group of faculae. Figs. 99-104 (see page 201) show the changes which took place in the great spot of September, 1870. They are from photographs by Mr. Rutherf urd of New York, and are borrowed from " The New Astron- omy " of Professor Langley, through the courtesy of his publishers. 188 THE SUN. 299. Spots within 15° or 20° of the sun's equator generally, on the whole, drift a little towards it, while those in higher latitudes drift away from it; but the motion is slight, and exceptions are fre- quent. In and around the spot itself the motion is usually inward towards ■ the centre, and downward at the centre. Not infrequently the frag- ments at the inner end of the penumbral filaments appear to draw off, move towards the centre of the spot, and then descend. Occa- sionally, though seldom, the motion is vigorous enough to be detected by the displacement of lines in the spectrum. 300. Duration. — The duration of the spots is very various, but, astronomically speaking, they are always short-lived phenomena, sometimes lasting for only a few days, more frequently, perhaps, for a month or two. In a single instance, a spot has been observed through as many as eighteen successive revolutions of the sun. 301. Distribution. — It is a significant fact that the spots are con- fined mostly to two zones of the sun's surface between 5° and 40° of latitude north and south. A few are found near the equator, none Fig. 105. — Distribution of Sun Spots in Latitude. beyond the latitude of 45°. Fig. 105 shows the distribution of sev- eral thousand spots as observed by Carrington and Sporer. Occasionally, what Trouvelot calls "veiled spots" are seen beyond the 45° limits — grayish patches surrounded by faculaj, which look as if a dark mass were submerged below the surface and dimly seen through a semi-transparent medium. NATURE OF THE SPOTS. 189 302. Theories as to the Nature of the Spots. — "We first mention (a) the theory of Sir William Herschel, because it still finds place in certain text-books, though certainly incorrect. His belief was that the spots were openings through two luminous strata, which he supposed to surround the central globe of the sun. This globe he supposed to be dark (and even habitable 1) . The outer stratum, the photosphere, was the brighter of the two, and the opening in it the larger, while the inner shell between it and the solid globe was of less luminous substance, and formed the penumbra. He thought the open- ing through these might be caused by volcanoes on the globe beneath. 303. (b) Another theory, now abandoned, was proposed inde- pendently both by Secchi and Faye about 1868. They supposed that the spots were openings in the photosphere caused by the bursting outward of the imprisoned gases underneath it ; the photosphere at that time being supposed to be liquid. They explained the darkness of the centre of the spot by the fact that a heated gas at a given temperature has a lower radiating power and sends out much less light than a liquid surface, or than clouds formed by the con- densation of the same material at even a lower temperature. This is true of gases at low pressure, but not of gases under great compression, such as must be the case within the body of the sun. Besides, if the gases possessed the small radiating power necessary to this theory, they would also possess small absorbing power, and therefore would be transparent ; the inner side of the photosphere on the opposite side of the sun would therefore be visible through the opening, so that the centre of such an eruption would not be dark, but, if anything, brighter than the general solar surface. Moreover, as we now know from the spectroscopic evidence, the motion at the centre of a spot is inward, not outward. 304. (c) Faye more recently has proposed and now maintains a theory which has numerous good points about it, and is accepted by many, viz.: that the spots are analogous to storms on the earth, being cyclones, due to the fact that the portions of the sun's surface near the equator make their revolution in a shorter time than those in higher latitudes. This causes a relative drift in adjacent portions of the photosphere, and according to him gives rise to vortices or whirl- pools like those in swiftly running water. The theory explains the distribution of the spots (which abound precisely in the regions where this relative drift is at the maximum) and many other facts, such as their " segmentation." According to it, however, all spots should be cyclonic, and the spiral motion of all the spots in the 190 THE SUN. southern hemisphere should be clock-wise, while in the northern hemi- sphere they should be counter-clock-wise. Now, as a matter of fact, only a very few of the spots show such spiral motions, and there is no such agreement in the general direction of the motion as the theory requires. Faye attempts to account for this by saying that we do not see the vortex itself, but only the cloud of cooler materials which is drawn together by the down-rushing vortex, itself hidden beneath this cloud. Still, it would seem that in such a case the cloud itself should gyrate. Moreover, the relative drift of the adjacent portions of the photosphere is too small to account for the phenomena satisfactorily. In the solar latitude of 20° two points separated by 1' of the sun's surface (123 miles) have a relative daily drift of only about four and one-sixth miles, insufficient to produce any sen- sible whirling. 305. (d) Secchi's later theory. He supposed the spots to be due to eruptions from the inner portions of the sun's surface, not in the spot, however, but only near it ; the spot itself being formed by the settling down upon the photosphere of materials thrown out by the eruption and cooled by their expansion and their motion through the upper regions. We have, however, in fact, as a usual thing, not a single eruption, but a ring of eruptions all around every large spot, all of them converging their bombardment, so to speak, upon the same centre, — a fact very difficult to explain if the spot origi- nates in the eruption, but not difficult to understand if the eruptions are the result of the spot. Perhaps the true explanation may be that when an eruption occurs at any point, the photosphere somewhere in the neighborhood settles down in consequence of the diminution of the pressure beneath, thus forming a " sink," so to speak, which is of course covered by a greater depth of cooler vapors above, and so looks dark. 306. (e) Mr. Lockyer, in his recent work on the chemistry of the sun, revives an old theory, first suggested by Sir John Herschel and accepted by the late Professor Peirce, that the spots are not formed by any action from within, but by cool matter descending from above, — matter very likely of meteoric origin ; but it is difficult to see how the distribution of the spots with reference to the sun's equator can be accounted for in this way. On the whole it is impossible to say that the problem of the origin of sun spots is yet satisfactorily solved. There is no question that PERIODICITY OF SUN SPOTS. 191 sun spots are closely associated with eruptions from beneath ; but which is cause and which effect, or whether both are due to some external action, remains undetermined. 307. Periodicity of Sun Spots. - In 1851 Schwabe of Dessau, by the comparison of an extensive series of observations running over nearly thirty years, showed that the sun spots are periodic, bring at times vastly more numerous than at others, with a roughly regular recurrence every ten or eleven years. This had been surmised by Horrebow more than a century before, though not proved. ^:::::±::::::::r::::::::::::::: = : t* -r- ■- \ n ■ -M I ' " "" ~ ■ ■ *3&I \ ^ ^zzk-i l 4 W80 1 X f *t \ -f i Ecrfes ?i - \ + v t V ' 1 f ^ H 2 V I ^ ,, h X -,Zx Z 4- aSl a s t \ 1 H i-t t t - - w ^ ., ^ A v--. t w 33 ^ t \ s,-~ --s \ it IT - - '* x -- '-- s-lXrhlki -iy> .7 si .is .at; .:sio sau _A t* 5 f 4% t. -(iff 4 3 1 fin 1 -iff ±S.V- - 3 3 IUUI BS3S ~t J I , £~I J.N f V x l i i ^r&niaiE 1± .. j/iLissajr^t \ I 31*3 I '*J _,£ 3 2 ji 3 t ± ^v 15 r V j J ijk 2 5 j w J X l -t -t— -*A jf ^2* 3 £ i ■^r,T ^ t % 2* % t h ^Z- it "r + T 2 v- .t S £ S- U \u I J f ±, \ i Sow SSi-/ § 7 S^t * z ±7 \j utJ ' 1 £2 \7 TO 5 £ no-- -2. - - S? - 7 " = _- Fig. 106. — Wolf's Sun-Spot Numbers. Subsequent study fully confirms this remarkable result of Schwabe. Wolf of Zurich has collected all the observations discoverable and finds a pretty complete record back to 1610. From these records is constructed the annexed diagram, Fig. 106. The ordinates of the curve represent what Wolf calls his " relative numbers," l which he has adopted as representing the spottedness. 1 This " relative number " is formed in rather an arbitrary manner from the observations which Wolf hunted up as the basis of his investigation. The formula is, r (the relative number) = k (10 g +/), in which g is the number of groups and isolated spots observed, f the total number of spots which can be counted in these groups and singly, while k is a coefficient which depends upon the observer and the size of his telescope : it is large for a small telescope 192 THE SUN. The average period is eleven and one-tenth years, but, as the figure shows, the spot maxima are quite irregular, both in time and as to the extent of spottedness. The last spot maximum occurred in 1883-84 (a year or two behind time), and we are now (1888) ap- proaching a minimum. During a maximum the surface of the sun is never free from spots, from twenty-five to fifty being frequently visible at once. During a minimum, on the other hand, weeks often pass without the appearance of a single one. 308. Possible Cause of the Periodicity.— The cause of this periodicity is not known. It has been attempted to connect it with planetary action. Some things in the Kew statistics of the sun spots look as if Venus, Mer- cury, and the Earth had something to do with it, the sun's surface being more spotted when these planets approach nearer; but the evidence is insufficient, or at least needs to be supplemented by further comparisons. Jupiter also has been suspected. His period is 11.86 years, which is not very different from the mean sun-spot period; but an examination of the different spot maxima show that some of them have occurred when he was near perihelion, and others when he was near aphelion ; and on the whole there is very little reason for supposing that he has any considerable influ- ence in the matter. Sir John Herschel suggested that it might be due to streams of meteors moving in an oval orbit with a period of about eleven years, and approach- ing so near at perihelion that numerous members of the meteoric group actu- ally fall into the sun ; but, as has been said before, the distribution of the spots would seem to contradict the idea. 309. Terrestrial Influence of the Sun Spots. — One influence of the sun spots upon the earth is perfectly demonstrated. When the spots are numerous, magnetic disturbances (the so-called magnetic storms) are most numerous and violent upon the earth, a fact not to be wondered at since violent disturbances upon the sun's surface have been in many individual cases immediately followed by mag- netic storms, with a brilliant exhibition of the Aurora Borealis. The nature and mechanism of the connection is as yet unknown, but of the fact there can be no question. The dotted lines in the figure of the sun-spot periodicity (Fig. 106) represent the magnetic storminess of the earth at the indicated dates ; and the correspondence between these curves and the curve of spottedness makes it impossible to doubt the connection. and not very persistent observer, and approaches unity the more likely the observer may be supposed to have noted every sun spot that appeared during the time covered by his observations. TERRESTRIAL INFLUENCE OE THE SUN SPOTS. 193 310. It has been attempted, also, to show that greater or less disturbance of the sun's surface, as indicated by the greater fre- quency of the sun's spots, is accompanied by effects upon the meteor- ology of the earth, upon its temperature, barometric pressure, storm- iness, and the amount of rain-fall. The researches of Mr. Meldrum of Mauritius with respect to the cyclones in the Indian Ocean appear to bear out the conclusion that there may be some such connection in that case, but the general results are by no means decisive. In some parts of the earth the rain-fall seems-to be greater during a spot maximum ; in others, less. As to the temperature, it is still uncertain whether it is higher or lower at the time of a spot maximum. The spots themselves are cooler (as Henry, Secchi, and Langley have shown) than the general surface of the photosphere ; but their extent is never sufficient to reduce the amount of heat radiated from the sun by as much as y^^ part. On the other hand, when the spots are most numerous, the generally disturbed condition of the photosphere would, as Langley has shown, necessarily be accompanied by an increased radiation. Dr. Gould considers that the meteorological records in the Argen- tine Republic between 1875 and 1885 show an indubitable connection between the wind currents and the number of sun spots. But the demonstration of such a relation really requires observations running through several spot periods. On the whole, it is now quite certain that whatever influence the sun spots exert upon terrestrial meteo- rology is very slight, if it exists at all. THE SOLAR SPECTRUM AND ITS REVELATIONS. 311. About 1860 the spectroscope appeared in the field as a new and powerful instrument of astronomical research, at once resolving many problems as to the nature and constitution of the heavenly bodies which before had not seemed to be even open to investigation. The essential part of the apparatus is either a prism or train of prisms, or else a diffraction grating, 1 which is capable of performing the same office of dispersing — that is, of spreading and sending in different directions — the light rays of different wave-lengths. If, with such a " dispersion piece," as it may be called (either prism or grating) , one looks at a distant point of light, as a star, he will see 1 The grating is merely a piece of glass or speculum metal, ruled with many thousand straight, equidistant lines, from 5000 to 20,000 in the inch. Usually the surface before ruling is accurately plane, but for some purposes the concave grat- ings, originated by Professor Rowland, are preferable. 194 THE SUN. instead of a point a long streak of light, red at one end and violet at the other. If the object looked at be not a point, but a line of light parallel to the edge of the prism or to the lines of the grating, then, instead of a mere colored streak without width, one gets a spectrum, a colored band of light, which may show markings that will give the observer most valuable information. (Physics, pp. 458-460.) For convenience' sake it is usual to form this line of light by admitting the light through a narrow "slit," which is at one end of a tube having at the other end an achromatic object- glass at such a distance that the slit is in its principal focus. This tube with slit and lens constitutes the " collimator," so called be- cause it is precisely the same as the instrument used in connection with the transit instrument to adjust its line of collimation (Article 60). Instead of looking at the spectrum with the naked eye, however, it is better in most cases to use a small telescope ; called the " view- telescope," to distinguish it from the large telescope, to which the spectroscope is often attached. Prism-Spectroscope ft Direct-Vision Spectroscope Fig. 107 — Different Forms of Spectroscope. 312. Construction of the Spectroscope. — The instrument, there- fore, as usually constructed, and shown in Fig. 107, consists of three parts, — collimator, dispersion-piece, and view-telescope; but in the direct-vision spectroscope, shown in the figure, the view-telescope is omitted. If the slit, S, be illuminated by strictly homogeneous light THE SPECTROSCOPE. 195 all of one wave-length, say yellow, the "real image" of the slit will be found at Y. If at the same time light of a different wave-length be also admitted, say red, a second image will be formed at R, and the observer will see a spectrum with two "bright lines," the lines being really nothing more than images of the slit. If light from a candle be admitted, there will be an infinite number of these slit-images, close packed, like the pickets in a fence, without interval or break, and we then get a continuous spectrum ; but if we look at sunlight or moonlight, we shall find a spectrum continuous in the maiu, but crossed by numerous dark lines, as if some of the " pickets" had been knocked off, leaving gaps. Fig. 108. — The Telespectroscope. 313. Integrating and Analyzing Spectroscope. — If we simply direct the collimator of a spectroscope towards a distant luminous object, every part of the slit receives light from every part of the object, so that in this case every elementary streak of the spectrum 196 THE STTN. is a spectrum of the entire body, without distinction of parts. A spectroscope used in this way is said to be an integrating instrument. If, however, we interpose a lens (the object-glass of a telescope) between the luminous object and the slit, so as to have in the plane of the slit a distinct, real image of the object, then the top of the slit, for instance, will be illuminated wholly by light from one part of the object, the middle of it by light from another point, and the bottom by light from still a third. The spectrum formed by the top of the slit belongs, then, to the light from that particular point of the object whose image falls upon that part- of the slit ; and so of the rest. We thus separate the spectra of the different parts of the object, and so optically analyze it. An instrument thus used is spoken of as an "analyzing spectroscope." The combined instru- ment formed by attaching a spectroscope to a large telescope for the spectroscopic observation of the heavenly bodies has been called by Mr. Lockyer a "telespectroscope." Fig. 108 shows the apparatus used by the writer for some years at Dartmouth College. For solar purposes a grating spectroscope is generally better than a prismatic, being less complicated and more compact for a given power. 314. Principles upon which Spectrum Analysis depends. — These, substantially as announced by Kirchhoff in 1858, are the three following : — 1st, A continuous spectrum is given by every incandescent body, the molecules of which so interfere with each other as to prevent their free, independent, luminous vibration ; that is, by bodies which are either solid or liquid, or, if gaseous, are under high pressure. 2d, The spectrum of a gaseous element, under low pressure, is discontinuous, made up of bright lines, and these lines are character- istic ; that is, the same substance under similar conditions always gives the same set of lines, and 'generally does so even under widely different conditions. 3d, A gaseous substance absorbs from white light passing through it precisely those rays of which its own spectrum consists. The spec- trum of white light which has been transmitted through it then ex- hibits a "reversed" spectrum of the gas ; that is, one which shows dark lines instead of the characteristic bright lines. Fig. 109 illustrates this principle. Suppose that in front of the slit of the spectroscope we place a spirit lamp with a little carbonate of soda and some salt of thallium upon the wick. We shall then get a spectrum showing the two yellow lines of sodium and the green line PRINCIPLES OP SPECTRUM ANALYSIS. 197 of thallium, bright. If now the lime-light be started right behind the lamp flame, we shall at once get the effect shown in the lower figure, — a continuous spectrum crossed by black lines just where the bright lines were before. Insert a screen between the lamp flame and the lime, and the dark lines instantly show bright again. Fig. 109.— Reversal of the Spectrum. 315. Chemical Constituents of the Sun. — By taking advantage of these principles we can detect the presence of a large number of well-known terrestrial elements in the sun. The solar spectrum is crossed by dark lines, which, with an instrument of high dispersion, number several thousand, and by ^proper arrangements it is possible to identify among these lines many which are due to the presence in the sun's lower atmosphere of known terrestrial elements in the state of vapor . To effect the com- parison necessary for this purpose, the spectroscope must be so arranged that the observer can have before him, side by side, the spectrum of sunlight and that of the substance to be tested. In order to do this, half of the slit is fitted with a little "comparison prism," so-called (Fig. 110), which reflects into it the light from the sun, while the other half of the slit receives directly JB"io. 110. — The Comparison Prism. 198 THE SUN. the light of some flame or electric spark. On looking into the eye- piece of the spectroscope, the observer will then see a spectrum, the lower half of which, for instance, is made by sunlight, while the upper half is made by light coming from an electric spark between two metal points, saj- of iron. Photography may also be most effectively used in these compari- sons instead of the eye. Fig. Ill is a rather unsatisfactory repro- duction, on a reduced scale, of a negative recently made by Pro- Vie. 111. Comparison of the Solar Spectrum with that of Iron. From a Negative by Prof. Trowbridge. fessor Trowbridge at Cambridge. The lower half is the violet por- tion of the spectrum of the sun, and the upper half that of the vapor of irou in an electric arc. The reader can see for himself with what absolute certainty such a photograph indicates the presence of iron in the solar atmosphere. A few of the lines in the photograph which do not show corresponding lines in the solar spectrum are due to impurities in the carbon, and not to iron. t 316. As the result of such comparisons we have the following list of sixteen elements, which are certainly known to exist in the sun, viz. : — Barium, Manganese, Calcium, Nickel, Chromium, Platinum, Cobalt, Silicon, Copper, Silver, Hydrogen, Sodium, Iron, Titanium, Magnesium, Vanadium. There is evidence, perhaps not quite conclusive, of the presence of the following : — Aluminium, Cadmium, Carbon, Lead, Molybdenum, Palladium, Uranium, Zinc. PRINCIPLES OF SPECTRUM ANALYSIS. 199 As to carbon, however, the spectrum is so peculiar, consisting of bands rather than lines, that it is very difficult to be sure, but the tendency of the latest investigations (of Eowland and Hutchins) is to establish its right to a place on the list. The wore recent researches have thrown much doubt on the presence of several substances which, a few years ago, were usually included in the list, as, for instance, strontium and cerium. It has been generally admitted also that the photographs of Dr. Henry Draper had demonstrated the presence of oxygen in the sun, represented in the solar spectrum, not by dark lines like other elements, but by certain wide, bright bands. The latest work, while it does not absolutely refute Dr. Draper's conclusion, appears however to turn the balance of evidence the other way. 317. It will be noticed that all the bodies named in the list, carbon alone excepted, are metals (chemically hydrogen is a metal), and that a multitude of the most important terrestrial elements fail to appear ; oxygen (?), nitrogen, chlorine, bromine, iodine, sulphur, phosphorus, arsenic, and boron are all missing. We must be cautious, however, as to negative conclusions. It is quite conceivable that the spectra of these bodies under solar conditions may be so different from their spec- tra as presented in our laboratories that we cannot recognize them ; for it is now quite certain that some substances, nitrogen, for instance, under different conditions, give two or more widely different spectra. Among the many thousand lines of the solar spectrum only a few hundred are so far identified. 318. Mr. Lockyer's Views. — Mr. Lockyer thinks it more prob- able that the missing substances are not truly elementary, but are decomposed or " dissociated" on the sun by the intense heat, and so do not exist there, but are replaced by their components ; he believes, in fact, that none of our so-called elements are really elementary, but that all are decomposable, and, to some extent actually decomposed in the sun and stars, and some of them by the electric spark in our own laboratories. Granting this, a crowd of interesting and remark- able spectroscopic facts find easy explanation. At the same time the hypothesis is encumbered with great difficulties and has not yet been finally accepted by physicists and chemists. For a full statement of his views the reader is referred to his " Chemistry of the Sun." 319. The Reversing Layer. — According to Kirchhoff's theory the dark lines are formed by the passing of light from the minute solid and liquid particles of which the photospheric clouds are supposed to 200 THE STJN. be formed, through vapors containing the substances which we recog- nize in the solar spectrum. If this be so, the spectrum of the gaseous envelope, which by its absorption forms the dark lines, should by itself show a spectrum of corresponding bright lines. The opportunities are of course rare when it is possible to obtain the spectrum of this gas- stratum alone by itself; but at the time of a total eclipse, at the moment when the sun's disc has just been obscured by the moon, and the sun's atmosphere is still visible beyond the moon's limb, if the slit of the spectroscope be carefully adjusted to the proper point, the observer ought to see this bright-line spectrum. The author succeeded in making this very observation at the Spanish eclipse of 1870. The lines of the solar spectrum, which up to the final obscuration of the sun had remained dark as usual (with the exception of a few belonging to the spectrum of the chromosphere), were suddenly " reversed," and the whole length of the spectrum was filled with brilliant-colored lines, which flashed out quickly and then gradually faded away, disappearing in about two seconds, — a most beautiful thing to see. Substantially the same thing has since then been several times observed. 320. The natural interpretation of this phenomenon is, that the formation of the dark lines in the solar spectrum is mainly, at least, produced by a very thin layer close down to the photosphere, since the moon's motion in two seconds would cover a thickness of only about 500 miles. It was not possible, how- ever, to be certain that all the dark lines were reversed, and in this uncer- tainty lies the possibility of a different interpretation. Mr. Lockyer doubts the existence of any such thin stratum. According to his views the solar atmosphere is very extensive, and those lines of iron, which correspond to the more complex combinations of its constituents, are formed only in the regions of lower temperature, high up in the sun's atmosphere. They should appear early at the time of an eclipse and last long, but not be very bright. Those due to the constituents of iron which are found only close down to the solar surface should be short and bright; and he thinks that the numerous bright lines observed under the conditions stated are due to such substances only. Observation needs to be directed to the special point to determine whether all of the dark lines are reversed at the edge of the sun, or only a few ; and if so, what ones. 321. Sun-Spot Spectrum. — This is like the general solar spectrum, except that certain lines are much widened, while certain others are thinned, and sometimes the lines of hydrogen become bright. It Is to be noticed that by far the larger proportion of the dark lines of any SUN SPOTS. 201 Fig. 99. — Sept. 19. Fig. 101. — Sept. 21. Fia. 103.— Sept 23. m m Fig. 102. —Sept. 22. mBm £« .. EiG. 104. — Sept. 26. The Great Sun Spot of 1870. 202 THE SUN. J 2* 43™ 2' 51» 2 h 46» FlQ. 112. The C line in the Spectrum of a Sun Spot, Sept. 22, 1870. given substance ai'e not affected at all in the spot spectrum, but only a certain few of them, a point which Mr. Lockyer considers very important. Not infrequently it happens that certain lines of the spectrum are crooked and broken in connection with sun spots, as shown by Fig. 112. Such phe- nomena are caused, according to Doppler's principle, 1 by the swift motion of matter towards or from the observer. In the particular case shown in the figure, hydrogen is the substance, and the greatest motion indicated was towards the observer at the rate of about 300 miles a second — an unusual velocity. These effects are most noticeable, not in the spots, but near them, usually just at the outer edge of the penumbra. The dark and apparently continuous spectrum which is due to the nucleus of a sun spot is not truly continuous, but under high dispersion is resolved into a range of extremely fine, close-packed, dark lines, separated by narrow spaces. At least this is so in the green and blue portions of the spectrum ; it is more difficult to make out this structure in the yellow and red. It appears to indicate that the absorbing medium which fills the hollow of a sun spot is gaseous, and not composed of precipitated particles like smoke, as has been suggested. 1 Doppler's principle is this : that when we are approaching, or approached by, a body which is emitting regular vibrations, then the number of waves received by us in a second is increased, and their wave-length correspondingly diminished ; and vice versa when the distance of the vibrating body is increasing. Thus the pitch of a, musical tone rises while we are approaching the sounding body, and falls as we recede ; in just the same way the " ref rangibility " of the rays, say of hydrogen, emanating from the sun is increased (the wave-length being shortened) whenever we are approaching it with a, speed which bears a sensible ratio to the velocity of light. Calling \ the wave-length of the ray when the observer and the luminous object are relatively at rest, and x' the wave-length as affected by their relative motion ; putting Falso for the velocity of light (about 186,330 miles per second), and s for the speed with which the observer and source of light approach each other, we have \V+s)- [If the distance is increasing instead of diminishing, the denominator will be ( V— s) .] With the most powerful spectroscopes motions of from one to two miles per second along the line of sight can thus be detected. CHROMOSPHERE AND PROMINENCES. 203 322. The Chromosphere. —The chromosphere is a region of the sun's gaseous envelope which lies close above the photosphere, the "■reversing layer," if it exists at all, being only the most dense and hottest part of it. The chromosphere is so called, because as seen for an instant, during a total solar eclipse, it is of a bright scarlet color, the color being due to the hydrogen which is its main constit- uent. It is from 5000 to 10,000 miles in thickness, and iu structure is very like a sheet of scarlet flame, not being composed of horizon- tal sheets, but of (approximately) upright filaments. Its appearance has been compared very accurately to that of " a prairie on fire"; but the student must carefully guard against the idea that there is any real "burning" in the case; i.e., any process of combination between hydrogen and some other substance. The temperature is altogether too high for any formation of hydrogen compounds at the sun's surface. t 323. The Prominences. — At a total eclipse, after the totality has fairly set in, there are usually to b,e seen at the edge of the moon's disc a number of scarlet, star-like objects, which in the telescope appear as beautiful, fiery clouds of various form and size. These are the so-called '■'■prominences ," which very non-committal name was given while it was still doubtful whether they were solar or lunar. Photography, in 1860, proved that they really belong to the sun, for the photographs taken during the totality showed that the moon obviously moves over them, covering those upon the eastern limb, and uncovering those upon the western. Their spectrum, first observed in 1868, is gaseous, i.e., bright-lined, the lines of hydrogen being especially conspicuous. There are, however, a number of other bright lines, — among them the violet H and K lines usually ascribed to calcium, and a yellow line (known as D s because it is near the two B lines of sodium), which is due to some unidentified element provisionally named " helium." In connection with this eclipse, Janssen, who observed it in In- dia, found that the lines of the prominence spectrum were so bright "that he was able to observe them the next day after the eclipse in full sunlight ; -and he also found that by a proper management of his instrument he could study the form and behavior of the promi- nences nearly as well without an eclipse as during one. Lockyer, in England, some time earlier bad come to similar conclusions from 204 THE SUN. theoretical grounds, and he practically perfected his discovery a few weeks later than Janssen, although without knowledge of what he had done. By a remarkable but accidental coincidence their discov- eries were communicated to the French Academy on the same day ; and in their honor the French have struck a medal bearing their united effigies. Fig. 113. Spectroscope Slit adjusted for Observation of tbe Promi- 324. How the Spectroscope makes the Prominences Visible.— The only reason we cannot see the prominences at any time is on account of the bright illumination of our own atmosphere. We can screen off the direct light of the sun ; but we cannot screen off the reflected sunlight coming from the air which is directly between us and the prominences themselves ; a light so brilliant that the prominences cannot be seen through it without some kind of aid. The spectrum of this air-light is, of course, just the same as that of the sun — a con- tinuous" spectrum with the same dark lines upon it. When, therefore, we arrange the apparatus as indicated in Fig. 113, pointing the telescope so that the image of the sun's limb just touches the slit of the spectro- scope, then, if there is a prominence at that point, we shall have in our spectroscope two spectra superposed upon each other ; namely, the spectrum of the air-illumination and that of the prominence. The latter is a spectrum of bright lines, or, if the slit is opened a little, of bright images of whatever part of the prominence may fall within the edges of the slit. Now, the brightness of these images is not affected by any increase of dispersion in the spectro- scope. Increase 1 of dispersion merely sets these images farther apart, without making them fainter. The spectrum of the aerial illu- mination, on the other hand, is made very faint by its extension ; and, moreover, it presents dark lines (or spaces when the slit is opened) precisely at the points where the bright images of the prominences fall. A spectroscope of dispersive power sufficient to divide the two E lines, attached to a telescope of four or five inches aperture, gives a 1 Too high dispersion injures the definition, however, because the lines in the spectrum of hydrogen are rather broad and hazy. DIFFERENT KINDS OF PROMINENCES. 205 very satisfactory view of these beautiful objects ; the red image cor- responds to the C line, and is by far the best for such observations, though the D 3 line or the F line can also be used. When the instru- ment is properly adjusted, the slit opened a little, and the image of the sun's limb brought exactly to the edge of the slit, the observer at the eye-piece of the spectroscope will see things about as we have attempted to represent them in Fig. 114; as if he were looking at the clouds in an evening sky through a slightly opened window- blind. 325. Different Kinds of Prominences ; Their Forms and Motions. — The prominences may be broadly divided into two classes, — the quiescent or diffused, and the eruptive or "metallic," as Secchi calls them, because they show in their spectrum the lines of many metals besides hydrogen. The former, illustrated by Fig. 115 (see p. 206), arc immense clouds, often 60,000 miles in height, and of corresponding horizontal dimensions, either resting upon the chromosphere or connected with it by slender stems like great banyan-trees. They are not very brilliant, and are composed almost entirely of hydrogen and "helium." They often remain nearly un- changed for days together as they pass over the sun's limb. They are found on all portions of the disc, at the poles and equator as well as in the spot zones. Some of them are clouds floating entirely detached from the sun's surface. Usually these clouds are simply the remnants of prominences which appear to have been thrown up from below, but in some cases they actually form and grow larger without any visible connection with the chromosphere — a fact of considerable importance, as showing in those regions the presence of hydrogen, invisible to our spectroscopes until somehow or other it is made to give out the rays of its -famil- iar spectrum. All the forms and motions of the prominences, it may be said further, seem to indicate the same thing — that they The Chromosphere and Prominences seen in the Spectroscope. 206 THE SUN. Clouds. Diffuse. JSi & £ ,21111! 1 ■ Filamentary. Stemmed. Horns. Pig. 115. Quiescent Prominences. Scale 75,000 Miles to the Inch. From "The Sun," hy Permission of D. Appleton & Co. EltUPTIVE PROMINENCES. 207 Vertical Filaments. Prominences Sept. 7, 1871, 12.30 p.m. Cyclone. Flame. Jets near Sun's Limb, Oct. 5, 1871. Fig. 116. Eruptive Prominences. From " The Sun." By Permission of D. Appleton & Co. 208 THE SUN. exist and move, not in a vacuum, but in a medium of density com- parable with their own, as clouds do in our own atmosphere. 326. The eruptive prominences, on the other hand, are brilliant and active, not usually so large as the quiescent, but at times enormous, reaching elevations of 100,000, 200,000, or even 400,000 miles. They are illustrated by Fig. 116. Most frequently they are in the form of spikes or flames ; but they present also a great variety of other fantastic shapes, and are sometimes so brilliant as to be visible with the spectroscope on the surface of the sun itself, and not merely at the limb. Generally prominences of this class are asso- ciated with active sun spots, while both classes appear to be con- nected with the faculse. The figures given are from drawings of individual prominences that have been observed by the author at different times. These solar clouds are most fascinating objects to watch, on ac- count of the beauty of their forms, and the rapidity of their changes. In the case of the eruptive prominences the swiftness of the changes is sometimes wonderful — portions can be actually seen to move, and this implies a real velocity of at least 250 miles a second, so that it is no exaggeration to speak of such phenomena as veritable ' ' explo- sions " : of course, in such cases the lines in the spectrum are greatly broken and distorted, and frequently a "magnetic storm" follows upon the earth, with a brilliant Aurora Borealis. The number visible at a single time is variable, but it is not very unusual to find as many as twenty on the sun's limb at once. 327. The Corona. — This is a halo, or ' ' glory," of light which sur- rounds the sun at the time of the total eclipse. From the remotest times it has been well known, and described with enthusiasm, as being certainly one of the most beautiful of natural phenomena. The portion of the corona nearest the sun is almost dazzlingly bright, with a greenish, pearly tinge which contrasts finely with the scarlet blaze of the prominences. It is made up of streaks and filaments which on the whole radiate outwards from the sun's disc, though they are in many places strangely curved and intertwined. Usually these filaments are longest in the sun-spot zones, thus giving the corona a more or less -quadrangular figure. At the very poles of the sun, however, there are often tufts of sharply defined threads. For the most part the streamers have a length not much exceed- ing the sun's radius, but some of them at almost every eclipse go PHOTOGRAPHS OF THE CORONA. 209 far beyond this limit. In the clear air of Colorado during the eclipse of 1878, two of them could be traced for five or six degrees, — a distance of at least 9,000000 miles from the sun. A most striking feature of the corona usually consists of certain dark rifts which reach straight out from the moon's limb, clear to the extremest limit of the corona. The corona varies much in brightness at different eclipses, and of course the details are never twice the same. Its total light under ordinary circumstances is at least two or three times as great as that of the full moon. 328. Photographs of the Corona. — While the eye can perhaps grasp some of its details more satisfactorily than the photographic plate can do, it is found that drawings of the corona are hardly to be trusted. At any rate, it seldom happens that the representations of two artists agree sufficiently to justify any confidence in their scientific W Fio. 117. — Corona of the Egyptian Eclipse, 1882. accuracy. Photographs, on the other hand, may be trusted as far as they go, though they may fail to bring out some things which are conspicuous to the eye. Fig. 1 1 7 is from the photograph of the Egyp- tian eclipse of 1882, when a little comet was found close to the sun. Of course, as in the case of the prominences, the only reason we cannot see the corona without an eclipsed sun is the illumination of the earth's atmosphere. If we could ascend above our atmosphere, and manage to exist and to observe there, we could see it by simply screening oft the sun's disc. 210 THE SUN. So long, however, as the brightness of the illuminated air is more than about sixty times that of the corona, it must remain invisible to the eye. Dr. Huggins has thought that it might be possible by means of photographs to detect differences of illumination less than fa (the limit of the eye's percep- tion), and so to obtain pictures of the corona at any time ; especially as it appears that the coronal light is far richer in ultra-violet rays (the photo- graphic rays) than the general sunlight with which the air is illuminated. His attempts so far, however, have yielded only doubtful success. 329. Spectrum of the Corona. — This was first definitely observed in 1869 during the eclipse which passed over the western part of the United States in that year. It was then found that its most remarka- ble characteristic is a bright line in the green, which the writer identi- fied as coinciding with the dark line at 1474 on the scale of Kirchhoff's map (X= 5316). This line was also observed by Harkness. This result was for a time very puzzling, since the dark line in question is given by Angstrom and other authorities as due to the spectrum of iron The mystery has since been removed, however, by the discovery that under high dispersion the line is double, and that the corona line coincides with the more refrangible of the two components, while the other one is the line due to iron. We have as yet been unable to identify with any terrestrial element the substance to which this line is due, but the provisional name " coronium " has been proposed for it. The recent researches of Griinewald make it somewhat probable that both coronium and helium are components of hydrogen, which (in line with Mr. Lockyer's speculations) is supposed to be partially decomposed under solar conditions. Besides this conspicuous green line, the hydrogen lines are also faintly visible in the spectrum of the corona ; and by means of a pho- tographic camera used during the Egyptian eclipse of 1882, it was found that the upper or violet portion of the spectrum is very rich in lines, among which H and K are specially conspicuous. There is also, through the whole spectrum, a faint continuous background, which, however, according to Mr. Lockyer's statements, is not of uniform brightness, but "banded." In it some observers have re- ported the presence of a few of the more conspicuous dark lines of the ordinary solar spectrum, but the evidence on this point is rather conflicting. If during the totality we look at the eclipsed sun with a diffrac- tion grating, or through a prism of high dispersive power, we see three rings which are really images of the corona. One of them, the bright- est and the largest, is the green ring due to the 1474 line ; the others NATURE OF THE CORONA. 211 are a red ring due to C, and a blue one due to the F line of hydrogen. . 330. Nature of the Corona. — It is evident that the corona is a truly solar and not merely an optical or atmospheric phenomenon, from two facts : first, the identity of detail in photographs made at widely separate stations. In 1871 , for instance, photographs were obtained at the Indian station of Bekul, in Ceylon, and in Java, three stations sepa- rated by many hundreds of miles ; but, excepting minute differences of detail, such as might be expected to have resulted from the changes that would naturally go on in the corona during the half -hour while the moon's shadow was travelling from Bekul to Java, all the photo- graphs agree exactly, which of course would not be the case if the corona depended in any way upon the atmospheric conditions at the observer's station. Second (but first historically) , the presence of bright lines in the spectrum of the corona proves that it cannot be a terrestrial or lunar phenomenon, by demonstrating the presence in the corona of a self- luminous gas, which observation fails to find either near to the moon or in our own atmosphere. It must, therefore, be at the sun. But while it is thus certain that the corona contains luminous gas, it also is very likely that finely divided solid or liquid matter may be pres- ent in the corona ; that is, fog or dust of some kind. 1 331 . The corona cannot be a true " solar atmosphere " in any strict sense of the word. No gaseous envelope in any way analogous to the earth's atmosphere could possibly exist there in gravitational equi- librium under the solar conditions of pressure and temperature. The corona is probably a phenomenon due somehow to the intense activity of the forces there at work ; meteoric matter, cometic matter, matter ejected from within the sun, are all concerned. That this matter is inconceivably rare is evident from the fact that in several cases comets have passed directly through the corona without experiencing the least perceptible disturbance of their motions. It is altogether probable that at a very few thousand miles above the sun's surface, its density becomes far less than that of the best vacuum we can make in an electric lamp. 1 This is indicated by the partial radial polarization of the light of the corona. 211 A ADDENDA TO CHAPTEK VHI. Art. 285. In 1889 Dun6r in Sweden made an admirable series of spectroscopic determinations of the displacement of the Fraun- hofer lines at the eastern and western edges of the sun, and the results -were not in accordance with those obtained by Mr. Crew (note, p. 179), but completely confirm the received laws of equa- torial acceleration (Art. 284). They give, however, a little slower rate of rotation than do the observations of the spots, — about 25£ days at the sun's equator, instead of 25.0. 316. The researches of Rowland have greatly increased the number of elements recognized as present in the sun. In 1890 he gave the following list of 36, whose presence he regarded as certainly established, and it is probable that a few more will ulti- mately be added. The elements are arranged in the list according to the intensity of the dark lines by which they are represented in the solar spectrum, while the appended figures denote the rank which each element would hold if the arrangement had been based on the number instead of the intensity of the lines. In the case of iron the number exceeds 2000. An asterisk denotes that the lines of the element indicated appear often or always as bright lines in the spectrum of the chromo- sphere (Art. 322). ♦Strontium, 23. Vanadium, 8. *Barium, 24. Carbon, 7. Scandium, 12. Yttrium, 15. Zirconium, 9. Molybdenum, 17. Lanthanum, 14. Niobium, 16. Palladium, 18. Neodymium, 13. *Calcium, 11. *Iron, 1. *Hydrogen, 22. *Sodium, 20. ♦Nickel, 2. ♦Magnesium, 19. ♦Cobalt, 6. Silicon, 21. Aluminium, 25. ♦Titanium, 3. ♦Chromium, 5. ♦Manganese, 4. Copper, 30. Zinc, 29. Cadmium, 26. ♦Cerium, 10. Glueinum, 33. Germanium, 32. Rhodium, 27. Silver, 31. Tin, 34. Lead, 35. Erbium, 28. Potassium, 36. 323. Since 1890 photography has been applied with great success to the study of the chromosphere and prominences. By utilizing the H and K lines, which, like the hydrogen lines, are always < reversed ' in the prominences, it is possible to photograph their forms through the opened slit of the spectroscope; and by an ingenious arrangement Professor Hale of Chicago succeeds in ADDENDA TO CHAPTER VIII. 211 B getting pictures of the chromosphere and prominences around the whole circumference of the sun at once. It should be noted that while, in the ordinary solar spectrum, H and K are broad dark bands, in the spectrum of the prominences they are thin bright lines just at the centre of the dark bands; also, that H is double; i.e., there is a strong hydrogen line within the limit of the dark band a little below the line of calcium which occupies the centre of the band; finally, that in the spectrum of the facvlce on the sun's surface the calcium lines H and K always show bright. 212 THE SUN. CHAPTER IX. THE SUN'S LIGHT AND HEAT : COMPARISON OF SUNLIGHT WITH ARTIFICIAL LIGHTS. — MEASUREMENT OP THE SUN'S HEAT, AND DETERMINATION OF THE "SOLAR CONSTANT." — PYR- HELIOMETER, ACTINOMETER, AND BOLOMETER. — THE SUN'S TEMPERATURE. — THEORIES AS TO THE MAINTENANCE OF THE SUN'S RADIATION, AND CONCLUSIONS AS TO THE SUN'S POSSIBLE AGE AND FUTURE DURATION. 332. The Sun's Light. — The Quantity of Sunlight. It is very easy to compare (approximately) sunlight with the light of a standard 1 candle ; and the result is, that when the sun is in the zenith, it illumi- nates a white surface about 60,000 times as strongly as a standard candle at a distance of one metre. If we allow for the atmospheric ab- sorption, the number would be fully 70,000. If we then multiply 70,000 by the square of 150,000 million (roughly the number of metres in the sun's distance from the earth) , we shall get what a gas engineer would call the sun's " candle power." The number comes out 1575 billions of billions (English); i.e., 1575 with twenty-four ciphers following. 333. One way of making the comparison is the following : Arrange mat- ters as in Fig. 118. The sunlight is brought into a darkened room by a mirror M, which reflects the rays through a lens L of perhaps half an inch in diameter. After the rays pass the focus they diverge and form on the screen S a disc of light, the size of which may be varied by changing the distance of the screen. Suppose it so placed that the illuminated circle is just ten feet in diameter; that is, 240 times the diameter of the lens. The illumination of the disc will then be less than that of direct sunlight in the ratio of 240 2 (or 57,600) to 1 (neglecting the loss of light produced by 1 A standard candle is a sperm candle weighing one-sixth of a pound and burn- ing 120 grains an hour. The French " Carcel burner," used as a standard in their photometry, gives just ten times the quantity of light given by this standard candle. An ordinary gas-burner consuming five feet of gas hourly gives a light equivalent to from twelve to fifteen standard candles. INTENSITY OP THE SUN's LUMINOSITY. 213 the mirror and the lens, a loss which of course must be allowed for). Now place a little rod like a pencil near the screen, as at P, light a standard candle, and move the candle back and forth until the two shadows of the Fig. 118. — Comparison of Sunlight with a Standard Candle. pencil, one formed by the candle, and the other by the light from the lens, are equally dark. It will be found that the candle has to be put at a dis- tance of about one metre from the screen ; though the results would vary a good deal from day to day with the clearness of the air. 334. "When the sun's light is compared with that of the full moon and of various stars, we find, as stated (Art. 259), that it is about 600,000 times that of the full moon. It is 7,000,000000 times as great as the light received from Sirius, and about 40,000,000000 times that from Vega or Arcturus. 335. The Intensity of the Sun's Luminosity. — This is a very different thing from the total quantity of its light, as expressed by its "candle power" (a surface of comparatively feeble luminosity can give a great quantity of light if large enough) . It is the amount of light per square inch of luminous surface which determines the intensity. Making the necessary computations from the best data obtainable (only roughish approximations being possible), it appears that the sun's surface is about 190,000 times as bright as that of a 214 THE SUN. candle flame, and about 150 times as bright as the lime of the calcium light. Even the darkest part of a solar spot outshines the lime. The intensely brilliant spot in the so-called "crater" of an electric arc comes nearer sunlight than anything else known, being from one-half to one-fourth as bright as the surface of the sun itself. But either the electric arc or the calcium light, when interposed between the eye and the sun looks like a dark spot on the disc. 336. Comparative Brightness of Different Portions of the Sun's Surface. — By forming a large image of the sun, say a foot in di- ameter, upon a screen, we can compare with each other the rays coming from different parts of the sun's disc. It thus appears that there is a great diminution of light at the edge, the light there, accord- ing to Professor Pickeriug's experiments, being just about one-third as strong as at the centre. There is also an obvious difference of color, the light from the edge of the disc being brownish red as com- pared with that from the centre. The reason is, that the red and yellow rays of the spectrum lose much less of their brightness at the limb than do the blue and violet. According to Vogel, the latter rays are affected nearly twice as much as the former. Por this reason, photographs of the sun exhibit the darkening of the limb much more strongly than one usually sees it in the telescope. 337. Cause of the Darkening of the limb. — It is due unques- tionably to the general absorption of the sun's rays by the lower por- tion of the overlying atmosphere. The reason is obvious from the figure (Fig. 119) . The thinner this atmosphere, other things being equal, the greater the ratio between the percentage of absorption at the centre and edge of the disc, and the more obvious the darkening of the limb. Attempts have been made to determine from the observed dif- brightness of centre and limb the total Unfortunately we have Fig. 119. Cause of the Darkening of the Sun's Lirah. ferences between the percentage of the sun's light thus absorbed, to supplement the observed data with some very uncertain assump- tions in order to solve the problem ; and it can only be said that it is probable that the amount of light absorbed by the sun's atmos- the sun's heat. 215 phere lies between fifty and eighty per cent ; i.e., the sun deprived of its gaseous envelope would probably shine from two to five times as brightly as now. It is noticeable also, as Langley long ago pointed out, that thus stripped, the "complexion" of the sun would be markedly changed from yellowish white to a good full blue, since the blue and violet rays are much more powerfully absorbed than those at the lower end of the spectrum. THE SUN'S HEAT. 338. It* Quantity; the "Solar Constant." By the "quantity of heat" received by the earth from the sun we mean the number of heat-units received in each unit of time by a square unit of surface when the sun is in the zenith. The heat-unit most employed by engineers is the calorie, which is the quantity of heat required to. raise the temperature of one kilogram of water one degree centigrade. It is found by observation that each square metre of surface exposed perpendicularly to the sun's rays receives from the sun each minute f rom twenty-five to thirty of these calories ; or rather it would do so if a considerable portion of the sun's heat were not stopped by the earth's atmosphere, which absorbs some thirty per cent of the whole, even when the sun is vertical, and a much larger proportion when the sun is near the horizon. This quantity, twenty-Jive calories 1 per square metre per minute (using the smaller of the values mentioned, which certainly is not too large), is known as the " Solar Constant." 339. Method of determining the "Solar Constant." — The method by which the solar constant is determined is simple enough in prin- ciple, though complicated with serious practical difficulties which affect its accuracy. It is done by allowing a beam of sunlight of known cross-section to shine upon a Jcnown weight of water (or other substance of known specific heat) for a known length of time, and 1 For many scientific purposes the engineering calorie is inconveniently large, and a smaller one is employed, which replaces the kilogram of water by the gram heated one degree — the smaller calorie being thus only T ^Vw of the engineering unit. As stated by many writers (Langley, for instance), the solar constant is the number of these small calories received per square centimetre of surface in a minute. This would make the number 2.5 instead of 25. It would perhaps be better to bring the whole down to the " c. g. s. system " by sub- stituting the second for the minute ; and this would give us for the solar constant, on the " c. g. s. system," 0.0417 (small) calories per square centimetre per second. 216 THE SUN. measuring the rise of temperature. It is necessary, however, to de- termine and allow for the heat received from other sources during the experiment, and for that lost by radiation. Above all, the absorb- ing effect of our own atmosphere is to be taken into account, and this is the most difficult and uncertain part of the work, since the atmospheric absorption is continually changing with every change of the transparency of the air, or of the sun's altitude. 340. Pyrheliometers and Actinometers. — The instruments with which these measurements are made, are known as " pyrheliometers " and "actinometers." Fig. 120 represents the pyr- heliometer of Pouillet, with which in 1838 he made his determination of the solar constant, at the same time that Sir John Herschel was experimenting at the Cape of Good Hope in practically the same way. They were the first apparently to understand and attack the problem in a reasonable manner. The pyr- heliometer consists essentially of a little cylin- drical box ab, like a snuff-box, made of thin silver plate, with a diameter of one decimetre and such a thickness that it holds 100 grams of water. The upper surface is carefully blackened, while the rest is polished as brill- iantly as possible. In the water is inserted the bulb of a delicate thermometer, and the whole is so mounted that it can be turned in any direction so as to point it directly towards the sun. It is used by first holding a screen between it and the sun for (say) five minutes, and watching the rise or fall of the mercury in the thermometer at ni. There will usually be some slight change due to the radiation of surrounding bodies. The screen is then re- moved, and the sun is allowed to shine upon Fio. 120. — Pouiiiet's Pyrhcliometer. the blackened surface for five minutes, the instrument being continually turned upon the thermometer as an axis, in order to keep the water in the calorimeter box well stirred. At the end of the five minutes the screen is replaced and the rise of the temperature noted. The difference between this and the change of the thermometer during the first five minutes will give us the amount by which a beam of sunlight one decimetre in diameter has raised the temperature of 100 grams of water in five minutes, and were it not for the troublesome corrections which must be made, would furnish directly the value of the solar constant. CORRECTION FOR ATMOSPHERIC ABSORPTION. 217 341. The second apparatus, Fig. 121, is the actinometer of Violle, which consists of two concentric metal spheres, the inner of which is blackened on the inside, while the outer one is brightly polished, the space between the two being filled with water at a known temperature, kept circulating by a pump of some kind. The thermoscopic body in this case, instead of being a box filled with water, is the blackened bulb of the thermometer T; and the obser- vations may be made either in the same way as with the pyrheliometer, or simply by noting the difference between the temperature finally attained by the ther- mometer T after it has ceased to rise in the sun's rays, and the temperature of the water circulating in the shell. 342. Correction for Atmospheric Absorption. — The correction for at- mospheric absorption is determined by making observations at various altitudes of the sun between zenith and horizon. If the rays were homogeneous (that is, all of one wave-length), it would be comparatively easy to deduce the true correction and the true value of the sqlar constant. In fact, however, the visible solar spectrum is but a small portion of the whole spectrum of the sun's radiance, and, as Langley has shown, it is necessary to determine the coefficient of absorption separately for all the rays of different wave-length. Fig. 121. — Violle's Actioometer. 343. The Bolometer. — This he has done by means of his " Bolometer," an instrument which is capable of indicating exceedingly minute changes in the amount of radiation received by an extremely thin strip of metal. This strip is so arranged that the least change in its electrical resistance due to any change of temperature will disturb a delicate galvanometer. The instrument is far more sensitive than any thermometer or even thermo- pile, and has the especial advantage of being extremely quick in its re- sponse to any change of radiation. Fig. 122 shows it so connected with a spectroscope that the observer can bring to the bolometer, B, rays of any wave-length he chooses. The rays enter through the collimator lens L, and are then refracted by the prism P to the reflector M, whence they are sent back to B. Langley has shown that the corrections for atmospheric absorption deduced by earlier observers are all considerably too small, and has raised the re- ceived value of the solar constant from 20 or 25, which was the value accepted a few years ago, to 30. We have, however, provisionally retained 218 THE SUN. the 25, as his new results, though almost certainly correct, have not yet been universally accepted, and perhaps need verification. 344. A less technical statement of the solar radiation may be made in terms of thickness of the quantity of ice which would be melted by it in a given time. Since it requires about eighty calories of heat to melt a kilo- gram of ice, it follows that twenf!y-five calories per minute per square metre would liquefy in an hour a sheet of ice one metre square and about nine- teen millimetres thick. Ac- cording to this the sun's heat would melt about 174 feet of ice annually on the earth's equator ; or 136£ feet yearly all over the surface of the earth, if the heat annually received were equally distributed in all latitudes. (See note at end of the chapter, page 227.) Langley's Spectro-Bolomeler, as used for Mapping the Energy of the Prismatic Spectrum. 345. Solar Heat expressed as Energy. — Since according to the known value of the "mechanical equivalent of heat" (Physics, p. 159) a horse-power corresponds to about 10 T 7 ^- calories per minute, it follows that each square metre of surface (neglecting the air-absorp- tion) would receive, when the s>tn is overhead, about two and one- third horse-power continuously. Atmospheric absorption cuts this down to about one and one-half horse-power, of which about one- eighth can be actually utilized bj T properly constructed machinery, as, for instance, the solar engines of Ericsson and Mouchot (see Langley's "New Astronomy"). In Ericsson's apparatus the re- flector, about 11 feet by 16 feet, collected heat enough to work a three-horse-power engine very well. Taking the earth's surface as a whole, the energy received during a year aggregates about sixty mile- tons for every square foot. That is to say, the heat annually re- ceived on each square foot of the. earth's surface, if employed in a Fig. 122. SOLAR RADIATION AT THE SUN'S SURFACE. 219 perfect heat engine, would be able to Jioist sixty tons to the height of a mile. 346. Solar Radiation at the Sun's Surface. — If, now, we estimate the amount of radiation at the sun's surface itself, we come to results which are simply amazing and beyond comprehension. It is necessary to multiply the solar constant observed at the earth (which is at a distance of 93,000000 miles from the sun) by the square of the ratio between 93,000000 and 433,250, the radius of the sun. This square is about 46,000 ; in other words, the amount of heat emitted in a minute by a square metre of the sun's surface is about 46,000 times as great as that received by a square metre at the earth. Car- rying out the calculations, we find that this heat radiation at the sur- face of the sun amounts to over a million calories per square metre per minute; that it is over 100,000 horse-power per square metre continuously acting ; that if the sun were frozen over completely to a depth of fifty feet, the heat emitted is sufficient to melt this whole shell in one minute of time; that if an ice bridge could be formed from the earth to the sun by a column of ice two and one-fourth miles square at the base and extending across the whole 93,000000 of miles, and if by some means the whole of the solar radiation could be concentrated upon this column, it would be melted in one second of time, and in between seven and eight seconds more would be dissipated in vapor. To maintain such a development of heat by combustion would require the hourly burning of a layer of the best anthracite coal from sixteen to twenty feet thicJc over the sun's entire surface, — a ton for every square foot of surface, — at least nine times as much as the consumption of the most powerful blast fur- nace in existence. At that rate the sun, if made of solid coal, would not last 6000 years 347. Waste of Solar Heat. — These estimates are of course based on the assumption that the sun radiates heat equally in all directions, and. there is no assignable reason why it should not do so. On this assumption, how- ever, so far as we can see, only a minute fraction of the whole radiation ever reaches a resting-place. The earth receives about 2200,000000 of the whole, and the other planets of the solar system, with the comets and the meteors, get also their shares ; all of them together, perhaps ten or twenty times as much as the earth. Something like 100,000000 of the whole seems to be utilized within the limits of the solar system. As for the rest, science cannot yet tell what becomes of it. A part, of course, reaches distant stars and other objects in interstellar space ; but by far the larger portion seems to be " wasted," accord- ing to our human ideas of waste. 220 TIJE SUN. 348. Experiments with the thermopile, first conducted by Henry at Princeton in 1845, show that the heat from the edges of the sun's disc, like the light, is less than that from the centre — according to Langley's measurements about half as much. The explanation evidently lies in its absorption by the solar atmosphere. 349. The Sun's Temperature. — While we can measure with some accuracy the quantity of heat sent us by the sun, it is different with its temperature, in respect to which we can only say that it must be very high — much higher than any temperature attainable by known methods on the surface of the earth. This is shown by a number of facts, for in- stance, by the great abun- dance of the violet and ultra-violet rays in the sunlight. Again, by the penetrat- ing power of sunlight; a large percentage of the heat from a common fire, for instance, being stopped by a plate of glass, while nearly the whole of the solar radiation passes through. The most impressive demonstration, however, follows from this fact; viz., that at the focus of a powerful burning-lens all known substances melt and vaporize, as in an electric arc. Now at the focus of the lens the limit of the temperature is that which would be produced by the sun's direct radiation at a point where the sun's angular diameter equals that of the burning-lens itself seen from the focus, as represented in Fig. 123. An object at F would theoreti- cally (that is, if there was no loss of heat conducted away by sur- rounding bodies and by the atmosphere) reach the same temperature as if carried to a point where the sun's angular diameter equals the angle LFL'. In the most powerful burning-lenses yet constructed a body at the focus is thus virtually carried up to within about 240,000 miles of the sun's surface, where its apparent diameter would be about 80°. Here, as has been said, the most refrac- tory substances are immediately subdued. If the earth were to ap- proach the sun as near as the moon is to us, she would melt and be vaporized. 350. Ericsson in 1872 made an exceedingly ingenious and interesting experiment illustrating the intensity of the solar heat. He floated a calor- EFFECTIVE TEMPERATURE. 221 imeter, containing about ten pounds of water, upon the surface of a large mass of molten iron by means of a raft of fire-brick, and found that the radiation of the metal was a trifle over 250 calories per minute for each square foot of surface ; which is only T hs P ait of the amount emitted by the same area of the sun's surface. He estimated the temperature of the metal at 3000° F. or 1649° C. 351. Effective Temperature. — The question of the sun's temper- ature is embarrassed by the fact that it has no one temperature ; the temperature at different parts of the solar photosphere and chromo- sphere must be very different. We evade this difficulty to some extent by substituting for the actual temperature, as the object of inquiry, what has been called the sun's " effective temperature"; that is, the temperature which a sheet of lampblack must have in order to radiate the amount of heat actually thrown off by the sun. (Phys- icists have taken the radiating power of lampblack as unity.) If we could depend upon the laws * deduced from laboratory experiments, by which it has been sought to connect the temperature of the body with its rate of radiation, the matter would then be comparatively simple: from the known radiated quantity of heat (in calories) we could compute the effective temperature in degrees. But at present it is only by a very unsatisfactory process of extrapolation that we can reach conclusions. The sun's temperature is so much higher than any which we can manage in our laboratories, that there is not yet much certainty to be obtained in the matter. Eosetti, the most recent investigator, whose results seem to be on the whole the most probable, obtains 10,000° C. or 18,000° F. for the effective tempera- ture. 352. Constancy of the Sun's Heat. — It is an interesting and thus far unsolved problem, whether the total amount of the sun's radia- tion varies perceptibly at different times. It is only certain that the variations, if real, are too small to be detected by our present means of observation. Possibly, at some time in the future, observations on a mountain summit above the main body of our atmosphere may decide the question. 1 A number of such laws have been formulated; for instance, the well-known law of Dulong and Petit (Physics, p. 470). The French physicists Pouillet and Vicaire, using this formula, have deduced values for the sun's effective temper- ature ranging from 1500° to 2500° C. Ericsson and Secchi, using Newton's law of radiation (which, however, is certainly inapplicable under the circumstances), put the figure among the millions. Zollner, Sporer, and Lane give values rang- ing from 25,000° to 50,000° C. 222 THE SUN. It is not unlikely that changes in the earth's climate such as have given rise to glacial and carboniferous periods may ultimately be traced to the condition of the sun itself, especially to changes in the thickness of the absorbing atmosphere, which, as Langley has pointed out, must have a great influence in the matter. Since the Christian era, however, it is certain that the amount of heat annually received from the sun has remained practically unchanged. This is inferred from the distribution of plants and animals, which is still sub- stantially the same as in the days of Pliny. 353. Maintenance of the Solar Heat. — The question at once arises, if the sun is sending off such an enormous quantity of heat annually, how is it that it does not grow cold ? (a) The sun's heat cannot be kept up by combustion. As has been said before, it would have burned out long ago, even if made of solid coal burning in oxygen. (6) Nor can it be simply a heated body cooling down. Huge as it is, an easy calculation shows that its temperature must have fallen greatly within the last 2000 years by such a loss of heat, even if it had a specific heat higher than that of any known substance. As matters stand at present, the available theories seem to be reduced to two, — that of Mayer, which ascribes the solar heat to the energy of meteoric matter falling on the sun ; and that of Helm- holtz, who finds the cause in a slow contraction of the sun's diameter. 354. Meteoric Theory of Sun's Heat. — The first is based on the fact that when a moving body is stopped, its mass-energy becomes molecular energy, and appears mainly as heat. The amount of heat developed in such a case is given by the formula 8339' in which Q is the number of calories of heat produced, M the mass of the moving body in kilograms, and V its velocity in metres per second ; the denominator is the " mechanical equivalent of heat" Physics, p. 159) multiplied by 2 g expressed in metres; i.e., 425 x 2x9.81. Now, the velocity of a body coming from any considerable distance and falling iuto the sun can be shown to be about 380 miles per second, or more than 610 kilometres. A body weighing one kilogram OBJECTIONS TO METEORIC THEORY OP SUN's HEAT. 223 would therefore, on striking the sun with this velocity, produce about 45,000000 calories of heat, [" (filOOOO) 2 "] [_ 8339 J This is 6000 times more than could be produced by burning it, even if it were coal or solidified hydrogen burning in pure oxygen. Now, as meteoric matter is continually falling upon the earth, it must be also falling upon the sun, and in vastly greater quantities, and an easy calculation shows that a quantity of meteoric matter equal to ^ of the earth's mass striking the sun's surface annually with the velocity of 600 kilometres per second would account for its whole radiation. 355. Objections to Meteoric Theory of Sun's Heat. — There can be no question that a certain fraction of the sun's heat is obtained in this way, but it is very improbable that this fraction is a large one ; indeed, it is hardly possible that it can be as much as one per cent of the whole. (1) The annual fall on the sun's surface of such a quantity of meteoric matter implies the presence near the sun of a vastly greater mass ; for, as we shall see hereafter, only a few of the meteors that approach the sun from outer space would strike the surface: most of them would act like the comets aud swing around it without touching. Now, if there were any considerable quantity of such matter near the sun, there would result dis- turbances in the motions of the planets Mercury and Venus, such as obser- vation does not reveal. (2) Professor Peirce has shown further that if the heat of the sun were produced in this way, the earth ought to receive from the meteors that strike her surface about half as much heat as she gets from the sun. Now the quantity of meteoric matter which would have to fall upon the earth to fur- nish us daily half as much heat as we receive from the sun, would amount to nearly fifty tons for each square mile. It is not likely that we actually get miwoom of that amount. It is difficult to determine the amount of heat which the earth actually does receive from meteors, but all observations indicate that the quantity is extremely small. The writer has estimated it, from the best data attainable, as less in a year than we get from the sun in a second. 356. Helmholtz's Theory of Solar Contraction. — We seem to be shut up to the theory of Helmholtz, now almost universally accepted ; namely, that the heat necessary to maintain the sun's radiation is principally supplied by the slow contraction of Us bulk, aided, however, 224 THE SUN. by the accompanying liquefaction and solidification of portions of its gaseous mass. When a body falls through a certain distance, gradu- ally, against resistance, and then comes to rest, the same total amount of heat is produced as if it had fallen freely, and been stopped instantly. If, then, the sun does contract, heat is necessarily produced by the process, and that in enormous quantity, since the attracting force at the solar surface is more than twenty-seven times as great as terres- trial gravity, and the contracting mass is immense. In this pro- cess of contraction each particle at the surface moves inward by an amount equal to the diminution of the sun's radius : a particle below the surface moves less and under a diminished gravitating force ; but every particle in the whole mass, excepting only that at the exact centre of the globe, contributes something to the evolution of heat. In order to calculate the precise amount of heat evolved by a given shrinkage it would be necessary to know the law of increase of the sun's density from the surface to the centre ; but Helmholtz has shown that under the most unfavorable conditions a contraction in the sun's diameter of about two hundred and fifty feet a year (125 feet in the sun's radius) would account for the whole annual output of heat. This contraction is so slow that it would be quite imperceptible to observation. It would require more than 9000 years to reduce the sun's diameter by a single second of arc ; and nothing much less would be certainly detectable by our measurements. If the con- traction is more rapid than this, the mean temperature of the sun must be actually rising, notwithstanding the amount of heat it is losing. Long observation alone can determine whether this is really the case or not. 357. Lane's Law. — It is a remarkable fact, first demonstrated by Lane of Washington, in 1870, that a gaseous sphere, losing heat by radiation and contracting under its own gravity, must rise in temperature and actually grow hotter, until it ceases to be a "perfect gas," either by beginning to liquefy, or by reaching a density at which the laws of perfect gases no longer hold. The kinetic energy developed by the shrinkage of a gaseous mass is more than sufficient to replace the loss of heat which caused the shrink- age. In the case of a solid or liquid mass this is not so. The shrinkage of such a mass contracting under its own gravity on account of the loss of heat is never sufficient to make good the loss ; but the temperature falls and the body cools. At present it appears that in the sun the relative proportions oi true gases and liquids are such as to keep the temperature nearly stationary, the liquid portions of the sun being of course the little drops which are supposed to constitute the clouds of the photosphere. FUTURE DURATION OP THE SUN. 225 358. Future Duration of the Sun. — If this shrinkage theory of the solar heat is correct (and there is every reason to accept it), it follows that in time the sun's heat must come to an end, and, lookino- back- wards, we see that there must have been a beginning. "We have not sufficient data to enable us to calculate the future duration of the sun with exactness, though an approximate estimate can be made. According to Newcomb, if the sun maintains its present radiation, it will have shrunk to half its present diameter in about 5,000000 years at the longest. Since when reduced to this size it must be about eight times as dense as now, it can hardly then continue to be mainly gaseous, and its temperature must begin to fall. Newcomb's conclusion, therefore, is that it is not likely that the sun can continue to give sufficient heat to support such life on the earth as we are now acquainted with, for 10,000000 vears from the present time. 359. Age of the Sun. — As to the past of the solar history on this hypothesis, we can be a little more definite. It is only necessary to know the present amount of radiation, and the mass of the sun, to com- pute how long the solar fire can have been maintained at its present intensity b}- the processes of condensation. No conclusion of geom- etry is more certain than this, — that the contraction of the sun to its present size, from a diameter even many times greater than Nep- tune's orbit, would have furnished about 18,000000 times as much heat as the sun now supplies in a year, and therefore that the sun cannot have been emitting heat at the present rate for more than 18,000000 years, if its heat has really been generated in this manner. But of course this conclusion as to the possible past duration of the solar system rests upon the assumption that the sun has derived its heat solely in this way; and moreover, that it radiates heat equally in all directions in space, — assumptions which possibly further investigations may not confirm. 360. Constitution of the Sun. — (a) As to the nature of the main . body or nucleus of the sun, we cannot be said to have certain knowl- edge. It is probably gaseous, this being indicated by its low mean density and its high temperature — ■ enormously high even at the sur- face, where it is coolest. At the same time the gaseous matter at the nucleus must be in a very different state from gases as we commonly know them in our laboratories, on account of the intense heat and the extreme condensation by the enormous force of solar gravity. The central mass, while still strictly gaseous, because observing the three 226 THE SUN. physical laws of Boyle, Daltou, and Gay Lussac, which characterize gases, would be denser than water, and viscous ; probably something like tar or pitch in consistency. 1 While this doctrine of the gaseous constitution of the sun is gener- ally assented to, there are still some who are disposed to consider the great mass of the sun as liquid. 361. (&) The photosphere is probably a shell of incandescent clouds, formed by the condensation of the vapors which are exposed to the cold of space. 362. (c) The photospheric clouds float in an atmosphere contain- ing, still uncondensed, a considerable quantity of the same vapors out of which they themselves have been formed, just as in our own atmosphere the air around a cloud is still saturated with water vapor. This vapor-laden atmosphere, probably comparatively shallow, consti- tutes the reversing layer, and by its selective absorption produces the dark lines of the solar spectrum, while by its general absorption it probably produces the darkening at the limb of the sun. But it will be remembered that Mr. Lockyer and others are disposed to question the existence of any such shallow absorbing stratum, considering that the absorption takes place in all regions of the solar atmosphere even to a great elevation. 363. (d) The chromosphere and prominences are composed of the permanent gases, mainly hydrogen and helium, which are min- gled with the vapors of the reversing stratum in the region near the photosphere, but usually rise to far greater elevations than do the vapors. The appearances are for the most part as if the chromo- sphere was formed of jets of heated hydrogen ascending through the interspaces between the photospheric clouds, like flames playing over a coal fire. 1 The law of Dalton (Physics, p. 181) is, that any number of different gases and vapors tend to distribute themselves throughout the space which they occupy in common, each as if the others were absent. The law of Boyle or Mariotte (Physics, p. 110) is, that at any given temperature the volume of any given amount of gas varies inversely with the pressure; i.e., pv=p'v'. The law of Gay Lussac (Physics, p. 185) is, that a gas under constant pressure expands in volume uniformly under uniform increment of temperature, so that V, = V (1 + at). This is not true of vapors in presence of the liquids from which they have been evaporated ; for instance, of steam in a boiler. CONSTITUTION OT 1 THE SUN. 227 364. (e) The corona also rests on the photosphere, and the peculiar green line of its spectrum (Art. 329) is brightest just at the surface of the photosphere, in the reversing stratum and in the chromosphere itself; but the corona extends to a far greater elevation than even the prominences ever reach, and seems to be not wholly gaseous, but to contain, besides the hydrogen and the mysterious " corouium," dust and fog of some sort, perhaps meteoric. Many of its phenom- ena are as yet unexplained, and since it can only be observed during the brief moments of total solar eclipses, progress in its study is necessarily slow. 364*. Note to Article 344. The total heat received by the earth from the sun in any given time is that intercepted by its diametrical cross-section, i.e, by the area of one of its great circles kept always perpendicular to the sun's rays. The quantity of ice which would be melted annually on this circular plane by the solar rays would be a sheet having a thickness of 166.5 metres or 546 feet (19°™ x 24 X 365J = 166.5""")- The thickness of the ice which could be melted in a year on a narrow 546" equatorial belt would be Z-T-, or 174", since such a belt intercepts the rays ■k that would otherwise fall on a diametrical strip of the same width upon the circular plane. If the sun's heat were uniformly distributed over the earth's whole surface, which equals four great circles, (4tt.R 2 ), it could melt a shell having a thick- ness of — , or 136'". 4 It is true that at the sea-level the solar-constaut is much diminished by atmospheric absorption ; and probably does not exceed fifteen calories per minute directly received from the sun's rays. But a large part of the solar heat absorbed by the atmosphere reaches the earth's surface indirectly, so that it must not be considered as lost to the earth, because not directly measurable by the actinometer. 228 ECLIPSES. CHAPTER X. ECLIPSES : FORM AND DIMENSIONS OF SHADOWS. LUNAE ECLIPSES. — SOLAR ECLIPSES. — TOTAL, ANNULAR, AND PAR- TIAL. — ECLIPTIC LIMITS AND NUMBER OF ECLIPSES IN A YEAR. — THE SAROS. — OCCULTATIONS. 365. The word eclipse (Greek IkXei^is) is strictly a medical term, meaning a faint or swoon. Astronomically it is applied to the dark- ening of a heavenly body, especially of the sun or moon, though some of the satellites of other planets besides the earth are also "eclipsed" from time to time. An eclipse of the moon is caused by its passage through the shadow of the earth ; an eclipse of the sun, by the interposition of the moon between the sun and the ob- server, or, what comes to the same thiug, by the passage of the moon's shadow over the observer. 366. Shadows. — If interplanetary space were slightly dusty, we should see, accompanying the earth and moon and each of the planets, a long black shadow projecting behind it and travelling with it. Geometrically speaking, this shadow of a body, the earth for instance, is a solid — not a surface. It is the space from which sunlight is excluded. If we regard the sun and other heavenly bodies as. truly spherical, these shadows are cones with their axes in the line joining the centres of the sun and the shadow-casting body, the point being always directed away from the sun, because the sun is always the larger of the two. 367. Dimensions of the Earth's Shadow. —The length of the shadow is easily found. In Fig. 124 we have from the similar triangles OED and ECa, OD : Ea: : OE : EG or I. OD is the dif- ference between the radii of the sun and the earth, = _R — r. Ea = r, and OE is the distance of the earth from the sun = A. / r \ 1 Hence i=Ax B — rJ 108.5 (The fractional factor is constant, since the radii of the sun and PENUMBRA. 229 earth are fixed quantities. Substituting the values of the radii, we find it to be 555.) This gives 857,200 miles for the length of the earth's shadow when A has its mean value of 93,000000 miles, re- M a__-— -""""" -—~"~~~ M' -~~ ., Fig. 124 — Dimensions of the Earth's Shadow. garding the earth as a perfect sphere and taking its mean radius. This length varies about 14,000 miles on each side of the mean as the earth changes its distance from the sun. The semi-angle of the cone (the angle ECb, or ECB in the figure) is found as follows. Since OEB is exterior to the triangle BEC, OEB=EBC+BCE, or BCE = OEB - EBC. Now, OEB is the sun's apparent semi-diameter as seen from the earth, and EBC is the earth's semi-diameter as seen from the sun, which is the same thing as the sun's horizontal parallax (Art. 83). Putting S for the sun's semi-diameter, and p for its parallax, we have — Semi-angle at O = S — p. 1 From the cone aCb all sunlight is excluded, or would be were it not for the fact that the atmosphere of the earth by its refraction bends some of the rays into this shadow. The effect is to make the shadow a little larger in diameter, but less perfectly dark. 368. Penumbra. — If we draw the lines Ba and Ab, crossing at C between the earth and the sun, they will bound the penumbra. Within this space a part, but not the whole, of the sunlight is cut off : an observer outside of the shadow, but within this cone-frustum, 1 Also, 1= given above. sin (5— p) , an expression sometimes more convenient than the one 230 ECLIPSES. which tapers towards the sun, would see the earth as a black body encroaching on the sun's disc. The semi-angle of the penumbra EG'a is easily shown to be S +p. 369. Although geometrically the boundaries of the shadow and penumbra are perfectly definite, they are not so optically. If a screen were placed at M (Fig. 124) perpendicular to the axis of the shadow, no sharply defined lines would mark the boundaries of either shadow or penumbra; near the edge of the shadow, the penumbra would be very nearly as dark as the shadow itself, only a mere speck of the sun being visible there ; and at the outer limit of the penumbra the shading would be still more gradual. 370. Eclipses of the Moon. — The axis of the earth's shadow is always directed to a point exactly opposite the sun. If, then, at the time of full moon, the moon happens to be near the ecliptic (that is, not far from one of the nodes of her orbit) , she will pass into the shadow and be eclipsed. Since, however, the moon's orbit is inclined about five and one-fourth degrees to the plane of the ecliptic, this does not happen very often (seldom more than twice a year) . Ordi- narily the moon passes north or south of the shadow without touch- ing it. Lunar eclipses are of two kinds, — partial and total : total when the moon passes into the shadow completely ; partial when she goes so far to the north or south of the centre of the shadow that only a portion of her disc is obscured. We may also have a " penumbral eclipse " when she passes merely through the penumbra without touching the shadow. In this case, however, the loss of light is so gradual and so slight, unless she almost grazes the shadow, that an observer would notice nothing unusual. 371. Size of the Earth's Shadow at the Point where the Moon crosses it. — Since EG in Fig. 125 is 857,000 miles, and the distance of the moon from the earth is on the average about 239,000 miles, CM must be 618,000 miles, and MN, the semi-diameter of the shadow at this point, will be f-^-f of the earth's radius. This gives MN= 2854 miles, and makes the whole diameter of the shadow a little over 5700 miles, about two and two-thirds times the diameter of the moon. But this quantity varies considerably. The shadow is sometimes more than three times as large as the moon, sometimes hardly more than twice its size. DURATION OF LUNAR ECLIPSE. 231 372. We may reach the same result in another way. Considering the triangle ECN, Fig. 125, we have the angular semi-diameter of the cross- section of the shadow where the moon passes through it, as seen from the earth, represented by MEN. But whence ENa = MEN + ECN; MEN = ENa- -ECN. Now ENa is the semi-diameter of the earth as seen from the moon ; that is, it is the moon's horizontal parallax, for which write P. Hence, substituting for ECN its value £ — p, we get MEN=P+p-S. MEN is called " the radius of the shadow." The mean value of P is 57' 2" ; of p, 8".8 ; and of S, 16' 2", which makes the mean value of MEN= 41' 9". Fig. 125. — Diameter of Earth's Shadow where the Moon crosses it. The mean value of the moon's apparent semi-diameter is 15' 40", the ratio between the semi-diameter of the moon and the radius of the shadow being about 2|, as before. In computing a lunar eclipse, this angular value for the " radius of the shadow," as it is called, is more convenient than its value in miles. It is customary to increase it by about -fa part in order to allow for the effect of the earth's atmosphere, the value ordinarily used being |J (P-f p— S). Some computers, however, use fj, and others |f. On account of the indis- tinctness of the edge of the shadow it is not easy to determine what precise value ought to be employed. 373. Duration of a Lunar Eclipse. — When central, a total eclipse of the moon may, all things favoring, continue total for about two hours, the interval from the first contact to the last being about two hours more. This depends upon the fact that the moon's hourly motion is nearly equal to its own diameter. The whole interval from first contact to last is the time occupied by the moon in moving from 232 ECLIPSES. a to d (Fig. 126). The totality lasts while it moves from b to c. The duration of a non-central eclipse varies, of course, according to the part of the shadow through which the moon passes. Fig. 126. — Duration of a Lunar Eclipse. 374. Lunar Ecliptic Limit. — The lunar ecliptic limit is the great- est distance from the node of the moon's orbit at which the sun can be consistently with having an eclipse. This limit depends upon the inclination of the moon's orbit, which varies a little, and also upon the radius of the shadow at the time of the eclipse and the moon's apparent semi-diameter, which quantities are still more variable. Hence we recognize two limits, the major and minor. If the dis- tance of the sun from the node at the time of full moon exceeds the major limit, an eclipse is impossible ; if it is less thau the minor, an eclipse is inevitable. The major limit is found to be 12° 15' ; the minor, 9° 30'. Since the sun passes over an arc of 12° 15' in less than thirteen days, it follows that an eclipse of the moon cannot possibly take place more than thirteen days before or after the time when the sun crosses the node. 375. In Fig. 127 let NE be the ecliptic, and NM the orbit of the moon, the point N being the node, and the angle at N the inclination of the moon's Fig. 127. — Lunar Ecliptic Limit. orbit. E is the centre of the earth's shadow. The sun, of course, is directly opposite, and its distance from the opposite node is equal to EN. M is the PHENOMENA OP A TOTAL LUNAR ECLIPSE. 233 centre of the moon. Call the semi-diameter of the moon S' ; then EM (the greatest possible distance between E and M which permits an eclipse) equals the sum of the semi-diameters of the moon and shadow, or S'+ (P +p — S), and the corresponding ecliptic limit EN is found by solving the spherical triangle MNE, having given ME and the angle at N, which is about 5J°. We must also know one other angle, and with sufficient approximation for such purposes we may regard the angle at M as a right angle. The solution will give the value of the ecliptic limit by assigning the proper values to the quantities involved. The limit is always very nearly eleven times EM, be- cause the inclination of the moon's orbit is nearly ^ of a " radian." 376. Phenomena of a Total Lunar Eclipse. — Half an hour or so before the moon reaches the shadow its eastern limb begins to be sensibly darkened, and the edge of the shadow itself, when it is first reached, looks nearly black by contrast with the bright parts of the moon's surface. To the naked eye the outline of the shadow appears reasonably sharp ; but with even a small telescope it is found to be indefinite and hazy, and with a large instrument and high magnifying power it becomes entirely indistinguishable. It is impossible to de- Fia. 128. — Light bent into Earth's Shadow by Refraction. termine the exact moment when the edge of the shadow reaches any particular point on the moon within half a minute or so. After the moon has wholly entered the shadow her disc is usually still distinctly visible, illuminated with a dull, copper-colored light, which is sunlight, deflected around the earth into the shadow by the refraction of our own atmosphere, or rather by that portion of our atmosphere which lies within ten or fifteen miles of the earth's sur- face. Since the ordinary horizontal refraction is 34', it follows that light which just grazes the earth's surface will be bent inwards by twice that amount, or 1° 8'. Now, the maximum "radius of the shadow" is only 1° 2'. In an extreme case, therefore, even when the moon is exactly central in the largest possible shadow, it receives some sunlight coming around the edge of the earth, as shown by Fig. 128. To an observer stationed on the moon, the disc of the earth would appear to be surrounded by a narrow ring of brilliant sunshine, colored with sunset hues by the same vapors which tinge 234 ECLIPSES. terrestrial sunsets, but acting with double power because the light has traversed a double thickness of our air. If the weather hap- pens to be clear at this portion of the earth (upon its rim as seen from the moon), the quantity of light transmitted through the at- mosphere is very considerable, and the moon is strongly illumi- nated. If, on the other hand, the weather happens to be stormy in this region, the clouds cut off nearly all the light. In the lunar eclipse of 1884 the moon was absolutely invisible to the naked eye, a very unusual circumstance on such an occasion. At the eclipse of Jan. 28, 1888, Pickering found that the photographic power of the centrally eclipsed moon was about t 400 000 °' that of the moon when uneclipsed. 377. Uses made of Lunar Eclipses. — In astronomical importance a lunar eclipse cannot be at all compared with a solar eclipse. It has its uses, however, a. Many dates in chronology are fixed by reference to certain lunar eclipses. For instance, the date of the Christian era is determined by a lunar eclipse which happened upon the night before Herod died. b. Before better methods were devised, lunar eclipses were made use of to some extent in determining the longitude. Unfortunately, as has been said (Art. 119), it is impossible to note the critical instants with any degree of accuracy, on account of the indefiniteness of the earth's shadow, c. The study of the spectrum of the eclipsed moon gives us some data as to the constitution of our own atmosphere. We are thus enabled to examine light which has passed through a greater thickness of air than is obtainable in any other way. d. The study of the heat radiated by the moon during the different phases of an eclipse gives us some important information as to the absorbs ing power and temperature of its surface. Observations have been made at Lord Rosse's observatory of all the recent lunar eclipses, with this end in view. e. Finally, at the time when the moon is eclipsed, it is possible to ob- serve its passage over small stars which cannot be seen at all when near the moon except at such a time. Observations of these star occultations made fit different parts of the earth, furnish the best possible data for computing the dimensions of the moon, its parallax, and for determining its precise position in its orbit at the time of observation. The eclipses of the last few years have been very carefully observed in this way by concert between the different leading observatories. 378. Computation of a Lunar Eclipse. — Since all the phases of a lunar eclipse are seen everywhere at the same absolute instant wherever the moon is above the horizon, it follows that a single com- putation giving the Greenwich times of the different phenomena is all that is needed, and can be made and published once for all. Each observer has merely to correct the predicted time by simply adding ECLIPSES OP THE SUN. 235 or subtracting his longitude from Greenwich in order to get the true local time. The computation is very simple, hut lies rather beyond the scope of this work. ECLIPSES OF THE SUN. 379. Dimensions of the Moon's Shadow.— By the same method as that used for the shadow of the earth (merely substituting in the formulae the radius of the moon for that of the earth) , we find that the length of the moon's shadow at any time is -L_ of its distance from the snn, and at new moon averages 232,150 miles. It varies not quite 4000 miles each way, and so ranges from 236,050 miles to 228,300.. The semi-angle of the moon's shadow is practically equal to the semi-diameter of the sun seen at the earth, or very nearly 16'. 380. The Moon's Shadow on the Earth's Surface. — Since the mean length of the shadow is less than the mean distance of the moon from the earth (which is 238,800 miles), it is obvious that on the average it will not reach to the earth. On account of the C~~ »» >to Swn. Fig. 129. — The Moon's Shadow on the Earth. eccentricity of the moon's qrbit however, our satellite is much of the time considerably nearer than this mean distance, and may come within 221,600 miles from the earth's centre, or about 217,650 miles from its surface. The shadow, also, under favorable circum- stances, may have a length of 236,050 miles. Its point may there- fore at times extend nearly 18,400 miles beyond the earth's surface. The cross-section of the shadow where the earth's surface cuts it (at o in Fig. 129) will then be 167 miles. This is the largest value possible. Of course, if the shadow strikes obliquely on the surface of the earth, as it must except when the moon is in the zenith, the shadow spot will be oval instead of circular, and the length of the oval along the earth's surface mav much exceed the true cross-section of the shadow. 381. The "Negative " Shadow. — Since the distance of the moou may be as great as 252,970 miles from the earth's centre, or nearly 236 ECLIPSES. 249,000 miles from its surface, while the shadow may be as short as 228,300 miles, we may have the state of things indicated by placing the earth at B in the figure. The vertex of the shadow, V, will then fall 20,700 miles short of the surface, and the cross-section of the " shadow produced" will have a diameter of 196 miles where the earth's surface cuts it. When the shadow falls near the edge of the earth, this cross-section may be as great as 230 miles. The shadow- spot which is formed by the intersection of the produced shadow- cone with the earth's surface is sometimes called the negative shadow, because in calculating an eclipse its radius comes out from the formulae as a minus quantity in case the shadow does not reach the earth. 382. Total and Annular Eclipses. — To an observer within the true shadow cone, that is, between Vand the moon in Fig. 129, the sun will be totally eclipsed ; but an observer in the produced cone beyond V will see the moon projected on the sun, leaving an un- eclipsed ring around it. He will have what is called an annular eclipse. These annular eclipses are considerably more frequent than total eclipses — nearly in the ratio of three to two. 383. The Penumbra and Partial Eclipses. — The penumbra can easily be shown to have a diameter on the line QD (Fig. 129) of very nearly twice the moon's diameter. We may take it as having an aver- age diameter at this point of 4400 miles ; but as the earth is often be- C F Fig. 130. —Width of the Penumbra of the Moon's Shadow. yond V, its cross-section at the earth is sometimes as much as 4800 miles. An observer situated within the penumbra observes a partial eclipse : if he is near the shadow cone, the sun will be mostly covered by the moon ; but if near the outer limit of the penumbra, the moon will only slightly encroach on the sun's disc. While, therefore, total and annular eclipses are visible as such only by an observer within the narrow path traversed by the shadow-spot, the same eclipse will be visible as a partial one everywhere within 2000 miles on VELOCITY OF SHADOW AND DURATION OF ECLIPSES. 237 either side of the shadow path ; and the 2000 miles is to he reck- oned perpendicularly to the axis of the shadow. When, for instance, the penumbra falls, as shown in Fig. 130, the distance BO measured along the earth's surface will be over 3000 miles, although BF is only 2000. 384. Velocity of the Shadow and Duration of Eclipses. — The moon advances along its orbit very nearly 2100' miles an hour, and were it not for the earth's rotation this is the rate at which the shadow would pass the observer. The earth, however, is rotating towards the east in the same general direction as that in which the shadow moves, and .its surface moves at the rate of about 1040 Fig. 131. — Track of the Moon's Shadow, Eclipse of July, 1878. miles an hour at the equator. An observer, therefore, on the earth's equator, with the moon near the zenith, would be passed by the shadow with a speed of about 1060 miles per hour (2100 - 1040) ; and this is its slowest velocity, which is about equal to that of a cannon-ball. In higher latitudes, where the velocity of the earth's rotation is less, the relative speed of the shadow is higher ; and where the shadow falls very obliquely, as it does when an eclipse occurs near sunrise or sun- 238 • ECLIPSES. set, the advance of the shadow along the earth's surface may become exceedingly swift, — as great as 4000 or 5000 miles per hour. Fig. 131, which we owe to the courtesy of the publishers of Langley's " New Astronomy,'' shows the track of the moon's shadow during the eclipse of July 29, 1878. 385. Duration of an Eclipse. — A total eclipse of the sun observed at a station near the equator under the most favorable conditions pos- sible (the shadow-spot having its maximum diameter of 167 miles), may continue total for seven minutes and fifty -eigM seconds. In lati- tude 40° the duration of totality can barely equal six and one-quarter minutes. The greatest possible excess of the radius of the moon over that of the sun is only 1' 19". An annular eclipse may last for 12 m 24 s at the equator. The maximum width of the ring of the sun visible around the moon is 1' 37". In the observation of an eclipse four "contacts" are noted: the first when the edge of the moon first touches the edge of the sun ; the second, when the eclipse becomes total or annular; the third, at the cessation of the total or annular phase ; and the fourth, when the moon finally leaves the disc of the sun. From the first contact to the fourth the duration may be a little over two hours. 386. The Solar Ecliptic Limits. — It is necessary, in order to have an eclipse of the sun, that the moon should encroach on the cone AOBD (Fig. 132), which envelops earth and sun. In this case the "true" angular distance between the centres of the sun and moon Fis. 132. — Solar Ecliptic Limits. — that is, their distance as seen from the centre of the earth — would be the angle ME 8 in the figure. This is made up of three angles : MEF, which equals the moon's semi-diameter, S 1 ; AES, the sun's semi-diameter, S ; and FEA. This latter angle is equal to the differ- PHENOMENA OF A SOLAR ECLIPSE. 239 ence between GFE and FAE. GFE is the moon's horizontal paral- lax (the semi-diameter of the earth seen from the moon) , and FAE or GAE is the sun's parallax. FEA, therefore, equals P—p; and the whole angle MES equals S + S' + P-p. This angle may range from 1° 34' 13" to 1° 24' 19", according to the changing distances ' of the sun and moon from the earth. The corresponding distances of the sun from the node, calculated in the same way as the lunar ecliptic limits (taking the maximum incli- nation of the moon's orbit as 5° 19' and the minimum as 4° 57', according to Neison), give 18° 31' and 15° 21' for the major and minor ecliptic limits. In order that an eclipse may be centred (total or annular) at any part of the earth, it is necessary that the moon should lie wholly inside the cone ACBD, as at M. In this case the angle WES will be S — 5" + P—p, and the corresponding major and minor central ecliptic limits come out 11° 50' and 9° 55'. 387. Phenomena of a Solar Eclipse. — There is nothing of special interest until the sun is mostly covered, though before that time the shadows cast by foliage begin to look peculiar. The light shining through every small interstice among the leaves, instead of forming a little circle on the earth, makes a little crescent — an image of the partly covered sun. Some ten minutes before totality the darkness begins to be felt, and the remaining light, coming as it does from the edge of the sun only, is much altered in quality, producing an effect very like that of a calcium light rather than sunshine. Animals are perplexed, and birds go to roost. The temperature falls a few degrees, and some- times dew appears. 1 We give herewith in "a table the different quantities which determine the dimensions of the shadows of the earth and moon, as well as the ecliptic limits and the duration of eclipses. Greatest. Least. Mean. 16' 18" 15' 45" 16' 02" Apparent semi-diameter of moon .... 16' 46" 14' 44" 15' 45' 7 9».05 8". 55 8".80 Horizontal parallax of the moon . . . . 61' 18" 53' 58" 57' 38" Inclination of moon's orbit .... 5° 19' 4° 57' 6° 8' 40" Sun's radius, 433,200 miles ; earth's (mean), 3956 ; moon's, 1081.5. 240 • ECLIPSES. In a few moments, if the observer is so situated that his view com- mands a distant western horizon, the moon's shadow is seen coming much like a heavy thunder-storm. It advances with almost terrifying swiftness until it envelops nim. For a moment the air appears to quiver, and on every white sur- face bands or '■'■fringes" alternately light and dark, appear. They are a few inches wide and from a foot to three feet apart, and on the whole seem to be parallel to the edge of the shadow. Probably they travel with the wind; but observations on this point are as yet hardly decisive. The phenomenon is not fully explained, but is probably due to irregular atmospheric refraction of the light coming from the indefinitely narrow strip of the sun's limb on the point of disappearing. 388. Appearance of the Corona and Prominences. — As soon as the shadow arrives, and sometimes a little before it, the corona and promi- nences become visible. The stars of the first three magnitudes make their appearance at the same time. The suddenness with which the darkness descends upon the ob- server is exceedingly striking ; the sun is so brilliant that even the small portion which remains visible up to within a very few seconds of the time of totality so dazzles the eye that it is not prepared for the sudden transition. In a few moments, however, the vision be- comes accustomed to the changed circumstances, and it is then found that the darkness is not really very intense. If the totality is of short duration, — that is, if the diameter of the moon exceeds that of the sun by less than a minute of arc, — the lower parts of the corona and chromosphere, which are very brilliant, give a light at least three or four times as great as that of the full moon. Since the shadow also in such a case is of small diameter, a large quantity of light is sent in from the surrounding air, where thirty or fort}' miles away the sun is still shining ; and what may seem remarkable, this intrusion of outside light is greatest when the sky is clouded. In such an eclipse there is not much difficulty in reading an ordinary watch-face. In an eclipse of long duration (say five or six minutes) it is much darker, and lanterns are necessary. 389. Observations of an Eclipse. — A total solar eclipse offers an opportunity of making an immense number of observations of great import- ance which are possible at no other time, besides certain others which can also be made during a partial eclipse. We mention (a) Times of the four contacts, and direction of the line joining the cusps during the partial phases. CALCULATION OF A SOLAR ECLIPSE. 241 These observations determine accurately the relative position of the sun and moon at the time, and so furnish the means for correcting the tables of their motion. (6) The search for intra-mercurial planets. It has been thought likely that there may be one or more planets between the orbit of Mercury and the sun, and during a total eclipse they would become visible, if ever. On the whole, however, the observations, so far made, negative the existence of any body of considerable size in this region, though in 1878, Professor Watson and Mr. Swift, it was thought, had discovered one, if not two, such planets, (c) Observations on the fringes, which have been described as show- ing themselves at the commencement of totality. Probably the phenomenon is merely atmospheric and of little importance, but it is not yet sufficiently understood, (d) Photometric measurements of the intensity of the light at different stages of the eclipse and during totality, (e) Telescopic observations of the details of the prominences and of the corona. (/) Spectroscopic observa- tions, both visual and photographic, upon the spectra of the lower atmos- phere of the sun, the prominences, and the corona, (jg) Observations with the polariscope upon the polarization of the light of the corona, for the purpose of determining the relation between the reflected and intrinsic light, and perhaps the size of the reflecting particles which are distributed through the corona, Qi) Photography, both of the partial phases and of the corona. 390. Calculation of a Solar Eclipse. — The calculation of a solar eclipse cannot be dealt with in any such summary way as that of a lunar eclipse, owing to the moon's parallax, which makes the times of contact and other phenomena different at every different station. The moon's apparent ,path in .ihe sky, relative to the centre of the sun, is not even a portion of a great circle, nor is it described with a uniform velocity. Moreover, since the phenomena of a solar eclipse admit of very accurate observations, it is necessary to take account of numerous little details which are of no import- ance in a lunar eclipse. Certain data for each solar eclipse hold good wherever the observer may be. These are calculated beforehand and published in the nautical alma- nacs ; and from them, with the knowledge of his geographical position, the observer can work out the results for his own station. But the calculations are somewhat complicated and lie beyond our scope. The reader is referred to any work on practical astronomy ; Chauvenet and Loomis treat the mat- ter very fully. 391. Number of Eclipses in a Year. — The least possible number is two, both central eclipses of the sun. The largest possible number is seven, Jive of the sun and two of the moon. The eclipses each year happen at two seasons (which may be called the "eclipse months"), half a year apart — about the times, of course, at which the sun in its annual path crosses the two nodes of the moon's orbit. If these nodes were stationary, the eclipse months would be always the same ; but 242 ECLIPSES. because the nodes retrograde around the ecliptic ouce in about nine- teen years, the eclipse months are continually changing. The time required by the sun in passing around from a node to the same node again is 346.62 days, which is sometimes called the " eclipse year." 392. Number of Lunar Eclipses. — Representing the ecliptic by a circle (Fig. 133) with the two opposite nodes A and a, it is easy to see first, that there can be but two lunar eclipses in a year (omitting for a moment one exceptional case) . The major lunar ecliptic limit is 12° 15' ; hence there is only a space of twice that amount, or 24° 30', between L and V, at each " node month," within which the occur- rence of a full moon might give a lunar eclipse. Now, in a syn- odic month the sun moves along the ecliptic 29° 6', while the node moves in the opposite direction 1° 31', giving the relative motion of the sun referred to the node equal to 30° 37'; i.e., the full- moon points on the circle would fall at a distance of 30° 37' from each other. Only one full moon, there- fore, can possibly occur within the lunar ecliptic limits each time that the sun passes the node. Since the minor ecliptic limit for the moon is only 9° 30', it may easily happen that neither of the full moons which occur nearest to the time when the sun is at the node will fall within the limit. There are accordingly many years which have no lunar eclipses. Three lunar eclipses, however, may possibly happen in one calendar year in the following way. Suppose the first eclipse occurs about Jan. 1, the sun passing the node about that time ; the second may then happen about June 25 at the other node, a. The first node, A, will run back during the year, so that the sun will encounter it again about Dec. 13 at A', and thus a third eclipse may occur in December of the same year. This actually occurred in 1787, the dates of the three lunar eclipses being Jan. 3, June 30, and Dec. 24. ,393. Number of Solar Eclipses. — Considering now solar eclipses, we find that there must inevitably be two. Twice the minor limit Ftg. 133. — Number of Eclipses Annually. FREQUENCY OF ECLIPSES. 243 (Art. 386) of a solar eclipse (15° 21') is 30° 42', which is more than the sun's whole motion in a month. One new moon, at least, there- fore, must fall within the limiting distance of the node, and two may do so, since in the figure, SS' is always greater than the distance between the points occupied by two successive new moons. If the two new moons in the two eclipse months happen to fall very near a node, the two full moons, a fortnight earlier and later, will both be very likely to fall outside the lunar limit. In that case the year will have only two eclipses, both solar and both central ; i.e., either total or annular. This was the case in 1886. Again, if in any year two full moons occur when the sun is very near the node, then since the major solar limit is 18° 31', it may happen, and often does, that there will be two partial solar eclipses, one a fortnight before, the other a fortnight after, each of the lunar eclipses, and so the year will have three eclipses in each eclipse month — six eclipses in all, two lunar and four solar. A fifth solar eclipse may also come in near the end of the year, if the node was passed about Jan. 15, in the same way that sometimes happens with a lunar eclipse : the year will then have seven eclipses. This was the case in 1823. The most usual number of eclipses is four or five. 394. Relative Frequency of Solar and Lunar. Eclipses. — Al- though, taking the whole earth into account, the solar eclipses are the most numerous, about in the proportion of four to three, it is not so with the eclipses visible at any given place. , A solar eclipse can be seen only from a limited portion of the globe, while a lunar eclipse is visible over considerably more than half the earth, either at the beginning or end, if not throughout its whole duration ; and this more than reverses the proportion between lunar and solar eclipses for any given station. 395. Recurrence of Eclipses, and the Saros. — It is not known how early it was discovered that eclipses recur at a regular interval of eighteen years and eleven and one-third days (ten and one-third days, if there happen to be five leap years in the interval) ; but the Chaldeans knew the period very well, and called it the Saros (which means "restitution" or " repetition"), and used it in predicting the recur- rence of these phenomena. It is a period of 223 synodic months, which is almost exactly equal to nineteen eclipse years. The eclipse year is 346 d .6201, and nineteen of them equal 6585 d .78, while 223 months equal 6585 d .32. 244 ECLIPSES. The difference is only T \ 6 T of a day (about 11 hours) iu which time the sun moves 28'. If, therefore, an eclipse should occur to-day at new moon, with the sun exactly at the node, then after 223 months (18 years 11 days) a new moon will occur again with the sun only 28' west of the node ; so that the circumstances of the first eclipse will be pretty nearly repeated. It would however occur about eight hours of longitude further west on the earth's surface, since the 223 months exceed the even 6585 days by ^npof a da y> or ? h 42™. 396. Number of Recurrences of a Given Eclipse. — It is usual to speak of eclipses recurring at this regular interval as " repetitions " of one and the same eclipse. A lunar eclipse is usually thus "repeated" 48 or 49 times. Beginning as a very small partial eclipse, with the sun about 12° east of the node, it will be a little larger at its next occurrence eighteen years later ; and after 13 or 14 repetitions the sun will have come so near the node that the eclipse will have become total. It will then be repeated as a total eclipse 22 or 23 times, after which it will become partial again with the sun west of the node, and after 13 more returns as a partial eclipse will finally dwindle away and disappear, having thus recurred regularly once in every 223 months during an interval of 865J years. The same thing happens with the solar eclipses, only since the solar ecliptic limit is larger than the lunar, a solar eclipse has from 68 to 75 re- turns, occupying some 1260 years. Of these about 25 are only partial eclipses, the sun being so near the ecliptic limit that the axis of the shadow does not reach the earth at all. The 45 eclipses in the middle of the period are central somewhere or other on the earth, about 18 of them being total, and about 27 annular. These numbers vary somewhat, however, in differ- ent cases. 397. It is to be noticed that the Saros exhibits not only a close commensurability of the synodic months with the eclipse years, but also with the nodical 1 and anomalistic months : 242 nodical months equal 6585.357 days ; 239 anomalistic months equal 6585.549 days. This last coincidence is important. The moon at the end of the Saros of 223 months not only returns very closely to its original position with respect to the san and the node, but also with respect to the line of apsides of its orbit. If it was at perigee originally, it will again be within five hours of perigee at the end of the Saros. If this were not so, the time of the eclipse might be displaced several hours 1 The nodical month is the time of the moon's revolution from one of its nodes to the same node again, and is equal to 27 d .21222 ; the anomalistic month is the time of revolution from perigee to perigee again, and equals 27 d .55460. See Arts. 454, 455. The nodical month is also called the draconitic month. NUMBER OF ECLIPSES IN A SINGLE SAROS. 245 by the perturbations of the moou's motion, to be considered later in Chap. XH. 398. Number of Eclipses in a Single Saros. — The total number is usually about seventy, varying two or three one way or the other, as new eclipses come in at the eastern limit and go out at the western. Of the 70, 29 are usually lunar and 41 solar ; and of the solar, 27 are centfal, 1 7 being annular and 10 total. (These numbers are necessarily only approximate.) It appears, therefore, that total solar eclipses, somewhere or other on the earth, are not very rare, there beino- about ten in eighteen years. Since, however, the shadow track averages less than 100 miles in width, each total eclipse is visible, as total, over only a very small fraction of the earth's whole surface — about ^ in the mean. This gives about one total eclipse in 360 years, in the long run, at any given station. The total solar eclipses visible in the United States during the present century are the following : — June 16, 1806, in New York and New England, duration 4£ minutes; Nov. 30, 1834, in Arkansas, Missouri, Alabama, and Georgia, duration 2 minutes ; July 18, 1860, in Washington Territory and Labrador, 3 minutes ; Aug. 7, 1869, in Iowa, Illinois, Kentucky, North Carolina, 2 J minutes ; July 29, 1878, in Wyoming, Colorado, Texas, 2 \ minutes; Jan. 11, 1880, in California, duration 32 seconds; Jan. 1, 1889, in California and Montana, 2£ minutes; May 27, 1900, from Louisiana to Virginia, 2 minutes. 399. Occultations of Stars. — In theory, and in the method of com- putation, the occultation of a star is precisely like a solar eclipse, except that the shadow of the moon projected by a star is a cylinder instead of a cone, since, compared with the distance of the sun, that of a star is infinite : moreover, the star is a mere point, so that there is no sensible penumbra. In other words, a star has neither parallax nor semi-diameter, and these circumstances somewhat simplify the formulae. As the moon moves always towards the east, the disappearance of the star always takes place at the eastern limb, and the reappearance at the western. In the first half of the lunation the eastern limb is dark and invisible, and the star vanishes without warning. The suddenness with which it vanishes and reappears has already been referred to (Art. 255) as proof of the non-existence of a lunar atmos- phere. Observations of this sort determine the moon's place with great accuracy, and when corresponding observations are made at 246 ECLIPSES. different places, thej- supply one of the best possible means of deter- mining their difference of longitude. In some cases observers have reported that a star, instead of disappearing instantaneously when struck by the moon's limb (faintly visible by earth- shine), has appeared to cling to the limb for a second or two before vanish- ing, and in a few instances they have even reported it as having reappeared and disappeared a second time, as if it had been for a moment visible through a rift in the moon's crust. Some of these anomalous phenomena have been explained by the subsequent discovery that the star was double, or had a faint companion. EQUABLE DESCRIPTION OF AREAS. 247 CHAPTER XL CENTRAL FORCES : EQUABLE DESCRIPTION OF AREAS. — AREAL, LINEAR, AND ANGULAR VELOCITIES. — KEPLER'S LAWS AND INFERENCES FROM THEM. — GRAVITATION DEMONSTRATED BY THE MOON'S MOTION. — CONIC SECTIONS AS ORBITS. — THE PROBLEM OF TWO BODIES. — THE "VELOCITY FROM INFINITY," AND ITS RELATION TO THE SPECIES OF ORBIT DESCRIBED BY A BODY MOVING UNDER GRAVITATION. — THE INTENSITY OF GRAVITATION. 400. A moving body left to itself, according to Newton's first law of motion (Physics, p. 27), moves on forever in a straight line with a uniform velocity. If we find a body so moving, we may, therefore, infer that it is either acted on by no force whatever, or, if forces are acting upon it, that they exactly balance each other. It has been customary with some writers to speak of a body thus moving " uniformly in a straight line " as actuated by a " projectile force" a very unfortunate expression, which is a survival of the Aristotelian idea that rest is more " natural " to matter than motion, and that when a body moves, some force must operate to keep it moving. The mere uniform rectilinear motion of a material mass in empty space implies no physical cause, and demands no explanation. Change of motion, either in speed or in direction — this alone implies force in operation, 401. If a bodj' moves in a straight line, with swiftness either increasing or decreasing, we infer a force acting exactly in the line of mo- tion, and accelerating or retarding it. If it moves in a curve, we know that some force is acting athwart the line of motion. If the velocity in the curve increases, we know that the direction of the force that acts is forward, like ab (Fig. 134), making an angle of less than 90° with the " line of motion" a t (the tangent to the path of the body) ; and vice versa, if the motion of the body is retarded. Fig. 134. — Curvature of an Onbit. 248 CENTRAL FORCES. If the speed does not alter at all, we know that the force must act along the line ac, exactly perpendicular to the line of motion. Here, also, we find many writers, the older ones especially, bringing in the " projectile force," and saying that when a body moves in a curve it does so under the action of two forces, one a force that draws it sideways, the other the " projectile force " directed along its path. We repeat ; this " pro- jectile force " has no present existence or meaning in the problem. Such a force may have put the body in motion long ago, but its function has ceased, and now we have only to do with the action of one single force, — the deflecting force, which alters the direction of the body's motion. Of course it is not intended to deny that the deflecting force may itself be the resultant of any number of forces all acting together ; but a single force act- ing athwart a body's line of motion is sufficient to cause it to describe a curvilinear orbit, and from such an orbit we can only infer the necessary existence of one such force. 402. Description of Areas. — (a) If a body is moving uniformly on ~ a straight line, and if we connect the points A,B, O, etc., Fig. 135, which it occupies at the end of successive units of time with any point what- ever, as 0, we shall have a series of triangles, AOB, etc., which will all be equal; since their bases AB, BO, etc., are equal and on the same straight line, and they have a common vertex at 0. Calling the line from A to its radius vector, and the " cen- fig. 135. tre," we may say, therefore, that Description of Areas in Uniform Motion. when a body is moving Undis- turbed by any force whatever, its radius vector, from any centre arbitrarily chosen, describes equal areas in equal times around that centre. The area enclosed in the triangle described by the radius vector in a unit of time is called the body's " areal (or "areolar") velocity," and in this case is constant. 403. (6) Moreover, any impulse in the line of the radius vector, either towards or from the centre, leaves unchanged both the plane of the body's motion and its areal velocity. Suppose a body moving uniformly on the line AG (Fig. 136) with such a velocity that it describes AB, BO, etc., iu successive units of time ; then, by the preceding section, the areal velocity will be con- DESCRIPTION 0¥ AREAS. 249 stant, and measured by the area of any one of the equal triangles AOB, BOG, or COL. Suppose, now, that at the body receives an " impulse " directed along the radius vector towards 0— a blow, for instance, which by itself would make the body describe OK in a unit of time. The body will now take a new path, which will carry it to the point D, determined by constructing the " parallelogram of Fig. 136. — Description of Areas under an Impulse directed towards a Centre. motions" CKDL, and thus combining the new motion OK with the former motion OL, according to Newton's second law (Physics, p. 27). The new areal velocity, measured by the triangle OCD, will be the same as before, as is easily shown. Triangle BOC= triangle COL, because BC = OL, and is their common vertex. Also, triangle COL =± triangle COD, because they have the common base OC, and their summits L and D are on a line which was drawn parallel to this base in constructing the parallelogram of motions. Hence triangle BOC= triangle OOD, and the areal velocity remains unchanged. . Also, as may be seen by following out the same reasoning with OK and CD', the same result would hold true if the impulse had been directed away from instead of towards it. 404. This result depends entirely on the fact that the impulse CK or CK' was exactly along the radius vector CO. If it had not been so, then in 250 CENTRAL FORCES. constructing the parallelogram of motions to find the points D and Tf, we should have had to draw LD or LD' not parallel to CO, and the two triangles BOC and COD would necessarily have been unequal. COD would be greater than BOC if CK were directed ahead of the radius vector, and less if behind it. As regards the plane of motion, the point D is on the plane OOL, because LD was drawn through L parallel to 00. OOL is a part of the plane which contains the triangles BOC and AOB, and hence OGD also lies in the same plane. 405. (c) From this obviously follows the important general prop- osition that wlien a body is moving under the action of a force always directed towards or from a fixed centre, the radius vector will describe equal areas in equal times; and the path of the body will all lie in one plane. Such a force constantly acting is simply equivalent to an indefinite number of separate impulses. Now if no single impulse directed along the radius vector can alter the areal velocity or plane of motion, neither can a succession of them. Hence the proposition follows. In case of a continuously acting force the orbit, however, will become a curve instead of being a broken line. Observe that this proposition remains true whether the force is attractive or repulsive, and that it is independent of the law of the force ; that is, the force may vary directly with the distance, or in- versely as the square of the distance, or as the logarithm of it, or in any conceivable way ; it may even be discontinuous, acting only at intervals and ceasing between times : and still the law holds good. 406. Conversely, if a body moves in this way, describing equal areas in equal times around a point, it is easily shown that all the forces acting upon the body must be directed toward that point. We, however, leave the demonstration to the student. Since the earth moves very nearly in this way in its orbit around the sun, we conclude that the only force of any consequence acting upon the earth is directed towards the sun. We sa}', " of any con- sequence," because „there are other small forces which do slightly modify the earth's motion, and prevent it from exactly fulfilling the law of areas. As a direct consequence of the law of equal areas we have certain laws with respect to the linear and angular velocities of a body mov- ing under the action of a central force. LAW OF LINEAR VELOCITY. 251 407. Law of Linear Velocity. — Suppose a body moving under the action of a force always directed towards S (Fig. 137), and let AB be a portion of its path which it de- scribes in a second. Draw the tangent Bb. Regarding the sector ASB as a triangle (which it will be, sensibly, since the curvature of the path in one second will be very small) the area of this tri-' angle will be l(ABxSb). Now AB, the distance travelled in a second, is the linear- velocity of the body (called linear because it is measured with the Fia. 137. Same units as any Other line; i.e., in Linear and Angular Velocities. miles or in feet per second) , and Sb is the distance from the centre of force to the " line of motion," as the tan- gent Bb is called. For Sb, p is usually written ; hence in every 2A part of the same orbit, F"(the velocity in miles per second_) = — , and is inversely proportional to p. If p were to become zero, V would become infinite, unless A were zero also. 408. Law of Angular Velocity. — • Referring again to the same figure, the area of ASB is equal to \ (AS X BS X sin ASB) , or A = %r 1 r 2 smw. If we draw r to the middle point of AB, then r x r 2 = t 9 , nearly, since in a second of time the distance would not change perceptibly as compared with its whole length. =A, and w = — — . * r Now a is the angular velocity of the body ; that is, the number of "radians" which it describes in a second of time, as seen from S, while r is the radius vector. \ 409. . In every case, therefore, of motion under a central force, I. The Areal velocity (acres per second) is constant; II. Tlie Linear velocity (miles per second) varies inversely as the distance from the centre of force to the body's line of motion at the moment, which line of motion is the tangent to the orbit at the point where the body happens to be ; III. The Angular velocity (radians, or degrees, per second) varies inversely as the square of the distance of the body from the centre of force. 252 CENTRAL. FORCES. 410. Tne student will remember that it was found by observation that the sun's angular velocity varies as the square of its apparent diameter, and from this (Art. 186) the law of equal areas was inferred as a fact with respect to the earth's motion. Newton was the first to point out that a body moving under the action of a central force must necessarily observe this law of areas, and, con- versely, that a body thus observing the law of areas must neces- sarily be under the control of a central force.' 411 . Circular Motion. — In the case of a body moving in a circle under the action of a central force, the force must be constant, and (Physics, p. 62) is given by the formula /=T' (a) r in which r is the radius of the circle and Fthe velocity, while/ is the central force measured by the " acceleration " of the body towards the centre ; that is, by the number of units of velocity which the force would generate in the body in a second of time ; just as the force of gravity is expressed by writing, gr = 9.81 metres. For many purposes it is desirable to have an expression which shall substitute for V (a quantity not given directly by observation ) the time of revolution, t, which is so given. Since V equals the circumference of the circle divided by t, or — — , we have at once, by * b substituting this value for T^in equation (a), This, of course, is merely the equivalent of equation (o), but is often more convenient. KEPLER'S LAWS. 412. In 1607-1620 Kepler discovered as facts, without an expla- nation, three laws which govern the motions of the planets, — laws which still bear his name. He worked them out from a discussion of the observations which Tycho Brahe had made through many preceding years. The three laws are as follows : — I. The orbit of each planet is an ellipse, with the sun in one of its foci. II. The radius vector of each planet describes equal areas in equal times. KEPLER'S LAWS. 253 III. The "Harmonic law," so-called. The squares of the periods of the planets are proportional to the cubes of their mean distances from the sun; i.e., tf : -l. But by Kepler's third law t 3 :t 1 2 ::r a :r l s , whence «,» = * r ) ■ 1 ,.a Substitute this value of tf in the preceding equation ; we have ft ~i* t* ri ~ r 2 ' which is the law of inverse squares. 416. Conversely, the harmonic law is just as easily shown to be a necessary consequence of the law of gravitation in the case of circular orbits. From Art. 411, Eq. (6), we have CORRECTION OF KEPLER'S THIRD LAW. 255 also, from the law of gravitation, ' /= -j, M being the mass of the sun. A' Hence, equating the two values of /, ¥=1+% and fi=i£* r 2 i 2 M Similarly for another planet, 2 4^ Whence, t 2 : tf = r> : r, s . The demonstration for elliptical orbits is a little more complicated, involving the "law of areas." It is given in all works on Theoretical Astronomy, and maybe found in Loomis's "Elements" of Astronomy," p. 134. 417. Correction of Kepler's Third law. — The "harmonic law' as it stands is not exactly true, though the difference is too small to appear in the observations which Kepler made use of in its discovery. It would be exactly true if the planets were mere particles of matter ; but as a planet's mass is a sensible, though a very small fraction of the sun's mass, it comes into account. The planet Jupiter, for instance, attracts the sun as well as is attracted by it. If at the distance r Jupiter is drawn towards the sun by a force which would give it in a second an acceleration expressed by — (the r 2 sun's mass being M~), then the sun in the same time is accelerated towards Jupiter by the quantity — (m bein g the mass of Jupiter ) . The rate at which the two tend to approach each other is therefore expressed by — i — Hence, in discussing the motions of the planet Jupiter around the centre of the sun, instead of writing /=— simply, „ j- j. s M+m we must put /= — — ; r 2 but (in the case of circular motion) /=— —-• Hence, we find fl ( M+ m) = 4 irV 8 ; or, as a proportion, t 2 (M+ m) : t x 2 (M+ mj = r 8 : rfi which is strictly true as long as the planet's motions are undisturbed. 256 CENTRAL FORCES. 418. Inferences from Kepler's Laws. — From Kepler's laws we are entitled to infer — First (from the second law) , that the force which retains the planets in their orbits is directed towards the sun. Second (from the first law), that on any given planet the force lohich acts varies inversely as the square of its distance from the sun. Third (from the harmonic law), that the force is the same for one planet as it would be for another in the same place ; or, in other words, the attracting force depends only on the mass and distance of the bodies concerned, and' is wholly independent of their physical con- ditions, such as their temperature, chemical constitution, etc. It makes no difference in the motion of a planet around the sun whether it be made of hydrogen or iron, whether it be hot or cold. 419. Verification of " Gravitation " by Means of the Moon's Mo- tion. — When the idea of gravita- tion first occurred to Newton (the apple story is probably apocry- phal), he endeavored to verify it by comparing the force which keeps the moon in her orbit with the force of gravity at the earth's surface, reduced in the proper proportion. For lack, however, of an accurate knowledge of the earth's dimen- sions, he failed at first, there being a discrepancy of about sixteen per cent. He had assumed a degree to be exactly sixty miles in length. Some years afterward, when Pie- ard's measure of the arc of a merid- ian in Northern France had been made and reported to the Royal Society, making a degree about sixty-nine miles long, he saw at once that the new value would reconcile the discrepancy ; and he resumed his unfinished work and completed it. Fig. 138. Verification of the Hypothesis of Gravitation by Means of the Motion of the Moon. 420. At the earth's surface a body falls about 193 inches in a second. The distance of the moon being very nearly sixty times the earth's radius, if gravity really varies inversely as the square of the VERIFICATION OF GRAVITATION. 257 distance, a stone at that distance from the earth should fall — or 1 „„ a.... *v_* :„ ,•«. v* *_ ,.« 193 inches !0 as far ; that is, it ought to fall ° " 1UUCB = 0.0535 inches, — 3600 3600 a little more than one-twentieth of an inch. Now the distance which the moon actually does fall towards the earth in a second, i.e., the deflection of its orbit from a straight line in a second of time, is easily found ; and if the force which keeps the moon in its orbit is really the same as that which makes bodies fall towards the centre of the earth, this deflection ought to come out equal to in -.0535. Let AE (Fig. 138) be the distance the moon travels in a second = _, where r is the T radius of the moon's orbit, and t the number of seconds in a month. Then, since AEF is a right-angled triangle, we have, AB : AE : : AE : AF (or 2 r) ; , AJ} AE 2 whence AB = 2r The calculation is easy enough, though the numbers are rather large. As a result it gives us AB = 0.0534 inches, which is practically equal to the thirty-six hundredth part of 193 inches. If the quantities did not agree in amount, the discrepancy would disprove the theory, and, as we have said, Newton loyally gave it up until he was able to show that the apparent discordance was the result of a mistake in the original data, and disappeared when the data were corrected. The agreement, however, does not establish the theory, but only renders it probable. It does not establish it completely, because it is conceivable that the agreement might be a case of acci- dental coincidence, while the forces might really differ as much in their nature as an electrical attraction and a magnetic. 421. Newton was not satisfied with merely showing that the prin- cipal motions of the planets and the moon could be explained by the law of gravitation ; but he went on to investigate the converse prob- lem, and to determine what must be the motions necessary under that law. He found that the orbit of a body moving around a central mass is not of necessity a circle, or even a nearly circular ellipse like the planetary orbits, but that it may be a conic section of any eccentricity whatever — a circle, ellipse, parabola, or even an hyperbola ; but it must be a conic. 258 CENTEAL FORCES. 422. For the benefit of those of our readers who are not acquainted with conic sections we give the following brief account of them (Fig. 139): — a. If a cone of any angle be cut perpendicularly to the axis, the section will be a circle — MN in the figure. b. If it be cut by a plane which makes with the axis an angle greater than the semi-angle of the cone, so that the plane of section cuts com- pletely across the cone (as EF) , the section is an ellipse ; the circle being merely a special case of the ellipse. Ellipses, of course, differ greatly in form, from those which are very narrow to the perfect circle. c. The parabola is formed by cutting the cone with a plane parallel to its side; i.e., making with the axis an angle equal to the semi-angle of the cone. RPO is such a plane. As all circles are alike in form, so are all parabolas, whatever the angle of the cone at V and wherever the point P is taken. If the cutting plane is thus situated, then, no matter what is the angle of the cone or the place where the cut is made, the curve will always be the same in shape, though of course its size will depend upon a variety of circumstances. We do not stop to prove the statement, at first a little surprising ; but it is true. d. If the cutting plane makes an angle with the axis of the cone less than the semi-angle at V, so that the cutting plane gets continually deeper and deeper into the cone, then the curve is an hyperbola; so called, because the plane in this case "shoots over" (iwip ySAXeiv) and intersects the " cone produced," cutting out of this second cone a curve precisely like the curve cut from the original, as at JETG'ir in the figure. The axis of the hyperbola lies outside of the curve itself, being the line HIT in the figure, and the "■centre" of the curve is also outside of the curve at the middle point of this axis. Fig. 139. — The Conies. THE CONIC SECTIONS. 259 423. Philosophically speaking there are therefore but two species of conic sections, — the ellipse and the hyperbola, with the parabola for a partition between them. (The. circle, as has been said before, is merely a special case of the ellipse.) Fig. 140 will give the reader Ellipse F" C pf B ^JT 'k c F'J r p y 1+ecosV / l, the curve is an hyperbola; if e = l, it is a parabola. 260 CENTRAL FORCES. 424. Problem of Two Bodies. — This problem, proposed and solved by Newton, is the following : — Given the masses of two spheres and their positions and motions at any moment; given, also, the law of gravitation : required their motion ever afterwards, and the data necessary to compute their place at any future time. The mathematical methods by which the problem is solved require the use of the calculus, and must be sought in works on analytical mechanics or theoretical astronomy. Some of the results, however, are simple and easily stated. 425. (1) In the first place the motion of the centre of gravity of the two bodies will not be affected by their mutual attraction, but it will move on uniformly through space, as if the bodies were united into one at that point, and their motions combined under the same laws which hold good in the case of the collision of inelastic bodies. The motion of this centre of gravity is most easily worked out graphically as follows : First, in Fig. 141, join the original places of the bodies A and £ by a straight line, and mark on it G, the place of the centre of gravity ; then take the positions A' and B' they would occupy at the end of a unit of time (if they did not attract each other), and mark the new position of the centre of gravity & on the line joining them. The line GG 1 connecting Fie. 141. — Motion of Bodies relative to their Centre of Gravity. the two positions of the centre of gravity will show the direction and rapidity of its motion ; with reference to this point the two bodies will have opposite motions proportional to their distances from it; that is, they will swing around this point as if on a rod pivoted there, and will either both move towards it along the rod, or from it, with speeds inversely proportional to their masses. These relative motions with respect to the centre of gravity are easily found by drawing through G a line parallel to A'B', and meas- uring off on it distances GA" and GB" respectively equal to G'A> and G'B 1 A A" and BB" will then be the two motions of A and B relative to their centre of gravity G. EFFECT OF MUTUAL ATTRACTION. * 261 426. The Effect of their Mutual Attraction. —This will cause them to describe similar conies around this centre of gravity ; the size of their two orbits being inversely proportional to their masses. The form of the orbits and dimensions will be determined by the combined mass of the two bodies, and by their velocities with respect to the common centre of gravity. 427. The Orbit of the Smaller relative to the Centre of the Larger. — It is convenient (though it is not necessary) to drop the consid- eration of the centre of gravity of the two bodies, and to consider the motion of the smaller one around the centre of the larger one. In reference to that point, it will move precisely as if its mass had been added to that of the larger body, while itself had become a mere particle. This relative orbit will in all respects be like the actual one around the centre of gravity, only magnified in the proportion of M + m to M ; i.e., if m is -jig- of M, the actual orbit around G will be magnified by ^-J to produce the relative orbit around M. 428. The Orbit determined by Projection. — Suppose that in the figure (Fig. 142) the body P is moving in the direction of the arrow. Q A ~ Fig. 142. — Elliptical Orbit determined by Projection. and is attracted by S, supposed to be at rest. P will thenceforward move in a conic, either in an ellipse or hyperbola, according to its velocity, as we shall see in a moment. S being at one focus of the curve, the other focus will be somewhere on the line P1SF, which makes the same angle with PQ that r (SP) does (since it is a prop- erty of the conies that a tangent-line at any point of the curve makes equal angles with the lines drawn from the two foci to that point). 262 CENTRAL FORCES. If we can find the place of the second focus F, or the length of the line PF in the figure, the curve can at once be drawn. Now, it can be proved, though the demonstration lies beyond our scope, that a, the semi-major axis of the conic, is determined by the equation V i = S-- -\ (Equation 1 ) in which r is the distance SP, V is the velocity, and /* is the attracting mass at S expressed in proper units. (See Watson's " Theoretical Astronomy," p. 49 ; only for h- he writes k 2 (1 + ro)). V, r, and u. being given, of course a can be found : we get a = /* o —■ m (Equation 2) Then by subtracting r from 2 a we shall get r', or the distance PF, if the curve is an ellipse. If it is a hyperbola, a will come out nega- tive ; and to find r' we must take r' = 2a + r and measure it off to F', on the other side of the line of motion. In either case, however, we easily find the other focus, and the line drawn through the foci will be the line of apsides ; a point half-way between the foci will be the centre of the curve, and any line drawn through this centre will be a diameter. Having the two foci and the major axis 2 a, i.e., AA', the curve can at once be drawn. 429. Expression for a in Terms of the " Velocity from Infinity," or " Parabolic Velocity." — The expression for a admits of a more con- venient and very interesting form. It is shown in analytical mechan- ics that if, under the law of gravitation, a particle falls towards an attracting body whose mass is fn, from one distance s to another dis- tance r, its velocity is given by the simple equation iv s = 2 /x (- - IV (Equation 3) * If the difference between s and r is called h, this equation becomes *\r r+hl *\r* + rh) Now if h is very small as compared with r, this gives V '-&> which is the same as the usual expression for the velocity of a falling body at the earth's surface, viz., V 2 = 2gh, 2g being replaced by the fraction — • EXPRESSION FOR a. 263 If in this equation s be made infinite, w does not also become infinite (that is, a body falling from an infinite distance towards the sun will not acquire an infinite velocity until it actually reaches the centre of the sun, and r becomes zero) ; but we get in this case «•=*/*. r This special value of w is usually called " the velocity from infinity for the distance r," or the " parabolic velocity" (for a reason which will appear very soon) . U is generally used as its symbol ; therefore U 2 =— - ; whence m = ^- . (Equation 4) Substituting this value of fi in equation 2, we get „ r ( IP 2 \U 2 ~ V' (Equation 5) 430. Relation between the Velocity and the Species of Conic described. — From equation 5 it is obvious how the velocity determines whether the orbit will be an ellipse or a hyperbola. If V 2 is less than U 2 , the denominator of the fraction will be positive, a will also be positive, and the curve will be an ellipse; i.e., if the velocity of the body P, at the distance r from the central body S, be less than the Velocity acquired by the bod}' falling from infinity to that point, the body will piove around S permanently in an ellipse. If, on the other hand, V 2 is greater than U 2 , the denominator will become negative, a will also come out negative, and the orbit will be a hyperbola. In this case P, after once moving past S at the peri- helion point, will go off never to return ; and it will recede towards a different region of space from that out of which it came, because the two legs of the hyperbola never become parallel. There will in this case be no permanent connection between the two bodies. They simply pass each other with a little graceful recognition of each other's presence by a curvature in their paths, and then part company forever. If V* exactly equals U 2 , the denominator of the fraction becomes zero, a comes out infinite, and the curve is a parabola. In this case, also, the body will never return ; but it will recede from the sun ulti- mately towards the same point on the celestial sphere as that from which it appeared to come, since the two legs of the parabola tend to parallelism. Obviously, if a body were thus moving in a parabola, the slightest increase of its velocity would transform the orbit into an 264 CENTRAL FORCES. hyperbola, and the least diminution into an ellipse; the bearing of which remark will become evident when we come to deal with comets. 431 . Again, since a = - (_— — j, all bodies having the same velocity V, at the same distance r from the centre of force, will have major axes of the same length for their orbits, no matter what may be the direction of their motion. They will have the same period also, the expression for the period being t =!^L. (Watson, p. 46, Equation 28.) Fig. 143. — Confocal Conies described under Different Velocities of Projection. If, therefore, a body moving around the sun were to explode at any point, all of its particles which did not receive a velocity greater than the " parabolic velocity" would come around to the same point again, and those which were projected with equal velocities would come around and meet at the same moment, however widely different their paths might be. GRAPHICAL CONSTRUCTIONS. 265 432. Fig. 143 represents the orbits which would be described by five bodies projected at with different velocities along the line OV, the distance OS or r being taken as unity, as well as the parabolic velocity U 3 . The squares of the velocities are assumed as given below, with the resulting values of a and r'. V\ = i ! ■« hence a x = f ; and r,'= -J. This places the empty focus at F x . For the next larger ellipse V 2 — i • n — i . r /_ i In the same way V 3 2 = f ; a s = 2 ; r 3 ' = 3. V i 2 = 1 ; a, = co ; r 4 ' = go . (Parabola.) V* = 2 ; „a 5 = - J ; r 6 ' = - 2. (Hyperbola.) 433. Fig. 144 shows how three bodies projected at P with equal velocities, but in different directions, indicated by the arrows, describe three different Fig. 144. — Ellipses of the Same Periodic Time. ellipses ; all, however, having the same period, and the same length of semi- major axis ; namely, a = 2 r ; V 2 being taken equal to ft/ 2 . For a fourth body, V 2 is taken as = \U 2 , and with the direction of motion perpendicular to r. This body will move in a perfect circle, a coming out 266 CENTRAL FORCES. equal to r, when V 2 = £ U 2 . In order to have circular motion, both conditions must be fulfilled; namely, F^must equal \U 2 , and the direction of motion must be perpendicular to the radius vector. 434. Velocity of a Planet at Any Point in its Orbit. — If AA (Fig. 145) be the major axis of a planet's orbit, and KK' the diameter of a circle de- scribed around S with A A' as radius, then the velocity of a planet at any point, N, on its orbit is equal to that which it would have acquired by falling to N from the point n on the circumference of the circle. The demonstration is not dif- ficult and may be found in No. 1426 of the "Astronom- ische Nachrichten." Fig. 145. — -Van der Kolk's Theorem. 435. Projectiles near the Earth. — A good illustration of the principles stated above is obtained by considering the motion of bodies projected horizontally from the top of a tower near the earth's surface, supposing the air to be removed so there will be no resistance to the motion. The "parabolic velocity" due to the earth's attraction equals 6.94 miles per second at the earth's sur- face ; i.e., a body falling from the stars to the surface of the earth, drawn by the earth's attraction only, would have acquired this velocity on reaching the earth's surface. First. If a body be projected with a very small velocity, it would fall nearly straight downwards. If the earth were concentrated at the point in its centre so that the body should not strike its surface, it would move in a very long narrow ellipse having the centre of the earth at the further focus, and woulcl return to the original point after an interval of 29.9 minutes. Second. With a larger velocity the orbit would be a wider ellipse with a longer period, C being still at the remoter focus. Third. V= UV\, or about 4.9 miles per second. In this case the orbit of Fig. 146. —Projectiles near the Earth. INTENSITY OF SOIM.R ATTRACTION. 267 the body would be a perfect circle, and the period would be 1 h. 24.7 m. It will be remembered that we found that if the earth's rotation were 17 times as rapid, thus completing a revolution in l h 24 m .7, the centrifugal force at the equator would become equal to gravity (Art. 154). Also, Art. 413 (4), this same time, l h 24 m .7, was found from Kepler's third law as the period of a satellite revolving close to the earth's surface. Fourth. V= U= 6.94 miles. In this case the projectile would go off in a parabola, never to return. Fifth. V > 6.94. In this case, also, the body would never return, but would pass off in a hyperbola. 436. Intensity of Solar Attraction. — The attraction between the sun 'and the eaith from some points of view looks like a very feeble action. It is only able, as has been before stated (Art. 278), to bend the earth out of a rectilinear course to the extent of about one-ninth of an inch in a second, while she is travelling nearly nineteen miles ; and yet if it were attempted to replace by bonds of steel the invisible gravitation which holds the earth to the sun, we should find the surprising result that it would be necessary to cover the whole surface of the earth with wires as large as telegraph wires, and only about half an inch apart from each other, in order to get a metallic connection that could stand the strain. This liga- ment of wires would be stretched almost to the breaking point. The attraction of the sun for the earth expressed as tons of force (not tons of mass, of course) is 3,600,000 millions of millions of tons (36 with seventeen ciphers) ; and similar stresses act through the apparently empty space in all -directions between all the different pairs of bodies in the universe. 436*. Note to Art. 429. — It is worth noting that U 2 (the square of the parabolic velocity at any point) is simply twice the gravitation potential due to the sun's attraction at that point. The " potential" may be defined as the energy which would be acquired by a mass of one unit, in falling to the point in question from a place where the potential (and attraction) is zero, i.e., from infinity (Physics, pp. 26 and 28). Now Jm F 2 is the general ex- pression for the kinetic energy of a mass, m, moving with velocity V ; if in this expression we make m.= 1, and V= U, we shall have, for the case in hand, Energy = J U 2 , which equals the Potential at the point. 268 THE PROBLEM OF THKEE BODIES. CHAPTER XII. THE PROBLEM OF THREE BODIES. — DISTURBING FORCES : LUNAR PERTURBATIONS AND THE TIDES. / 437. The problem of two bodies is completely solved ; but if, instead of two spheres attracting each other, we have three'or more, given completely in respect to their positions, masses, and velocities, the general problem of finding their subsequent motions and predict- ing their positions at any future date transcends the present power of our mathematics. This problem of three bodies is in itself just as determinate and capable of solution as that of two. Given the initial data, — that is, the positions, masses, and motions of the three bodies at a given instant, — then their motions for all the future, and the posi- tions they will occupy at any given date, are absolutely predeter- mined. The difficulty of the problem lies simply in the inadequacy of our present mathematical methods, and it is altogether probable that some time in the future this difficulty will be overcome — very possibly by the invention of new functions and numerical tables which shall bear some such relation to our present tables of loga- rithms, sines, etc. as these do to the old multiplication table of Pythagoras. 438. But while the general problem of three bodies is thus intract- able, all the special cases of it which arise in the consideration of the moon's motion and in the motions of the planets have been solved by special methods of approximation. Newton himself led the way ; and the strongest proof of the truth of his theory of gravitation lies in the fact that it not only accounts for the regidar elliptic motions of the heavenly bodies, but also for the apparent irregularities of these motions. 439. The Disturbing Force. —In the case where two bodies are revolving around their common centre of gravity, and the third body is either very much smaller than the central one, or very remote, the "WHY THE STTN DOES NOT TAKE AWAY THE MOON. 269 motion of the two will be but slightly modified by the action of the third ; and in such a case the small differences between the actual motion and the motion as it would be if the third body were not present, are technically called " disturbances " and " perturbations," 1 and the force which produces them is called the " disturbing force." This disturbing force is not the attraction of the disturbing body, but only a component of that attraction , and usually only a small fraction of it,, (^"£jie disturbing force of the attracting body depends upon the differ- ence of its attraction upon the two bodies it disturbs; difference either in amount or in direction, or in both. For instance, if the sun attracted "the earth and moon exactly alike (i.e. , equally and along parallel lines) , it would not disturb their relative motions in the least, no matter how powerful its attraction might be. The sun's maxi- mum disturbing force on the moon, as we shall see, is only about one eighty-ninth of the earth's attraction ; and yet the sun's attraction for the moon is actually much greater than that of the earth. Since the sun's mass is 330,000 times that of the earth, and its distance just about 389 times that of the moon from the earth, its attraction on the moon equals the earth's attraction X -^—j = 2.18 ; i.e., the sun's attraction on the moon is more than double that of the earth. 440. Why the Sun does not take the Moon away from the Earth. — If at the time of new moon, when the moon is between the earth and sun, the sun attracts the moon more than twice as much as the earth does, it is a natural question why the sun does not draw the moon away entirely, and rob us of our satellite. It would do so if it were the case of a "tug of war" ; that is, if earth and sun were fixed in space, pulling opposite ways upon the moon between them. But it is not so ; neither sun nor earth has any foothold, so to speak ; but all three bodies are free to move, like chips floating on water. The sun attracts the earth almost as much as he does the moon, and both earth and moon fall towards him freely ; though of course this 1 The student will bear in mind that these terms (" perturbations " and " dis- turbances") are mere figures of speeeh; that philosophically the purely ellipti- cal motion of two mutually attracting bodies alone in space is no more " regular " than the (at present) incomputable motion of three or more attracting bodies. We have in mind a theologian of some note who once maintained that the " per- turbations" in the solar system are a consequence of Adam's fall. Hence the caution. 270 THE PROBLEM OP THREE BODIES. falling motion towards the sun is continually combined with whatever other motion the earth or moon possesses. The only effective dis- turbance is produced by the fact that, in the case considered, the new moon, being nearer the sun than the earth is by about -j^g- part of the whole distance, falls towards the sun a trifle faster than the earth, and so on that account the curvature of its orbit toward the earth is, for the time being, diminished. At the half-moon the two bodies are equally attracted towards the sun, but on converging lines; and so as they fall towards the sun they approach each other slightly ; and for this reason, at quadrature, the moon's orbit is a little more curved towards the earth than it would be otherwise. 441. Diagram of the Disturbing Force. — A very simple diagram enables us to find graphically the disturbing force produced by a third body. (What follows applies verbatim et literatim to either of the two diagrams of Fig. 147.) LFl Fig. 147. — Determination of the Jlisturomg Force by Graphical Construction. Let E be the earth, M the moon, and S the disturbing body (the sun in this case) ; and let the sun's attraction on the moon be repre- sented by the line MS. On the same scale the attraction of the sun on the earth will be represented by the line EG, G being a point so taken that EG : MS = MS 2 : ES 2 ; that is, — The sun's attraction on the earth is to the sun's attraction on the moon as the square of the sun's distance from the moon is to the square of the sun's distance from the earth, according to the law of gravitation. (MS has to do double duty in this proportion : in the first ratio it represents a force ; in the second, a distance.) From this proportion EG = MS x ^-^. ES 2 ' DIAGRAM Off THE DISTURBING FORCE. 271 In figure (a) the moon is nearer to the sun than the earth is, and so EG comes out less than MS. In figure (ft) the reverse is the case, and therefore in this case EG is larger than MS. Now if the force represented by the line MS were parallel and equal to that represented by EG, there would be no disturbance, as has been said. If, then, we can resolve the force MS into two components, one of which is equal and parallel to EG, this component will be in- nocent and harmless, and the other one will make all the disturbance. To effect this resolution, draw through M the line MK parallel and equal to EG. Join KS, and draw ML parallel and equal to it. ML is then the disturbing force on the same scale as MS; i.e., the line ML shows the true direction of the disturbing force, and in amount the disturbing force is equal to the sun's attraction for the moon mul- tiplied by the fraction I __ )• The diagonal of the parallelogram ^MSj MLS K is MS, which represents the resultant of the two forces MK and ML, that form its sides. For the sake of clearness the lines which represent forces in the figures are indicated by herring-bone markings. 442. At first it seems a little strange that in figure (6) the dis- turbing force should be directed away from the sun; but a little reflection justifies the result. If E and M were connected by a rod, and the .E-end of the rod were pulled towards the right more swiftly than the ilf-end, it is easy to see that the latter would be relatively thrown to the left, as the figure shows. 443. The sun is the only bodj< that sensibly disturbs the moon. The planets, of course, act upon the moon to disturb it, but their mass is so small compared with that of the sun, and their distances so great, that in no case is their direct action sensible. It is true, however, that some of the lunar perturbations are affected by the existence of one or two of the planets. While they cannot disturb the moon directly, they do so indirectly: they disturb the earth in her orbit sufficiently to make the sun's action different from what it would be if the planets did not exist. In this way the planets Venus, Mars, and Jupiter make themselves felt in the lunar theory. There are also a few small disturbances that depend upon the fact that the earth is not a perfect sphere. 444. Since the distance of the sun is nearly 400 times that of the moon from the earth, the construction of the disturbing force ML, 272 THE PKOBLEM OF THREE BODIES. Fig. 147, admits of considerable simplification. It is only necessary to drop tlie perpendicular MP upon the line that joins the earth and the sun, and take the point L upon this line, so that EL equals three times EP. The line ML so determined will then very approximately (but not exactly) be the true disturbing force. To prove this relation, let MS, in Fig. 147, be D, ES= R, ME = r, and EP = p, also R = D + p, very nearly, p being negative. when MS>ES. EG MS 2 Z) s was taken equal to MS x -=-^s = ™' EiS Jti Now, EL=GS=(Ei>-EG) = R-- = — l p- ^ (J ^ y • Developing this expression, we have D*+2Dp + p* Since p is very small as compared with D, all the terms except the first nearly vanish both in numerator and denominator, and we have EL = Z> 2 = 3p (very nearly). 445. Resolution of the Disturbing Force into Components. — In discussing the effect of the disturbing force it is more convenient to resolve it into three components known as the radial, the tangential, and the orthogonal. The first of these acts in the direction of the ^iwUiismltfsisuistiMumiMiuw Fig. 148.— Radial and Tangential Components of the Disturbing Force. radius vector, tending to draw the moon either towards or from the earth. The second, the tangential, operates to accelerate or retard the moon's orbital velocity. Fig. 148 exhibits these two components at different points of the moon's orbit. COMPONENTS OF THE DISTURBING FORCE. 273 The orthogonal component has no existence in cases where the disturbing body lies in the plane of the disturbed orbit ; but when- ever it lies outside of that plane, the disturbing force ML will gen- erally also lie outside of the orbit-plane, and will have a component tending to draw the moving body out of the plane of its orbit. The motion of the moon's node and the changes of the inclination of its orbit are due to this component of the sun's disturbing force, which sould not be conveniently represented in the figure. 446. The radial force in the case of the moon's orbit is a maxi- mum at syzygies and quadratures ; in fact, at quadratures the whole disturbing force is radial, the tangential and orthogonal components both vanishing. At syzygies (new moon and full moon) the radial force is negative; that is, it draws the moon from, the earth, dimin- ishing the earth's attraction by about one eighty-ninth l of its whole amount. At quadrature or half-moon the radial force is positive; and since L then falls at E, it is represented by the line QE, and is just half what it is at syzygies ; that is, it equals about one one hundred and seventy -eighth of the earth's attraction. It becomes zero at four points 54° 44' on each side of the line of syzygies. This angle is found from the condition that the disturbing force M y L v etc., in Fig. 148, must be perpendicular to the radius EM 1 at this point, which gives us EP X : P^-.-.P^-.P^. But P 1 L 1 = 2EP 1 ; therefore P 1 M 1 2 = 2 EP*, and %^-= tan M X EL Y = V2. EP X H7. The tangential component starts at zero at the time of full moon, rises to a maximum at the critical angle of 45° (having at that point a value of -j-^ of the earth's attraction), and disappears again at quadratures. During the first and third quadrants this force is negative; that is, it retards the moon's motion ; in the second and fourth it is positive and accelerates the motion. !At syzygies ML = NL = 2xEN (Fig. 147); but EN = ^£. Therefore ML— — of the sun's attraction on the moon. Now the sun's attraction is 2.18 389 2X2 18 1 times the earth's ; hence NL = the earth's attraction multiplied by — — - — = ^—> 274 THE PROBLEM OF THREE BODIES. 448. Lunar Perturbations. — So far it has been all plain sailing, for nothing beyond elementary mathematics is required in determin- ing the disturbing force at any point in the moon's orbit ; but to determine what will be the effect of this continually varying force after the lapse of a given time, upon the moon's place in the slcy is a problem of a very different order, and far beyond our scope. The reader who wishes to follow up this subject must take up the more extended works upon theoretical astronomy and the lunar theory. A few points, however, may be noted here. 449. In the first place, it is found most convenient to consider the moon as never deviating from an elliptical orbit, but to consider the orbit itself as continually changing in place and form, writhing and squirming, so to speak, under the disturbing forces ; just as if the orbit were a material hoop with the moon strung upon it like a bead and unable to get away from it, although she can be set forward and backward in her motion upon it. 450. In the next place, it is found possible to represent nearly all the perturbations by periodical formulas — the same values recurring over and over again indefinitely at regular intervals. This is because the sun, moon, and earth keep coming back into the same, or nearly the same, relative positions, and this leads to recurring values of the disturbing force itself, and also of its effects. 451. Third, the number of these separate perturbations which have to be taken account of is very large. In the computation of the moon's longitude in the American Ephemeris about seventy dif- ferent inequalities are reckoned in, and about half as many in the computation for the latitude. Theoretically the number is infinite, but only a certain number produce effects sensible to observation. It is of no use to compute disturbances that do not displace the moon as much as one-tenth of a second of arc ; i.e., about 500 feet in her orbit. 452. Fourth, in spite of all that has been done, the lunar theory is still incomplete, or in some way slightly erroneous. The best tables yet made begin to give inaccurate results after fifteen or twenty years, and require correction. The almanac place of the moon at present is not unfrequently "out" as much as 3" or 4" of arc ; i.e., about three or four miles. Astronomers are continually at LUNAR PERTURBATIONS. 275 work on the subject, but the computations by our present methods are exceedingly tedious and liable to numerical error. 453. The principal effects of the sun's disturbing action on the moon are the following : — First: Effect on Length of Month. —Since the radial component of. the disturbing force is negative more than half the way round (54° 44' on each side of the line of syzygies) and is twice as great at syzygies as the positive component is at quadrature, the net result is that, taking the whole month through, the earth's attraction for the moon is lessened by nearly ^ part. The effect of this is to in- crease the mean distance of the moon from the earth, and to make the month about an hour longer than it would be if the sun exerted no disturbing force. 454. Second : The Revolution of the Line of Apsides'. — This is due mainly to the radial component of the disturbing force. When the moon comes to apogee at the time of new or full moon, the diminution of the earth's effective attraction for the moon causes it to move on farther than ,it would otherwise do before turning the corner, so to speak, the con- sequence being that the line of apsides advances in the line of the moon's motion. When the moon passes perigee at that time, the effect is reversed, and the apsides regress; but since the disturbing force is greater at apogee than at perigee, in the long run the advancing motion " ai ~2L« predominates. When perigee or apogee is FlG ' 149 ' -,,.... j. 7 .it j. Advance of the Apsides of passed at the time or quadrature, the line or ap- thc Moon . 8 j bit sides is also disturbed ; but the disturbances thus produced exactly balance each other in the long run. The net result, as has been stated before (Art. 238), is that the line of apsides completes a direct revolution once in about nine years (8.855 years — Neison) . It does not move forward steadily and uniformly, but its motion is made up of alternate advance and regression. Fig. 149 illustrates this motion of the moon's apsides. For a fuller discussion of the subject, see HerschePs "Outlines of Astron- omy,'' sections 677-689 ; or Airy's " Gravitation," pp. 89-100. 455. Third: The Regression of the Nodes. — The orthogonal com- ponent generally (not always) tends to draw the moon towards the plane 276 THE PROBLEM OP THKEE BODIES. Of the ecliptic. Whenever this is the case at the time when the moon is passing a node, the effect (as is easily seen from Fig. 150) of such a force PiO x , acting upon the moon at P x , is to shift the node back- ward from N r to N 2 , the moon taking the new path P^N^. As the moon is approaching the node, the inclination of its orbit is also in- creased ; but as the moon leaves the node, it is again diminished, the path N 2 P 2 being bent at P 2 back to PJ> 2 , parallel to PjOj : so that while by both operations the node is made to recede from Nj to N 3 , the inclination suffers very little change, if the orthogonal component remains the same on both sides of the node. Since the orthogonal component vanishes twice a year, — when the sun is at the nodes of the moon's orbit, — and also twice a month, — when she is in quadrature, — the rate at which the nodes regress J>2 Fig. 150. — Regression of the Nodes of the Moon's Orbit. is extremely variable. In the long run it makes its backward revo- lution once in about nineteen years (Arts. 249 and 391). [18.5997 years. — Neison. J See Herschel's " Outlines of Astronomy," section 638 seqq. 456. Fourth: The Evection. — This is an irregularity which at the maximum puts the moon forward orbackward aboutU°(l° 16'27".01 Neison), and has for its period the time which is occupied by the sun in passing from the line of apsides of the moon's orbit to the same line again ; i.e., about a year and an eighth. This is the largest of the moon's perturbations, and was earliest discovered, having been detected by Hipparchus about 150 years B.C., and afterwards more fully worked out by Ptolemy, though of course without any under- standing of its cause. It was the only lunar perturbation known to the ancients. It depends upon the alternate increase and decrease of the eccentricity of the moon's orbit, which is always a maximum when EVECTION. 277 the sun is passing the moon's line of apsides, and a minimum when the sun is at right angles to it. This inequality may affect the time of an eclipse by nearly six hours, making it anywhere from three hours early to three hours late, as compared with the time at which it would otherwise occur ; it was this circumstance which called the attention of Hipparchus to it. See Herschel's " Outlines of Astronomy,'' sections 748 seqq. 457. Fifth : The Variation. — This is an inequality due mainly to the tangential component of the disturbing force. It has a period of one month, and a maximum amount of 39' 30". 70, attained when the moon is half-way between the syzygies and quadratures, at the so-called " octants." At the first and third octants the moon is 39^-' ahead of her mean place (about an hour and twenty minutes) ; at the second and fourth she is as much behind. This inequality was de- tected by Tycho Brahe, though there is some reason for believing that it had been previously discovered by the Arabian astronomer, Aboul Wefa, five centuries earlier. This inequality does not affect the time of an eclipse, being zero both at the syzygies and quadra- tures, and therefore was not detected by the Greek astronomers. See Herschel's " Outlines of Astronomy," sections 705 seqq. 458. Sixth : The Annual Equation. — The one remaining ine- quality which affects the moon's place by an amount visible to the naked eye, is the so-called "annual equation." When the earth is nearer the sun than its mean distance, the sun's disturbing force is, of course, greater than the meau, and the month is lengthened a little; during'that half of the year, therefore, the moon keeps falling behindhand ; and vice versa during the half when the sun's distance exceeds the mean. The maximum amount of this inequality is 11' 9". 00, and its period one anomalistic year. See Herschel's " Outlines of Astronomy," sections 738 seqq. There remains one lunar irregularity among the multitude of lesser ones, which is of great interest theoretically, and is still a bone of contention among mathematical astronomers ; namely, — 459. Seventh : The Secular Acceleration of the Moon's Mean Mo- tion. — It was found by Halley, early in the last century, by a com- parison of ancient with modern eclipses, that the month is now 278 THE PROBLEM OP THREE BODIES. certainly shorter than it was in the days of Ptolemy, and that the shortening has been progressive, apparently going on continuously, — in scecula sceculorum, —whence the name. In 100 years the moon, according to the results of Laplace, gets in advance of its mean place about 10" ; and the advance increases with the square of the time, so that in a thousand years it would gain nearly 1000", and in 2000 years 4000", or more than a degree. The moon at present is supposed to be just about a degree in advance of the posi- tion it would have held if it had kept on since the Christian era with precisely the rate of motion it then had. If this acceleration were to continue indefinitely, the ultimate result would be that the moon would fall upon the earth, as the quickened motion corresponds to a shortened distance. 460. It was nearly 100 years after Halley's discovery before Laplace found its explanation in the decreasing eccentricity of the earth's orbit. Under the action of the other planets this orbit is now growing more nearly circular, without, however, changing the length of its major axis. Thus its area becomes larger, and the earth's average distance from the sun becomes greater (although the mean distance, technically so-called, does not change, the "mean distance" being simply half the major axis). As a result of this rounding up of the earth's orbit, the average disturbing force of the sun is therefore dimin- ished, and this diminution allows the month to come nearer the length it would have if there were no sun to disturb the motion ; that is to say, the month keeps shortening little by little, and it will continue to do so until the eccentricity of the earth's orbit begins to increase again, some 25,000 years hence. 461. But the theoretical amount of this acceleration, about 6" in a century, does not agree with the value obtained by comparing the most ancient and modern eclipses, which is about 12" ; and this value, again, does not agree with the one derived by comparing modern observations of the moon with those made by the Arabians about a thousand years ago, which, according to recent investigations by Professor Newcomb, indicate an acceleration of only about 8". So long as the actual acceleration was considered to be 12", it was gener- ally supposed that the discrepancy between the theoretical and observed result is due to a retardation of the earth's rotation by the friction of the tides, and a consequent lengthening of the day. Evidently if the day and the seconds become a little longer, there will be fewer of them in each month or year, and the apparent effect of such a change would be to shorten all really constant astronomical periods by one and the same percentage. THE TIDES. 279 As matters stand to-day it is hardly possible to assert with confidence that there is any real discrepancy to be accounted for between the theoret- ical and observed values, the latter being considerably uncertain. In New- comb's '.' Popular Astronomy" (pp. 96-102) there will be found an interesting and trustworthy discussion of the subject. Questions like this, and those relating to the remaining discrepancies between the lunar tables and the observed places of our satellite, lie on the very frontiers of mathematical astronomy, and can be dealt with only by the ablest and most skilful analysts. THE TIDES. 462. Just as the disturbing force due to the sun's attraction affects the motions of the moon in her orbit, so the disturbing forces due to the attractions of the moon and sun acting upon the fluids of the earth's surface produce the tides. These consist of the regular rise and fall of the water of the ocean, the average interval between successive high waters being 24 h 51 ra , which is precisely the same as the average interval between two successive passages of the moon across the meridian. This coincidence, maintained indefi- nitely, of itself makes it certain that there must oe some causal con- nection between the moon and the tides. As some one has said, the odd 51 minutes is the moon's r ' ear-mark." Ls 463. Definitions. — When the water is rising, it is '■'■flood" tide ; when falling, it is " ebb." It is " high water" at the moment when the tide is highest, and " low water" when it is lowest. "Spring tides" are the highest tides of the mouth (which occur near the times of new and full moon), while "neap tides" are the smallest, which occur when the moon is in quadrature. The relative heights of the spring and neap tides are about as 7 to 4. At the time of spring tides the interval between the corresponding tides of suc- cessive days is less than the average, being only about 24 h 38'", and then the tides are said to "prime." At neap tides the interval is 25 h 6 m , which is greater than the mean, and the tides " lag." The "establishment" of a port is the mean interval between the time of high water at that port and the next preceding passage of the moon across the meridian. At New York, for instance, this "establishment" is 8 h 13 m , although the actual interval varies about 22 minutes on each side of the mean at different times of the month. That the moon is largely responsible for the tides is also shown by the fact that the tides, at the time when the moon is in perigee, are nearly twenty per cent higher than those which occur when she is in 280 THE TIDES. apogee. The highest tides of all happen when a new or fuU moon occurs at the time the moon is in perigee, especially if this occurs about January 1st, when the earth is nearest to the sun. Since, as we shall see, the " tide-raising" force varies inversely as the -cube of the distance, slight variations in the distance of the moon and sun from the earth make much greater variations in the height of the tide — greater nearly in the ratio of 3 to 1. 464. The Tide-Raising Force. — • This is the difference between the attractions of the sun aud moon (mainly the latter) on the main body of the earth, and the attractions of the same bodies on parti- cles at different parts of the earth's surface. The tide-raising force is but a very small part of the whole attraction. The amount of this disturbing force for a particle at any point on the earth's surface can be found approximately by the same geo- D G/ To Moon He Fiq. 151. — The Moon's Tide-Raising Force on the Earth. metrical construction which was used for the lunar theory (Art. 441). Draw aline from the moon through the centre of the earth. At the points A and B, Fig. 151, where the moon is directly over head or under foot, the tide-raising force is directly opposed to gravity, and equals nearly -fa of the moon's whole attraction, since the line Aa represents the disturbing force on the same scale as the line from A to the moon represents the moon's attraction, and this line, AM, is about sixty times the earth's radius, while Aa is just double it, be cause Ca has to be taken equal to 3 x CA (Art. 444) . Since the moon's mass is only about -fa of the earth's, and its dis- tance is sixty radii of the earth, this lifting force under the moon, expressed as a fraction of the earth's gravity, equals 1 V J V 1 1 i.e., a body weighing four thousand tons loses about one pound of its weight when the moon is over head or under foot. At^Z> and E, anywhere on the circle of the earth's surface which is 90° from A and B, the moon's disturbing force increases the THE TIDE-RAISING FORCE. 281 weight of a body by just half this amount, the disturbing force being measured by the lines DC and EC. At a point F, situated anywhere on a circle drawn around either A or B with a radius of 54° 44', the weight of a body is neither increased nor decreased, but it is urged towards A or JB with a horizontal force expressed by the line Ff. which force is equal to about Tnshnnns of its weight. In the same way the tidal forces at G and H are expressed by the lines Gg and Hh. 465. The same result for the lifting-force directly under the moon may be obtained more exactly as follows. The distance from the moon to the centre of the earth is sixty times the earth's radius, and therefore the distance from the moon to the points A and B respectively will be 59 and 61. The moon's attraction at A, C, and B, expressed as fractions of the earth's gravity, will be as follows : — i Attraction of moon on particle at A = g x Jg5= 0.0000035910 x Si- Attraction of moon on particle at C = g X £k = 0.0000034723 x g- Attraction of moon on particle at B = g x j$2 = 0.0000033593 x 9- Hence A - C = 0.0000001187 g = g. y 8,424,000 y C-B = 0.0000001 130 g = g. 8,835,000 This is more correct than the preceding, which is based on an approxima- tion that considers the moon's distance as very large compared with the earth's radius, while it is really only sixty times as great, and sixty is hardly a "very large " number in such a case. Attempts have been made to observe directly the variations in the force of gravity produced by the moon's action, but they are too small to be detected with certainty by any experimental method yet contrived. Both Darwin and Zollner found that other causes which they could not get rid of produced disturbances more than sufficient to mask the whole action of the moon. 466. It is worth while to note in this connection that the maximum lifting-force due to the attraction of a distant body varies inversely as the cube of its distance, as is easily shown, thus : — calling D the distance of the disturbing body from the earth's centre, and r the earth's radius, we have Attraction at A = ; attraction at C = — -• (D-ry IP Tide-raising force at A = 3f\ —\ = m\ ^^ T ?, Z>r ~ ^ — r x \ l(Z»-r)» D 2 ) (D 2 (D 2 -2Dr + r 2 )) = MS °- Dr - r * I = Mi |Tl,nearly. when r is a small fraction of D. 282 THE TIDES. 467. It is very apt to puzzle the student that the moon's action should be a lifting force at B as well as at A (Fig. 151). He is likely to thiuk of the earth as fixed, and the moon also fixed and attracting the water upon the earth, in which case, of course, the moon's attraction, while it would decrease gravity at A, would increase it at B. The two bodies are not fixed, however. Let him think of the three particles at A, C, and B, Fig. 152, as unconnected with each other, and falling freely towards the moon; then it is obvious that they would ■ -pr " 7 separate ; A would fall faster than Fio.i52.-The Statical Theory of the Tides. C, and C than B. Now imagine them connected by an elastic cord. It is obvious that they will still draw apart until the tension of the cord prevents any further separation. Its tension will then measure the •' lifting force" of the moon which tends to draw both the par- ticles ..1 and B away from C. 468. The Sun's Action. — This is precisely like that of the moon, except that the sun's distance, instead of being only sixty times the earth's radius, is nearly 23,500 times that quantity. Since the tide- raising power varies as the cube of the distance inversely, while the attracting force varies only with the inverse square, it turns out that although the sun's attraction on the earth is nearly 200 times as great as that of the moon, its tide-raising power is only about two- fifths as much. When the sun is over head or under foot, his dis- turbing force diminishes gravity by about l8 co TnTTTT- 469. Statical Theory of the Tides. — If the earth were wholly composed of water, and if it kept always the same face towards the moon (as the moon does towards the earth), so that every particle on the earth's surface was always subjected to the same disturbing force from the moon, then, leaving out of account the sun's action, a per- manent tide would be raised upon the earth, distorting it into a lemon-shaped form with the point towards the moon. It would be permanently high water at the points A and B (Fig. 152) directly under the moon, and low water all around the earth on the circle 90° from these points, as at D and E. The difference of the level of the water at A and D would in this case be about two feet. THE PRIMING AND LAGGING OF THE TIDES. 283 The sun's action would produce a similar tide superposed upon the lunar tide and having about two-fifths of the same elevation. If the two tide summits should coincide, the resulting elevation of the high water would be the sum of the two separate tides. If the sun were 90° from the moon, the waves would be in opposition, and the height of the tide would be decreased, the solar tide partly filling up the depression at the low water due to the moon's action. Suppose now the earth to be put in rotation. It is easy to see that these tidal waves would tend to move over the earth's surface, following the moon and sun at a certain angle dependent on the inertia of the water, and with a westward velocity precisely equal to that of the earth's eastward rotation, — about a thousand miles an hour at the equator. But it is also evident that on account of the varying depth of the ocean, and the irregular form of the shores, the tides could not maintain this motion, and that the actual result must become exceedingly complicated. In fact, the statical theory be- comes utterly unsatisfactory in regard to what actually takes place, and it is necessary to depend almost entirely upon the results of observation, using the theory merely as a guide in the discussion of the observations. Yet while this statical theory of the tides worked out by Newton is eertainly inadequate, and in some respects incorrect, it easily furnishes the explanation of some of the most prominent of the peculiarities of the tides. 470. The Priming and Lagging of the Tides. — About the time of new and full moon, as has been stated before (Art. 463), the interval between the corresponding tides of successive days is about thirteen minutes less than the average of 24 h 51 m , while a week later it is about as much longer. The reason is found in the combination of the solar and lunar tides. L Priming and Lagging of the Tide. On the days of new and full moon the two tides coincide, and the tide wave has its crest directly under the moon, or rather at the normal distance behind the moon which corresponds to the " establishment " of the port of observation. 284 THE TIDES. At quadrature the crest of the solar tide will be just 90° from the crest of the lunar wave, but it will leave the summit of the combined wave just where it would be if there were no solar wave at all : evidently there is no possible reason why the smaller wave at will be small. The tides which occur twelve hours later will be small at the northern station, then situated at C, and large at the southern station, then at C. For a port on the equator at E or Q there will be no such difference. In the Atlantic Ocean the difference is hardly noticeable, because, as we shall see very soon, the tides in that ocean are mainly (not entirely) due to tide waves propagated into it from the Pacific and Indian Oceans around the Cape of Good Hope. 472. The Wave Theory of the Tides. — If the earth were entirely covered with deep water, except a few little islands projecting here and there to serve for observing stations, the tide waves would run N n^?# S!|j^yl jgrc ip^\ Wkc' *^^^ Fio. 155. — The Diurnal Inequality. FREE AND FORCED OSCILLATIONS. 285 around the globe regularly ; and if the depth of the ocean were over thirteen miles, the tide crests, as can be shown, would follow the moon at an angle of just 90°. It would be high water just where the statical theory would give low water. If the depth were (as it really is) much less than thirteen miles, the tide wave in the ocean could not keep up with the moon, and the result would be a very compli- cated one. The real state of the case is still worse. The continents of North and South America, with the southern antarctic continent, make a barrier almost complete from pole to pole, leaving only a narrow passage at Cape Horn ; and the varying depth of the water, and the irregular contours of the shores are such that it is quite impossible to determine by theory what the course and character of the tide wave must be. We must depend upon observation ; and observations are inadequate, because, with the exception of a few islands, our only possible tide stations are on the shores of continents where local circumstances largely control the phenomena. 473. Free and Forced Oscillations. — If the water in the ocean is suddenly disturbed (as for instance, by an earthquake), and then left to itself, a " free" wave will be formed, which, if the horizontal dimensions of the wave are large as compared with the depth of the water, will travel at a rate depending solely on the depth^ The veloc- ity of such a free wave is given by the formula v = -\/gh ; that is, it is equal to the velocity acquired by a body in falling through half the depth of the ocean. Thus a depth of 25 feet gives a velocity of 19 + miles per hour. 100 10,000 " " " 40,000 " " " 67,200 (12| miles) 90,000 " " ' " 39 " 388 " 775 " 1000 " 1165 Observations upon the waves caused by certain earthquakes in South America and Japan have thus informed us that between the coasts of those countries the Pacific averages between two and one- half and three miles in depth. 474. Now, as the moon in its diurnal motion passes across the American continent each day, and comes over the Pacific Ocean, it starts such a "parent" wave in the Pacific, and the wave once started moves on nearly (but not exactly) like an earthquake wave. Not exactly, because the velocity of the earth's rotation being about 286 THE TIDES. 1050 miles an hour at the equator, the moon runs relatively westward faster than the wave can naturally follow, and so for a while slightly accelerates it. A second tidal wave is produced daily twelve hours later when the moon passes underneath. The tidal wave is thus, in its origin, a forced oscillation, while in its subsequent travel it is pretty nearly a free one. 475. Co-Tidal Lines. — These are lines drawn upon the surface of the ocean connecting those places which have their high water at the same moment of Greenwich time. They mark the crest of the tide wave for each hour of Greenwich time ; and if we could draw them with certainty upon the globe, we should have all necessary information as to the motion of the wave. Unfortunately we can obtain no direct knowledge as to the position of these lines in mid- ocean ; we only get a few points here and there on the coasts and on islands, so that a great deal necessarily remains conjectural. Fig.' 156 is a reduced copy of such a map, borrowed with some modifica- tions from that given in Guyot's " Physical Geography." 476. Course of Travel of the Tidal Wave. — On studying the map we find that the main or " parent " wave starts twice a day in the Pacific, off Callao, on the coast of South America. This is shown on the chart by a sort of oval " eye " in the co-tidal lines, just as a mountain summit is shown on a topographical chart by an " eye " in the contour lines. From this point the wave travels northwest through the deepest water of the Pacific at the rate of about 850 miles per hour, reaching Kamtchatka in about ten hours. To the west and southwest the water is shallower and the travel slower, — only 400 to 600 miles per hour, — so that the wave arrives at New Zealand in about twelve hours. Passing on by Australia, and com- bining with the small wave which the moon raises directly in the Indian Ocean, the resultant tide crest reaches the Cape of Good Hope in about twenty-nine hours, and enters the Atlantic. Here it combines with the tide wave, twenty-four hours younger, which bas "backed" into the Atlantic around Cape Horn, and it is modified also by the direct tide produced by the moon's action upon the waters of the Atlantic. The resultant tide crest then travels northward through the Atlantic at the rate of nearly 700 miles per hour. It is about forty hours old when it first reaches the coast of the United States in Florida, and our coast is so situated that it arrives at all the principal ports within two or three hours of that time. It is forty-one or forty-two hours old when it arrives at New York and Boston. To reach London it has to travel around the northern end of Scotland and through the North Sea, and is nearly sixty hours old when it arrives at that port and the ports of the German Ocean, — Hamburg, itc. In the great oceans there are thus three or four tide crests travelling CO-TIDAL LINES. 287 e>_ ©_ 288 THE TIDES. simultaneously, following each other nearly in the same track, but with continual minor changes, owing to the variations in the relative positions of the sun and moon and their changing distances and declinations. If we take into account the tides in rivers and sounds, the number of simultaneous tide crests must be at least six or seven ; that is, the high water at the extremity of its travel, up the Amazon River, for instance, must be at least three or four days old, reckoned from its birth in the Pacific. 1 477. Tides in Rivers. — The tide wave ascends a river at a rate which depends upon the depth of the water, the amount of friction, and the swiftness of the stream. It may, and generally does, ascend until it comes to a rapid, where the velocity of the water is greater than that of the wave. In shallow streams, however, it dies out earlier. Contrary to what is usually supposed, it often ascends to an elevation far above that of the highest crest of the tide wave at the river's mouth. In the La Plata and Amazon it goes up to an elevation at least one hundred feet above the sea-level. The velocity of the tide wave in a river seldom exceeds ten or twenty miles an hour, and is usually less. » 478. Height of Tides. — In mid-ocean the difference between high and low water is usually between two and three feet, as observed on isolated deep-water islands in the Pacific ; but on the continental shores the height is usually much greater. As soon as the tide wave Fig. 157. — Increase iu Height of Tide on approaching the Shore. touches bottom, so to speak, the velocity is diminished and the height of the wave is increased, something as in the annexed figure (Fig. 157). Theoretically the height varies inversely as the fourth root of the depth. Thus, where the water is 100 feet deep, the tide wave should be twice as high as at the depth of 1600 feet. Where the configuration of the shore forces the wave into a corner, it sometimes becomes very high. At the head of the Bay of Fundy, tides of seventy feet are not uncommon, and an altitude of 100 feet is said to be occasionally attained. At Bristol, England, in the mouth of the Severn the tide rises fifty feet, and sometimes ascends the river (as it also does the Seine, in France, and 1 We are greatly indebted to Loomis's discussion of the subject in his " Ele- ments of Astronomy." V REFLECTION AND INTERFERENCE. 289 the Amazon) as a breaking wave, called the " bore " or " eiger " (French, mascaret), -with a nearly vertical front five or six feet in height, crested with foam, and very dangerous to small vessels. On the east coast of Ireland, opposite to Bristol, the tide ranges only about two feet. In mid-ocean the water has no progressive motion, but near the land it has, running in at the flood to fill up the bays and cover the flats, and then running out again at the ebb. The velocity of these tidal currents must not be confounded with that of the tide wave itself. 479. Reflection and Interference. — The tide wave when it reaches the shore is not entirely destroyed, especially.if the coast is bold and the water deep ; but is partly reflected, and the reflected wave goes back into the ocean to meet and modify the new tide wave which is coming in. Of course, in such a case we get " interferences," so that on islands in the Pacific only a few hundred miles apart we find great differences in the heights of the tides. At one place the direct waves and the waves reflected from the shores of Asia and South America may conspire to give a tide of three or four feet, or nearly double its normal value, while at another they nearly destroy each other. There are places, also, which are reached by tides coming by two different routes. Thus on the east coast of England and Scotland the tide waves come both around the northern end of Scotland and through the Straits of Dover. In some places on this coast we have, therefore, a tide of nearly doubly height, while at others not very far away there will be hardly any tide at all; and at intermediate points there are sometimes four distinct high waters in twenty-four hours. As a consequence of this reflection and interference of the tide waves it follows that if the tide-raising power were suddenly abolished, the tides would not immediately cease, but would con- tinue to run for several days, and perhaps weeks, before they gradually died out. y 480. Effect of the Varying Pressure of the Barometer, and of the Wind. — "When the barometer at a given port is lower than usual, the level of the water is generally- higher than the average, at the rate of about one foot for every inch of the mercury in the barometer ; and vice versa when it is higher than usual. "When the wind blows into the mouth of a harbor, it drives in the water of the ocean by its surface friction, and may raise the water several feet. In such cases the time of high water, contrary to what might at first be supposed, is delayed, sometimes as much a? fifteen or twenty minutes. This result depends upon the fact that the water runs into the harbor for a longer time than it would do if the wind were not blow- 290 THE TIDES. ing. The normal depth of the water on the bar is reached before the predicted time, so that at the predicted time the water is deeper than it would be if there were no wind, but the maximum, depth is not attained until some time later. Of course, the results are the opposite when the wind blows out of the harbor : the time of high water comes earlier, and the depth of water on the bar at the pre- dicted time of high water is less than it otherwise would be. 481. Tides in Lakes and Inland Seas. — These are small and diffi- cult to detect. Theoretically, the range between high and low water in a land-locked sea should bear about the same ratio to the rise and fall of the tide in mid-ocean that the length of the sea does to the diameter of the earth. Variations in the direction of the wind and the barometric pressure cause continual oscillations in the water-level which, even in a quiet lake, are much larger than the true tides; so that it is only by taking a long series of obser- vations, and discussing them with reference to the moon's position in the sky, that it is possible to separate the real tide from the effects of other causes. In Lake Michigan, at Chicago, a tide of about one and three-quarters inches has thus been detected, the " establishment " of the port being about thirty minutes. In Lake Erie, at Buffalo and Toledo, the tide is about three-quarters of an inch. On the coasts of the Mediterranean the tide averages about eighteen inches, attaining a height of three or four feet at the head of some of the bays. 482. The Rigidity of the Earth. — Sir W. Thomson has endeav- ored to make the tides the criterion of the rigidity of the earth's core. Evidently if the solid parts of the earth were fluid, there would be no observable tide anywhere, since the whole surface would rise and fall together. If the earth were semi-solid, so to speak (that is, viscous, and capable of yielding more or less to the forces tending to change its form) , the tides would be observable, but to a less degree than if the earth's core were rigid. And with this further peculiarity — since a viscous body requires time to change its form, waves of short period would be observable upon the semi-solid earth nearly to their full ex- tent, while those of long period would almost entirely disappear, owing to the slow yielding of the earth's crust. Now the actual tide wave, as observed, is really made up of a multitude of component tide waves of different periods, ranging from half a day upwards. According to the "principle of forced vibrations" every regularly recurring periodic change in the forces which act on the surface of the ocean must produce a tide of greater or less magnitude, and of exactly corresponding period. EFFECT OP TIDES ON THE EARTH'S ROTATION. 291 We have, for instance, the semi-diurnal, solar, and lunar tides ; then the two monthly tides due to the change in the moon's distance and declination, and the two annual tides due to the changes of the sun's distance and declination, not to speak of the nineteen-year tide due to the revolution of the moon's nodes. A thorough analytical discussion of thirty-three years' tidal ob- servations at different parts of the world has been made under the direction of Sir W. Thomson by Mr. George Darwin, with the result that not only do the short waves show themselves, but the waves of long period are found to manifest themselves with almost their full theoretical value. Thomson's conclusion is that the earth as a whole " must be more rigid than steel, but perhaps not quite so rigid as glass." This result is at variance with the prevalent belief of geolo- gists that the core of the earth is a molten mass, and has led to much discussion which we cannot deal with here. 483. Effect of the Tides on the Earth's Rotation. — If the tidal motion consisted merely in the upward and downward motion of the particles of the ocean to the extent of two feet or so twice a day, it would involve a very trifling expenditure of energy ; and this is the case with the mid-ocean tide. But near the land this almost insensi- ble mere oscillatory motion is transformed into the bodily travelling of immense masses of water, which flow in upon the shallows and then out again to sea with a great amount of fluid friction ; and this involves the expenditure of a very considerable amount of energy which is dissipated as heat. From what sources does this energy come? The answer is that it must be derived mainly from the earth's energy of rotation, and the necessary effect is to diminish that energy by lessening the speed of the rotation. Compared with the earth's whole stock of rotational energy, however, the loss of it by tidal friction, even in a century, is very small, and the effect on the length of the day is extremely slight. The reader will recall the remarks upon the subject of the secular accel- eration of the moon's mean motion a few pages back (Art. 461). While it is certain that the tidal friction tends to lengthen the day, it does not follow that the day really grows longer. There are counteracting causes : — for example, the earth's jadiation of heat into space, and the consequent shrinkage of her volume. — As matters stand we do not know whether, as a fact, the day is really longer or shorter than it was a thousand years ago-. The 292 THE TIDES. change, if any has really occurred, can hardly be as great as j-^ of a second. J 484. Effect of the Tide on the Moon's Motion. — Not only does the tide diminish the earth's energy of rotation directly by the tidal friction, but, theoretically, it also communicates a minute portion of that energy to the moon. It will be seen that a tidal wave, situated as in Fig. 158, would slightly accelerate the moon's motion, the attraction of the moon by the tidal protuber- ance F being slightly greater than that of the tide wave at F' — a difference tending to draw it along in its orbit a little, thus increasing the major axis of the moon's orbit. The tendency is therefore to make the moon recede from the earth, and to lengthen the month. Upon this interaction between the tides and the motions of the earth and moon Professor George Darwin "has founded his theory of " tidal evolution " ; namely , that the satellites of a planet, having separated from it millions of years ago, have been made to recede to their present dis- tances by just such an action. An excellent popular statement of the theory will be found in the closing chapter of Ball's " Story of the Heavens." The original papers of Mr. Darwin in the " Philosophical Transactions " are of course intensely mathematical. Fig. 158. Effect of the Tide on the Moon's Motion. THE PLANETS: THEIR MOTIONS. 293 CHAPTER XIII. THE PLANETS : THEIR MOTIONS, APPARENT AND REAL. — THE PTOLEMAIC, TYCHONIC, AND OOPERNIOAN SYSTEMS. — THE ORBITS AND THEIR ELEMENTS. — PLANETARY PERTURBATIONS. 485. For the most part, the stars keep their relative configurations unchanged, however much they alter their positions in the sky from hour to hour. The "dipper" remains always a "dipper" in what- ever part of the heavens it may be. But while this is true of the stars in general, certain of the heavenly bodies, and among them those that are the most conspicuous of all, form an exception. The sun and moon continually change their places, moving eastward among the stars ; and certain others, which to the eye appear as very brilliant stars, also move, 1 though not in quite so simple a way. 486. These bodies were named by the Greeks the "planets" ; that is, " wanderers." The ancient astronomers counted seven of them. They reckoned the sun and moon, and in addition Mercury, Venus, Mars, Jupiter, and Saturn. Venus and Jupiter are at all times more brilliant than any of the fixed stars. Mars at times, but not usually, is nearly as bright as Jupiter ; and Saturn is brighter than all but a very few of the stars. Mercury is also bright, but seldom seen, because always near the sun. At present the sun and moon are not reckoned as planets ; but the roll includes, in addition to the five other bodies known by the ancients, the earth itself, which Copernicus showed should be counted among them, and also two new bodies of great magnitude (though inconspicuous because of their distance) which have been discovered in modern times ; then there is in addition a host of so-called " aste- roids" which circulate in the otherwise vacant space between the planets Mars and Jupiter. 1 Whan we speak of the motion of the planets, the reader will understand that the diurnal motion is not taken into account. We speak of their motions among the stars. 294 THE PLANETS. 487. The list of the plauets in the order of distance from the sun stands thus at present : Mercury, Venus, the Earth, Mais, Jupiter, Saturn, Uranus, and Neptune ; and between Mars and Jupiter, in the place where a planet would naturally be expected to revolve, there are at present known nearly 300 little planets, which probably repre- sent a single one, somehow " spoiled in the making," so to speak, or burst into fragments. The planets are all dark bodies, shining only by reflected sun- light, — globes which, like the earth, revolve around the sjin in orbits nearly circular, moving all in the same direction, and (with some exceptions among the asteroids) nearly in the common plane of the ecliptic and sun's equator. All of them but the inner two and the asteroids are also attended by " satellites." Of these the earth has one (the moon), Mars two, Jupiter four, Saturn eight, Uranus four, and Neptune one ; i.e., so far as at present known ; for although it is hardly probable, it is not at all impossible that others may yet be found. 488. Relative Distances of Planets from the Sun: Bode's Law. — There is a curious approximate relation between the distances of the planets from the sun, which makes it easy to remember them. It is usually known as Bode's Law, because Bode first brought it prominently into notice in 1772, though Titius of Wittenberg seems to have discovered and enunciated it some years earlier. The law is this : Write a series of 4's. To the second 4 add 3 ; to the third add 3x2, or 6 ; to the fourth, 3 X 4, or 12 ; and so on, doubling the added number each time, as in the accompanying scheme. 4 4 4 3 _6 4 7 10 « 9 © The resulting numbers, divided by 10, are pretty nearly the true mean distances of the planets from the sun, in terms of the radius of the earth's orbit. In the case of Neptune, however, the law breaks down utterly, and is not even approximately correct. For the present, at least, the law is to be regarded as a mere coincidence, there being so far no reasonable explanation of any such numerical relation. The general expression for the nth term of the series is 4 + 3 X 2( n_2 > ; but it does not hold good of the first term, which is simply 4, instead of being 5J, i.e., (4 + 3 x 2 -1 ), as it should be. 4 4 4 4 4 4 12 24 48 96 192 384 16 [28] 52 100 196 388 S ® % h ¥ W TABLE OP NAMES, DISTANCES, AND PERIODS. 295 489. Table of Names, Distances, and Periods. Name. Symbol. Distance. Bode. DlIT. Sid. Period. Syn. Pekiod. Mercury .... Earth 9 © 2 0.387 0.723 1.000 1.523 0.4 0.7 1.0 1.6 -0.013 + 0.023 0.000 -0.077 881 or 3m 224.7 d or 7J m 365J d or ly 687 d or ly 10] m 116<» 5841- "7801,., Mean Asteroid • 2.650 2.8 -0.150 3y.l to 8y.0 various Jupiter .... Uranus .... Neptune .... 5.202 9.539 19.183 30.054 5.2 10.0 19.6 38.8 + 0.002 - 0.461 -0.417 -8.746! lly.9 29y.5 84y.O 164y.8 3991 3781 3701 367^1 The column headed "Bode" gives the distance according to Bode's law; the column headed " Diff.," the difference between the true distance and that given by Bode's law. Ju piter % __--"'' Fig. 159. — Plan of the Orbits of the Planets inside of Saturn. 296 THE PLANETS. 490. Fig. 159 shows the smaller orbits of the system (including the orbit of Jupiter) drawn to scale, the radius of the earth's orbit being taken as one centimetre. On this scale the diameter of Saturn's orbit would be 19 cm .08, that of Uranus would be 38 cm .36, and that of Neptune, 60 cm .ll. The nearest fixed star on the same scale would be about a mile and a quarter away. It will be seen that the orbits of Mercury, Mars, and Jupiter are quite distinctly "out of centre " with respect to the sun. This is intentional and correct. The dotted half of each orbit is that which lies below, i.e., south of, the plane of the ecliptic. The place of perihelion of each planet's orbit is marked with a P. The orbits of five of the asteroids, includ- ing the nearest and the most remote, as well as the most eccentric, are also given. Periods. — The sidereal period of a planet is the time of its revolu- tion around the sun from a star to the same star again, as seen from the sun. The synodic period is the time between two successive con- junctions of the planet with the sun, as seen from the earth. The sidereal and synodic periods are connected by the same relation as the sidereal and synodic months (Art. 232) ; namely, — 1 = 1-1 S P E' in which E, P, and S are respectively the periods of the earth and of the planet, and the planet's synodic period, and the numerical differ- ence between — and — is to be taken without regard to sign. Th" P E two last columns of the table in Article 489 give the approximate / periods, both sidereal and synodic, for the different planets. 491. Apparent Motions. — As viewed from a distant point on the line drawn through the sun, perpendicular to the plane of the eclip- tic, the planets would be seen to travel in their nearly circular orbits with a regular motion. As seen from the earth the apparent motion is much more complicated, being made up of their real motion around the sun combined with an apparent motion due to the earth's own movement. 492. Law of Relative Motion. —The motion of a body relative to the earth can be very simply stated. It is always the same as if the body had, combined with its own motion, another motion, identical with that of the earth, but reversed. COMBINATION OF EARTH'S AND PLANET'S MOTIONS. 297 The proof of this is simple. Let E, Fig. 160, be the earth, and P the planet, its direction and distance being given by the line EP. Let E have a motion which will take it to E' in a unit of time, and P a motion which will take it to P' in the same time. Then at the end of a unit of time the distance and direction of P from E will be given by the line E'P'. But if we suppose E to remain at rest, and give to P a motion Pe equal to EE' but Flo. 160. — The Relative Motions of Two Bodies. opposite iu direction, and combine this motion with PP' by drawing th*» parallelogram of motions, we shall get P" for the resulting place of P at the end of the unit of time ; and because the line EP" is parallel and equal to E'P' (as follows from the construction), the -point P", as seen from E. would occupy, in the celestial sphere, precisely.the same position as P' seen from E' ; since all parallel lines pierce the sphere at one and the same opti- cal point (Art. 7). If, therefore, the earth moves in a circle, every body really at rest will appear to move in a circle of the same size as the earth's orbit, but keeping in such a part of its circle as always to have its motion precisely opposite to the earth's own real motion at the moment. We shall have occasion to use this principle very frequently. 493. Effect of the Combination of the Earth's Motion with that of the Planet. — As a consequence, the apparent or "geocentric" motion of a planet must be made up of two motions, — that of a body moving once a year around the circumference of a circle equal to the earth's Pig. 161. Geocentric Motion of Jupiter from 1708 to 1720. (Cassini.) 298 THE PLANETS. orbit, while the centre of this circle itself goes around the sun upon the real orbit of the planet, and with a periodic time equal to that of the planet. Jupiter, for instance, appears to move as in Fig. 161, making eleven loops in each revolution, the smaller circle having a diameter of about one-fifth that of the larger one, upon which its centre moves, since the diameter of Jupiter's orbit is about five times that of the earth's. 494. Direct and Retrograde Motions of the Planets and Stationary Points. — As a consequence of this looped motion we have the pecu- liar back-and-forth movement of the planets among the stars which Superior Conjunction Opposition Fia. 162. — Planetary Configurations and Aspects. has been described. Starting from the time when the sun is between us and the planet, — the time of superior conjunction, 1 as it is called, because the planet is then above the sun, i.e., further from the earth, 1 We give Fig. 162 to illustrate the meaning of the different terms, Opposition, Quadrature, Inferior and Superior Conjunction, and Greatest Elongation. E is the position of the earth, the inner circle being the orbit of an inferior planet, while the outer circle is the orbit of a superior planet. In general, the angle PES (the angle at the earth between lines drawn from the earth to the planet and to the sun) is the planet's elongation at the moment. For a superior planet it can have any value from zero to 180° ; for an inferior it has a maximum value that the planet cannot exceed, depending upon the diameter of its orbit. MOTrOK OF THE PLANETS. 299 — the planet moves eastward among the stars for a certain time, con- tinually increasing its longitude (and also its right ascension) until at last its apparent motion slackens and it becomes stationary. The dis- tance of this stationary point from the sun depends upon the size of the planet's orbit compared with that of the earth. Then it reverses its motion and moves, westward, or "retrogrades," for a while, the middle of the arc of retrogression being passed at the time when the earth and planet are in line with the sun, arid on the same side of it. If the planet is one of the outer ones, it will then be opposite to the sun in the sky like the full moon, and is said to be " i)i opposition." If the planet is one of the inferior planets (Venus or Mercury), it will then be in " inferior conjunction," as it is called, between the earth and sun. After the planet has completed its arc of retrogression, it again becomes stationary, turns upon its course, and once more advances eastward among the stars, until the synodic period is completed by its re-arrival at superior conjunction. Both in the number of degrees passed over, and in the time spent in this motion, the eastward or "direct" motion always exceeds the retrograde. In the case of the remoter planets the excess is small — from 3° to 10° ; in the case of the nearest ones, Mars and Venus, it is from 16° to 18°. As observed with a sidereal clock, all the planets come later to the meridian each night when moving direct, since their right ascension is then increasing ; but vice versa, of course, when they are retrograding. 495. Motion of the Planets with Respect to the Sun's Place in the Sky. Change of Elongation. — The visibility of a planet depends mainly upon its angular distance, or "elongation," from .the sun, be- cause when near the sun the planet will be above the horizon only by day, and cannot usually be seen. As regards their motions, consid- ered from this point of view, there is a marked difference between the inferior planets and the superior. 496. Behavior of a Superior Planet. — The superior planets drop always steadily westward with respect to the sun's place in the heavens, continually increasing their western elongation, or decreasing their eastern : they therefore invariably come earlier to the meridian every successive night, as observed by a time-piece keeping solar time. Beginning at superior conjunction, the planet is then moving eastward among the stars with its greatest speed ; but even then its eastward motion 300 THE PLANETS. is not so great as the sun's, and so the planet relatively falls westward. After a while it will have fallen behind by 90°, and will then be in westernquad- rature, and on the meridian at sunrise ; at the end of half its synodic period it will have lost 180°, and will be just opposite the sun at sunset, being then at its least possible distance from the earth, and at its greatest brilliance. At this time the difference between the times of its daily culminations is also the greatest possible, and may be as much (in the case of Mars) as six minutes, by which amount it arrives at the meridian earlier each successive night. After opposition the planet is higher in the sky each night at sunset until it reaches eastern quadrature, when it is 90° east of the sun, and therefore on the meridian at sunset. Thence it drops back, falling more and more slowly westwards towards the sun, until the synodic period is completed by a new conjunction. 497. Motion of an Inferior Planet. — The inferior planets appear, on the other hand, to vibrate across the sun, moving out equal dis- tances on each side of it, but making the westward swing much quicker than the eastern. The reason of this difference is obvious from Fig. 162. Matters take place with respect to the earth, sun, and planet as if the earth were at rest, and the planet revolving around the sun once in a syn- odic (not sidereal) period. Now, since the distance between the points of greatest elongation, Fand V, is less through inferior con- junction J, than from V around to V through C, the time ought to be correspondingly shorter, as it is. At superior conjunction the planet is moving eastward faster than the sun. Accordingly, it creeps out to the east from the sun's rays, becoming visible in the twilight as an evening star. As long as its direct motion is greater than the sun's it keeps receding from it until it reaches its "greatest eastern elongation," as it is called, which could in no case be greater than 90°, even if the planet's distance from the sun were almost equal to that of the earth. (In the case of Venus it is actually about 47°, while for Mercury it ranges from 18° to 28°. ) Then as its eastern motion slackens the sun begins to overtake it, and when the planet becomes really stationary as regards its motion among the stars, it appears to be slipping westward towards the sun at the rate of about a degree a day. At the stationary point it begins really to retrograde, and adds its motion to the sun's advance, so that from that point it rushes swiftly towards the inferior conjunction. It passes this and runs out quickly on the western side, becoming a morning star, and reach- ing its western elongation in just the same number of days that it took to drop from eastern elongation to inferior conjunction. When the elongation has been gained, the planet turns around to pursue the sun, gradually gains upon it, and at last overtakes it again at the next superior conjunction, hav- ing completed its synodic period. MOTIONS IN LATITUDE. 301 499. Motions in Latitude. —If the planets' orbits lay precisely in the same plane with each other and with the earth's orbit, they would always keep in the ecliptic. But in fact, while they never go far Fig. 163. — Loops of Regression. from that circle, they do deviate north or south to the extent of 5° or 6°, and Mercury sometimes as much as 8° ; their paths among the stars are consequently loops and kinks like those of Fig. 163, which represent forms actually observed. 500. The Ptolemaic System. — The ancient astronomers, for the most part, never doubted the fixity of the earth, and its position in the centre of the celestial universe, though there are some reasons to think that Pythagoras may have done so. Assuming this and the actual diurnal revolution of the heavens, Ptolemy, who flourished at Alexandria about 140 a.d., worked out the system which bears his name. His McydXij 2wra£is (or Almagest in Arabic) was for fourteen centuries the authoritative " Scripture of Astronomy." He showed that all the apparent motions of the planets could be accounted for by supposing each planet to move around the circumference of a circle called the "epicycle," while the centre of this circle, sometimes called the " fictitious planet," itself moved on the circumference of another and larger circle called the " deferent." It was as if the real planet was carried on the end of a crank-arm which turned around the fictitious planet as a centre, in such a way as to point towards or from the earth at times when the planet is in line with the sun. In the case of the superior planets the revolution in the epicycle was made once a year, so that the " crank-arm " was always parallel to the line joining earth and sun, while the motion around the deferent occupied what we now call the planet's period. Fig. 164 represents the Ptolemaic System, except that no attention is paid to dimensions, the " deferents " being spaced 302 THE PLANETS. at equal distances. It will be noticed that the epicycle-radii which carry at their extremities the planets Mars, Jupiter, and Saturn are all always parallel to the line that joins the earth and sun. In the case of Venus and Mercury this was not so. Ptolemy supposed that the deferent circles for these planets lay between the earth and the sun, and that the "fictitious planet" in both Fig. 164. — The Ptolemaic SyBtem. cases revolved in the deferent once a year, always keeping exactly between the earth and the sun : the motion in the epicycle in this case was completed in the time oE the planet's period, as we now know it. He ought to have seen that, for these two planets, the deferent was really the orbit of the sun itself, as the ancient Egyptians are said to have understood. 501. To account for some of the irregularities of the planets' motions it was necessary to suppose that both the deferent and epicycle, though circular, are eccentric, the earth not being exactly in the centre of the deferent, nor the " fictitious planet " in the exact centre of the epicycle. In after times, when the knowledge of the planetary motions had become more accurate, the Arabian astronomers added epicycle upon epicycle until the system became very complicated. King Alphonso of Spain is said to have remarked to the astronomers who presented to him the Alphonsine tables of the planetary motions, which had been computed under his orders, that " if he had been present at the creation he would have given some good advice." 502. Some of the ancient astronomers attempted to account for the plan- etary and stellar motions in a mechanical way by means of what were called the " crystalline spheres." The planet Jupiter, for instance, was supposed to THE COPERNICAN SYSTEM. 303 be set like a jewel on the surface of a small globe of something like glass, and this itself was set in a hollow made to fit it in the thick shell of a still larger sphere which surrounded the earth. Thus the planets were suppoi;. - 1 and carried by the motions of these invisible crystalline spheres ; but this idea, though prevalent, was by no means universally accepted. 503. Copernican System.— Copernicus (1473-1543) asserted tb ; diurnal rotation of the earth on its axis, and showed that it wou d fully account for the apparent diurnal revolution of the stars. Vie also showed that nearly all the known motions of the planets cojld be accounted for by supposing them to revolve around the sun, ffith the earth as one of them, in orbits circular, but slightly out of c'atve. His system, as he left it, was nearly that which is accepted t ;-day, and Fig. 159 may be taken as representing it. He was, hf wever, obliged to retain a few small epicycles to account for certaii of the irregularities. So far, no one dared to doubt the exact circularity of celestial orbits. It was metaphysically improper that heavenly bodi s should move in any but perfect curves, and the circle was regard d as the only perfect one. It was left for Kepler, some sixty-five y ars later than Copernicus, to show that the planetary orbits are elliptical, and to bring the system substantially into the form in which we kno .r it now. 504. Tychonic System. — Tycho Brahe, who came between ' opernicus and Kepler, found himself unable to accept the Copernican syst m for two reasons. One reason was that it was unfavorably regarded by the clergy, and he was a good churchman. The other was the scientific ob : iction that if the earth moved around the sun, the fixed stars all ought to ap; ear to move in a corresponding manner (Art. 492), each star describing anm ally an oval in the heavens of the same apparent dimensions as the earth's orbit itself, seen from the star. Technically speaking, they ought to hav an "annual parallax." His instruments were by far the most accurate tb t had so far been made, and he could detect no such parallax ; hence he ■ oncluded, not illogically, but incorrectly, that the earth must be at rest. B i rejected the Copernican system, placed the earth at the centre of the unft ,rse, according to the then received interpretation of Scripture, made the sun revolve around the earth once a year, and then (this was the peculiarity of h' j system) made all the planets except the earth revolve around the sun. This theory just as fully accounts for all the motions o the planets as the Copernican, but breaks down absolutely when it encou iters the aberra- tion of light, and the annual parallax of the stars, which /e can now detect with our modern instruments, although Tycho could n t with his. The Tychonic system never was generally accepted ; the & pernican was very soon firmly established by Kepler and Newton. 304 THE PLANETS. 505. Elements of a Planet's Orbit. — These elements are the numerical quantities which must be given in order to describe the orbit with precision, and to furnish the means of finding the planet's place in the orbit at any given time, whether past or future. They are seven in number, as follows : — 1. The semi-major axis, a. 2. The eccentricity, e. 3. The inclination to the ecliptic, *. 4. The longitude of the ascending node, JJ. 5. The longitude of perihelion, -k. 6. The epoch, E. 7. The period P, or daily motion ju. 506. Of these, the first five pertain to the orbit itself, regarded as an ellipse lying in space with one focus at the sun, while two are necessary to determine the planet's place in the orbit. The semi-major axis, a (CA in Fig. 165), defines the Size of the Fig. 165. — The Elements of a Planet's Orbit. orbit, and may be expressed either in "astronomical units" (the earth's mean distance from the sun is the astronomical unit) or in miles. The Eccentricity defines the orbit's Form. It is a mere numerical quantity, being the fraction - (usually expressed decimally), obtained by dividing the distance between the sun and the centre of the orbit by the semi-major axis. In some computations it is convenient to use, instead of the decimal fraction itself, the angle which has e for its sine; i.e., <£ = sin _1 e. ELEMENTS OF A PLANET'S ORBIT. 305 The third element, i, is the Inclination between the plane of the planet's orbit and that of the earth. In the figure it is the angle KNO, the plane of the ecliptic being lettered EKLM. The fourth element, & (the Longitude of the ascending node), defines what has been called the "aspect" of the orbit-plane ; i.e., the direction in which it faces. The line of nodes is the line NIP in the figure, the intersection of the two planes of the orbit and ecliptic ; and the angle TSN is the longitude of the ascending node, the line Sf being the line drawn from the sun to the first of Aries. The planet, moving around its orbit in the plane ORBT, and in the direc- tion of the arrow, passes from the lower or southern side of the plane of the ecliptic to the northern at the point n, which, as seen from 8, is in the same direction as N. The fifth, and last, of the elements which belong strictly to the orbit itself is -k, the so-called Longitude of the perihelion, which de- fines the direction in which the major axis of the ellipse (the line PA) lies on the plane ORBT. It is not strictly a longitude, but equals the sum of the two angles Q, and o> ; i.e., TSN (in the plane of the ecliptic) + NSP (in the plane of the orbit). It is quite sufficient to give to alone, and in the case of cometary orbits this is usually done. 507. If we regard the orbit as an oval wire hoop suspended in space, these five elements completely define its position, form, and size. The plane of the orbit is fixed by the elements numbered three and four, the position of the orbit in this plane by number five, the .form of it by number two, and finally its magnitude by number one. The student will recollect that the general equation of curves of the second degree (the conies) in analytical geometry contains five constants, and therefore that number of data is enough to define such a curve completely. 508. To determine where the planet will be at any subsequent date we need two more elements. Sixth. The Periodic Time, — we must have the sidereal period, P, or else the mean daily motion, fi, which is simply 360° divided by the number of days in P. Seventh. And finally ; we must have a starting-point, the " Epoch," so-called ; i.e., the longitude of the planet as seen from the sun, at some given date, usually Jan. 1st, 1850 or 1900, or else some precise date at which the planet passed the perihelion or node. 306 THE PLANETS. 509. If it were not for perturbations caused by the mutual interaction between the planets, these elements would never change, and could be used directly for computing the planet's place at any date in the past or in the future; bat, excepting a and P, they do change on account of such interac- tion, and accordingly it is usual to add in tables of the planetary elements, columns headed A£, Air, At, and Ae, giving the amount by which the quantities &, it, i, and e respectively change in a century. 510. If Kepler's harmonic law were strictly true, we should not need both a and P, because we should have (Earth's Period) 2 : P 2 : : l 3 : a 8 , or P = a\ P being expressed in years and a in astronomical units. But since the exact form of the equation is P^ (1 + m,) : P./ (1 + m 2 ) : : af : af (Art. 417), it is necessary, in cases where the highest attainable accuracy is required, to regard P and a as independent, and give them both in the tables. 511. Geocentric Place. — Our observations of a planet's place are necessarily "geocentric," or earth-centred; they give us, when prop- erly corrected for refraction and parallax, the planet's right ascension and declination as seen from the centre of the earth, and from them, if desired, the corresponding geocentric longitude and latitude are easily obtained by the method explained in Article 180. It often happens that we want the place at some moment of time when the planet could not be directly observed, as, for instance, in the day time. If we have a series of observations of the planet made about that time, the place for the exact moment is readily deduced by a process of interpolation, axta with an accuracy actually exceeding that of any single observation of/'tfie series. Graphically it is done by simply plotting the observations Actually made. Suppose, for instance, we want the right ascension of Mars for 8 a.m. on June 3, and have meridian-circle observations made at 10 o'clock p.m. on June 1, at 9 h 55 m on June 2, at 9 h 50 m on June 3, and so on. "We first lay off the times of observation as abscissas along a horizontal line taken as the time-scale, and then lay off the observed right ascensions as ordinates. at points corresponding to the times. Then we draw a smooth curve through the points so determined, and from this curve we can read off directly, the right ascension corresponding to any desired moment: similarly for the declination. Of course, what can be done graphically can be done still more accurately by computation. 512. Heliocentric Place. — The heliocentric place of a planet is the place as seen from the sun ; and when we know the longitude of the node of a planet's orbit and its inclination, as well as the planet's dis- DETERMINATION OF A PLANET'S PERIOD. 307 tance from the sun, this heliocentric place can at once be deduced from the geocentric by a trigonometrical calculation. The process is rather tedious, however, and its discussion lies outside the scope of this work. (The reader is referred to Watson's " Theoretical Astronomy," p. 86. An elementary geometrical treatment of the reduction is also given in Loomis's "Treatise on Astronomy," p. 211.) 513. Determination of the Period of a Planet. — This can be done in two ways : First. By observation of its node-passage. When the planet is passing its node, it is in the plane of the ecliptic, and the earth being also always in that plane, the planet's latitude, both geocentric and heliocentric, will be zero, no matter what may be the place of the earth in its orbit. " (At any other point of the planet's orbit except the node its apparent latitude would not be thus independent of the earth's place, but would vaiy according to its distance from the earth.) If, then, we observe the planet at two successive passages of the same node, the interval between the moments when the latitude becomes zero will be .the planet's period, — exactly, if the node is stationary ; very approximately, even if the node is not absolutely stationary, as none of the nodes actually are. There are two difficulties with this method. (a) In the case ©f"U*anus the period is eighty-four years, and in that of Neptune 164 years — toolong to wait. (6) Since the orbits all cross the ecliptic at a very small angle, so that the latitude remains near zero for a number of days, it is extremely difficult to determine the precise minute and second when it is exactly zero ; and slight errors in the declinations observed will produce great errors in the result. 514. Second. By the mean synodic period of the planet. The synodic period is the interval between two successive oppositions or conjunctions of the planet, the opposition being the moment when the planet's longitude differs from that of the sun by 180°. This angle between the planet and sun cannot well be measured directly, but we can make with the meridian circle a series of observations both of the planet's right ascension and declination for several days before and after the date of opposition, and reduce the observations to latitude and longitude. The sun will be observed, of course, at noon, and the planet near midnight ; but from the solar observations we can deduce the longitudes of the sun corresponding to the exact moments when the planet was observed. From these we find the difference of longitude between the planet and the sun at the time of each planetary observation ; and finally from these differences 308 THE PLANETS. of longitude, we find the precise moment when that difference was exactly 180°, or the moment of opposition. This can be ascertained within a very few seconds of time if the observations are good. Since the orbits are not strictly circular, the interval between two successive observations will not be the mean synodic period, but only an approximation to it ; but when we know it nearly, we can compare oppositions many years apart, and by dividing the interval by the known number of entire synodic periods (which is easily determined when we know the approximate length of a single period) we get the mean synodic period very closely, — especially if the two oppositions occur at about the same time of the year. Having the synodic period, the true sidereal period at once follows from the equation P E S 515. To find the Distance of a Planet in Terms of the Earth's Distance. — When we know the planet's sidereal period, this is easily done by means of two observations of the planet's "elongation" *k Fig. 166. — Determination of the Distance of a Planet from the Sun. taken at an interval equal to its periodic time. The " elongation" of a planet is the difference between its longitude and that of the TO FIND THE DISTANCE OF A PLANET. 309 sun, and a series of meridian-circle observations . of sun and planet will furnish these differences of longitude for any selected moment included within the term of observation. To find the distance of the planet Mars, for instance, we must therefore have two observations separated by an interval of 687 days. Suppose the earth to have been at A (Fig. 166) at the moment of the first observation. Then at the time of the second observation she will be at the point C, the angle ASG being that which the earth will describe in the next 43|- days, which is the difference between two complete years (or 730|- days) and the 687-day interval between the two observations. The angles SCM and SAM are the " elongations" of the planet from the sun, and are given directly by the observations. The two sides 8A and SO are also given, being the earth's distance from the sun at the dates of observation. Hence we can easily solve the quadrilateral, and find the length of SM, as well as the angle ASM. This angle determines the planet's heliocentric longitude at M, since we know the direction of SA, the longitude of the earth at the time of observation. The student can follow out for himself the process by which, from two elongations of Venus, SA V and SB V, observed at an interval of 225 days, the distance of Venus from the sun (of SV) can be obtained. 516. In order that this method may apply with strict accuracy it is necessary that at the moment of observation M should be in the same plane as A, S, and C ; that is, at the node. If it is not so, the process will give us, not the true distance of the planet itself from the sun, but that of the " pro- jection " of this distance on the plane of , ' ' . ----> the ecliptic ; i.e., the distance from the sun to the point m (Fig. 167), where the per- pendicular from the planet would strike ^f that plane. But when we have determined Fia _ 167 , Am and the angle mAM, the planet's geo- centric latitude, we easily compute Mm; and from Sm and Mm we get the true distance SM and the heliocentric latitude of the planet MSm. 517. From a series of pairs of observations distributed around the planet's orbit it would evidently be possible to work out the orbit completely. It was in this way that Kepler showed that the orbits of the planets are ellipses, and deduced their distances from 310 THE PLANETS. the sun ; and his third, or harmonic law, was then discovered simply by making a comparison between the distances thus found and the corresponding periods. 518. Mean Distance of an Inferior Planet by Means of Observa- tions of its Greatest Elongation. — By observing from the earth the greatest elongation SEV (Fig. 168) of one of the inferior planets, its distance from the sun can very easily be deduced if we regard the orbit as a circle ; for the triangle SVE will be right-angled at V, and SV= SE x sin SEV. In the case of Venus the orbit is so nearly circular that the method answers very well, the greatest elonga- tion never differing much from 47°. Mercury's orbit is so eccentric that the distance thus obtained from a single elongation might be very wide of the true mean distance. Since the great- est elongation, SEM, varies all the way from 18° to 28°, it would be neces- sary to observe a great many elongations, and take the average result. Fig. 168. Distance of an Inferior Planet determined by Observations of its Greatest Elongation. 519. Deduction of the Orbit of a Planet from Three Observations. — When one has command of a great number of observations of a planet running back many years, and can select such as are conven- ient for his purpose, as Kepler could from Tycho's records, it is comparatively easy to find the elements of a planet's orbit ; but when a new planet is discovered, the case is different. The problem first arose practically in 1801, when Ceres, the first of the asteroids, was discovered by Piazzi in Sicily, observed for a few weeks and then lost in the sun's rays at conjunction, before other astronomers could be notified of the discovery, in those days of slow communication, made slower and more uncertain by war. Gauss, then a young man at Gottingen, attacked the problem, and invented the method which, with slight modifications, is now univer- sally used in such cases. We do not propose to enter into details, but simply say that three absolutely accurate observations of a planet's right ascension and declination are sufficient to determine its orbit. Three observations, made only as accurately as is now possible, with intervals of two or PLANETARY PERTURBATIONS. 311 three weeks between them, will give a very good approximation to the orbit ; and it ean then be corrected by further observations. 520. " Since there are five independent variables in the general equation (in space) of a conic having a given focus — the sun — it is necessary to have five conditions in order to determine them. Three are given by the observations themselves; viz., the directions of the body as seen from the earth at three given instants; a fourth is supplied by the 'law of equal areas,' since the sectors included between the three radii vectores must be proportional to the elapsed times ; finally, the fifth is imposed by the require- ment that the changes in the speed of the body must correspond to the vari- ations in the length of the radius vector, in accordance with the known intensity of the sun's attraction." (The student is referred to Gauss's "Theoria Motus," or to Watson's " Theoretical Astronomy," or to Oppolzer's great work on " The Determina- tion of Orbits," for the full development of the subject. ) 521. Planetary Perturbations. — The attraction of the planets for each other disturbs their otherwise elliptical motion around the sun. As in the case of the lunar theory the disturbing forces are, however, always relatively small, but not for the same reason. The sun's disturbing force is small because its distance from the moon is nearly four hundred times that of the earth. In the planetary theory the disturbing bodies are often nearer to the disturbed than is the sun itself, as, for instance, in the disturbance of Saturn by Jupiter at certain points of their orbits ; but the mass of the disturbing body in no case is as great as t fa o P art °/^ e sun's mass, and for this reason the disturbing force arising from planetary attraction is never more than a small fraction of the sun's attraction. The greatest disturbing force which occurs in the planetary system (except in the case of some of the asteroids) is that of Jupiter on Saturn at the time when the planets are nearest : it then amounts to T jj of the sun's attraction. When these two planets are most remote from each other, it amounts to ^ 7 . There is no other case where the disturbing force is as much as -^^ of the sun's attraction (again excepting the asteroids disturbed by Jupiter). 522. In any special case the disturbing force can be worked out on precisely the same principles that lie at the foundation of the diagram by which the sun's disturbing force upon the moon was found (Art. 441, Fig. 147) ; but the resulting diagram will look very different, because the disturbing body is relatively very near the disturbed orbit. 312 THE PLANETS. The planetary perturbations which result from the "integration" of effects of the disturbing forces, i.e., from their continual action through long intervals of time, divide themselves into two great classes, — the Periodic and the Secular. 523. Periodic Perturbations. — These are such as depend on the positions of the planets in their orbits, and usually run through their course in a few revolutions of the planets concerned. For the most part they are very small. Those of Mercury never amount to more than 15", as seen from the sun. Those of Venus may reach about 30", those of the earth about 1', and those of Mars about 2'. The mutual disturbances between Jupiter and Saturn are much larger, amounting respectively to 28' and 48' ; while those of Uranus are again small, never exceeding 3', and those of Neptune are not more than half as great as that. In the case of the asteroids, which are powerfully disturbed by Jupiter, the periodical perturbations are enormous, sometimes as much as 5° or 6°. 524. Long Inequalities. — The periodic inequalities of the planets are so small, because, as a rule, there is a nearly complete compensation effected at every few revolutions, so that the accelerations balance the retardations. The line of conjunction falls at random in different parts of the orbits, and when this is the case, no considerable displacement of either planet can take place. But when the periodic times of two planets are nearly com- mensurable, their line of conjunction will fall very near the same place in the two orbits for a considerable number of years, and the small unbalanced disturbance left over at each conjunction will then accumulate in the same direction for a long time. Thus, five revolutions of Jupiter roughly equal two of Saturn ; and still more nearly, seventy-seven of Jupiter equal thirty- one of Saturn, in a period of 913 years. From this comes the so-called "long inequality" ol Jupiter and Saturn, amounting to 28' in the place of Jupiter and 48' in that of Saturn, and requiring more than 900 years to complete its cycle. In the case of the earth and Venus there is a similar "long inequality" with a period of 235" years, amounting, however, to less than 3" in the posi- tions of either of the planets ; between Uranus and Neptune there is a similar inequality with a period of over 4000 years, but this also is very small. 525. Secular Inequalities. —These are inequalities which depend not on the position of the planets in their orbits, but on the relative position of the orbits themselves, with reference to each other, — the way, for instance, in which the lines of nodes and apsides of two neighboring orbits lie With reference to each other. Since the plane- tary orbits change their positions very slowly, these perturbations, SECULAR INEQUALITIES. 313 although in the strict sense of the word periodic also, are very slow and majestic iu their march, and the periods involved are such as stagger the imagination. They are reckoned in myriads and hun- dreds of thousands of years. From year to year they are insignifi- cant, but with the lapse of time become important. 526. Secular Constancy of the Periods and Mean Distances. — It is a remarkable fact, demonstrated by Lagrange and La Place about 100 years ago, that the mean distances and periods are en- tirely free from all such secular disturbance. They are subject to slight periodic inequalities having periods of a few years, or even a few hundred years : but in the long run the two elements never change. They suffer no perturbations which depend on the position of the orbits themselves, but only such as depend on the positions of the planets in their orbits. 527. Revolution of the Nodes and Apsides. — The nodes and peri- helia, on the other hand, move on continuously. The lines of apsides of all the planets (Venus alone excepted) advance, and the nodes of all without exception (except possibly some of the asteroids), regress on the ecliptic. The quickest moving line of apsides — that of Saturn's orbit — completes its revolution in 67,000 years, while that of Neptune requires 540,000. The swiftest line of nodes is that of Uranus, which completes its circuit in less than 37,000 years, while the slowest — that of Mercury — requires 166,000 years. 528. The Inclinations of the Orbits. —These are all slowly chang- ing — some increasing, and others decreasing; but as La Place and Leverrier have shown, all the changes are confined within narrow limits for all the larger planets : they oscillate, but the oscillations are never extensive. It is not certain that this is so with the asteroids, some of which have inclinations to the ecliptic of 25° and 30°: it is possible that some of these inclinations may change by a very considerable amount. 529. The Eccentricities. — These also are slowly changing in the same way as the inclinations, some increasing and some decreasing ; and their changes also are closely restricted. The periods of the alter- nate increase and decrease are always many thousand years in length. The asteroids are again to be excepted ; the eccentricities of their orbits may change considerably. 314 THE PLANETS. 530. Stability of the Planetary System. — About the end of the eighteenth century La Place and Lagrange succeeded in proving that the mutual attraction of the planets could never destroy the system, nor even change the elements of the orbit of any one of the larger planets to an extent which would greatly alter its physical condition. The nodes and apsides revolve continuously, it is true, but that change is of no importance. The distances from the sun and the periods do not change at all in the long run ; while the inclinations and eccentricities, as has just been said, confine their variations within narrow limits. 531. The "Invariable Plane" of the Solar System. — There is no reason, except the fact that we live on the earth, for taking the plane of the earth's ovbit (the plane of the ecliptic) as the fundamental plane of the solar system. There is, however, in the system an " invariable plane," the position of which remains forever unchanged by any mutual action among the planets, as was discovered by La Place in 1784. This plane is defined by the following conditions, — that if from all the planets perpendiculars be drawn to it (i.e., to speak technically, if the planets be "projected " upon it), and then if we multiply each planet's mass by the area which the planet's pro- jected radius vector describes upon this plane in a unit of time, the sum of these products will be a maximum. The ecliptic is inclined about 1J° to this invariable plane, and has its ascending node nearly in longitude 286°. 532. La Place's Equations for the Inclinations and Eccentrici- ties. — La Place demonstrated the two following equations, viz. : (1) 2(mVaXe 2 ) = C. (2) 2 (mv^x tan 2 i) = C". Equation (1) may be thus translated : Multiply the mass of each planet by the square root of the semi-major axis of its orbit, and by the square of its eccen- tricity; add these products for all the planets, and the sum will be a constant quantity C, which is very small. It follows that no eccentricity can become very large, since e 2 in the equation is essentially positive : there can therefore be no counterbalancing of positive and negative eccentricities; and if the eccentricity of one planet increases, that of some other planet or planets must correspondingly decrease. The second equation is the same, merely substituting tan 2 i for e 2 , i being the inclination of the planet's orbit to the invariable plane. The constant in this case also is small, though of course not the same as in the preceding equation. These two equations, taken in connection with the invariability of the periods and the major axes of the planetary orbits, have been called the " Magna Charta " of the stability of the solar system. LA place's equations. 315 533. It does not follow that because the mutual attractions of the planets cannot seriously derange the system it is, therefore, of ne- cessity securely stable. There are many other conceivable actions which might end in its ultimate destruction ; such, for instance, as that of a resisting medium, or the entrance into the system of bodies coming from without. 316 THE PLANETS. CHAPTER XIV. THE PLANETS : METHODS OP FINDING THEIR DIAMETERS, MASSES, ETC. — THE " TERRESTRIAL PLANETS " AND ASTEROIDS. — INTRA-MERCURIAL PLANETS AND THE ZODIACAL LIGHT. In discussing the individual peculiarities of the planets, we have to consider a multitude of different data ; for instance, their diameters, their masses, and densities, their aodal rotation, their surface-markings, their reflecting power or " albedo," and their satellite systems. 534. Diameter. — The apparent diameter of a planet is ascertained by measurement with some kind of micrometer (Art. 73) . For this purpose the "double-image" micrometer has an advantage over the wire micrometer because of the effect of irradiation. When we bring two wires to touch the two limits of the planet in the field of view of the telescope, Fig. 169, a, the bright image of the planet is always measured too large, because every bright object appears to extend itself somewhat into the dark sur- b ^ — ._ rounding space, by its physiologi- cal action upon the retina of the eye. This is known as irradiation — well exemplified at the time of FlG ' 169- new moon, when the bright cres- Micrometer Measures of a Planet's Diameter. cent appears to be much larger than the " old moon " faintly visi- ble by earth-shine. With small instruments this error is often considera- ble, varying with the personal equation of the observer, but it may be reduced to some extent by using as bright an illumination of the field of view as the object will bear. With the double-image micrometer, the observer in measuring has to bring in contact two discs of equal brightness, as in Fig. 169, b ; and in this case the irradiation almost vanishes at the point of contact. The diameter thus measured is, of course, only the apparent diam- eter, to be expressed in seconds of arc, and varies with every change of distance. To get the real diameter in linear units, we have Real diameter = ^ x D ", 206265 EXTENT OF SURFACE AND VOLUME. 317 in which A is the distance of the planet from the earth, and D" the diameter in seconds of arc. If A is given only in astronomical units, the diameter comes out, of course, in terms of that unit. To get the diameter in miles, we must know the value of this unit in miles ; that is, the sun's distance from the earth. 535. Extent of Surface and Volume. — Having the diameter, the surface, of course, is proportional to its square, and is equal to the earth's surface multiplied by ( - j, in which s is the semi-diameter of the planet and p that of the earth. The volume equals ( - ] in terms of the earth's volume. (The stu- dent must be on his guard against confounding the volume or bulk of a planet with its mass.) The nearer the planet, other things being equal, the more accurately the above data can be determined. The error of 0".l in measuring the apparent diameter of Venus, when nearest, counts for less than thirteen miles in the real diameter of the planet ; while in Neptune's case it would correspond to more than 1300 miles. The student must not be surprised, therefore, at finding considerable discrepan- cies in the data given for the remoter planets by different authorities. 536. Mass of a Planet which has a Satellite. — In this case its mass is easily and accurately found by observing the period arid distance of the satellite. We have the fundamental equation {M+m)=^y in which M is the mass of the planet, m that of its satellite, r the radius of the orbit of the satellite, and t its period. The formula is derived as follows : From the law of gravitation the accel- erating force which acts on the satellite is given by the equation „ M+m r 2 (Art. 417), in which M is the mass of the planet and m that of the satellite. From the law of circular motion (Art. 411, Eq. 6) we have -^ 2 U)< M and finally 318 THE PLANETS, whence (equating the two values of /) we have *-" = *"(?)' (jr+«0= 4 «*(£). This demonstration is strictly good only for circular orbits ; but the equation is equally true, and can be proved for elliptical orbits, if for r we put a, the semi-major axis of the satellite's orbit. For many purposes a proportion is more convenient than this equa- tion, since the equation requires that M, m, r, and t be expressed in properly chosen units in order that it may be numerically true. Con- verting the equation into a proportion, we have (Jf+m):(J«i + m,)=£: % or, in words, the united mass of a body and its satellite is to the united mass of a second body and its satellite as the cube of the distance of the first satellite divided by the square of its period is to the cube of the distance of the second satellite divided by the square of its period. This enables us at once to compare the masses of any two bodies which have attendants revolving around them. The mass of the moon is so considerable as compared with that of the earth (about fa) that it will not do to neglect it ; but in all other cases the satellite is less than l0 1 00 of the mass of its primary, and need not be taken into account. 537. Examples. — (1) Required the mass of the sun compared with that of the earth. The proportion is (S + earth) : (E + moon) = ( 93000000 ) 8 : C 239000 )". v ' K ' (365J) 2 (27.4)' 2 The quantities in the last term of the proportion are of course the distance and period of the moon ; and it is to be remembered that for the period of the moon we must use, not the actual sidereal period, but the period as it would be if the moon's motion were undisturbed, — a period about an hour shorter. (2) Compare the mass of the earth with that of Jupiter, whose remotest satellite has a period of 16| days, and a distance of 1,160,000 miles. We have (E + moon) : (/ + satellite) = f 239000 )" : ( 1160000 ) 8 , ' (27.4)2 (16f) 2 MASS OF A PLANET WHICH HAS NO SATELLITE. 319 which gives the mass of Jupiter about 310 times as great as that of the earth and moon together. 538. It is customary to express the mass of a planet as a certain fraction of the sun's mass, and the proportion is simply Sun: Planet = — •.-, T 2 t 2 whence Planet's mass = Sun's mass *(f*(f)' where T and R are the planet's period and distance from the sun . Since R and r can both be determined in astronomical units without any necessity for knowing the length of that unit in miles, the masses of the planets in terms of the sun's mass are independent of any knowl- edge of the solar parallax. But to compare them with the earth, we must know this parallax, since the moon's distance from the earth, which enters into the equations, is found by observation in miles or in radii of the earth, and not in astronomical units. In order to make use of the satellites for this purpose we must determine by micrometrical observations their distances from the planets and their periods. 539. Mass of a Planet which has no Satellite. — When a planet has not a satellite, the determination of its mass is a very difficult and troublesome problem, and can be solved only by finding some pertur- bation produced by the planet, and then ascertaining, by a sort of " trial and error" method, the mass" which would produce that pertur- bation. Venus disturbs the earth and Mercury, and from these per- turbations her mass is ascertained. Mercury disturbs Venus, and also one or two comets which come near him, and in this way we get a rather rough determination of his mass. 540. Density. — The density of a body as compared with the earth is determined simply by dividing its mass by its volume ; i.e., _ m Density = -r-: . For example, Jupiter's diameter is about eleven times that of the earth (i.e. I - ]= 11), so that his volume is ll 8 , or 1331 times the earth's. His mass, derived from satellite observations, is about 316 times the earth's. The 320 THB PLANETS. density, therefore, equals T y&, or about 0.24, of the earth's density, or about 1} times that of water, the earth's density being 5.58 (Art. 171). 541. The Surface Gravity. — The force of gravity on a planet's surface as compared with that on the surface of the earth is important in giving us an idea of its physical condition. If r is the radius of the planet in terras of the earth's radius, then Surface gravity, or G, = — -^ 2 , = -^ a X ;7 i.e., it equals the planet's density, multiplied by its diameter expressed in terms of the earth's diameter. For Jupiter, therefore, G = 51|= 11 X density = 11 X 0.24 = 2.64 nearly. That is, a body at Jupiter would weigh 2.6 times as much as at the earth's surface. 542. The Planet's Oblateness. — The "oblateness" or "polar- compression " is the difference between the equatorial and polar diameters divided by the equatorial diameter. It is, of course, determined, when it is possible to determine it at all, simply by micrometric measurements of the difference between the greatest and least diameters. The quantity is always very small and the observa- tions delicate. 543. The Time of Rotation, when it can be determined, is found by observing the passage of some spot visible in the telescope across the central line of the planet's disc. In reducing the observations to find the interval between such transits, account has to be taken of the continual change in the direction of the line which joins the planet and the earth, and also of the variations in the distance, which will alter the time taken by light in coming to the earth from the body. 544. The Inclination of the Axis is deduced from the same obser- vations which are used in obtaining the rotation period. It is neces- sary to determine with the micrometer the paths described by differ- ent spots as they move across the planet's disc. It is possible to ascertain it with accuracy for only a very few of the planets : Mars, Jupiter, and Saturn are the only ones that furnish the needed data. 545. The Surface Peculiarities and Topography of the surface are studied by the telescope. The observer makes drawings of any SPECTEOSCOPIC PECULIARITIES AND ALBEDO. 321 markings which he may see, aud by their comparison is at last able to discriminate between what is temporary and what is permanent on the planet. Mars alone, thus far, permits us to make a map of its surface. 546. Spectroscopic Peculiarities and Albedo. — The character- istics of. the planet's atmosphere can be to some extent studied by means of the spectroscope, which in some few cases shows the pres- ence of water-vapor and other absorbing media, by dark bands in the planet's spectrum. The "albedo," or reflecting power of a planet's surface is determined by photometric observations, com- paring it with a real or artificial star, or with some other planet. 547. The Satellite System of a Planet. — The principal data to be ascertained are the distances and periods of the satellites, and the observations are made by measuring the apparent distances and directions of the satellites from the centre of the planet with the wire micrometer (Art. 73). Observations made at the times when the satellite is near its elongation are especially valuable in determining the distance. If the planet and earth were at rest, the satellite's path would appear to be an ellipse, unaltered in dimensions during the whole series of observa- tions ; but since the earth and planet are both moving, it becomes a compli- cated problem to determine the satellite's true orbit from the ensemble of observations. 548. With the exception of the moon and the outer satellite of Saturn, all the satellites of the planetary system move almost exactly in the plane of the equator of the primary ; and all but the moon and the seventh satellite of Saturn (Hyperion) move in orbits almost circular. La Place and Tisse- rand have shown that the equatorial protuberance of a planet compels any satellite which is not very remote from its primary to move nearly in the equatorial plane, but the almost perfect circularity of the orbits is not yet explained. When there are a number of satellites in a system, interesting problems arise in connection with their mutual disturbances ; and in a few cases it becomes possible to determine a satellite's mass as compared with that of its primary. In several instances satellites show peculiar variations in their brightness, which are supposed to indicate that they make an axial rotation in the time of one revolution around the primary, in the same way as our moon does. 549. Humboldt's Classification of the Planets. — Humboldt has divided the planets into two groups : the terrestrial planets, so called, 322 THE PLANETS. and the major planets. The terrestrial planets are Mercury, Venus, the earth, and Mars. They are bodies of the same order of magni- tude, ranging from 3000 to 8000 miles in diameter, not very differ- ent in density (though Mercury is much denser than either of the others), and are probably roughly alike in physical consti- tution, and covered with water and air. But we hasten to say that the differences in the amount of heat and light which they receive from the sun, and in the force of gravity upon their surfaces, and probably in the density of their atmospheres, are such as to bar any positive conclusions as to their being the abode of life resembling the forms of life with which we are acquainted on the earth. 550. The four major planets, Jupiter, Saturn, Uranus, and Nep- tune, are much larger bodies (ranging in diameter between 30,000 and 90,000 miles), are much less dense, and so far as we can make Fig. 170. — Relative Sizes of the Planets. out, present to us only a surface of cloud, and may not have anything solid about them. There are some reasons for suspecting that they are at a high temperature ; in fact, that Jupiter is a sort of semi-sun; but this is by no means yet certain. As for the multitudinous asteroids, the probability is that they represent a single planet of the terrestrial group which, as has been intimated, failed for some reason in its evolution, or else has MERCURY. 323 been broken to pieces. All of them united would not make a planet one-half the mass of the earth. Fig. 170 shows the relative sizes of the different planets. In what follows, all the numerical data, so far as they depend on the solar parallax, are determined on the assumption that that paral- lax is 8". 80, and that the sun's mean distance is 92,897000 miles. MERCURY. 551. There is no record of the discovery of the planet. It has been known from remote antiquity ; and we have recorded observa- tions running back to B.C. 264. For a time the ancient astronomers seem to have failed to recognize it as the same body on the eastern and western sides of the sun, so that the Greeks had for a time two names for it, — Apollo when it was morning star, and Mercury when it was evening star. According to Arago, the Egyptians called it Set and Horus, and the Hindoos also gave it two names. It is so near the sun that it is comparatively seldom seen with the naked eye ; but when near its greatest elongation it is easily enough visible as a brilliant star of the first magnitude low down in the twi- light, perhaps not quite so bright as Sirius, but certainly brighter than Arcturus. It is usually visible for about a fortnight at each elonga- tion, and is best seen in the evening at such eastern elongations as occur in March and April. In Northern Europe it is much more difficult to observe than in lower latitudes, and Copernicus is said never to have seen it. Tycho, however, obtained a considerable number of observations. 552. It is exceptional in the solar system in a great variety of ways. It is the nearest planet to the sun, receives the most light and heat, is the swiftest in its movement, and (excepting some of the asteroids) has the most eccentric orbit, with the greatest inclination to the ecliptic. It is also the smallest in diameter (again excepting asteroids) , has the least mass, and the greatest density of all the planets. 553. Distance, Light, and Heat. — Its mean distance from the sun is 36,000000 miles, but the eccentricity of its orbit is so great (0.205), that the sun is seven and one-half millions of miles out of its centre, and the actual distance of the planet from the sun ranges all the way from 28,500000 to 43,500000, while its velocity in its orbit 324 MERCURY. varies from thirty-five miles a second at perihelion to only twenty-three at aphelion. On the average it receives 6.7 times as much light and heat as the earth ; but the heat received at perihelion is to that at aphelion in the ratio of 9 to 4. For this reason there must be two seasons in its year due to the changing distance, even if the equator of the planet is parallel to the plane of its orbit, which would preclude seasons like our own. If the planet's equator is inclined at an angle like the earth's, then the seasons must be very complicated. 554. Period. — The sidereal period is very nearly 88 days, and the synodic period, or the time from conjunction to conjunction again, is about 116 days. The greatest elongation ranges from 18° to 28°, and occurs about twenty-two days before and after the inferior conjunction, or about thirty-six days before and after the superior conjunction. The planet's arc of retrogression is about 12° (consid- erably variable), and the stationary point is very near the greatest elongation. 555. Inclination. — The inclination of the orbit to the ecliptic is about 7°, but the greatest geocentric latitude (that is, the planet's greatest distance from the ecliptic as seen from the earth) is never quite so great. 556. Diameter, Surface, and Volume. — The apparent diameter ranges from 5" to about 13", according to its distance from us ; the least distance from the earth being about 57,000000 miles (93 — 36), while the greatest is about 129,000000 (93 + 36). The real diameter is very near 3000 miles, not differing from that more than fifty miles either way. It is not easy to measure, and the " probable error" is perhaps rather larger than would have been expected. With this diameter, its surface is \ of the earth's, and its volume i. 1 557. Mass, Density, and Surface Gravity. — Its mass is very diffi- cult to determine, since it has no satellite, and the values obtained by La Place, Encke, Leverrier, and others, range all the way from \ of the earth's mass to -£%. A recent, and perhaps the most reliable, determination, although the largest of all, is that of Backlund, by means of the perturbations of Encke's comet. He makes its mass about \ that of the earth. This gives TELESCOPIC APPEARANCE AND PHASES. 325 us a mean density about 2| times that of the earth, or nearly 12£ times that of water, a little less than that of the metal mercury. So far as known, the earth stands next to it in density, but with a wide interval, so that Mercury is altogether exceptional in this respect, as has been said before. The result, however, is liable to a considerable error, on account of the uncertainty of the planet's mass, Leverrier's estimate of the mass and density being only about one-half as great as Backhand's. The superficial gravity is about f that at the earth's surface if we accept "Backlund's mass. 558. Its Albedo, or reflecting power, as determined by Zollner is very low — only 0.13, somewhat inferior to that of the moon. In 1878 Mr. Nasmyth observed the planet in the same field of view with Venus; and although Mercury was then not much more than half as far from the sun as Venus, and therefore four times as brightly illuminated, it appeared to be less luminous in the telescope. "Venus was like silver, Mercury like zinc or lead." In the proportion of light given out at its different phases, it behaves like the moon, flashing out strongly near the full, as if it had a surface of the same rough structure as that of our satellite. t559. Telescopic Appearance and Phases. *— Seen by the telescope, the planet looks like a little moon, showing phases precisely similar to those of our satellite. At inferior conjunction the dark side is towards us ; at superior conjunction the illuminated surface. At Fig. 171. — Phases of Mercury and Venus. the greatest elongation it appears like a half- moon. Between supe- rior conjunction and greatest elongation it is gibbous, while between inferior conjunctions and the elongations it shows the crescent phase. Fig. 171 illustrates the phases of both Mercury and Venus. For the most part, Mercury can be observed only by daylight; but when proper precautions are taken to screen the object-glass of the telescope from 326 MERCURY. the direct sunlight, the observation is not difficult. The surface presents very little of interest. There are no markings well enough defined to give us any trustworthy determination of the planet's rotation, or of its geog- raphy. Occasionally the terminator (i.e., the line which separates the illu- minated and unilluminated portions of the planet) appears to be a little irregular, instead of a true oval, as it should be ; and at times when the phase is crescent-form, the points, or cusps, are a little blunted ; from this some as- tronomers have inferred the existence of high mountains upon the planet. Schroter, a German astronomer, the contemporary of the elder Herschel, deduced from his observations a rotation period for the planet of 24 h. 5 m. ; but later observers, with instruments certainly far more perfect, have not been able to verify his results, and they are now considered as of little weight. 560. Atmosphere. — The evidence upon this subject is not con- clusive. Its atmosphere, if it has one, must, however, be much less dense than that of Venus. No ring of light is seen surrounding the disc of the planet when it enters the limb of the sun at the time of a transit, while in the case of Venus such a ring, due to the atmos- pheric refraction, is very conspicuous. On the other hand, Huggins and Vogel, who have examined the spectrum of the planet, report that certain lines in the spectrum, due to the presence of water- vapor, were decidedly stronger than in the spectrum of the air (illu- minated by sunshine), which formed the background for the planet, making it probable that it has an atmosphere containing water-vapor like the atmosphere of the earth, but probably less extensive and dense. 561. Transits. — Usually at the time of inferior conjunction the planet passes north or south of the sun, the inclination of its orbit being 7° ; but if the conjunction occurs when the planet is very near its node, it will cross the disc of the sun and be visible upon it as a small black spot — not, however, large enough to be seen without a telescope, as Venus can under similar circumstances. At this time we have the best opportunity for measuring the diameter of the planet ; but unless special precautions are taken, the measured diameter under these circumstances is likely to be too small, on account of the irradia- tion of the surrounding background, which encroaches upon the planet's disc. Since the planet's nodes are in longitudes 227° and 47°, and are passed by the earth on May 7 and November 9, the transits can occur only near those days. If the orbit of the planet were strictly circular, the "transit limit" (corresponding to an ecliptic limit) INTERVAL BETWEEN TRANSITS. 327 would be 2° 10'; but at the May transits the planet is near its aphelion and much nearer the earth than ordinarily, so that the limit is diminished, while the November limit is correspondingly increased. The May transits are in fact only about half as numerous as the November transits. 562. Interval between Transits. — Twenty-two synodic periods of Mercury are pretty nearly equal to 7 years ; 41 still more nearly equal 13 years ; and 145 almost exactly equal 46 years. Hence, after a November transit, a second one is possible in 7 years, prob- able in 13 years, and practically certain in 46. For the May tran- sits the repetition after 7 years is not possible, and it often fails in 13 years. The transits of the present century are the following: — May Transits. — 1832, May 5 ; 1845, May 8 ; 1878, May 6 ; 1891, May 9. November Transits. — 1802, Nov. 9; 1815, Nov. 11; 1822, Nov. 5; 1835, Nov. 7; 1848, Nov. 10; 1861, Nov. 12; 1868, Nov. 5; 1881, Nov. 7; 1894, Nov. 10. The first transit of Mercury ever observed was by Gassendi, Nov. 7, 1631. The transits of Mercury are of no particular astronomical importance, except as giving accurate determinations of the planet's place, by means of which its orbit can be determined. Newcomb has also recently made an investigation of all the recorded transits, for the purpose of testing the uniformity of the earth's rotation. They indicate no perceptible change in the length of the day. VENUS. 563. The next planet in order from the sun is Venus, the brightest and most conspicuous of all, the earth's twin sister in magnitude, density, and general constitution, if not also in age, as to which we have no knowledge. Like Mercury, it had two names among the Greeks, — Phosphorus as morning star, and Hesperus as evening star. It is so brilliant that it is easily seen by the naked eye in the daytime, for several weeks when near its greatest elongation ; sometimes it is bright enough to catch the eye at once, but usually it is seen by day- light only when one knows precisely where to look for it. (It is not, however, the " Star of Bethlehem," though it has of late been frequently taken for it.) 564. Distance, Period, and Inclination of Orbit. — Its mean dis- tance from the sun is 67,200000 miles. The eccentricity of the 328 venus. orbit is the smallest in the planetary system (only 0.007) , so that the greatest and least distances of the planet from the sun differ from the mean only 470,000 miles each waj-. Its orbital velocity is twenty- two miles per second. Its sidereal period is 225 days, or seveu and one-half months, and its synodic period 584 days — a year and seven mouths. From supe- rior conjunction to elongation on either side is 220 days, while from inferior conjunction to elongation is only 71 or 72 days. The arc of retrogression is 16°. The inclination of its orbit is only 3£°. 565. Diameter, Surface, and Volume. — The apparent diameter ranges from 67" at the time of inferior conjunction to only 11" at the superior. This great difference depends, of course, upon the enor- mous change in the distance of the planet from the earth. At inferior conjunction the planet is only 26,000000 miles from us (93 — 67). No other body ever comes so near the earth except the moon, and occasionally a comet. Its greatest distance at superior conjunction is 160,000000 miles (93 + 67), so that the ratio between the greatest distance and the least is more than 6 to 1. The real diameter of the planet is 7700 (± 30) miles. Its sur- face, as compared with that of the earth, is ninety-five per cent; its volume ninety-two per cent. 566. Mass, Density, and Gravity. — By means of the perturba- tions she produces upon the earth, the mass of Venus is found to be seventy-eight per cent of the earth's ; hence her density is eighty-six per cent, and her superficial gravity eighty-three per cent of the earth's. 567. Phases. — The telescopic appearance of the planet is strik- ing on account of her great brilliance. When about midway between greatest elongation and inferior conjunction she has an apparent diameter of 40", so that, with a magnifying power of only forty-five, she looks exactly like the moon four days old, and of precisely the same apparent size. Very few persons, however, would think so on their first view through the telescope, for a novice always underrates the apparent size of a tele- scopic object : he instinctively adjusts his focus as if looking at a picture only a few inches away, instead of projecting the object visually into the sky. MAXIMUM BRIGHTNESS. 329 According to the theory of Ptolemy, Venus could never show us more than half her illuminated surface, since according to his hypoth- esis she was always between us and the supposed orbit of the sun (Art. 500). Accordingly, when in 1610 Galileo discovered that she exhibited the gibbous phase as well as the crescent, it was a strong argument for Copernicus. Galileo announced his discovery in a curious way, by publishing the anagram, — "Haec immatura a me jam frustra leguntur ; o. y." Some months later he furnished the translation, — " Cynthise figuras aeinulatur mater amorum," which is formed by merely transposing the letters of the anagram. His object was to prevent any one from claiming to have anticipated him in this discovery, as had been done with respect to his discovery of the sun spots. ifiQ. 172. — Telescopic Appearances of Venue. Fig. 172 represents the disc of the planet as seen at four points in its orbit. 1, 3, and 5 are taken at superior conjunction, greatest elongation, and near inferior conjunction respectively, while 2 and 4 are at interme- diate points. 568. Maximum Brightness. — The planet attains its maximum brilliance thirty-six days before and after inferior conjunction, at a 330 venus, distance of about 38° or 39° from the sun, when its phase is like that of the moon about five days old. It then casts a strong shadow, and, as has already been said, is easily visible by day with the naked eye. 569. Surface Markings. — These are not at all conspicuous. Near the limb of the planet, which is always much brighter than the central parts (as is also the case with Mercury and Mars), they can never be well seen, although sometimes when Venus was in the crescent phase, intensely bright spots have been reported near the cusps, as at a and 6 in No. 4, Fig. 172. These may perhaps be ice-caps like those which are seen on Mars. Near the " terminator," which ie less brilliant and less sharply defined than the limb, irregular darkish shadings are sometimes seen, such as are indicated by the dotted lines in the figures, but without any distinct outline. They may be continents and oceans dimly visible, or they may be mere atmospheric objects ; observations do not yet decide. 1 570. Rotation of the Planet. — From the observation of such markings Schroter deduced the rotation period of 23 h 21™, which has since been partially confirmed by one or two other observers, and may be approximately correct. De Vico concluded that the planet's equator makes an angle of about 54° with the plane of its orbit. This, however, is extremely doubtful. If the bright spots which have been referred to iu Art. 569 are really polar ice-caps, the equator cannot be very greatly inclined. (See Clerke's " History of Astronomy," p. 299.) No sensible difference has been ascertained between the different diam- eters of the planet. If it were really as much flattened at the poles as the earth is, there should be a difference of 0".2 between the polar and equato- rial diameters as measured at the time of the planet's transit. 571. Mountains. — From certain irregularities occasionally ob- served upon the terminator, and especially from the peculiar blunted form of one of the cusps of the crescent, various observers have con- cluded that there are numerous high mountains upon the surface of the planet. Schroter assigned to some of those near the southern pole the extravagant altitude of twenty-five or thirty miles, but the evidence is entirely insufficient to warrant any confidence in the con- clusion. 572. Albedo. — According to Zollner the Albedo of the planet is 0.50, which is about three times that of the moon, and almost four EVIDENCES OP ATMOSPHERE. §31 times that of Mercury. It is, however, exceeded by the reflecting power of the surfaces of Jupiter and Uranus, while that of Saturn appears to be about the same. This high reflecting power probably indicates that the surface is mostly covered with cloud, as few rocks or soils could match it in brightness. 573. Evidences of Atmosphere. — When the planet is near the sun, the horns of the crescent extend notably beyond the diameter, and when very near the sun, a thin line of light has been seen by several observers, especially Professor Lyman of New Haven, to complete the whole circumference. This is due to refraction of sun- light by the planet's atmosphere, a phenomenon still better seen as the planet is entering upon the sun's disc at a transit, when the black disc is surrounded by a beautiful ring of light. From the ob- servations of the transit of 1874, Watson concluded that the planet's atmosphere must have a depth of about fifty-five miles, that of the earth being usually reckoned at forty miles. Other observers in different ways have come to substantially the same results. Its atmosphere is probably from one and a half to two times as extensive and dense as our own, and the spectroscope shows evidence of the presence of water-vapor in it. Lights on Dark Portion. — Many observers have also reported faint lights as visible at times on the dark portion of the planet's disc. These cannot be accounted for by reflection, but must origi- nate on the planet's surface ; they recall the Aurora Borealis and other electrical manifestations on the earth. 574. Satellites. — No satellite is known, although in the last century a number of observers at various times thought they had found one. In most cases they observed small stars near the planet, which we can now identify by computing the place occupied by the planet at the date of observation. It is not, however, impossible that the planet may have some very minute and near attendants like those of Mars, which may yet be brought to light by means of the great telescopes of the future, or by pho- tography. Of course the extreme brilliance of the planet, and the fact that the necessary observations can be made only in strong twilight, render the discovery of such objects, if they exist, very difficult. 575. Transits. — Occasionally Venus passes between the earth and the sun at inferior conjunction, giving us a so-called '■'■transit." 332 venus. She is then visible ('even to the naked eye) as a black spot on the disc, crossing it from east to west. As the inclination of the planet's orbit is nearly 3£°, the " transit limit " is small (about 4°) , and the transits are therefore very rare phenomena. The sun passes the nodes of the orbit on June 5 and December 7, so that all transits must occur on or near those dates. When Venus crosses the sun's disc centrally, the duration of the transit is about eight hours. Taking the mean diameter of the sun as 32', or ^fj of a circumference, and the planet's synodic period as 584 days, the geocentric duration of a central transit should be f^X^X 584 d , which equals 0.332 days, or 7 h 58 m - When the transit track lies near the edge of the disc, the duration is of course correspondingly shortened. 576. Recurrence of Transits. — Five synodic, or thirteen sidereal, revolutions of Venus are very nearly equal to eight years, the differ- ence being only a little more than one day ; and still more nearly, in fact almost exactly, 243 years are equal to 152 synodic, or 395 side- real, revolutions. If, then, we have a transit at any time, we may have another at the same node eight years earlier or later. Sixteen years before or after it would be impossible, and no other transit can occur at the same node until after the lapse of two hundred and thirty-floe or two hundred and forty-three years. If the planet crosses the sun nearly centrally, the transit will not be accompanied by another at an eight-year interval, but the planet will pass either north or south of the sun's disc, at the conjunctions next preceding and following. If, however, as is now the. case, the transit path is near the northern or southern edge of the sun, then there will be a companion transit across the opposite edge of the disc eight years before or after. Thus, if we have a pair of June transits, separated by an eight-year interval, it will be followed by another pair at the same node in 243 years ; and a pair of December transits will come in about halfway between the two pairs of June tran- sits. After a thousand years or so from the present time the transits will cease to come in pairs, as they have been doing for 2000 years. 577. Transits of Venus have occurred or will occur on the following December 7, 1631. ) June 5, 1761. } December 4, 1639. > June 3, 1769. \ December 9, 1874. > June 8, 2004 December 6, 1882. > June 8, 2004. \ > June 6, 2012. J MABS. 333 The special interest in these transits consists in the use that has been made of them for the purpose of finding the sun's parallax, a subject which will be discussed later on (Chap. XVI.). The first observed transit, in 1639, was seen by only two persons, — Horrox and Crabtree, in Eng- land. The four which have occurred since then have been extensively observed in all parts of the world where they were visible, by scientific expeditions sent out for the purpose by different nations. The transits of 1769 and 1882 were visible in the United Fig. 173. States. Fig. 173 shows the track of Venus across Transit of Venus Tracks, the sun's disc at the two transits of 1874 and 1882. MARS. This planet is also prehistoric as to its discovery. It is so con- spicuous in color and brightness, and in the extent and apparent capriciousness of its movement among the stars, that it could not have escaped the notice of the very earliest observers. 578. Orbit. — Its mean distarice from the sun is 141,500000 miles, but the eccentricity of the orbit is so considerable (0.093) that the distance varies about 13,000000 miles. The light and heat which it receives from the sun is somewhat less than half of that received by the earth. The inclination of its orbit is small, 1°51'. The planet's sidereal period is 687 days, or V 10J mo , which gives it an average orbital velocity' of fifteen miles per second. Its synodic period is 780 days, or 2 y l| mo . It is the longest in the solar system, that of Venus (584 days) coming next. Of the 780 days, it moves eastward during 710, and retrogrades during 70, through an arc of 18°. 579. At opposition its average distance from the earth is 48,600- 000 miles (141,500000 miles miuus 92,900000 miles). When the opposition occurs near the planet's perihelion, this distance is re- duced to 35,050,000 miles ; if near aphelion, it is increased to over 61,000000. At conjunction the average distance from the earth is 234,400000 miles (141,500000 plus 92,900000). The apparent diameter and brilliancy of the planet, of course, vary enormously with these great changes of distance. If we put R for the planet's distance from the sun, and A for its distance from the earth, its brightness, neglecting the correction for phase, should 334 MAES. equal . We find from this that, taking the brightness at conjunction as unity (at which time the planet is about as bright as the pole-star), it is more than twenty-three times brighter at the average opposition, and fifty- three times brighter if the opposition occurs at the planet's perihelion. At an unfavorable opposition Mars, as has been said, may be 61,000000 miles distant, and its brightness then is only about twelve times as great as at conjunction, — the difference between favorable and unfavorable oppositions being more than four to one. These favorable oppositions occur always in the latter part of August (at which date the sun passes the line of apsides of the planet), and at intervals of fifteen or seventeen years. The last was in 1877, and the next will be in 1892. A reference to Fig. 159 will show how great is the difference between the planet's opposition distance from the earth under varying circumstances. 580. Diameter, Surface, and Volume. — The apparent -diameter of the planet ranges from 3". 6 at conjunction, to 25". at a favorable opposition. Its real diameter is very closely 4200 miles, — the error may be twenty miles one wa}- or the other. This makes its surface 0.28, and its volume 0.147 (equal to \) of the earth's. 581. Mass, Density, and Gravity. — Observations upon its satel- lites give its mass as 1 compared with that of the earth. This makes its density 0.73 and superficial gravity 0.38 ; that is, a body which weighs 100 pounds on the earth would have a weight of 38 pounds on the surface of Mars. 582. Phases. — Since the orbit of the planet is outside that of the earth, it never comes between us and the sun, and can never show the crescent phase,; but at quadrature enough of the unilluminated portion is turned towards the earth to make the disc clearly gibbous like the moon three or four days from full. Fig. 174 shows its maximum phase accurately drawn to scale. fig. 174. 5 83 - The " Albedo " of the Planet. — Accord- Greatest Phase of Ma™. in S to Zollner's observations this is 0.26, which is considerably higher than that of the moon (|) , and just double that of Mercury. 584. Rotation. — The planet's time of rotation is 24 h 37 m 22".67. This very exact determination has been made by Kaiser and Bak- INCLINATION OF THE PLANET'S EQUATOR. 335 huyzen, by comparing drawings of the planet which were made more than 200 years ago by Huyghens with others made recently. It is obvious that observations made a few days or weeks apart will give the time or rotation with only approximate accuracy. Knowing it thus approximately, we can then determine, without fear of error, the whole mimber of rotations between two observations separated by a much longer interval of time. This will give a second and closer approximation to the true period ; and with this we can carry our reckoning over centuries, and thus finally determine the period within a very minute fraction of a second. The number given is not uncer- tain by more than ^ of a second, if so much. 585. The Inclination of the Planet's Equator to the Plane of its Orbit. — This is very nearly 24° 50' (26° 21' to the ecliptic), not very different from the inclination of the earth's equator ; so far, therefore, as depends upon that circumstance, its seasons should be substantially the same as our own. 586. Polar Compression. — There is a slight but sensible flatten- ing of the planet at the poles. The earlier observers found for the polar compression values as large as ^, and even ^. These large values, however, are inconsistent with the existence of any extensive surface of liquid upon the planet, and more recent observations of the writer show the polar compression to be about -^^ ; which is almost exactly what would be expected from a planet constituted as we suppose Mars to be. 587. Telescopic Appearance and Surface-Markings. — The fact that we are able to determine the time of rotation so accurately of course implies the existence of identifiable markings upon the sur- face. Viewed through a powerful telescope', the planet's disc, as a whole, is ruddy, — in fact, almost rose color, — very bright around the limb, but not at the " terminator," if there is any considerable phase. The central portions of the disc present greenish and purplish patches of shade, for the most part not sharply defined, though some of the markings have Fig. 175. — Telescopic Views of Mars. 336 MARS. outlines reasonably distinct. On watching the planet for only a few hours even, the markings pass on across the disc, and are replaced by others. Some of them are permanent, and recur at regular in- tervals with the same form and appearance, while others appear to be only clouds which for a time veil the surface below, and then clear away. By comparing drawiugs made when the same side is turned towards the earth, it is possible in a short time to ascertain what features really belong to the planet's geography. The polar ice-caps, brilliant white patches near the poles, form a marked feature in the planet's telescopic appearance. These are be- lieved to be ice-caps from tho fact that the one which is near the pole that happens to be turned towards the sun continually diminishes in size, while the other increases, the process being reversed with the seasons of the planet. 588. The principal peculiarity of the surface of Mars appears to be the way in which land and water are intermeshed. There seem to be few great oceans and continents, but there are narrow arms of the sea, like the Baltic and the Eed Sea, penetrating and dividing the Fig. 176. — Schiaparolli's Map of Mars. land masses . As to the nomenclature of the principal ' ' Areographic " features, Mr. Proctor in his map has for the most part assigned the names of astronomers who have taken special interest in the study of ATMOSPHERE. 337 the planet. Schiaparelli, on the other hand, has taken his names mostly from classical geography. Fig. 176 is a reduced copy of his map of the planet, drawn on Mercator's Projection. The student will remember that in maps of this sort the polar regions are ex- tended beyond their due proportion, and also that the parts near the poles can never be seen from the earth nearly as well as those near the equator, and hence that our knowledge of the details in that part of the planet is very limited. (See note to Art. 588, p. 347.) t589. Atmosphere. — At one time it was supposed that the atmos- phere of the planet is very dense, but more recent observations have shown that this cannot be the case. The probability is that its density is considerably less than that of our own atmosphere. Dr. Huggins has found with the spectroscope strong indications of the presence of aqueous vapor. Although the planet receives from the sun less than half the amount of heat and light per unit of surface that the earth does, the climate appears, for some reason not yet discov- ered, to be much more mild than would be expected. So far as we can judge, the water on the planet is never frozen except very near the poles. The earth, as seen from Mars, would present much more extensive snow-caps. 590. Satellites. — -There are two satellites, which were discovered in August, 1877, by Professor Hall at Washington with the then new 26-inch telescope. They are exceedingly minute, and can be seen only with the most powerful instruments. The outer one, Deimos, is at a distance of 14,600 miles from the centre of the planet, and has a period of 30 h 18 m , while the inner one, Phobos, is at a distance of only 5800 miles, and its month is but 7 h 39 m long, not one-third of the day of Mars. Owing to this fact it rises in the west every night for the ' ' Marticoli " (if there are any_p_eople there) and sets in the east, after about 5^ h . — -—~ Lieimos does not do this ; it rises in the east like other stars, bat its orbital eastward motion among the stars is so nearly equal to its diurnal motion westward, that it is nearly 132 hours between two successive risings. This is more than four of its months, so that it undergoes all its changes of- phase four times in the interval. Of course, both the satellites are frequently eclipsed, — the inner one at two full moons out of three ; and it also transits across the sun's disc with corresponding frequency, as seen from some point or other on the planet's surface. 338 MAES. Their orbits appear to be exactly circular, and they move exactly in the plane of the planet's equator ; and they keep so, maintained in their relation to the equator by the action of the " equatorial bulge" upon the planet. 591. As givers of moonlight they do not amount to much. Their diameters are too small to be measured with any micrometer; but from their apparent "magnitude" (i.e., brightness), as seen from the earth, and assuming that their surfaces have the same reflective power as that of the planet, Professor Pickering has estimated the diameter of Phobos, which is the larger one, as about seven miles, and that of Deimos, as five or six. The light given by Phobos to the inhabitants of Mars would be about -£$ of our moonlight ; that of Deimos about 1 ^ . The period of Phobos is by far the shortest period in the solar sys- tem. Next to it is that of Mimas, the inner satellite of Saturn, which, however, is nearly three times as long, — 22^ h . This rapid- Fiq. ITT. — Orbits of the Satellites of Mars. ity of revolution raises important questions as to the theory of thp development of the solar system, and requires modification of the views which had been held up to the time of their discovery. If the nebular hypothesis is true, a shortening of the satellite's period, or a lengthening of the planet's day, must have occurred since the satellite came into being, since that hypothesis will not account for the existence of a satellite having a period shorter than the diurnal rota- tion of its primary. Fig. 177 is a diagram of the satellite orbits as THE ASTEROIDS, OR MINOR PLANETS. 339 they appeared from the earth in 1888. It is reduced from the American Nautical Almanac for that year. THE ASTEROIDS, OR MINOR PLANETS. t 592. These are a group of small planets circulating in the space between Mars and Jupiter. The name "asteroid" was suggested by Sir William Herschel early in the century, when the first ones were discovered. The later term planetoid is preferred by many. It was very early noticed that there is a break in the series of the distances of the planets from the sun. Kepler, indeed, at one time thought he had discovered the true law 1 and the real reason why the planets' distances are what they are. This theory of his was broached twenty-two years, however, before he discovered the harmonic law, and he probably abandoned it when he discovered the elliptical form of the planets' orbits. At any rate, in later life he suggested that it was likely that there was a planet between Mars and Jupiter too small to be seen. The impression that such a planet existed gained ground when Bode published the law which bears his name in 1772, and it was still further deepened when, nine years later, in 1781, Uranus was discovered, and the distance of the new planet was found to con- form to Bode's law. An association of twenty-four astronomers, mainly German, was immediately formed to look for the missing planet, who divided the zodiac between them and began the work. 1 His supposed law was as follows : Imagine the sun surrounded by a hollow spherical shell, on which lies the orbit of the earth. Inside of this shell inscribe a regular icosahedron (the twenty-sided regular solid), and within that inscribe a second sphere. This sphere will carry upon it the orbit of Venus. Inside of the sphere of Venus inscribe an octahedron (the eight-sided solid), and the sphere which fits within it will carry Mercury's orbit. Next, working outwards from the earth's orbit, circumscribe around the earth's sphere a. dodecahedron, circum- scribing around it another sphere, and this will carry upon it the orbit of Mars. Around the sphere of Mars circumscribe the tetrahedron, or the regular pyra- mid. The corners of this solid project very far, so that the sphere circumscribed around the tetrahedron will be at a very great distance from the sphere of Mars. It carries the orbit of Jupiter. Finally, the cube, or hexahedron, circumscribed around the orbit of Jupiter, gives us in the same way the orbit of Saturn. We thus obtain a series of distances not enormously incorrect (though by no means agreeing with fact even as closely as does Bode's law) ; and, moreover, the theory had the great advantage to Kepler's mind of accounting for the fact that there are (so far as was then known) but seven planets, there being possible but Jive regular solids. 340 THE MINOR PLANETS. Singularly enough, however, the first discovery was made, not by a member of this association, but by Piazzi, the Sicilian astronomer of Palermo, who was then engaged upou an extensive star-catalogue. On January 1, 1801, he observed a seventh-magnitude star which by the next evening had unquestionably moved, and kept on moving. He observed it carefully for some six weeks, when he was taken ill ; before he recovered, it had passed on towards superior conjunction, and was lost in the rays of the sun. He named it Ceres, after the tutelary goddess of Sicily. When at length the news reached Germany in the latter part of March it created a great excitement, and the problem now was to rediscover the lost planet. The association of planet-hunters began the search in Septem- ber, as soon as its elongation from the sun was great enough to give any prospect of success. During the summer Gauss devised his new method of computing a planetary orbit, and computed the ephemeris of its path. Very soon after receiving his results, Baron Von Zach rediscovered Ceres on December 31, and Dr. Olbers on the next day, just one year after it was first found by Piazzi. |593. In March, 1802, Dr. Olbers, who in looking for Ceres had carefully examined the small stars in the constellation of Virgo, on going over the ground again, found a second planet, which he named Pallas, a body of about the same brightness as Ceres. Two having now been found, and Pallas having a very eccentric and much inclined orbit, he conceived the idea that they were fragments of a broken planet, and that other planets of the same group could probably be found by searching near the intersection of their two orbits. Juno, the third, was discovered by Harding at Lilienthal (Schroter's obser- vatory) in 1804 ; and Vesta, the largest and brightest of the whole group (sometimes visible to the naked eye), was found by Olbers himself in 1807. The search was kept up for several years after this, but no more planets were found because they did not look after small enough stars. The fifth, Astraea, was discovered in 1845 by Hencke, an amateur astronomer who for fifteen years had been engaged in studying the smaller stars in hopes of just the reward he captured. In 1846 no asteroid was found (the discovery of Neptune was glory enough for that year), but in 1847 three more were brought to light ; and since then not a year has passed without adding from one to twenty to the number. The list at the end of 1888 counts up 281, and there is no prospect of diminution in the rate of discovery, though the new ones are mostly very small, — stars of the twelfth METHOD OF SEARCH. 34X and thirteenth magnitudes, which require a large telescope to make- them even visible. All the brighter ones have evidently been already picked up. They have been discovered by comparatively a few observers. Four per- sons have found more than 20 each : Palisa, of Vienna, who stands far in advance of all others, has alone discovered 68, and Dr. Peters of Clinton, N.Y., 52 ; Luther of Diisseldorf, 24 ; and the late Professor Watson, 22. The German astronomers have thus far discovered 102 ; the American, 78 ; the French, 66 ; the English, 19 ; and the Italians, 16. 594. Method of Search. — The asteroid-hunter selects certain portions of the sky, usually near the ecliptic, and prepares charts covering two or three square degrees, on which he sets down all the stars his telescope will show. This is a very laborious operation, which in the future is likely to be very much facilitated by photog- raphy. The chart once made, he goes over the ground from time to time, comparing every object with the map and looking out for interlopers. If he finds a star which is not on his chart, the prob- ability is that it is a planet, though it may be a variable star which was invisible when the chart was made. He very soon settles the question, however, by measuring with the micrometer the distance of the new star from some of its neighbors, keeping up the process foi an hour or two. If it is a planet, it will move pereeptibly in that length of time. Of course, great care must be taken to be sure that it is a new planet, and not one of the multitude already known. Generally it is possible to decide very quickly which of the known planets will be in the neighborhood, and a rough computation will commonly decide at once whether the planet is new or not. Not always, however, and mistakes in this regard are not very unusual. These minor planets are all named, the names being derived from mythology and legend. They are also designated by numbers, and the symbol for each planet is the number written in a circle. Thus, for Ceres the symbol is (T) ; for Hilda, (H) ; and so on. A full list of them, with the elements of their orbits, is published yearly in the " Annuaire du Bureau des Longitudes," Paris. 595. Their Orbits. — The mean distance of the different asteroids from the sun varies greatly, and of course the periods are corre- spondingly different. Medusa, (us), has the smallest mean dis- tance (213, or 198,000000 miles), and the shortest period, 3? 40 d . 342 THE MINOR PLANETS. Thule, (279), is the remotest, with a mean distance of 4.30 or 400,- 000000 miles, and a period of 8 y 313 d . According to Svedstrup, the mean distance of the "mean asteroid" is 2.65 (246,000000 miles), and its period about 4| years. Its distance from the earth at time of opposition would be, of course, 1.65, or 153,000000 miles. The inclinations of their orbits average about 8°; but Pallas, (2), has an inclination of 35°, and Euphrosyne, ©, of 26^°. Several of the orbits are extremely eccentric. Mthr&, @), has an almost cometary eccentricity of 0.38, and nine or ten others have eccentricities exceeding 0.30. They are distributed quite un- equally in the range of distance, there being, as Kirkwood has pointed out, very few at such distances that their periods would be exactly commensurable with that of Jupiter. 596. Diameter and Surface. — Very little is known as to the real size of these bodies. The four first discovered, and a few of the newer ones, show a minute but sensible disc in very powerful tele- scopes. Vesta, which is the brightest of the whole group, and is just visible to the naked eye when near opposition, may be from two hun- dred to four hundred miles in diameter. Pickering, by photometric measurements (assuming the reflecting power of the planet's surface to be the same as that of Mars), finds a diameter of 319 miles. The other three of the original four are perhaps two-thirds as large. As for the rest, it is hardly possible that any one of them can be as much as 100 miles in diameter, and the smallest, such as are now being discovered in such numbers, are probably less than ten miles through, — nothing more than "mountains broke loose." The sur- face area of one of the smaller ones would hardly make a small county. 597. Mass, Density, etc. — As to the individual masses and densi- ties we have no certain knowledge. It is probable that the density does not differ much from the density of the crust of the earth, or the mean density of Mars. If this is so, the mass of Vesta might pos- sibly be as great as ls ^ 00 or at j, 00 of the earth. On such a planet the force of superficial gravity would be about ^ to ^ of gravity on the earth, and a body projected from the surface with a velocity of about 2000 feet a second — that of an ordinary rifle-ball — would fly off into space and never return to the planet, but would cir- culate around the sun as a planet on its own account. On the smallest asteroids, with a diameter of about ten miles', it would be AGGREGATE MASS. 343 quite possible to throw a stone from the hand with velocity enough to send it off into space. 598. Aggregate Mass. — Although we can only estimate very roughly the masses of the individual members of the flqck, it is pos- sible to get some more certain knowledge of their aggregate mass. Leverrier from the motion of the line of apsides of the orbit of Mars demonstrated that the whole amount of matter thus distributed in the space between Mars and Jupiter cannot exceed about one-fourth of the mass of the earth. This quantity of matter collected-- into a body of the same density as Mars would make a planet of about 5000 miles in diameter. The united masses of those which are already known would make only a very small fraction of such a body. Up to August, 1880, the united bulk of the asteroids then discovered was estimated at ^ uls part of the earth's bulk, with a mass probably about -g-fo-Q °f the earth's. Presumably, therefore, the number of these bodies remaining undiscovered is exceedingly great — to be counted by thousands, if not by millions. Most of them, of course, must be much smaller than those which are already known. 599. Forms, Variations of Brightness, and Atmosphere. — We have no definite knowledge on this point, but Dr. Olbers observed in the case of Vesta certain fluctuations in her brightness which seemed to him to indicate that she is not a globe, but an angular mass, — a splinter of rock. This, however, is not confirmed by the more recent photometric observations of Muller or Pickering. Miiller examined seven of the asteroids, and found their changes of bright- ness very regular. Four of them, one of which is Vesta, behaved precisely like Mars, as if their surfaces were comparatively smooth, while three others, Ceres and Pallas among them, behaved like the moon and the planet Mer- cury, as if having a rough surface with very little atmosphere, and nearly cloudless. Some of the earlier observers reported evidences of an exten- sive halo around them. Later observations do not confirm it, and it is not likely that they carry much air with them. 600. Origin. — With respect to this we can only speculate. Two views have been held, as has been already intimated. One is, that the material, which according to the nebular hypothesis ought to have been concentrated to form a single planet of the class to which the earth belongs, has failed to be so collected, and has formed a flock of small separate masses. It is now very generally believed that the 344 THE MINOR PLANETS. matter which at present forms the planets was once distributed in rings, like the rings of Saturn. If so, this ring next outside of Mars would necessarily suffer violent perturbations from the nearness of the enormous planet Jupiter, and so would be under very different conditions from any of the other rings. This, as Peirce has shown, might account for its breaking up into many fragments. The other view is that a planet about the size of Mars has broken to pieces. It is true, as has been often urged, that this theory in its original form, as presented by Olbers, cannot be correct. No single explosion of a planet could give rise to the present assemblage of orbits, nor is it possible that even the perturbations of Jupiter could have converted a set of orbits originally all crossing at one point (the point of explosion) into the present tangle. The smaller orbits are so small that however turned about they lie wholly inside the larger, and cannot be made to intersect them. If, however, we admit a series of explosions, this difficulty is removed ; and if we grant an explosion at all, there seems to be nothing improbable in the hypothesis that the fragments formed by the bursting of the parent mass would carry away within themselves the same forces and reactions which caused the original bursting ; so that thi*j- themselves would be likely enough to explode at some time in their later history. At present opinion is divided between these two theories. 601 . The number of these bodies already known is so great, and the prospect for the future is so indefinite, that astronomers are at their wits' end how to take care of this numerous family. To compute the orbit and ephemeris of one of these little rocks is more laborious (on account of the great perturbations produced by Jupiter) than to do the same for one of the major planets ; and to keep track of such a minute body by observation is far more difficult. Until recently, the German Jahrbuch has been publish- ing the ephemeiides of such as came within the range of observation each year ; but this cannot be kept up much longer, and the probability is that hereafter only the larger ones, or those which present some remarkable peculiarity in their orbits, will be followed up. One little family of them, however, is "endowed." Professor Watson, at his death, left a fund to the American National Academy of Sciences to bear the expense of taking care of the twenty-two which lie discovered. INTRA-MERCURIAL PLANETS AND THE ZODIACAL LIGHT. It is very probable, indeed almost certain, that there are masses of matter revolving around the sun within the orbit of Mercury. INTRA-MERCURIAL PLANETS. 345 602. Motion of the Perihelion of Mercury's Orbit. — Leverrier, in 1859, from a discussion of all the observed transits of Mercury, found that the perihelion of its orbit has a movement of nearly 38" a century. This is more than can be accounted for by the action of the known planets, and, so far as known, could be explained only by the attraction of a planet, or ring of small planets, revolving inside this orbit nearly in its plane, with a mass about half as great as that of Mercury itself. We say "so far as known," because an alternative hypothesis has been proposed, viz., that the law of gravitation, though strictly true for bodies at rest is not absolutely so for bodies in motion ; that when bodies are moving towards each other the attraction is less by a minute fraction than if they were at rest. The hypothesis is known as the electro-dynamic theory of gravi- tation, but has at present very little to support it. If, however, it were true, then the peculiar motion of the apsides of Mercury's orbit would be a necessary consequence. Subsequent investigations by a number of mathematicians have fully confirmed Leverrier's results ; Mercury's orbit is beyond ques- tion affected as it would be if there were an intra-Mercurial planet, or a number of them. 603. Dr. Lesoarbault's Observation : Vulcan. — A certain country physician, living some eighty miles from Paris, Dr. Lescarbault, on the pub- lication of Leverrier's result, announced that he had actually seen this planet crossing the sun nine mouths before, on the 26th of March of that year, 1859. He was visited by Leverrier, who became satisfied of the genuineness of his obsei-vations, and the doctor was duly congratulated and honored as the discoverer of " Vulcan," which name was assigned to the supposed new planet. An interesting account of the matter may be found in Chambers' " Descriptive Astronomy " ; and in many of the works published -from twenty to twenty-five years ago, as well as in some more recent ones, " Vulcan " is assigned a place in the solar system, with a distance of about 13,000000 miles and a period of 19+ days. Lescarbault described it as having an apparent diameter of about 7", which would make it over 2500 miles in diameter. 604. Nevertheless, it is nearly certain that Vulcan does not exist. There are various opinions which we need not here discuss as to the ex- planation of this pseudo-discovery. But the planet, if real, ought since 1859 to have been visible on the sun's face at certain definite times which Leverrier calculated and published; and it has never been seen, though very carefully looked for. Small, round, dark objects have from time to €546 INTRA-MERCTJRIAL PLANETS. time been indeed reported on the sun's disc, which in the opinion of the observers at the time were not sun spots ; but most of these observations were made by amateurs with comparatively little experience, with small telescopes, and with no measuring apparatus by which they could certainly determine whether or not the spot seen moved like a planet. In most of these cases photographs or simultaneous observations made elsewhere by astronomers of established reputation, and having adequate apparatus, have proved that the problematical "dots" were really nothing but ordinary small sun spots, and the probability is that the same explanation applies to the rest. 605. Eclipse Observations. — A planet large enough to be seen distinctly on the sun by a 2£-inch telescope, such as Lescarbaralt used, would be a conspicuous object at the time of a solar eclipse, and most careful search has been made for the planet on such occa- sions ; but so far, although stars of the third and fourth magnitudes, and even of the fifth, have been clearly seen by the observers within _ a few degrees of the eclipsed sun, no planet has been found. One apparent exception occurred iu 1878. During the eclipse of that year, Professor Watson observed two starlike objects (of the fourth magni- tude), which he thought at the time could not be identified with any known stars consistently with his observations. Mr. Swift, also, at the same eclipse, reported the observations of two bright points very near the sun ; but these from his statement could not (both) have been identical with Watson's stars. Later investigations of Dr. Peters have shown that the assumption of a very small and very likely error in Professor Watson's circle-readings (which were got in a very ingenious, but rather rough way, without the use of graduations) would enable his stars to be identified with $ and £ Cancri, and it is almost certain that these were the stars he saw. Mr. Swift's observations remain unexplained. With this exception, the eclipse observations all give negative results, and astronomers generally are now disposed to consider the « Vulcan question " as settled definitely and ad- versely. 606. At the same time it is extremely probable that there are a number, and perhaps a very great number, of intra- Mercurial aste- roids. A body two hundred miles in diameter near the sun would have an angular diameter of only about £", as seen from the earth, and would not be easily visible on the sun's disc, except with very large telescopes. It would not be at all likely to be picked up acci- dentally. Objects with a diameter of not more than forty or fifty miles would be almost sure to escape observation, either at a transit or during a solar eclipse. THE ZODIACAL LIGHT. 347 607. Zodiacal Light. — This is a faint, soft beam of light that extends both ways from the sun along the ecliptic. In the evening it is best seen in February, March, and April, because the portion of the ecliptic which lies east of the sun's place is then most nearly perpendicular to the western horizon. During the autumnal months the zodiacal light is best seen in the morning sky for a similar reason. In our latitudes it can seldom be traced more than 90° or 100° from the sun ; but at high elevations within the tropics it is said to extend entirely across the sky, forming a complete ring, and there is said to be in it at the point exactly opposite to the sun a patch a few degrees in diameter of slightly brighter luminosity, called the " Gegenschein " or " counter-glow." The portions of this object near the sun are reasonably bright, and even conspicuous at the proper seasons of the year ; but the more distant portions in the neighborhood of the " counter-glow " are so extremely faint that it is only possible to observe them at a distance from cities and large towns, in places where the air is free from smoke, and where the darkness of the sky is not affected by the general illumination due to gas and electric lights. 608. The cause of the phenomenon is not certainly known, but at pres- ent the theory most generally accepted attributes it to sunlight reflected by myriads of small meteoric bodies which are revolving around the sun nearly in the plane of the ecliptic, forming a thin, flat sheet like one of Saturn's rings, and extending far beyond the orbit of the earth. It may be that the denser portion of this meteoric ring within the orbit of Mercury is the cause of the motion of the perihelion of that planet which Leverrier detected ; it is for this reason that we deal with the subject here rather than in connection with meteors. While this theory, however, is at present more generally accepted than any other, it cannot be said to be established. Some are disposed to consider the zodiacal light as a mere extension of the sun's corona, whatever that may be. 608*. (Note to Art. 588.) The Canals of Mars. — According to Schiaparelli (and his observations are at least partially confirmed by others) a most characteristic feature of the planet's surface is a series of long, straight, narrow "canals" connecting the larger bodies of water. These canals were first seen and recognized in 1877. In 1881 they were seen again, and at that time nearly all of them double. If there is not some fallacy in the observation, the problem as to the nature of these canals, and the cause of their " gemination," is a very important and perplexing one. The ob- servations of 1890, 1892 and 1894 have placed the existence of the so-called "canals" beyond doubt, and have confirmed their gemination; but they have not furnished any satisfactory explanation of the phenomena. 347 A ADDENDA TO CHAPTER XIV. 557. More recent determinations of the mass of. Mercury by Haerdtl, Harkness, and Backhand himself, all concur in showing that the value given in the text is much too great, and that the true value probably lies between ^ and ^ of the mass of the earth. This makes the density of Mercury somewhat less than that of the earth. 559. In 1889 Schiaparelli, the Italian astronomer, announced that he had discovered certain permanent markings upon the sur- face of Mercury, and that from them he had ascertained that the planet rotates on its axis only once during its orbital period of 88 days; it keeps the same face always turned towards the sun, behaving in this respect, just as the moon does towards the earth. Owing to the eccentricity of the orbit, however, the planet has a large ' libra- tion ' (Art. 250), amounting to nearly 23£° on each side of the mean position ; i.e., seen from a favorable position on the planet's surface, the sun instead of rising and setting daily, as with us, would oscillate about 74° back and forth in the sky, every 88 days. This asserted discovery is extremely important, and has excited great interest : it still waits confirmation by other astronomers, as the observations are very difficult ; but there is little doubt that it is correct. 570. Schiaparelli has not been able to determine with certainty the rotation of Venus; but his observations are sufficient to show that the period of Schroeter given in the text (23 h 21 m ) is very likely wrong, and that the planet probably rotates, like Mercury, only once in an orbital revolution ; keeping the same face always towards the sun. 589. The observations of Campbell in 1894 show no sensible difference between the spectrum of Mars and that of the moon, and indicate that the planet's atmosphere must be extremely rare. 593. In the Astron. Jahrbuch for 1895 the asteroids known and definitely 'numbered' count up 390. Several others have been dis- covered, but not sufficiently observed to determine their orbits. During 1892 a new method of search was introduced. By photo- graphing a portion of the heavens with a camera of wide field, mounted equatorially and moved by clock-work, pictures are obtained in which any planets present can be easily distinguished by their motion during the two or three hours during which the exposure of the plate is continued ; while the images of stars are round, if the clock-work runs correctly, the planets are apparently elongated. Wolf of Heidelberg and Charlois of Nice have been specially successful in this method of discovery. 348 THE MAJOR PLANETS. CHAPTER XV. THE PLANETS CONTINUED. — THE MAJOR PLANETS: JUPITER, SATURN, URANUS., AND NEPTUNE. JUPITER. 609. While this planet is not so brilliant as Venus at her best, it stands next to her in this respect, being on the average about five times brighter than Sirius, the brightest of the fixed stars. Jupiter, moreover, being a "superior" planet, is not confined, like Venus, to the neighborhood of the sun, but at the time of opposition is the chief ornament of the midnight sky. 610. Orbit. — The orbit presents no marked peculiarities. The mean distance of the planet from the sun is 483,000000 miles. The eccentricity of the orbit being nearly -fa (0.04825) ; the greatest and least distances vary by about 21,000000 miles each way, making the planet's greatest and least distances from the sun 504,000000 and 462,000000 miles respectively. The average distance of the planet from the earth at opposition is 390,000000, while at conjunc- tion it is 576,000000 miles. The minimum opposition distance is only 369,000000, which is obtained when the opposition occurs about October 6, Jupiter being in perihelibn when its heliocentric longi- tude is about 12°. At an aphelipn opposition (in April) the distance is 42,000000 miles greater; that is, 411,000000. The relative brightness of Jupiter at an average conjunction and at the nearest and most remote oppositions is respectively as the numbers 10, 27, and 18. The average brightness at opposition is, therefore, more than double that at conjunction ; and at an October opposition the planet is fifty per cent brighter than at an April one. The differences are considerable, but far less important than in the case of Mars, Venus, and Mercury. The inclination of the orbit to the ecliptic is small, — only 1° 19'. 611. Period. — The sidereal period is 11.86 years, and the synodic is 399 days (a number easily remembered) , a little more than a year and a month. The planet's orbital velocity is about eight miles a second. JUPITER. 349 612. Dimensions. — The planet's apparent diameter varies from 50" at an October opposition (or 45£" at an April one) to 32" at conjunction. The form, however, of the planet's disc is not truly circular, the polar diameter being about -fa part less than the equa- torial, so that the eye notices the oval form at once. The equa- torial diameter in miles is 88,200, the polar being 83,000. ' Its mean 1 diameter, therefore, is 86,500, — almost eleven times that of the earth. This makes its surface 119 times, and its volume 1300 times, that of the earth. It is by far the largest of the planets in the system ; in fact, whether we regard its bulk or its mass, larger than all the rest put together. 613. Mass, Density, etc. — Its mass is very accurately known, both by the motions of its satellites, and the perturbations of the asteroids. It is yw^s °^ * ne sun ' s mass, or very nearly 316 times that of the earth. Comparing this with its volume, we find its density 0.24, less than \ the density of the earth, and almost precisely the same as that of the sun. Its mean superficial gravity comes out 2.64 times that of the earth ; that is, a body on Jupiter would weigh 2-| times as much as upon the surface of the earth ; but on account of the rapid rotation of the planet and its ellipticity there is a very con- siderable difference between the force of gravity at the equator and at the pole, amounting to \ of the equatorial gravity. (On the earth the difference is only y^j-. ) 614. Phases and Albedo. — Its orbit is so much larger than that of the earth that the planet shows no sensible phases, even at quadra- ture, though at that time the edge farthest from the sun shows a slight darkening. The reflecting power, or Albedo, of the planet's surface is very high, — 0.62 according to Zollner, that of white paper being only 0.78. The centre of the disc of this planet (and the same is also true of Saturn) is considerably brighter than the limb — just the reverse, as will be remembered, from the condition of things upon the moon, and upon Mars, Venus, and Mercury. This peculiarity of a darkened limb, in which Jupiter resembles the sun, has sug- 1 The mean diameter of an oblate spheroid is 2g j" , not2-±— Of the three 3 ^ axes of symmetry which cross at right angles at the planet's centre, one is the axis of rotation, and both the others are equatorial. 350 JUPITER. gested the idea that it is to some extent self-luminous. This, how- ever, is not a necessary consequence, as a nearly transparent atmos- phere overlying a uniformly reflecting surface would produce the same effect. The light which the planet emits, if it emits any, must be very feeble as compared with sunlight, since the satellites, when they are eclipsed by entering the shadow, become totally invisible. 615. Axial Rotation. — The planet rotates on its axis in about 9 h 55 m . The time can be given only approximately, not because it is difficult to find and observe distinct markings on the planet's disc, but simply because different results are obtained from differ- ent spots, according to their nature and their distance from the planet's equator. Speaking generally, spots near the equator indi- cate a shorter day than those in higher latitudes, and certain small, sharply defined, bright, white spots, such as are often seen, give a quicker rotation than the dark markings in the same latitude. For instance, a white spot observed near the equator in 1886 for several months gives 9 h 50 m 4", while another one near latitude 60° gave 9 h 55 ra 12". The great red spot has given values ranging from 9 h 55™ 34".9 (in 1879) to 9 h 55™ 40 s .7 (in 1886),— the time of rotation as determined in each case, being certainly accurate within half a second. The progressive increase has been regular and unmistakable, and is not due to any possible uncertainty in the observations. 616. The Axis of Rotation and the Seasons. — The plane of the equator is inclined only 3° to that of the orbit, so that as far as the sun is concerned there can be no seasons. The heat and light received from the sun by Jupiter are, however, only about ^ as intense as the solar radiation at the earth, its distance being 5.2 times as great. 617. Telescopic Appearance. — Even in a small telescope the planet is a beautiful object. "When near opposition a magnifying power of only 40 makes its apparent size equal to that of the full moon (though, as remarked in connection with Venus, no novice would receive that impression), and with a telescope of 8 or 10 inches aperture, and with a magnifying power of 300 or 400, the disc is covered with an infinite variety of beautiful and interesting details which rapidly shift under the observer's eye in consequence of the planet's swift rotation. The picture is rich in color, also, browns and reds predominating, in contrast with olive-greens and occasional THE GREAT RED SPOT. 351 purples ; but to bring out the colors well and clearly requires large instruments. For the most part the markings are arranged in streaks more or less parallel to the planet's equator, as shown by Fig. 178. With a small telescope the markings usually reduce to two dark and comparatively well-defined belts, one on each side of the equator, occupying about the same regions of latitude that the trade-wind zones do upon the earth ; and very likely in Jupiter's case similar aerial currents have something to do with the appearance, though upon Jupiter, as has been already said, the solar heat is a compara- Fig. 178. — Telescopic Views of Jupiter. tively unimportant factor. The markings upon the planet are almost, if not entirely, atmospheric, as is proved by the manner in which they change their shapes and relative positions. They are cloud forms. It is hardly probable that we ever see anything upon the solid surface of the planet underneath, nor is it even certain that the planet has anything solid about it. In Fig. 178, the upper left-hand figure is from a drawing by Trouvelot made in February, 1872 ; the second is by Vogel in 1880. The small one below repre- sents the planet as seen in a .small telescope. 618. The Great Red Spot. — While most of the markings on the planet are evanescent, it is not so with all. There are some which 352 JTTPITBE. are at least "sub-permanent," and continue for years, not without change indeed, but with only slight changes. The " great red spot " is the most remarkable instance so far. It seems to have been first observed by Prof. C. W. Pritchett of Glasgow, Missouri, in July, 1878, as a pale, pinkish, oval spot some 13" in length by 3" in width (30,000 miles by 7000). Within a few months it had been noticed by a considerable number of other observers, though at first it did not attract any special attention, since no one thought of it as likely to be permanent. The next year, however, it was by far the most conspicuous object on the plauet. It was of a clear, strong brick-red color, with a length fully one-third the diameter of the planet and a width about one fourth of its length. For two or three years it remained without much change : in 1882-83 it gradually faded out : in 1885 it had become a pinkish oval ring, the central part being apparently occupied with a, white cloud. In 1886 it was again a little stronger in color, and the same in 1887, — an object not diffi- 8 4 Pig. 179. — Jupiter's " Red Spot." From Drawings by Mr. Denning. 1880-85. cult to see with a large telescope, but the merest ghost of what it was in 1880. The present year (1888) its appearance is about the same as in 1887, — perhaps a little paler. During the ten years its form and size have varied very little. It lies at the southern edge of the southern equatorial belt, in lati- TEMPERATURE AND PHYSICAL CONSTITUTION. 353 tude about 35°, and for some reason the belt seems to be " notched out " for it, so that there has been always a narrow white streak separating the belt from the spot. Even when the spot was palest and hard to see its place was always evident at once from the indentation in the outline of the belt. Such phenomena suggest abundant matter for speculation which would be out of place here. It must suffice to say that no satisfactory explanation of the phenomena has yet been presented. The unquestionable fact before mentioned (Art. 615), that the time of rotation of the spot has changed by more than 5 s in the ten years, greatly complicates the subject. Fig. 179, from the drawings of Mr. Denning, represents the appearance of the spot at four different dates; viz., 1, 1880, Nov. 19; 2, 1882, Oct. 30; 3, 1884, Feb. 6; 4, 1885, Feb. 25. 619. Temperature and Physical Constitution. — The rapidity of the changes upon the visible surface implies the expenditure of a con- siderable amount of heat, and since the heat received from the sun is too small to account for the phenomena which we see, Zollner, thirty years ago, suggested that it must come from within the planet, and that in all probability Jupiter is at a temperature not much short of incandescence, — hardly yet solidified to any considerable extent. Mr. Proctor has given special currency to these views among English readers. The idea that Jupiter might be such a " semi-sun " is not at all new. Buffon, Kant, Nasmyth, and Bond all entertained and discussed it ; but it is only since the investigations of Zollner that it has become an accepted item of scientific belief. (See Gierke's " History of Astronomy," p. 335 seqq.) 620. Atmosphere. — As to the composition of the planet's atmos- phere, the spectroscope gives us rather surprisingly little information. We get from the planet a good solar spectrum with the solar lines well marked, but there are no well-defined absorption bands due to the action of the planet's atmosphere. There are, however, some shadings in the lower red portion of the spectrum that are probably thus caused. The light, for the most part, seems to come from the upper surface of the planet's envelope of clouds without having penetrated to any depth. / 'T ■f 621. Satellite System. — Jupiter has four satellites, — the first heavenly bodies ever discovered — the first revelation of Galileo's telescope. His earliest observation of them was on Jan. 7, 1610, and in a very few weeks he had ascertained their true character, and determined their periods with an accuracy which is surprising when we consider his means of observation. The number of the heavenly 354 JUPITER. bodies was now no longer seven, and the discovery excited among churchmen and schoolmen a great deal of angry incredulity and vituperation. Galileo called them " the Medicean stars." They are now usually known as the first, second, etc., in the order of distance from the primary, but they also have names which are some- times used; viz., Io, Europa, Ganymede, and Callisto. Their rela- tive distances range between 262,000 and 1,169,000 miles, being very approximately 6, 9, 15, and 26 radii of the planet. The distance of the first from the surface of the planet is almost exactly the same as that of our own moon from the surface of the earth. Their sidereal periods range between l d 18^ h and 16 d 16J h (accurate values in distances and periods are given in the table in the Appendix) . The orbits are almost exactly circular, and lie in the plane of the planet's equator. The satellites slightly disturb each other's motions, and from these disturbances their masses can be ascertained in terras of the planet's mass. The third, which is much the largest, has a mass of about TT j5j of the planet's, a little more than double the mass of our own moon. The mass of the first satellite appears to, be a little less than A as much. The second is somewhat larger than the first, and the fourth is about half as large as the third; i.e., it has about the mass of our own moon. The densities of the first and fourth appear to be not very different from that of the planet itself, while the densities of the second and third are considerably greater. 622. Relation between Mean Motions and Longitudes of the Satellites. — In consequence of their mutual interaction a curious relation (discovered by La Place) exists between the mean motions of the first three satellites. The mean motion is of course 360° divided by T ( T being the satellite's period). It appears that the mean motion of the first plus twice the mean motion of the third equals three times that of the second, or r, t s r, A similar relation holds for their longitudes : L l + 2L a = SL,+ 18(P; so that they cannot all three come into opposition or conjunction with the sun at once. These relations are permanently maintained by their mutual attractions : exactly in the long run, though there are slight perturbations produced by the fourth satellite which disturb the arrangement slightly for short periods. The fourth satellite does not come into any such arrange- ment. jtjpiter's satellites. 355 623. Diameters, etc. — The diameter of the first satellite is a little more than 2400 miles ; the second is almost exactly the size of our own moon, i.e., between 2100 and 2200 miles ; and the third and fourth have diam- eters, respectively, of 3600 and 3000 miles, the third, Ganymede, being much larger than either of his sisters. W.hen Jupiter is in opposition, the fourth satellite is sometimes nearly 10£' away from the planet, or £ of the moon's diameter ; and in very clear air can be seen by a sharp eye without telescopic aid. The third, though much larger, never goes more than 6' from the planet, and it is perhaps doubtful whether it is ever seen with the naked eye, unless when the fourth happens to be close beside it. A good opera-glass will easily show them all as minute points\of light. ) ] \ 624. Brightness. — Since the sunlight of Jupiter .is only ^ as in- tense as ours, the moonlight made by the 1 satellites is decidedly inferior to our own, although their reflective power appears to be higher than that of the lunar surface. They differ among themselves considerably in this respect. The fourth satellite is of an especially dark complexion, so that it often looks perfectly black when it passes between us and the planet, and is projected on the disc. The others, under similar circumstances, show light or dark according as they have a dark or light portion of the planet for a background. Even the fourth, when crossing the disc, is always seen bright while very near the planet's limb. 625. Markings upon the Satellites. — The satellites show sensible discs when viewed with a large telescope, and all of them but the second sometimes show dark markings upon the surface. These markings, however, are only visible under the most favorable circumstances, and it has not been possible to determine whether they are atmospheric or really geographical, iior has it yet been possible to deduce from them the satellites' periods of rotation. 626. Variability. — Galileo noticed variations in the brightness of the satellites at different times, and subsequent observers have confirmed his result. In the case of the fourth satellite there seems to be a regular variation depending upon the place of the satellite in its orbit, and suggest- ing that in its axial rotation it behaves like our own moon, keeping always the same side next its primary. In addition it shows other irregular changes in its luminosity: so also do the other satellites according to nearly all authorities, though it is singular that one or two of the best observers do not find any such irregularity indicated by their instrumental 1 photometric observations. 627. Eclipses and Transits. — The satellites' orbits are so nearly in the plane of the planet's orbit that, excepting the fourth, they all pass through the shadow of the planet, and suffer eclipse at every i Clerke's " History of Astronomy," p. 339. 356 JUPITER. revolution. At conjunction, also, they cast their shadows upon the planet, and these shadows can easily be seen in the telescope as black dots on the planet's disc, the satellites themselves, which cross the disc about the same time, being much more difficult to observe. The fourth satellite escapes eclipse when Jupiter is far from the node of its orbit. Thus, during 1888 and in the first half of 1889, there are no eclipses of Callisto at all. Exactly at opposition or conjunction the planet's shadow lies straight behind it out of our sight, so that we cannot at that time Lii — f§p ■^WSIiiifli,r • 4 ■ " : 112° .J-—- ^ = - 2-^r^x" -J L J ^ ' y Shadow of 3 r 3 Fig. 180. — Eclipses of Jupiter's Satellites, at Western Elongation. observe the eclipses of the satellites, but only their transits across the disc. Before and after these times, however, the shadow lies one side of the planet. When the planet is at quadrature and the condition of things is as represented in Fig. 180 (which is drawn to scale), the shadow projects so far to one side of the planet that the whole eclipse of all the satellites, except the first, takes place clear of the planet's disc, — both the disappearance and reappearance of the satellite being visible. 628. "Equation of light." — The most important use that has been made of these eclipses has been to ascertain the time required by light in traversing the distance between us and the sun, the so- THE EQUATION OB" LIGHT. 357 called " equation of light." It was in 1675 that Roemer, the Danish astronomer (the inventor of the transit instrument, meridian circle, and prime vertical instrument, — a man nearly a century in advance of his day) , found that the eclipses of the satellites showed a peculiar variation in their times of occurrence, which he explained as due to the time taken by light to pass through space. His bold and original suggestion was rejected by most astronomers for more than fifty years, — until long after his death, — when Bradley's discovery of aberration (Art. 225) proved the correctness of his views. 629. If the planet and earth remained at an invariable distance the eclipses of the satellites would recur with unvarying regularity (their disturbances being very slight) , and the mean interval could be determined, and the times tabulated. But if we thus predict the times of eclipses for a synodic period of the planet, then, begin- ning at the time of opposition, it will be found that as the planet recedes from the earth, the eclipses fall constantly more and more behindhand, and by precisely the same amount for all four of the satellites. The difference between the tabulated and observed time continues to increase until the planet is near conjunction, when the eclipses are more than sixteen minutes late. From the insufficient observa- tions at his command, Roemer made the difference twenty-two minutes. After the conjunction, the eclipses quicken their pace and exactly make up all the loss ; so that when opposition is reached once more, they are again on time. It is easy to see from Fig. 181 that at opposition the planet is nearer the earth than at conjunction by just twice the radius of the earth's orbit ; i.e. ,JB—JA = 2SA. The whole apparent retardation of the eclipses between opposition and conjunction, should therefore be exactly twice the time required for light to come from the sun to the earth. This time is very nearly 500 seconds, or 8 m 20". Fig. 181. Determination of the Equation of Light. 358 JUPITER. Early in the century Delambre, from all the satellite eclipses of -which he could then secure observations, found it to be 493". A few years ago a redetermination by Glasenapp of Pulkowa made it 501", from fifteen years' observation of the eclipses of the first satellite. Probably this value is much nearer the truth than Delambre's. 630. Photometric Observations of the Eclipses. — The eclipses are gradual phenomena, the obscuration of the satellite proceeding con- tinuously from the time it first strikes the shadow of the planet until it entirely vanishes. The moment at which the satellite seems to disappear depends, therefore, on the state of the air and of the observer's eye, and upon the power of his telescope. The same is true of the reappearance ; so that the observations are doubtful to the extent of from half a minute for the first satellite (which moves quickly), to a full minute for the fourth. Professor Pickering has proposed to substitute for this comparatively indefinite moment of disappearance or reappearance, the instant when the satellite has lost or regained just half its normal light, and he determines this instant by a series of photometric comparisons with one of the neighboring uneclipsed satellites, or with the planet itself. These comparisons are made with a special photometer devised for the purpose, and planned with reference to rapid reading : by merely turning a small button, the observer is immediately able to make the image of the uneclipsed satellite appear to be of the same brightness as the satellite which is disappearing, and the observations can be repeated very rapidly with the help of special contrivances for recording the times and readings. It is found that this instant of " half-brightness " can be deduced from the set of photometric readings with an error not much exceeding a second or two. Observations of this kind have now been going at Cambridge (U. S.) 1 for several years. A similar plan has also been devised by Cornu, and is being carried out at the Paris Observatory under his direction. A series of such observations covering the planet's whole period of twelve years, ought to give us a much more accurate determination of the light- equation than we now have. 631. Until 1849 our only knowledge of the velocity of light was obtained by observations of Jupiter's satellites. By assuming as 1 Professor Pickering has more recently (August, 1888) applied photography to these observations with most gratifying success. A series of pictures is taken, each with an exposure of 10 s , the time being recorded on a chronograph, and they determine with great precision the moment when the satellite's brightness had any special value, say fifty per cent of its maximum. SATUBN. 359 known the earth's distance from the sun, the velocity of light follows when we know the time occupied by light in coming from the sun. At present, however, the case is reversed : we can deter- mine the velocity of light by two independent experimental methods, and with a surprising degree of accuracy ; and then, knowing the velocity and the light-equation, we can deduce the distance of the sun. SATURN. 632. The Orbit and Period. — Saturn is the remotest of the ancient planets, its mean distance from the sun being 9.54 astro- nomical units, or 886,000000 miles. This distance varies by nearly 50,000000 miles on account of the eccentricity of its orbit (0.056) , which is a little greater than that of Jupiter. Its nearest approach to the earth at a December opposition (the longitude of its perihelion being 90° 4') is 744 millions of miles, and its greatest distance at a May conjunction is 1028 millions. It is so far from the sun that these changes of distance do not so greatly affect its apparent brightness, as in the ease of the nearer planets, the whole range of variation from this cause being less than two to one ; that is, at the nearest of all oppositions, the planet is not twice as bright as at the remotest of all conjunctions. The changing phases of the rings make quite as great a difference as the variations of distance. The orbit is inclined to the ecliptic about 2\°. The sidereal period of the planet is twenty-nine and one-half years, the synodic period being 378 days. The planet itself is unique among the heavenly bodies. The great belted globe carries with it a retinue of eight satellites, and is sur- rounded by a system of rings unlike anything else in the universe so far as known, the whole constituting the most beautiful and most interesting of all telescopic objects. 633. Diameter, Volume, and Surface. — The apparent mean diam- eter of the planet varies from 20" to 14" according to the distance. We say mean diameter because this planet is more flattened at the pole than any other, its ellipticity being nearly ten per cent, though different observers vary somewhat in their results. The equatorial diameter of the planet is about 75,000 miles, and its polar about 68,000, the mean being very nearly 73,000, or a little more than nine times that of the earth. Its surface is therefore about eighty-two times, and its volume 760 times that of the earth. 360 SAtURff. 634. Mass, Density, and Gravity. — Its mass is only ninety-five times the earth's mass, from which follows the remarkable fact that the density of Saturn is only one-eighth that of the earth, or only about Jive-sevenths that of water. It is by far the least dense of all the planets. The superficial gravity is 1.2. 635. Axial Rotation. — It revolves upon its axis in about 10 h 14" 1 according to a determination of Professor Hall, made in 1876 by means of a white spot which suddenly appeared upon its surface, and continued visible for some weeks. Although the surface of the planet is beautifully marked with belts which often show delicate rose-colored tints, it is seldom that any well-defined markings present themselves by which the rotation can be determined. The inclination of the axis is about 28°. 636. Surface, Albedo, and Spectrum. — As in the case of Jupiter, the edges of the disc are not quite so brilliant as the central por- tions, so that the belts appear to fade out near the limb. These belts are less distinct and less variable than those of Jupiter ; and are arranged as shown in Fig. 182, with a very brilliant zone at the equator, though the engraving much exaggerates the contrast. The planet's pole is marked by a darkish cap of greenish hue. According to Zollner, the Albedo, or reflecting power of the surface is 0.52, almost precisely the same as that of Venus, but a little infe- rior to that of Jupiter. The spectrum of the planet is the solar spec- trum without any evidence of the presence of water-vapor, so far as can be made out, but with certain unexplained dark bands in the red and orange similar to those observed in the spectrum of Jupiter. The darkest of these bands, however, are not seen in the spectrum of the ring ; this might have been expected, since the ring probably has but little atmosphere. 637. The Rings. — The most remarkable peculiarity of Saturn is his ring-system. The planet is surrounded by three, thin, flat, con- centric rings like circular discs of paper pierced through the centre. Two of them are bright, while the third, the one nearest to the planet, is dusky aud comparatively difficult to see. They are generally re- ferred to by Struve's notation as A, B, and C, A being the exte- rior one. For nearly fifty years this appendage of Saturn was a complete enigma to astronomers. Galileo, in 1610, saw with his little tele- Saturn's rings. 361 scope that the planet appeared to have something attached to it on each side, and he announced the discovery that "the outermost planet is triple," — "ultimam planetam tergeminam observavi." Fia. 182. — Saturn and bis Rings. Not long afterwards the rings were edgewise to the earth so that they became invisible to him ; and in his perplexity he inquired " whether Saturn had devoured his children, according to the legend." Huy- 362 SATURN. ghens, in 1655, was the first to solve the problem and explain the true structure of the rings. Cassini, 1 twenty years later, discovered that the ring was double, — composed of two concentric portions with a narrow black rift of division between them. The third, or dusky ring, C, is an American discovery, and was first brought to light by W. C. Bond at Cambridge, U. S., in Novem- ber, 1850. About two weeks later, but before the news had been published in England, it was also discovered independently by Dawes. For a while there was some question whether it was not really a new formation ; but an examination of old drawings shows that Herschel and several other astronomers had previously seen it where it crosses the planet, although without recognizing its character. 638. Dimensions of the Rings. — The outer ring, A, has an exterior diameter of 168,000 miles, and is a little more than 10,000 miles wide. The division between it and ring, B, is about 1600 miles in width, and apparently perfectly uniform all around. Ring B is about 16,500 miles wide, and is much brighter than A, especially at its outer edge. At the inner edge it becomes less brilliant, and is joined without any sharp line of demarcation by ring C, which is sometimes known as the "gauze" or " crape " ring, because it is only feebly luminous and is semi-transparent, allowing the edge of the planet to be seen through it. The innermost ring is nearly, per- haps not quite, as wide as the outer one, A. There is thus left a clear space of from 9000 to 10,000 miles in width between the planet's equator and the inner edge of the gauze ring, the whole ring system having an external diameter of 168,000 miles, and a width of between 36,000 and 37,000. The thickness of the rings is very small indeed, probably not ex- ceeding 100' miles. If we were to construct a model of them on the scale of 10,000 miles to the inch, so that the outer one would be nearly seventeen inches in diameter, the thickness of an ordinary sheet of writing paper would be about in due proportion. This extreme thin- ness is proved by the appearances presented when the plane of the riug is directed towards the earth, as it is once in every fifteen years. 1 In consequence of a misunderstanding of some expressions used by Ball, an English astronomer who observed Saturn in 1665-66, the discovery of the division between the rings was for a time attributed to him, and statements to that effect will be found in a number of important books. The original drawings belonging to his paper in the Philosophical Transactions have, however, recently been found, and show that he did not see the division at all, nor, indeed, even under stand that the appendage was a ring. PHASES OF THE RINGS. 363 At that time the ring becomes invisible except to the most powerful telescopes. 639. Phases of the Rings. — The rings are parallel to the equator of the planet, which is inclined about 27° to its orbit, and about 28° to the plane of the ecliptic, the two nodes of the ring being in longi- tude 168° and 348°, in the constellations of Aquarius and Leo. Now in the planet's revolution around the sun, the plane of the planet's equator and of the rings always keeps parallel to itself (as shown in Fig. 183) , just as does the plane of the earth's equator. Twice, therefore, in the planet's revolution, when the plane of the ring Fig. 183. — The Phases of Saturn's Rings. passes through the earth, we see it edgewise ; and twice at its maxi- mum width, when it is at the points half-way between the nodes. The angle of inclination being 28°, the apparent width of the ring at the maximum is just about half its length. The last disappearance of the rings was in October, 1891; the next will be in the summer of 1907. Near the time of disappearance the ring appears simply as a thin needle of light projecting on each side of the planet to a distance nearly equal to its diameter. Upon this the satellites are threaded like beads when they pass between us and the planet. 640. Irregularities of Surface and Structure. — When the rings are edgewise we find that there are notable irregularities upon them. They are not truly plane, nor quite of even thickness throughout. The same thing is indicated by certain peculiarities sometimes reported in the form of the shadow cast by the planet on the rings ; but caution must be used in accepting and interpreting such observations, because illusions 364 SATURN. are very apt to occur from the least indistinctness of vision or feebleness of light. The writer has usually found that the better the seeing, the fewer abnormal appearances were noted, and the experience of the Washington observers is the same. It can hardly be doubted that the details of the rings are continu- ally changing to some extent. Thus the outer ring, A, is occasion- ally divided into two by a very narrow black line known as " Encke's division," although more usually there is merely a darkish streak upon it, not amounting to a real " crack" in the surface. 641. Structure of the Rings. — It is now universally admitted that the rings are not continuous sheets of either solid or liquid matter, but are composed of a swarm of separate particles, each a little independent moon pursuiug its own path around the planet. The idea was suggested loug ago, by J. Cassini in 1715, and by Wright in 1750, but was lost sight of until Bond revived it in con- nection with his discovery of the dusky ring. Professor Benjamin Peirce soon afterwards demonstrated that the rings could not be con- tinuous solids ; and Clerk Maxwell finally showed that they can be neither solid nor liquid sheets, but that all the known conditions would be answered by supposing them to consist of a flock of separate and independent bodies, moving in orbits nearly circular and in one plane, — in fact, a swarm of meteors. 642. Stability of the Ring. — If the ring were solid it would cer- tainly not be stable, and the least disturbance would bring it down upon the planet; nor is it certain that even the swarm-like structure makes it forever secure. It is impossible to say positively that the rings may not after a time be broken up. A few years ago there was much interest in a speculation which Struve published in 1851. All the measures which he could obtain up to that date appeared to show that a change was actually in progress, and that the inner edge of the ring was extending itself towards the planet. His latest series of measurements (in 1885) does not, however, confirm this theory. They show no considerable change since 1850, and the measurements of other observers agree with his in this respect. The researches of Professor Kirkwood of Indiana make it probable that the divisions in the ring are due to the perturbations produced by the satel- lites. They occur at distances from the planet where the period of a small body would be precisely commensurable with the periods of a number of the satellites. It will be remembered that similar gaps are found in the distribution of the asteroids, at points where the period of an asteroid would be commensurable with that of Jupiter. satuen's satellites. 365 I 643. Satellites. — Saturn has eight 1 of these attendants. The largest of them was discovered by Huyghens in 1655. It appears as a star of the ninth magnitude, and is easily observable with a three- inch telescope. Four others were discovered by Cassini before 1700, two by Sir William Herschel near the end of the last century, and one, Hyperion, the latest addition to the planet's family, by Bond of Cambridge, in September, 1848, and independently by Las- sell at Liverpool two days later. The range of the system is enormous. Iapetus has a distance'of 2,225000 miles, with a period of 79 days, nearly as long as that of Mercury. There is a remarkable variation iu the brightness of this satellite. On the western side of the planet it is fully twice as bright as upon the eastern, which practically demonstrates that, like our own moon, it keeps the same face towards the planet at all times, one-half of its surface being much more brilliant than the other. Mimas, the nearest and smallest of the satellites, coasts around the edge of the ring at a distance from it of only 34,000 miles, or 118,000 from the planet's centre, having a period of only 22^ hours. This satellite is so small and* so near the planet that it pan be seen only by very large telescopes and under favorable conditions. Titan, as its name suggests, is by far the largest of the family. Its distance is about 770,000 miles, and its period a little less than 16 days. It is probably 3000 or 4000 miles in diameter, and accord- ing to Stone, its mass is ^V<7 of Saturn's. 644. Peculiar Behavior of Hyperion. — Hyperion has a distance of 934,000 miles, and a period of 21£ days. Under the action of Titan its orbit is rendered considerably eccentric, and its line of apsides always keeps itself in the line of conjunction with Titan, retrograding in a way which at first seemed to defy theoretical explanation, but turns out to be only a " new case in celestial mechanics," and a necessary result of the disturb- ance by Titan. The orbit of Iapetus is inclined about 10° to the plane of the rings, but all 1 Until Herschel's time it was customary to distinguish the satellites as first,' second, etc., in order of distance from the planet ; but as Herschel's new satellites • were within the orbits of those which were known before, their discovery con- fused matters, and the confusion became worse confounded when the eighth appeared. They are now usually designated by names assigned by Sir John Herschel as follows, beginning with the most remote, namely : IapStus (Hype- rion), Titan ; Rhea, Dione, Tethys ; Enceladus, Mimas. It will be noticed that these names, leaving out Hyperion, which was undiscovered when they were assigned, form a line and a half of a regular Latin pentameter. 366 URANTJS. the other satellites move exactly in their plane, and all the five inner ones move in orbits sensibly circular. The orbits of Iapetus, Hyperion, and Titan have a slight eccentricity. It is not at all impossible or even improbable that other minute satellites may yet be discovered in the great gap between Titan and Iapetus. URANUS. 645. As the satellites of Jupiter were the first heavenly bodies to be "discovered," so Uranus was the first "discovered" planet, all the other planets then knowu having been known from prehistoric antiquity. On March 13, 1781, the elder Herschel, in sweeping over the heavens systematically with a seven-inch reflector made by himself, came upon an object which, by its disc, he saw at once was not an ordinary star. In a day or two he had ascertained that it moved, and announced the discovery as that of a comet. After a short time, however, it became obvious from the computations of Lexell, that its orbit was nearly circular, that its distance was enormous, and that its path did not at all resemble that ordinarily taken by a comet ; and within a yeai; its planetary character was recognized and it was formally admitted as a new member of the solar system. The name of Uranus, suggested by Bode, finally pre- vailed over other appellations (Herschel himself called it the Georgium Sidus, in honor of the king) , with the symbol # or § . The former is still generally used by English astronomers. The discovery of a new planet, a thing then utterly unprecedented, caused great excitement. The king knighted Herschel, gave him a pension, and furnished him with the funds for constructing his great forty-foot reflector of four feet aperture, with which he afterwards discovered the two inner satellites of Saturn. It was found on reckoning back from the date of Herschel's discovery that the planet had been several times before observed as a star by astronomers who narrowly missed the honor which fell to the more fortunate and diligent Herschel. Twelve such observations had been made by Lemonnier alone. 646. Orbit. — The mean-distance of Uranus from the sun is very nearly 18QQ_millions of miles, and the eccentricity a trifle less than that of Jupiter's orbit, amounting to about 83,000000. The inclina- tion of the orbit to the plane of the ecliptic is very slight, only 46'. The planet's periodic time is 84 years, and the synodic period (from opposition to opposition) 369 d 8 h . The orbital-velocity is 4^ miles per second. APPEARANCE AND MAGNITUDE OP URANUS. 367 647. Appearance and Magnitude. — Uranus is distinctly visible to the naked eye on a dark night as a small star of the so-called sixth magnitude. It is so remote, its orbit having a diameter more than 19 times that of the earth's, that there is very little change in its appearance, and it makes no practical difference whether it is at opposition or quadrature. In the telescope it shows a sea-green disc of about 4" in apparent diameter, corresponding to a real diameter of 32,000 miles. Its sur- face is about 16 times, and its volume about 66 times greater than that of the earth, so that the earth compares in size with Uranus about as the moon does with the earth. The mass of Uranus is,14.6 times that of the earth, and its density and surface-gravity are respec- tively 0.22 and 0.90. 648. Albedo and Light. — The reflecting power of the planet's surface is very high, its albedo, according to Zollner, being 0.64, even exceeding that of Jupiter. It is to be remembered, however, that sunlight at Uranus is only ^-g as intense as at the earth, and only about jJj as intense as at Jupiter ; so that the disc of the planet does not appear in the telescope even nearly as bright as a piece of Jupiter's disc of the same apparent size. The greenish blue tint of the planet is accounted for by the fact that its spectrum shows certain conspicu- ous dark bands in its lower portion, bands perhaps identical with those which are visible in the spectrum of Saturn, but much more intense. The F line is also specially prominent in the spectrum of Uranus. These facts probably indicate a dense atmosphere. 649. Polar Compression, Belts, and Rotation. — The disc of the planet shows a decided ellipticity — - about j 1 ^ according to the Prince- ton observations of 1883, which agree nearly with those of Schiapa- relli. There are also sometimes visible upon the planet's disc certain extremely faint bands or belts, much like the belts of Jupiter viewed with a very small telescope. "What is exceedingly singular, however, is that the trend of these belts seems to indicate a plane of rotation not coinciding with the plane of the satellites' orbits. Nearly all the observers who have seen them at all find that they are inclined to the satellites' orbit-plane at an angle of from 15° to 40°. Now unless there is some error in Tisserand's investigations upon the motions of satellites, it is certain that the plane of these orbits must of necessity nearly coincide with the planet's equator. Probably the error lies in judging the direction of the belts, which at the best are at the very limit of visibility. 368 URANUS. One or two observers have assigned to the planet rotation periods ranging from 9 h to 12 h ; but it cannot be said that any determination of this element yet made is to be trusted. 650. Satellites. — The planet has four satellites; viz., Ariel, Umbriel, Titania, and Oberon ; Ariel being the nearest to the planet. The two brightest of them, Oberon and Titania, were discovered by Sir "William Herschel a few years after the discovery of the planet. He observed them sufficiently to obtain a reasonably correct determi- nation of their distances and periods. It is not certain that he saw either of the other two, though he thought be had found six satellites in all, and a few years ago a popular writer on astronomy actually credited the planet with eight satellites, — the four whose names have been given, and four others which Herschel supposed he had seen. Ariel and Umbriel were first certainly discovered by Lassell in 1851, and have since been satisfactorily observed by numerous large telescopes. They are telescopically the smallest bodies in the solar system, and the most difficult to see. In real size, they are, of course, much larger than the satel- lites of Mars or many of the asteroids, very likely measuring from 200 to 500 miles in diameter ; but they are ten times as far away as the asteroids, and illuminated by a sunlight not ^ as brilliant as theirs. 651. Satellite Orbits. — -The orbits of the satellites are sensibly circu- lar, and all lie in one plane, which, as has been said, ought to be, and prob- ably is, coincident with the plane of the planet's equator. They are very close-packed also, Oberon having a distance of only 375,000 miles, with a period of 13 d ll", while Ariel has a period t>f 2 d 12", at a distance of 120,000 miles. Titania, the largest and brightest of them, has a distance of 280,000 miles, somewhat greater than that of the moon from the earth, with a period of 8 d 17 h . Under favorable circumstances this satellite can be just seen with a telescope of eight or nine inches aperture. 652. Plane of Revolution. — The most remarkable thing about this satellite system remains to be mentioned. The plane of their orbits is inclined 82°. 2 to the plane of the ecliptic, and in that plane they revolve backwards; or we may say, what comes to the same thing, that their orbits are inclined to the ecliptic at an angle of 97°.8, in which case their revolution is to be considered as direct. When the line of nodes of their orbit plane passes through the earth, as it did in 1840 and 1882, the orbits are seen edgewise and appear as straight lines. On the other hand, in 1861, they were seen almost in plan NEPTUNE. 369 as nearly perfect circles, and will be seen so again in 1903. The year 1882-83 was a specially favorable time for determining the inclination of the orbits and the position of the nodes, as well as for measuring the polar compres- sion of the planet. NEPTUNE. 653. The discovery of this planet is justly reckoned as the ({greatest triumph of mathematical astronomy. Uranus failed to move precisely in the path which the computers predicted for it, and was misguided by some unknown influence to an extent which a keen eye might almost see without telescopic aid. The difference between its observed place and that prescribed for it had become in 1845 nearly as much as the " intolerable" quantity of 2' of arc. Near the bright star Vega there are two little stars which form with it a small equilateral triangle, the sides of the triangle being about 1|° long. The northern one of the two little stars is the beautiful double-double star e Lyrse, and can be seen as double by a keen eye without a telescope, the two companions being about 3J' apart. Now the distance between the computed place of Uranus and its actual position was, when at its maxi- mum, just a little more than half of the distance between these components of e Lyrse, that only a keen eye can separate. One would almost say that such a minute discrepancy between observation and theory was hardly worth minding, and that to consider it " intolerable " was what a Scotchman would call " sheer pernickittyness." But just these minute discrepancies constituted the data which were found sufficient for calculating the position of a hitherto unknown planet, and bringing it to light. Leverrier wrote to Galle, in substance : " Direct your telescope to a point on the ecliptic in the constellation of Aquarius, in longitude 326°, and you will find within a degree of that place a new planet, looking like a star of about the ninth magnitude, and having a perceptible disc.'' The planet was found at Berlin on the night of Sept. 23, 1846, in exact accordance with this prediction, within half an hour aftei the astronomers began looking for it, and only about 52' distant from the precise point that Leverrier had indicated. 654. So far as the mathematical operations are concerned, the honor is to be equally divided between two then young men, — Leverrier of Paris, and Adams of Cambridge, England. Each took up the problem, and by perfectly independent and considerably dif- ferent methods arrived at substantially the same solution, and each 370 NEPTUNE. promptly communicated the result (Adams some months earlier than Leverrier) to a practical astronomer provided with the necessary apparatus for actually detecting the planet. Adams, who was then a graduate of three years' standing, a fellow and a tutor in his college, communicated his results to Challis, his professor of astronomy at Cambridge, in the autumn of 1845. Challis at once con- sulted Airy, the Astronomer Royal, but between them the matter rather lay in abeyance for some months, until a notice appeared of a preliminary paper by Leverrier, which indicated that he also had reached substantially the same conclusions as Adams. Then, at the urgent suggestion of Airy, Challis decided to begin the search at once, and to capture the planet by siege, so to speak. If he had had such star-maps as we now possess of the regions where the planet lay concealed, it would have been comparatively an dasy operation ; but as he had not, he decided to go over a space 10° wide by 30° long, and to go over it three times. The positions of all fixed stars would of course be the same at each of the three observations, but a planet would change its place in the meantime, and so would be surely detected. He began his work on July 29, including in his sweep all stars down to the tenth magnitude. When, on Oct. 1, he learned of the actual discovery of the planet, he had recorded the positions of something over 3000 stars, and was preparing to map them. He had already secured, as it turned out, three observations of the planet on Aug. 4, Aug. 12, and Sept. 29, and of course it was only the question of a few weeks more or less when the discussion of the observations would have brought the planet to light. But while this rather deliberate process was going on in England, Lever- rier had revised his work, making a second approximation, and had commu- nicated his results to Galle, at Berlin, substantially as above indicated. The Berlin astronomers had the great advantage of a new star-chart by Bremiker, covering that very region of the sky, and therefore did not need to enter upon any such tedious campaign as that begun by Challis. In less than • half an hour they found a new star, not indicated on the map, and showing a sensible disc, just as Leverrier had predicted ; and within twenty-four hours its motion proved it to be the planet. 655. Computed Elements Erroneous. — Both Adams and Leverrier, besides computing the planet's position in the sky had deduced elements of its orbit, and a value for its mass, which turned out to be considerably erroneous. The reason was that they had assumed that the mean distance of the new planet from the sun would follow Bode's law, a supposition which, as it turned out, is not even roughly true, although it was entirely warranted by the existing facts, since all the then known planets, not excepting Uranus, obey it with reasonable OLD OBSERVATIONS OF NEPTUNE. 371 exactness. This assumption of an erroneous mean distance of thirty- eight astronomical units, instead of the true distance of thirty, carried with it errors in all the other elements of the orbit ; and the computed elements are so wide of the truth that great authorities have main- tained that the actual Neptune was not at all the Neptune of Leverrier and Adams, but an entirely different planet; and even that the discovery was a " happy accident." It was not an accident at all, however. While the data and methods employed were not competent to determine the planet's orbit accurately, they were sufficient to determine the direction of the unknown body, which was the one thing needed to insure its discovery. The computers informed the searchers precisely where to point their telescopes, and could do so again were a new case of the same kind to appear. 656. Old Observations of Neptune. — After a few weeks' observa- tion of the new planet it became possible to compute an approximate orbit ; and reckoning back by means of this approximate orbit, the approximate place on any given date for many years preceding could be found. On examining the observations of stars made by different astronomers in these regions of the sky, there were found several instances in which they had observed the planet; a star of the ninth magnitude in the proper place for Neptune being recorded in their star-catalogues, while the place is now vacant. These old observations, thus recovered, were of great use in deter- mining the planet's orbit with accuracy. 657. The Orbit of Neptune. — The planet's mean distance from the sun is a little more than 2800,000000 of miles, instead of being over 3600,000000, as it should be according to Bode's law. The orbit, instead of being considerably eccentric, as it appeared to be from the computation of Adams and Leverrier, is more nearly circular than any other in the system except that of Venus, its eccentricity being only 1 9 . Even this small fraction, however, makes a varia- tion of over 50,000000 of miles in the planet's distance from the sun at different parts of its orbit. The inclination of the orbit is about If . The period of the planet is about 164 years, instead of 217, as it should have been according to Leverrier's computed orbit. The orbital velocity is about 3^ miles a second. 658. The Solar System as seen from Neptune. —At Neptune's distance the sun itself has an apparent diameter of only a little more than 1' of arc, — only about the diameter of Venus when nearest us, and too small to be seen as a disc by the eye, if there are eyes on Neptune. The light and heat received from it are only -^ part of 372 NEPTUNE. what we get at the earth. Still, we must not imagine that, as com- pared with starlight or even moonlight, the Neptunian sunlight is feeble. Assuming Zollner's estimate that sunlight at the earth is 618,000 times as intense as the light of the full moon, we find that the sun, even at Neptune, gives a light equal to 687 full moons. This is about seventy-eight times the light of a standard candle at one metre distance, or about the light of a thousand candle power electric arc at a distance of 10| feet — abundant for all visual purposes. In fact, as seen from Neptune, the sun would look very much like a large electric arc-lamp at a distance of a few feet. We call special attention to this, because erroneous statements are not unfrequently met with that " at Neptune the sun would be only a first magnitude star." It would really give about 44,000000 times the light of such a star. 659. From Neptune the four terrest.; a.1 planets would be hopelessly invisible, unless with powerful telescopes and by carefully screening off the sunlight. Mars would never reach an elongation of 3° from the sun ; the maximum elongation of the earth would be about 2°, and that of Venus about 1|°. Jupiter, attaining an elongation of about 10°, would probably be easily seen somewhat as we see Mercury.- Saturn and Uranus would be conspicuous, though the latter is the only planet of the whole system that can be better seen from Neptune than it can be from the earth. 660. The Planet itself. — Neptune appears in the telescope as a small star of between the eighth and ninth magnitudes, absolutely invisible to the naked eye, though easily seen with a good opera-glass. It shows a greenish disc, having an apparent diameter of about 2".6, which varies very little, since the entire range of variation in the planet's distance from us is only about -^ of the whole. The real diameter of the planet is about 35,000 miles (but the probable error of this must be fully 500 miles) ; the volume is about 85 times that of the earth. Its mass, as determined by means of its satellite, is about 17 times that of the earth, and its density 0.20. The planet's albedo, according to Zollner, is about forty-six per cent, a trifle lower than that of Saturn and Venus, and considerably below that of Jupiter and Uranus. There are no visible markings upon its surface, and nothing is known as to its rotation. The spectrum of the planet appears to be precisely like that of Uranus. The light is so feeble that the ordinary lines of the solar spectrum are difficult to make out, but there are a number of heavy, dark bands, which indicate the existence of a dense atmosphere, through which the light, reflected by the cloud-covered surface of the planet, is transmitted, — an atmosphere which appears to be identically the NEPTUNE'S SATELLITE. 373 same on both Uranus and Neptune, while some of its constituents are probably present in Jupiter and Saturn, as shown by the principal dark band in the red. It is not possible as yet to identify the substance which produces these bands. It will be seen that Uranus and Neptune form a " pair of twins " very much as the earth and Venus do ; being nearly alike in magni- tude, density, and other characteristics. 661. Satellite. — Neptune has one satellite, discovered by Lassell within a month after the discovery of the planet itself. Its distance is 223,000 miles, and its period is 5 d 21 h 2 m .7. Its orbit is inclined 34° 53', and it moves backward in it; i.e., clockwise, from east towards the west, like the satellites of Uranus. It is a very small object, appearing of about the same brightness as Oberon, the outer satellite of Uranus. From its brightness, as compared with that of Neptune itself, we estimate that its diameter is about the same as that of our own moon, though perhaps a little larger. Since Neptune is so far from the sun, and the planet has no other satellite of any size (none certainly comparable with this one), its motion must be practically undisturbed and very nearly uniform. It has been therefore pro- posed to make it a test of the uniformity of other motions in the solar system, such as the length of the day and the length of the month. It revolves rapidly enough to admit of very precise observations by large telescopes. It is, of course, possible that the planet has other undiscovered satellites, but if so, they must be very minute. 662. Trans-Neptunian Planet. — Perhaps the breaking down of Bode's law at Neptune may be regarded as an indication that the system terminates with him, and that there is no remoter planet. If such a planet exists, how- ever, it is sure to be found sooner or later, either by means of its disturbing action upon Uranus and Neptune, or else by the methods of the asteroid hunters, although, of course, its slow motion will render its detection in this way difficult. In 1877, Professor Todd, from a graphical investigation of the outstand- ing differences between the computed and observed places of Uranus (after allowing for Neptune's action), concluded that an undiscovered planet very probably exists, and that its longitude was then about 170°. He made a careful but unsuccessful search for it with the Washington telescope, going over a region about 40° long by 2° wide with a power of 400, hoping to recognize the planet by its disc. 663. In the Appendix, we give tables containing the most accurate data of the planetary system at present available, but with renewed cautions to the student that these data are of very different degrees of accuracy. 374 THE PLANETS. The distances (in astronomical units), and the periods of the planets (except perhaps some of the asteroids) are known with extreme precision ; probably the very last figure of the table may be trusted. The other elements of their orbits are also known very closely, if not quite so precisely as the distances and periods. The masses, in terms of the sun's mass, stand next to the orbit elements in order of precision, with an error probably not exceeding one per cent (except, however, in the case of Mercury, the mass of which remains still very uncertain). The ratio of the earth's mass to the sun's is however less accurately known, being at present subject to an uncertainty of at least two per cent. This is because its determination involves a knowledge of the solar parallax (Art. 278*), the cube of which appears in the formula for the ratio of the masses. Of course all the masses of the planets expressed in terms of the earth's mass are subject to the same uncertainty in addition to all other possible causes of error. When we come to the diameters, volumes, and densities of the planets, the data become less and less certain, as has been pointed out before. For the nearer and larger planets, say Venus, Mars, and Jupiter, they are reason- ably satisfactory, for the remoter ones less so, and the figures for the density of the distant planets, — Mercury, Uranus, and Neptune, for instance, — are very likely subject to errors of ten or twenty per cent, if not more. 664. We borrow from Herschel's " Outlines of Astronomy " the following illustration of the relative magnitudes and distances of the members of our system. "Choose any well-levelled field. On it place a globe two feet in diameter. This will represent the sun ; Mercury will be represented by a grain of mustard-seed on the circumference of a circle 164 feet in diameter for its orbit ; Venus, a pea on a circle of 284 feet in diameter ; the Earth also, a pea on a circle of 430 feet ; Mars, a rather large pin's-head on a circle of 654 feet; the asteroids, grains of sand in orbits of 1000 to 1200 feet; Jupiter, a moderate-sized orange in a circle nearly half a mile across; Saturn, a small orange on a circle of four-fifths of a mile ; Uranus, a full-sized cherry or small plum upon the circumference of a circle more than a mile and a half; and finally Neptune, a good-sized plum on a circle about two miles and a half in diameter. ... To imitate the motions of the planets in the above-men- tioned orbits, Mercury must describe its own diameter in 41 seconds ; Venus, in 4° 14"; the Earth, in 7 minutes; Mars, in 4 m 48 s ; Jupiter, in 2" 56°; Saturn, in 3* 13° ; Uranus, in 2" 16" ; and Neptune, in 3 b 30°." We may add that on this scale the nearest star would be on the opposite side of the globe, at the antipodes, 8000 miles away. ADDENDUM TO CHAPTER XV. 374 A 621. On Sept. 9, 1892, Mr. Barnard at the Lick Observa- tory discovered a fifth satellite of Jupiter. It is extremely small, certainly not exceeding a hundred miles in diameter, and so near the planet that it is exceedingly difficult as a telescopic object, — quite beyond the possibility of observa- tion by any instruments of less than 18 or 20 inches aperture. The distance of the satellite from the planet's centre is about 112,500 miles, and its period ll h 57.4"'; its orbit is of course almost exactly in the plane of the planet's equator, and very nearly circu- lar (see Art. 548). DETERMINATION OF THE SUN'S DISTANCE. 375 CHAPTER XVI. THE DETERMINATION OF THE STJN's HORIZONTAL PARALLAX AND DISTANCE. — TRANSITS OP VENUS AND OPPOSITIONS OP MARS. — GRAVITATIONAL METHODS. — DETERMINATION BY MEANS OF THE VELOCITY OP LIGHT. 665. This problem, from some points of view, is the most funda- mental of all that are encountered by the astronomer. It is true that it is possible to deal with many of the subjects that present themselves in the science without the necessity of employing any units of length and mass but those that are purely astronomical, leaving for subse- quent determination the relation between these units and the more familiar ones of ordinary life : we can get, so to speak, a map of the solar system, correct in proportion, though without a scale of miles. But to give the map its real meaning and use, we must find the scale finally, if not at first, and until this is done we can form no true con- ceptions of the actual dimensions, masses, and distances of the heav- enly bodies. The problem of finding the true value of the astronomical unit is difficult, because of the great disproportion between the size of the earth and the distance of the sun. The relative smallness of the earth limits the length of our available "base line," which is less than j-j-jTnr part of the distance to be determined by it. It is as if a person confined in an upper room with a wide prospect were set to determine the distance of objects ten miles or more away, without going outside the limits of his single window. It is hopeless to look for accurate results by direct methods, such as answer well enough in the moon's case, and astronomers are driven to indirect ones. 666. Historical. — Until nearly 1700 no even reasonably accurate knowledge of the sun's distance had been obtained. Aristarchus, by a very ingenious though inaccurate method, had found, as he thought, that the distance of the sun was nineteen times as great as that of the moon (it is really 390 times as great), and Hipparchus, combining this determination of Aristarchus with his own knowledge of the moon's distance, estimated the 376 DETERMINATION OP THE SUN'S DISTANCE. sun's parallax at 3', which is more than twenty times too large. This value was accepted by Ptolemy, and remained undisputed for twelve centuries, until Kepler, from Tycho's observations of Mars, satisfied himself that the sun's parallax could not exceed 1' ; i.e., that the sun's distance must be at least as great as twelve or fifteen millions of miles. Between 1670 and 1680 Cassini proposed to determine the solar parallax by observations of Mars ; for by that time the distance of Mars from the earth at any moment could be very accu- rately computed in astronomical units, so that the determination of the par- allax of Mars would make known that of the sun. Observations in France, compared with observations made in South America by astronomers sent out for the purpose, showed that the parallax of Mars could not exceed 25", or that of the sun, 10"- Cassini set it at 9".5, corresponding to a distance of 86,-000000 of miles, — giving the first reasonable approach to the true dimen- sions of the solar system. In 1677, and more fully in 1716, Halley explained how transits of Venus might be utilized to furnish a far moi-e accurate determination of the solar parallax than was possible by any method before used. He died before the transits of 1761 and 1769 occurred, but they were both observed, the first not very satisfactorily, but the second with perfect success, and in the most widely separated parts of the globe. The results, however, were by no means as accordant as had been expected. Various values of the sun's par- allax were deduced, ranging all the way from 8|" to 9", according to the observations used, and the way they were treated in the discussion. Towards the end of the century, La Place adopted and used for a while the value 8".81, while Delambre preferred 8".C. 667. In 1822-24 Encke collected all the transit observations that had been published, and discussed them as a whole in an extremely thorough manner, deducing as a final result from the two transits of 1761 and 1769, 8". 5776, corresponding to a distance of 95} millions of miles. The decimal is very imposing, and the discussion by which it was obtained was unquestionably thorough and laborious, so that his value stood unquestioned and classical for many years. The first note of dissent was heard in 1854, when Hansen, in publishing certain researches upon the motion of the moon, an- nounced that Encke's parallax was. certainly too small ; he after- wards fixed the figure at 8". 97, but the correction of certain numeric cal errors in his work reduced this result to 8". 92. Three or four years later Leverrier found a value of 8".95 from the so- called lunar equation of the sun's motion ; and in 1862 Poucault, from a new determination of the velocity of light, combined with the constant of aber- ration, got the value 8". 86. A re-discussion of the old transit of Venus observations was then made by Stone, of England, who deduced from them METHODS OF FINDING THE SOLAR PARALLAX. 377 a value of 8".943. The value of 8". 95 was adopted by the British Nautical Almanac, and used in it until the issue of 1882. The corresponding dis- tance of 91} millions of miles found its way into numerous text-books, and, though known to be erroneous, still holds its place in some of them. In 1867 Newcomb made a discussion of all the data then avail- able, and obtained the value 8". 848 (or 8". 85 practically), which value is now (1888) used in all the Nautical Almanacs except the French. They use 8". 86, the result of an investigation published by Leverrier in 1872. 668. The observations of Grill on the planet Mars, in 1877, and the new determinations of the velocity of light by Michelson and Newcomb in this country, as well as the investigations of Neison and others upon the so-called "parallactic inequality" of the moon, all point, however, to a somewhat smaller value. Professor New- comb says (in 1885), " All we can say at present is that the solar parallax is probably between 8". 78 and 8". 83, or if outside these limits, that it can be very little outside." It is not, however, thought worth while to change the constant used in the almanacs until the final reduction of the transits of 1874 and 1882 has been made, and until certain experiments and investigations now in progress have been finished. The difference between 8".80 and 8".S5 is of no prac- tical account for almanac purposes, and the change would involve alterations in a number of the tables. Accepting Clarke's value of the earth's equatorial radius (Art. 145), viz., 6,378 206.4° or 3963.3 miles, we find that a solar parallax of 8".75 corresponds to 23,573 radii of the earth = 93,428000 miles. 8".80 " " 23,439 " " " " =92,897000 " 8".85 " " 23,307 " " " " =92,372000 " 8".90 " " 23,196 " " " " =91,852000 " 669. Methods of finding the Solar Parallax and Distance. — We may classify them as follows : — I. Ancient Methods. (A) Method of Aristarchus [0]. (B) Method of Hipparchus [0]. II. Geometrical and Trigonometrical Methods, in which we attempt to find by angular measurements the parallax, either of the sun itself or of one of the nearer planets. 378 DETERMINATION OP THE SUN'S DISTANCE. (A) The direct method [0]. (B) Observations of the displacement of Mars among the stars at the time of opposition. (a) Declination observations from two or more stations in widely different latitudes made with meridian circles or micrometer [25]. (6) Observations made at a single station near the equator, by measuring the distance of the planet east or west from neighboring stars, using the heliometer [90]. (C) Declination observations of Venus [20]. (D) Observations of one of the nearer asteroids in the same way as Mars. (a) Meridian observations at two stations in widely different latitudes [20]. (6) Heliometer observations at an equatorial station [75]. (E) Observations of the transits of Venus at widely separated • stations. (a) Observations of the contacts. (1) Halley's method — the "method of durations" [40]. (2) Delisle's method — observation of absolute times [50]. - (6) Heliometer measurements of the position of the planet on the sun [75]. (c) Photographic methods — various [20 to 75]. III. Gravitational Methods. (A) By the moon's parallactic inequality [70]. , (-B) By the lunar equation of the sun's motion [40]. (0) By the perturbations produced by the earth on Venus and Mars [70] ; (ultimately [95]). iV. By the Velocity of Light, combined with (A) The light equation [80] . (-B) The constant of aberration [90]. The figures in brackets at the right are intended to express roughly the relative value of the different methods, on the scale of 100 for a method which would insure absolute accuracy. 670. Of the Ancient Methods, that of Aristarchus is so ingenious and simple that it really deserved to be successful. When the moon is exactly at the half phase, the angle at M (Fig. 184) must be just GEOMETRICAL METHODS. 379 90°, and the angle AEM must equal MSE. If, then, we can find how much shorter the arc NM (from new to half moon) is than MF (from half moon to full), half the difference will measure AM, and give the angle at S. Aristarchus concluded that the first quarter of the month was just about twelve hours shorter than the second, so that AM was measured by six hours' motion of the ^^r$L moon, or a little less than 4°. Hence he found Sj0, the distance of the sun, to be about nineteen times EM — a value absurdly fio. 184. Wrong, since SE is in fact Aristarchus' Method of Determining the Sun's Distance. nearly 390 times EM. The real difference between the two quarters of the month is only about hdlf an hour, instead of twelve hours. The difficulty with the method 's that, owing to the ragged and broken character of the lunar su face, it is impossible to observe the moment of half moon with sufficient accuracy. 671. The estimate of Hipparchus was based upon the erroneous calcu- lation of Aristarchus that the sun's distance is 19 times the moon's, and the solar parallax, therefore, fV of the moon's parallax. The " radius of the earth's shadow," where the moon cuts it at a lunar, eclipse, is given, as Hipparchus knew, by the formula p=P+p — S (Art. 372), or P + p = p + S. Assuming that P = 19 p, we have 20^? = p + S. Now S, the sun's semi-diameter, is about 15' ; and from the duration of lunar eclipses Hipparchus found p to be about 40' ; hence he obtained for p, the . solar parallax, a value a little less than 3', which, as has been already men- tioned, was accepted by Ptolemy, and by succeeding astronomers for more than 1500 years. (Wolf's " History of Astronomy," p. 175.) 672. Of the Geometrical Methods, A, the "direct method" con- sists in observing the sun's apparent declination with the meridian circle at two stations widely differing in latitude, just as we observe the moon when finding its parallax (Art. 239). Theoretically, . observations of this sort might give the value of the sun's parallax within £" or so, but the method is practically worthless, because the errors of observation are large as compared with the quantity to be determined. The sun's limb is a very bad object to point on, and besides, its heat disturbs the adjustments of the instrument, thus rendering the observations still more inaccurate. 380 DETERMINATION OF THE SUN'S DISTANCE. 673. The first of the two methods of observing the planet Mars is precisely the same as this direct method of observing the sun; but the distance of Mars at a "near opposition" is only a little more than \ that of the sun, so that any error of observation affects the final result by only about \ as much; and, moreover, Mars is a very good object to observe, so that the errors of observa- tion themselves are much lessened. The planet's distance from the earth having been found in astronomical units by the method of Art. 515, the determination of its distance in miles will fix the value of this unit, and so give us directly the sun's distance and parallax. The method requires two observers working at a distance from each other with different instruments, which is a serious disadvantage. For some unexplained reason, observations of this sort seem almost inva- riably to give too large a result for the solar parallax, averaging between 8". 90 and 8".98. The red color of the planet may possibly have something to do with this by affecting the astronomical refraction. This method, in 1680, was the first to give a reasonable approximation to the sun's true distance, as has been mentioned before. The planet Venus can be observed in the same way, and has been once so observed by Gillis, 1849-52, at Santiago, Chili, in co-operation with the Washington observers, but the result was not very satisfactory. 674. Heliometer Observations of Mars (Method 6) . — It is pos- sible, however, for a single observer to obtain better results than can be got by two or more using the preceding method. Suppose that the orbital motion of Mars is suspended for a while at oppo- sition, and that the planet is on or near the celestial equator; Fig. 185. — Effect of Parallax on the Right Ascension of Mars. and also that the observer is at a station, 0, on the earth's equator. When Mars is rising at M„ Fig. 185, the horizontal parallax OM e C depresses the planet ; that is, he appears from to be further east than he would if seen from C, the centre of the earth ; so that the parallax then increases the planet's right ascen- sion. Twelve hours later, when he is setting, the parallax will HELIOMETER OBSERVATIONS OF MARS. 381 throw him towards the west, diminishing his right ascension by the same amount. If, then, when the planet is rising, we measure care- fully its distance west of a star S, which is supposed to be just east of it (the distance Jf 6 $ in Fig. 186) , and then measure the distance M W S from the same star again when it is setting, the difference will give us twice the horizontal parallax. The earth's rotation will have performed for the observer the function of a long journey in transporting him from one station to another 8000 miles away in a straight line. 675. Of course the observations are not practically limited to the moment when the planet is just rising, nor is it necessary that the star measured from should be exactly east or west of the planet. Measures,, from a number of the neighboring stars, Si, S 2 , S s , and $4 would fix the positions M, and M w with more accuracy than meas- ures from S alone. Nor will the planet stop in its orbit to be ob- served, nor will it have a declina- tion of zero, nor can the observer command a station exactly on the earth's equator. But these varia- tions from the ideal conditions do not at all affect the principles in- volved ; they simply complicate the calculations slightly without com- promising their accuracy. The method has the very great advantage that all the observations are, made by one person, and with one instrument, so that, as far as can be seen, all errors that could affect the result are very thoroughly eliminated. Micrometric Comparison of Mars with Neigh- boring Stars. 676. The most elaborate determination of the solar parallax yet made by this method is-that of Mr. Gill, who was sent out for the purpose by the Royal Astronomical Society in 1877 to Ascension Island in the Atlantic Ocean. His result, from 350 sets of measurements, gives a solar parallax of 8".783±0".01o, — a result probably very close to the truth, though pos- sibly a little small. In 1892 and 1894 favorable oppositions of Mars will occur again, and it is quite likely that the observations will be repeated on a scale even more extensive. 382 DETERMINATION OP THE SUN'S DISTANCE. Venus cannot be observed in this way, since either her rising or setting is in the daytime, when the small stars cannot be seen near her; but the nearer asteroids can be utilized by this method. As they are more distant, however, than Mars, a given error in observation produces a larger final error in the result. 677. The Heliometer. — The heliometer, the instrument employed in these measures, is one of the most important of the modern instruments of precision. As its name implies, it was first designed to measure the diameter of the sun, but it is now used to measure any distance ranging from a few minutes up to one or two degrees, which it does with the same accu- racy as that with which the filar micrometer measures distances of a few seconds. It is a " double image " micrometer, made by dividing the object- glass of a telescope along its diameter, as shown in Fig. 187. The two halves are so mounted that they can slide by each other for a distance of three I or four inches, the separation of the centres being - j accurately measured by a delicate scale, or by a micrometer screw operated and read by a suitable arrangement from the eye-end. The instrument is mounted equatorially with clock-work, and the M m M tu ^ e can ^ e turne ^ in its cradle so as to make * © ® O * ne ^ me °^ division of the lenses lie in any desired Sl So S s direction. When the centres of the two halves of Fio. 187.— The Heliometer. the object-glass coincide, the whole acts as a single lens, giving but one image of each object or star in the field of view. As soon as the centres are separated, each half of the object-glass forms its own image. To measure the distance from Mars to a star, the telescope tube is turned so that the line of centres points in the right direction, and then the semi- lenses are separated until one of the two images of the star comes exactly in the centre of one of the images of Mars ; this can be done in two positions of the semi-lens A with respect to B, as shown by the figure. We may either make S (the image of the star formed by semi-lens B) coincide with Mi formed by A, or make S. t coincide with M . The whole distance from 1 to 2 then measures twice the distance between M and S. 678. Transit of Venus Observations. — At the time when Venus passes between us and the sun, her distance from the earth is only some 26,000000 of miles, so that her horizontal parallax is nearly four times as great as that of the sun itself. At this time her appar- ent displacement upon the sun's disc, due to a change of the observ- er's station upon the earth, is the difference between her own parallax due to this displacement, and that of the sun itself ; and this differ- ence is greater than the sun's parallax nearly in the ratio of 3 to halley's method. 383 1, or, more exactly, of 723 to 277. The object, then, of the observa- tions of a transit is to obtain in some way a measure of the angular displacement of Venus on the sun's disc, corresponding to the known distance between the observer's stations upon the earth. 679. Halley's Method. —The method proposed by Halley, who in 1677 brought to notice the great advantages presented by a transit of Venus for determining the sun's parallax, was as follows : Two sta- tions are' chosen upon the earth's surface, as far separated in latitude as possible. From them we observe the duration of the transit; that is, the interval of time between its beginning and end, both of which must be visible at both stations. If the clock runs correctly during the few hours during which the transit lasts, this is all that is necessary. We do not need to know its error in reference to Greenwich time, nor even in respect to the local time, except roughly. This was a great advantage of the method in those days, before the era of chro- nometers, when the determination of the longitude of a place was a very difficult and uncertain operation. The observa- tion to be made is simply to note the clock time at which "contact" occurs, there being four of these contacts, — two exterior and two internal, at the points marked 1, 2, 3, 4, in Fig. 188. Halley depended mainly on the two internal contacts, which he supposed could be observed with an error not exceeding one second of time. 680. Computation of the Parallax. — Having the durations of the transits at the two stations, and knowing the hourly angular motion of Venus, we have at once and very accurately the length jof the two Fig. 188. Contacts in a Transit of Venus. D {Earth) Fio. 189. — Halley's Method. chords described by Venus upon the sun, expressed in seconds of arc. We also know the sun's semi-diameter in seconds, and hence 384 DETERMINATION OF THE SUN'S DISTANCE. in the triangles (Fig. 189) Sab and Sde, we can compute the length (in seconds still) of Sb and Se, the difference of which, be, is the displacement sought, due to the distance between the stations on the earth. 1 The virtual base line is, of course, not the distance between B and E as a straight line, because that line is not perpendicular to the line of sight from the earth to Venus, nor to the plane of the planet's orbit, but the true value to be used is easily found. Calling this base line /?, we have r being the radius of the earth. The rotation of the earth, of course, comes in to shift the places of E and B during the transit, but this can easily be allowed for. 681 . The Black Drop. — Halley expected, as has been said, that it would be possible to observe the instant of internal contact within ^^^^^^^^^^^^ a single second of time, but he reckoned with- H^HHMflj^^fl out his host. At the transits of 1761 and ^^^^^^^^^^^ 1769, at most of the stations the planet at II the time of internal contact showed a " liga- ^^ ment" or "black drop," like Fig. 190, instead i of presenting the appearance of a round disc I I neatly touching the edge of the sun ; and the Fig. wo.— The Black Drop, time of real contact was thus made doubtful by 10 s or 15". This "ligament" depends upon the fact that the optical edge of the image of a bright body is not, and in the nature of things cannot be, abso- lutely sharp in the eye or in the telescope. In the eye itself we have irradiation. In the telescope we have the difficulty that even in a perfect instrument the image of a luminous point or line has a certain width (which 1 In order that the method may be practically successful, it is necessary that the transit track should lie near the edge of the sun's disc, for two reasons. It is desirable that the duration should not be more than three or four hours, while for a central transit it lasts eight hours (Art. 575). Moreover, if the two chords were near the centre of the disc, any small error in the length of either chord would produce a great error in the computed distance between them. When they lie as in the figure (which has been the case in all recent transits), the reverse is true : a considerable error in the observed length of one of the chords affects their computed distance only slightly. DBLISLE S METHOD. 385 with a given magnifying power is less for a large instrument). Moreover a telescope is usually more or less imperfect, and practically adds other defects of definition, so that whenever the limbs of two objects approach each other in the field of view of a telescope we have more or less distortion due to the over- lapping of the two "penum- bras of imperfect definition,'' — the same sort of effect that is obtained by putting the thumb and finger in contact, holding them up within two or three inches of the eye and then separating them : as they separate, a "black ligament" will be seen between them. With modern telescopes, and by great care in preventing the sun's image from being too bright, so as to diminish irradi- ation in the eye as far as pos- sible, the black drop was re- duced to reasonably small pro- portions in 1874 and 1882, and practice beforehand with an " artificial transit " enabled the observer in some degree to allow for its effect. But a new difficulty appeared, from which there seems to be absolutely no way of escape, — the planets atmosphere causes it to be surrounded by a luminous ring as it enters upon the sun's disc, and thus renders the time of the contact uncertain by at least five or six seconds. In both the transits of 1874 and 1882, differences of that amount continually appeared among the results of the best observers. Fig. 191 shows the appearances due to this cause as observed by Vogel in 1882. Fig. 191. - Atmosphere of Venus as seen during a Transit. (Vogel, 1882.) 682. Delisle's Method. — Halley's method requires the use of polar stations, uncomfortable and hard to reach, and also that the weather should permit the observer to see both the beginning and end of the transit. Delisle's method, on the other hand, utilizes two stations near the equator, taken on a line roughly parallel to the planet's motion. It requires also that the observers should know their longitude accurately, so as to be able to determine the Greenwich time at any moment ; but it does not require that they should see both the beginning and end of the transit ; observations of either phase can be utilized : and this is a great advantage. Suppose, then (Fig. 192), that the 386 DETERMINATION OF THE SUN V S DISTANCE. observer W on one side of the earth notes the moment of internal contact in Greenwich time, the planet then being at V v When E notes the contact (also in Greenwich time) the planet will be at V 2 , and the angle at D will be the angular diameter of the earth as seen *Jr*\ Fig. 192. — Dellsle's Method. from D ; i.e., simply twice the sun's parallax. Now the angle D is at once determined by the time occupied by Venus in passing from Vi to V 2 , since in 584 days (the synodic period) she moves completely round from the line DW to the same line again. If the time from Vi to V, were twelve minutes, we should find the angle' at D to be about 18". 683. Heliometer Observations. — Instead of observing simply the times of contact, and leaving the rest of the transit unutilized, as in the two preceding methods, it is possible to make a continuous series of measurements of the distance and direction of the planet from the nearest point of the sun's limb. These measurements^are best made with the heliometer (Art. 677), and give the means of determining the planet's apparent position upon the sun's disc at any moment with extreme precision. Such sets of measurements, made at widely separated stations, will thus furnish accurate determinations of the apparent displacement of the planet on the sun's disc, corresponding to known distances on the earth, aud so will give the solar parallax. During the transit of 1882 extensive series of observations of this sort were made by the German parties, two of which were in the United States, — one at Hartford, Conn., and the other at San Antonio, Texas. The results have not yet (August, 1888) been published, but they will soon appear, and it is understood that they are considerably more accordant than those obtained by any other method of observation. 684. Photographic Observations. — The heliometer measurements cannot be made very rapidly. Under the most favorable circum- stances a complete set requires at least fifteen minutes, so that the TRANSIT OF VENUS PHOTOGRAPHY. 387 whole number obtainable during the seven or eight hours of the transit is quite limited. Photographs, on the other hand, can be made with great rapidity (if necessary, at the rate of two a minute) , and then after the transit we can measure at leisure the position of the planet .on the sun's disc as shown upon the plate. At first sight this method appears extremely promising, and in 1874 great reliance was placed upon it. Nearly all the parties, some fifty in number, were provided with elaborate photographic apparatus of various kinds. On the whole, however, the results, upon discussion, appear to be no more accordant than those obtained by other methods, so that in 1882 the method was generally abandoned, and used only by the American parties, who employed an apparatus having some peculiar advantages of its own. 685. English, German, and French Methods. — In 1874, the English pasties used telescopes of six or seven inches aperture, and magnified the image of the sun formed by the object-glass by a combination of lenses applied at the eye-end. There were no special appliances for eliminating the distortion produced by the enlarging lenses, nor for ascertaining the exact orientation of the picture (that is (- the direction of the image upon the plate with reference to north and south), nor for determining its scale. The Germans and Russians employed a nearly similar apparatus, but with the important difference that at the principal focus of the object-glass they inserted a plate of glass ruled with squai-es. These squares are photo- graphed upon ..the image of the sun, and furnish a very satisfactory means of determining the scale and distortion, if any, of the image. The object- glasses used by the English and the Germans had a focal length of seven or eight feet. The French employed object-glasses with a focal length of some fourteen feet, the telescope being horizontal, while the rays of the sun were reflected into it by a plane mirror ; instead of glass plates they used the old-fashioned metallic daguerreotype plates, in order to avoid any possible " creeping " of the collodion film, which was feared in the more modern wet- plate process. The French plates furnish, however, no accurate orientation of the picture. 686. The American Apparatus. — The Americans used a similar plan, with some modifications and additions. The telescope lenses employed were five inches ..in diameter and forty feet in focal length, so that the image directly formed upon the plate was about 4} inches in diameter, and needed no enlargement. The telescope was placed horizontal and in the meridian, its exact direction being determinable by a small transit instrument which was mounted in such a manner that it could look into the photograph tele- scope, as into a collimator, when the reflector was removed. The reflector itself was a plane mirror of unsilvered glass driven by clock-work. Fig. 193 DETERMINATION OF THE SUN'S DISTANCE. shows the arrangement of the apparatus. In front of the photographic plate, and close to it, was supported a glass plate ruled with squares called the " reticle plate," and in the narrow space between this and the photograph Fig. 193- — American Apparatus for Photographing the Transit of VenuB. plate was suspended a plumb-line of fine silver wire, the image of which appeared upon the plate, and gave the means of determining the orientation of the image with extreme precision. If the reflec- tor were, and would con- tinue to be, perfectly plane through the whole opera- tion, the method could not fail to give extremely accurate results; but the measurements and discus- sion of the observations seem to show that this mirror was actually dis- torted to a considerable extent by the rays of the sun. On the whole the American plates do not appear to be much more trustworthy than those Pre. 194. - Photograph of Transit of Venna. obtained by other meth- ods. Fig. 194 is a re- duced copy of one of the photographs made at Princeton during the transit of 1882. The black disc near the middle, with a bright spot in the centre, GRAVITATIONAL METHODS. 389 is the image of a metal disc cemented to the reticle to mark the centre lines of the reticle plate ; 192 plates were taken during the transit, and at some of the stations where the weather was good the number was much greater nearly 300 in some cases. The difficulties to be encountered are numerous. Photographic irradiation, or the spread of the image on the plate, slight distortion of the image by the lenses or mirrors employed, irregularities of atmospheric refraction, uncer- tainty as to the precise scale of the picture, — all these present themselves in a very formidable manner. It is obvious why this should be so, when we recall that on a four-inch picture of the sun's disc, titbits °^ an mcn corre- sponds to about ^ of a second of arc, and the whole uncertainty as to the solar parallax does not amount to as much as that. An image of the sun, therefore, in which the position of Venus upon the sun's disc cannot be determined accurately without an error exceeding xttVto °f an inch, is of very little value. Imperfections that would be of no account whatever in plates taken for any other purpose make them practically worthless for this. Still there is reason to hope that considering the enormous number of photographs made in 1874 and 1882 (certainly not less than 5000 in all), the result to be obtained from such a mass of material will prove to be worth something. Gravitational Methods. — These are too recondite to permit any full explanation here. We can only indicate briefly the principles involved. 687. (1) The first of these methods is by the moon's parallactic inequality, an irregularity in the moon's motion which has received this name, because by means of it the sun's parallax can be deter- mined. It depends upon the fact that the sun's disturbing action upon the moon differs sensibly from what it would be if its distance, instead of being less than 400 times that of the moon from the earth, were infinitely great. The disturbing action upon the half of the moon's orbit which lies nearest the sun is greater than on the opposite half of the orbit. The retarding action of the tangential force, therefore, during the first quarter after new moon, is perceptibly greater than the acceleration produced during the second quarter (Art. 447) , so that at the first and third quarters respectively, the moon is a little more than 2' behind and ahead of the place she would occupy if the tangential forces were equal in all four quadrants of the orbit — as they would be if the sun's distance were infinite. This puts the moon about four minutes of time behindhand at the first quarter, and as much ahead at the third ; and if the centre of the moon could be observed within a fraction of a second 390 DETERMINATION OP THE SUN'S DISTANCE. of arc ( as it could if she were a mere point of light like a star) , the observations would give a very accurate determination of the sun's distance. The irregularities of the moon's limb, however, and the worse fact, that at the first quarter we observe the western limb, while at the third quarter it is the eastern one which alone is observable, make the result somewhat uncertain, though the method certainly ranks high. 688. (2) The "lunar equation of the sun's motion" is, it will be remem- bered, an apparent slight monthly displacement of the sun, amounting to about 6".3, and due to the fact that both earth and moon revolve around their common centre of gravity. It is generally made use of (Art. 243) to deter- mine the mass of the moon as compared with that of the earth, using as a datum the assumed known distance of the sun; but if we consider the mass of the moon as known (determined by the tides, for instance), then we can find the sun's parallax 1 in terms of the lunar equation. The method is not a good one, since the solar parallax, 8". 8, is greater than the quantity by means of which it is determined. 689. (3) The third method (by the earth's perturbations of Venus and Mars) is one of the most important of the whole list. It de- pends upon the principle that if the mass of the earth, as compared with that of the sun, be accurately known, then the distance of the sun can be found at once. The reader will remember that in Art. 278 the mass of the sun was found by comparing the distance which the earth falls towards the sun in a second (as measured by the curvature of her orbit) with the force of gravity at the earth's surface ; and in the calculation the sun's distance enters as a neces- sary datum. Now, if we know independently the sun's mass com- pared with the earth's, the distance becomes the only unknown quan- tity, and can be found from the other data. In the same way as in Art. 536 we have in which S and E are the masses of the sun and earth, D is the mean 1 Putting L for the maximum value of the lunar equation (about 6".3 of arc), P for the sun's parallax, and R and r for the distance of the moon and the semi- diameter of the earth respectively, we have the equation GRAVITATIONAL METHODS. 391 distance of the earth from the sun, and T the number of seconds in a vear. Also we have for the force of gravity at the earth's surface, in which r is the earth's radius. Dividing the preceding equation by this, we get S+E = WID* E gT 2 \ r- If we put — = M 9 this becomes E »=(*±±),™ In this equation everything in the second term is known when we have once found M, or the ratio between the masses of the sun and earth ; g is found by pendulum observations on the earth, T is the length of the year in seconds, and r is the earth's radius. Now, the disturbing force of the earth upon its next neighbors, Mars and Venus, depends directly upon its mass as compared with the sun's mass, and the ratio of the masses can be determined when the perturbations have been accurately ascertained ; though the cal- culation is, of course, anything but simple. But the great beauty of the method lies in this, that as time goes on, and the effect of the earth upon the revolution of the nodes and apsides of the neighbor- ing orbits accumulates, the determination of the earth's mass in terms of the sun's becomes continually and cumulatively more precise. Even at present the method ranks high for accuracy, — so high that Lever- rier, who first developed it, would have nothing to do with the transit of Venus observations in 1874, declaring that all such old-fashioned ways of getting at the sun's parallax were relatively of no value. The method is the " method of the future" and two or three hundred years hence will have superseded all the others, — unless indeed it should appear that bodies at present unknown are interfering with the movements of our neighboring planets, or unless it should turn out that the law of gravitation is not quite so simple as it is now supposed to be. 690. The Physical Method. — The physical, or " photo-tachy-metri- cal" method, as it has been dubbed, depends upon the fundamental 292 DETERMINATION OF THE SUN'S DISTANCE. assumption that light travels in interplanetary space with the same velocity as in vacuo. This is certainly very nearly true, and prob- ably exactly so, though we cannot yet prove it. By the recent experiments of Michelson and Newcomb in this country, following the general method of Foucault, the velocity of light has been ascertained with very great precision and may be taken as 299,860 kilometres, or 186,330 miles, with a probable error which cannot well be as great as twenty-five miles either way. 691. Sun's Distance from the Equation of Light. — (1) " The equation of light " is the time occupied by light in travelling between the sun and earth, and is determined by observation of the eclipses of Jupiter's satellites (Art. 629). By simply multiplying the veloc- ity of light by this time (49 9 s ± 2 s ) we have at once the sun's dis- tance ; and that independent of all knowledge as to the earth's dimen- sions. The reader will remember, however, that the determination of this "light-equation" is not yet so satisfactory as desirable on account of the indefinite nature of the eclipse observations involved. 692. From the Constant of Aberration. — (2) When we know the velocity of light we can also derive the sun's distance from the "constant of aberration," and this constant, 20". 492, derived from star observations (Art. 225), is known with a considerably higher percentage of accuracy than the light-equation. Calling the constant a, we have U tan a = — i V where U'\s the velocity of the earth in its orbit, and Fthe velocity of light. Now U equals the circumference of the earth's orbit; divided by the length of the year ; i.e., 2jrD T ' U = hence tan a = and J? = tana (7T). 2jrZ> VT On the whole it seems likely at present that the value of the sun's dis- tance thus derived is the most accurate of all. Using a = 20".492 and sun's distance from constant of aberration. 393 V= 186,330 miles, we have D = 92,975,500 miles, and taking the earth's equatorial radius as 3963.296 miles (Clarke, 1878), we get 8"-793 as the sun's equatorial horizontal parallax. 693. The reader will notice that the geometrical methods give the distance of the sun directly, apart from all hypothesis or assumption, except as to the accuracy of 'the observations themselves, and of their necessary corrections for refraction, etc. : the gravitational methods, on the other hand, assume the exactness of the law of gravitation ; and the physical method assumes that light travels in space with the same velocity as in our terrestrial experiments. The near accord- ance of the results obtained by the different methods shows that these assumptions must be very nearly correct, if not absolutely so. 394 COMETS. CHAPTER XVII. COMETS : THEIR NUMBER, MOTIONS, AND ORBITS. — THEIR CON- STITUENT PARTS AND APPEARANCE. — THEIR SPECTRA. — THEIR PHYSICAL CONSTITUTION, AND ORIGIN. 694. From time to time bodies of a very different character from the planets make their appearance in the heavens, remain visible for some weeks or months, move over a longer or shorter path among the stars, and then vanish. These are the Comets, or " hairy stars," as the word means, since the appearance of such as are bright enough to be visible to the naked eye is that of a star surrounded by a hazy cloud, and usually carrying with it a streaming trail of light. Some of them have been magnificent objects, — the nucleus, or central star, as brilliant as Venus and visible even by day, while the cloudy head was nearly as large as the sun itself, and the tail extended from the horizon to the zenith, — a train of shining sub- stance long enough to reach from the earth to the sun. The major- ity of comets, however, are faint, and visible only with a telescope. 695. Superstitions. — In ancient times these bodies were regarded with great alarm and aversion, being considered from the astrological point of view as always ominous of evil. Their appearance was supposed to presage war, or pestilence, or the death of princes. These notions have survived until very recent times with more or less vigor, but, it is hardly necessary to say, without the least reason. The most careful research fails to show any effect upon the earth produced by a comet, even of the largest size. There is no observable change of temperature or of any meteoro- logical condition, nor any effect upon vegetable or animal life. 696. Number of Comets.— Thus far we have on our lists about 650 different comets. About 400 of these were recorded previous to 1600, before the invention of the telescope, and must, of course, have been bright enough to attract the attention of the naked eye. Since that time the number annually observed has increased very greatly, for only a few of these bodies, perhaps one in five, are visible without telescopic aid. Their total number must be enor- mous. Not unfrequently from five to eight are discovered in a DESIGNATION OP COMETS. 395 single year, and there is seldom a day when one or more is not in sight. While telescopic comets, however, are thus numerous, brilliant ones are comparatively rare. Between 1500 and 1800 there were, according to Newcomb, 79 visible to the naked eye, or about one in three and three-fourths years. Humboldt enumerates 43 within the same period as conspicuous; during the first half of the present century there were 9 such, and since 1850 there have been 11. Since, and including, 1880 we have had 7, — a remarkable number for so short a time, — and two of them, the principal comet of 1881 and the great comet of 1882, were unusually fine ones. In August, 1881, for a little time two comets were conspicuously visible to the naked eye at once and near together in the sky, a thing almost if not quite unprecedented. 697. Designation of Comets. — The more remarkable ones gen- erally bear the name of their discoverer, or of some astronomer who made important investigations relating to them, — as for instance, Halley's, Encke's, and Donati's comets. They are also designated hy the year of discovery, with a Roman number indicating the order of perihelion passage. A third method of designation is by year and letter, the letters denoting the order in which the comets of a given year were discovered. Thus Donati's comet was both comet F and comet VI, 1858. Comet I is, however, not necessarily comet A, though it usually is so. In some cases the comet bears the name of two discoverers. Thus the Pons-Brooks comet of 1883 is a comet which was discovered by Pons in 18i2, and at its return in 1883 was discovered by Brooks. 698. The Discovery of Comets. — As a rule these bodies are first seen by comet-hunters, who make a business of searching for them. For such purposes they are usually provided with a telescope known as a "comet- seeker," having an aperture of from four to six inches, with an eye-piece of low power, and a large field of view. When first seen, a comet is usually a mere roundish patch of faintly luminous cloud, which, if really a comet, will reveal its true character within an hour or two by its motion. Some observers have found a great number of these bodies. Messier discovered thirteen between 1760 and 1798, and Pons twenty-seven between 1800 and 1827. 1 699. Duration of Visibility, and Brightness. — The time during which they are visible differs very much. The great comet of 1811 396 COMETS. was observed for seventeen months, the longest time on record. The comet of 1861 was observed for a year, and the great comet of 1882 for five months. In some cases, when a comet does not happen to be discovered until it is receding from the sun, it is seen only for a week or two. As to their brightness they also differ widely. The great majority can be seen only with a telescope, although a considerable number reach the limit of naked-eye vision at that part of their orbit where they are most favorably situated. A few, as has been said above, become conspicuous; and a very few, perhaps four or five in a century, are so brilliant that they can be seen by the naked eye iu full sunlight, as was the case with the great comets of 1843 and 1882. 700. Their Orbits. — The ideas of the ancients as to the motions of these bodies were very vague. Aristotle and his school believed them to be nothing but earthly exhalations inflamed in the upper regions of the air, and therefore meteorological phenomena rather than astronomical. Ptolemy accordingly omits*all notice of them in the Almagest. Tycho Brahe was the first to show that they are more distant than the moon by comparing observations of the comet of 1577 made in different parts of Europe. Its position among the stars at any moment, as seen from his observatory at Uranienburg, was sensibly the same as that observed at Prague, more than 400 miles to the south. It followed that its distance must be much greater than that of the moon, and that its real orbit must he of enormous size, cutting through interplanetary space in a manner abso- lutely incompatible with the old doctrine of the crystalline spheres. He supposed the path to be circular, however, as befitted the motion of a celestial body. Kepler supposed that comets moved in straight lines ; and he seems to have been half disposed to consider them as living creatures, travelling through space with will and purpose, " like fishes in the sea." Hevelius first, nearly a hundred years later, suggested that the orbits are probably parabolas, and his pupil Doerf el proved this to be the case in 1681 for the comet of that year. The theory of gravitation had now appeared, and Newton soon worked out and published a method by which the ele- ments of a comet's orbit can be determined from the observations. Imme- diately afterwards Halley, using this method and computing the parabolic orbits of all the comets for which he could find the needed observations, ascertained that a series of brilliant comets having nearly the same orbit had appeared at intervals of about seventy-five years. He concluded that these were different appearances of one and the same comet, the orbit not being really parabolic but elliptical, and he predicted its return, which actually occurred in 1759 — the first of " periodic comets." DETERMINATION OF A COMET S ORBIT. 397 701. Determination of a Comet's Orbit. — Strictly speaking, the orbit of a comet being always a conic section, like that of a planet, requires only three perfect observations for its determination ; but it seldom happens that the observations 1 can be made so accurately as to enable us to distinguish an orbit truly parabolic from one slightly hyperbolic, or from an ellipse of long period. The plane of the orbit and its perihelion distance, can be made out with reason* DoW--'' Fig. 195. The Close Coincidence of Different Species of Conietary Orbits within the Earth's Orbit. able accuracy from such observations as are practically obtainable, but the eccentricity, and the major axis with its corresponding period, can seldom be determined with much precision from the data obtained at a single appearance of a comet. The reason is that a comet is visible only in that very small 1 Observations for the determination of a comet's place are usually made with an equatorial, by measuring the apparent distance between the comet and some neighboring " comparison star " with some form of micrometer, as indicated in Art. 129. If the star's place is not already accurately known, it is afterwards specially observed with the meridian circle of some standard observatory ; this observation of comparison stars forms quite an item in the regular work of such an institution. 398 COMETS. portion of its orbit which lies near the earth and sun, and, as the figure shows (Fig. 195), in this portion of the orbit, the long ellipse, the parabola, and the hyperbola almost coincide. Moreover, from the diffuse nature of a comet it is not possible to observe it with the same accuracy as a planet. Comets which really move in parabolas or hyperbolas visit the sun but once, and then recede, never to return ; while those that move in ellipses return in regular periods, unless disturbed. It will be understood, that in a catalogue of comets' orbits, those which are indicated as parabolic are not strictly so. All that can be said is that during the time while the comet was visible,' its position did not deviate from the parabola given by an amount sensible to observation. The chances are infinity to one against a comet's moving exactly in a parabola, since the least retardation of its velocity would render the orbit elliptical, and the least acceleration, hyperbolic, according to the principles explained in Article 430. 702. Relative Numbers of Parabolic, Elliptical, and Hyperbolic Comets. — The orbits of about 270 comets have been thus far com- puted. Of this number about 200 are sensibly parabolic, and six appear to be hyperbolic, although the eccentricity exceeds unity by so small a quantitj' as to leave the matter somewhat doubtful. There are also a number of comets which, according to the best computa- tions, appear to have orbits really elliptical, but with periods so long that their elliptical character cannot be positively asserted. About fifty have orbits which are certainly and distinctly oval ; and twenty- six of these have periods which are less than one hundred years. Tliirteen of these periodic comets have already been actually observed at more than one return. As to the rest of the twenty-six, some of them are expected to return again within a few years, and some of them have been lost, — either in the same way as the comet of Biela, of which we shall soon speak, or by having their orbits so changed by perturbations that they no longer come near enough to the earth to be observed. Table III. of the Appendix gives the elements of these thirteen comets taken from the " Annuaire du Bureau des Longitudes" for 1888. It will be observed that the shortest period is that of Encke's comet, which is only three and one-half years, while the period of Halley's comet exceeds seventy-six. There are three comets with computed periods ranging between seventy and eighty years, whose returns are looked for within the next forty years. There is also one comet with a period of thirty-three years which is due to return in 1899. RECOGNITION OF ELLIPTIC COMETS. 399 703. Fig. 196 shows the orbits of several of the comets of short period, — from three to eight years. (It would cause confusion to insert all of them.) It will be seen that in every case the comet's orbit comes very near to the orbit of .Jupiter, and when the orbit crosses that of Jupiter, one of the nodes is always near the place of apparent in- tersection (the node being marked on the comet's orbit by a short cross-line). If Jupiter were at that point of its orbit at the time when the comet was passing, the two bodies would really be very near to each other. The fact, as we shall see, is a very significant one, pointing to a connection be- tween these bodies and the planet. It is true for all the comets whose periods are less than eight years — for those not inserted in the diagram as well as those that are. The orbits of the seventy-five- year comets are similarly related to the orbit of Neptune, and the thirty-three-year comet passes very close to the orbit of Uranus. Fig. 196. — Orbits of Short-period Comets. 704. Recognition of Elliptic Comets. — Modern observations are so much more accurate than those made two centuries ago that it is now sometimes possible to determine the eccentricity and period of an elliptic comet by means of the observations made at a single appearance. Still, as a general rule, it is not safe to pronounce upon the ellipticity of a comet's orbit until it has been observed at least twice, nor always then. A comet possesses no "personal identity," so to speak, by which it can be recognized merely by looking at it, — no personal peculiarities like those of the planets Jupiter and Saturn. It is identifiable only by its path. When the approximate parabolic elements of a new comet's orbit have been computed, we examine the catalogue of preceding comets to see if we can find others which resemble it ; that is, which have nearly the same incli- 400 COMETS. nation and longitude of the node with the same perihelion distance and peri- helion longitude. If so, it is probable that we have to do with the same comet in both cases. But it is not certain, and investigations, often very long and intricate, must be made to see whether an elliptical orbit of the necessary period can be reconciled with the observations, after taking into account the perturbations produced by planetary action. These perturbations are extremely troublesome to compute, and are often very great, since the comets not unfrequently pass near to the larger planets. In some such cases the orbit is completely altered. Even if the result of this investigation appears to show that the comets are probably identical, we are not yet absolutely safe in the conclusion, for we have what are known as — 705. Cometary Groups. — These are groups of comets which pursue nearly the same orbits, following along one after another at a greater or smaller interval, as if they had once been united, or had come from some common source. The existence of such groups was first pointed out by Hoek of Utrecht in 1865. The most remarkable group of this sort is the one composed of the great comets of 1668, 1843, 1880, and 1882, and there is some reason to suspect that the little comet visible on the picture of the corona of the Egyptian eclipse (Art. 328) also belongs to it. The bodies of this group have orbits very peculiar in their extremely small perihelion distance (they actually go within half a million miles of the sun's surface) , and yet, although their elements are almost identical, they cannot possibly all be different appearances of one and the same comet. So far as regards the comets of 1668 and 1843, considered alone, there is nothing absolutely forbidding the idea of their identity: perturbations might account for the differences between their two orbits. But the comets of 1880 and 1882 cannot possibly be one and the same; they were both observed for a considerable time and accurately, and the observations of both are absolutely inconsistent with a period of two years or anything like it. In fact, for the comet of 1882 all of the different computers found periods ranging between 600 and 900 years. There are about half a dozen other such comet-groups now known. 706. Perihelion Distances. — These vary greatly. Eight comets have a perihelion distance less than six millions of miles ; about seventy-four per cent of all that have been observed lie within the earth's orbit ; about twenty-four per cent lie outside, but within twice the earth's distance from the sun ; and six comets have been observed with a perihelion distance exceeding that limit. ORBIT PLANES. 401 A single one, the comet of 1729, had a perihelion distance exceeding four astronomical units, — as great as the mean distance of the remoter asteroids. It must have been an enormous comet to be visible from such a distance. It is one of the six hyperbolic comets. Obviously, however, the distribution of comets as determined by observa- tion, depends not merely on the existence of the comets themselves, but upon their visibility from the earth. Those comets which approach near the orbit of the earth have the best chance of being seen, because their conspicuous- ness increases as they approach us, so that we must not lay too much stress on the apparent crowding of the perihelia within the earth's orbit. The perihelia are not distributed equally in all directions from the sun, but more than sixty per cent are within 45° of what is called " the sun's way" ; i.e., the line in space along which the sun is travel- ling, carrying with it its attendant systems. 707. Orbit Planes. — The inclinations of the comets' orbits range all the way from 0° to 90°. The ascending nodes are distributed all around the ecliptic, with a decided tendency, however, to cluster in two regions having a longitude of about 80° and 270°. 708. Direction of Motion. — With the two exceptions of Halley's comet, and the comet of the Leonid meteors (Art. 786) , the elliptical comets which have periods less than one hundred years all move in the dire ction of the planets ; and the same is true of the six hyper- bolic comets. Of the other comets, a few more move retrograde than direct, but there is no decided preponderance either way. 709. It is hardly necessary to point out that the fact that the comets move for the most part in parabolas, and that the planes of their orbits have no evident relation to the plane of the planetary motions, tends to indicate (though it falls short of demonstrating) that they do not in any proper sense belong to the solar system itself, but are merely visitors from interstellar space. They come towards the sun with almost precisely the velocity they would have if they had simply dropped towards it from an infinite distance, and they leave it with a velocity which, if no force but the sun's attrac- tion operates upon them, will carry them back to an unlimited distance, or until they encounter the attraction of some other sun. With one remarkable exception, their motions appear to be just what might be expected of ponderable masses moving in empty space under the law of gravitation. 402 COMETS. 710. Acceleration of Encke's Comet. — The one exception referred to is in the ease of Eucke's comet which, since its first discovery in- the last century (it was not, however, discovered to be a periodic comet until 1819), has been continually quickening its speed and shortening its period at the rate of about two hours and a half in each revolution ; as if it were under the action of some resistance to its motion. No perturbation of any known body will account for such an acceleration, and thus far no reasonable explanation has been suggested as even possible, except the one mentioned — the resistance of an interplanetary medium which retards its motion just as air retards a rifle bullet. At first sight it seems almost paradoxi- cal that a resistance should accelerate a comet's speed ; but referring to Article 429 we see that since the semi-major axis of a comet's orbit is given by the equation rf U 2 2\U 2 - V any diminution of J 7 " will also diminish a; and it can be shown that this reduction in the size of the orbit will be followed by an increase of velocity above that which the body had in the larger orbit. It is accelerated by being thus allowed to drop nearer to the sun, and gains its speed in moving .inwards under the sun's attraction. 711. Another action of such a retarding force is to diminish the eccen- tricity of the body's orbit, making it more nearly circular. If the action were to go on without intermission, the result would be a spiral path winding in- ward towards the sun, upon which the comet would ultimately fall. For many years the behavior of Encke's comet was quoted as an absolute demonstra- tion of the existence of the " luminiferous ether." Since, however, no other comets show any such action (unless perhaps Winnecke's 1 comet — No. 5 in the table in the appendix), and moreover, since according to the investiga- tions of Von Asten and Backlund the" acceleration of Encke's comet itself seems suddenly to have diminished by nearly one-half in 1868, there remains much doubt as to the theory of a resisting medium. It looks rather more probable that this acceleration is due to something else than the luminiferous ether — perhaps to some regularly recurring encounter of the comet with a cloud of meteoric matter. The fact that the planets show no such effect is not surprising, since, as we shall see, they are enormously more dense- than any comet, so that the resistance that would bring a comet to rest within a 1 Oppolzer, in 1880, found that according to his computations Winnecke's comet was accelerated precisely in the same way as Encke's, but by less than half the amount. His result, however, is not confirmed by the recent work of Hardtl, who finds no acceleration at all. PHYSICAL CHAEACTEK1STICS OF COMETS. 403 single year would not sensibly affect a body like our earth in centuries. The "resisting medium," if it exists at all, must have much less retarding power than the residual gas in one of Crookes's best vacuum tubes. 712. Physical Characteristics of Comets. — The orbits of these bodies are now thoroughly understood, and their motions are calcu- lable with as much accuracy as the nature of the observations permit ; but we find in their physical constitution some of the most perplex- ing and baffling problems in the whole range of astronomy, — appar- ent paradoxes which as yet have received no satisfactory explanation. While comets are evidently subject to gravitational attraction, as shown by their orbits, they also exhibit evidence of being acted upon by powerful repulsive forces emanating from the sun. While they shine, in part at least, by reflected light, they are also certainly self- luminous, their light being developed in a way not yet satisfactorily explained. They are the bulkiest bodies known, in some cases thousands of times larger than the sun or stars ; but they are "airy nothings," and the smallest asteroid probably rivals the largest of them in actual mass. 713. Constituent Parts of a Comet. — (a) The essential part of a comet — that which is always present and gives it its name — is the coma or nebulosity, a hazy cloud of faintly shining matter, which is usually nearly spherical or oval in shape, though not always so. (6) Next we have the nucleus, which, however, is not found in all comets, but commonly makes its appearance as the comet approaches the sun. It is a bright, more or less star-like point near the centre of the coma, and is the object usually pointed on in determining the comet's place by observation. In some cases the nucleus is double or even multiple ; that is, instead of a single nucleus there may be two or more near the centre of a comet. Perhaps' three comets out of fo.ur present a nucleus during some portion of their visibility. (c) The tail or train, is a streamer of light which ordinarily ac- companies a bright comet, and is often found even in connection with a telescopic comet. As the comet approaches the sun, the tail follows it much as the smoke and steam from the locomotive trail after it. But that the tail does not really consist of matter simply left behind in that way, is obvious from the fact that as the comet recedes from the sun, the train precedes it instead of following. It is always directed away from the sun, though its precise position and form is to some extent determined by the comet's motion. There is abundant evidence that it is a material substance in an exceedingly tenuous 404 COMETS. condition, which in some way is driven off from the comet and then repelled by some solar action. (See also Art. 736.) (d) Envelopes and Jets. — In the case of a very brilliant comet, its head is often veined by short jets of light which appear to be con- Naked-eye View of Donati's Comet, Oct. 4, 1858. (Bond.) timially emitted by the nucleus; and sometimes instead of jets the nucleus throws off a series of concentric envelopes, like hollow shells, one within the other. These phenomena, however, are not usually observed in telescopic comets to any marked extent. DIMENSIONS OF COMETS. 405 714. Dimensions of Comets. — The volume of a comet is often enormous — sometimes almost beyond conception, if the tail be in- cluded in the estimate of bulk. As a general rule the head or coma of a telescopic comet is from 40,000 to 100,000 miles in diameter. A comet less than 10,000 miles in diameter is very unusual ; in fact, such a comet would be almost sure to escape observation. Many, however, are much larger than 100,000 miles. The head of the comet of 1811 at one time measured nearly 1,200000 miles, — more than forty per cent larger than the diameter of the sua itself. The comet of 1680 had a head 600,000 miles across. The head of Donati's comet of 1858 was 250,000 miles in diameter. The head of the great cornet of 1882 was not so bulky as many others, having had a diameter of only 150,000 miles ; but its tail was at one time 100,000000 miles in length. 715. Contraction of a Comet's Head as it approaches the Sun. — It is a very singular fact that the head of a comet continually and regu- larly changes its diameter as it approaches to and recedes from the sun ; and what is more singular yet, it contracts when it approaches the sim, instead of expanding, as one would naturally expect it to do under the action of the solar heat. No satisfactory explanation is known. Perhaps the one suggested by Sir John Herschel is as plausible as any, — that the change is optical rather than real; that near the sun a part of the cometary matter becomes invisible, having been evaporated, perhaps, by the solar heat, just as a cloud of fog might be. The change is especially conspicuous in Encke's comet. When this body first comes into sight, at a distance of about 130,000000 miles from the sun, it has a diameter of nearly 300,000 miles. When it is near the peri- helion, at a distance from the sun of only 33,000000 miles, its diameter shrinks to 12,000 or 14,000 miles, the volume then being less than rrhvn °f what it was when first seen. As it recedes it expands, and resumes its original dimensions. Other comets show a similar, but usually less strik- ing, change. 716. Dimensions of the Nucleus. — This has a diameter ranging in different comets from 6000 or 8000 miles in diameter (Comet III,. 1845) to a mere point not exceeding 100 miles. Like the head, it also undergoes considerable and rapid changes in diameter, though its changes do not appear to depend in any regular way upon the comet's 406 COMETS. distance from the sun, but rather upon its activity at the time. They are usually associated with the development of jets and envelopes. 717. Dimensions of a Comet's Tail. — The tail of a large comet, as regards simple magnitude, is by far its most imposing feature. The length is seldom less than 10,000000 to 15,000000 miles ; it frequently reaches from 30,000000 to 50,000000, and in several cases has been known to exceed 100,000000. It is usually more or less fan-shaped, so that at the outer extremity it is millions of miles across, being shaped roughly like a cone projecting behind the comet from the sun, and more or less bent like a horn. The volume of such a train as that of the comet of 1882, 100,000000 miles in length, and some 200,000 miles in diameter at the comet's head, with a diameter of 10,000000 at its ex- tremity, exceeds the bulk of the sun itself by more than 8000 times. 718. The Mass of Comets. — While the volume of comets is enormous, their masses appear to be insignificant. Our knowledge in this respect is, however, thus far entirely negative; that is, while in many cases we are able to say positively that the mass of a particular comet cannot have exceeded a limit which can be named, we have never been able to fix a lower limit which we know it must have reached ; it has in no case been possible to perceive any action what- ever produced b\ - a comet on the earth or any other body of the planetary system, from which we can deduce its mass ; and this, although they have frequently come so near the earth and other planets that their own orbits have been entirely transformed, and if their masses had been as much as TwViro' °' * ue earth's, they would have produced very appreciable effects upon the motion of the planet which disturbed them. Lexell's comet of 1770, Biela's comet on more than one occasion, and sev- eral others, have come so near the earth that the length of their periods of revolution have been changed by the earth's attraction to the extent of several weeks, but in no instance has the length of the year been altered by a single second. One might be tempted to think that comets were pos- sessed of matter without attracting power ; but attraction is always mutual, and since the comets move according to the law of gravitation, and them- selves suffer perturbation from attraction, there is no escape from the con- clusion that, enormous as they are in volume, they contain very little matter. Some have gone so far as to say that a comet properly packed could be car- ried about in a hat-box or a man's pocket, which, of course, is an extravagant assertion. The probability is that the total amount of matter in a comet of any size, though very small as compared with its bulk, is yet to be estimated DENSITY. -407 at many millions of tons. The earth's mass (Art. 132, 4) is expressed in tons by 6 with twenty-one ciphers following (6000 millions, of millions, of millions of tons). A body, therefore, weighing only one-millionth as much as the earth would contain 6000 millions of millions of tons, which is very nearly equal to the mass of the earth's atmosphere. 719. The late Professor Peirce based his estimate of a comet's mass upon the extent of the nebulous envelope which it carries with it, assuming (what may be doubted, however) that this envelope is gaseous, and is held in equilibrium by the attraction of solid matter in and near the nucleus ; and on this assumption he came to the conclusion that the matter in and near the nucleus of an average comet must be equivalent in mass to an iron ball as much as 100 miles in diameter. This would be about xinnnnr °f t,le earth's mass. While this esti- mate is not intrinsically improbable, it cannot, however, be relied upon. We simply do not know anything about a comet's mass, ex- cept that it is exceedingly small as compared with that of the earth. 720. Density. — This must necessarily be almost iu conceivably small. If a comet 40,000 miles in diameter has a mass equal to i 5 J 6 6 of the earth's mass, its mean density is a little less than •g-inrtf of that of the air at the earth's surface, — much lower than that of the best air- pump vacuum. Near the centre of the comet the density would proba- bly be greater than the mean ; but near its exterior very much less. As for the density of its tail, when such a comet has one, that, of course, must be far lower yet, and much below the density of the residual gas left in the best vacuum we can make by any means known to science. This estimate of the density of a comet is borne out by the fact that small stars can be seen through the head of a comet 100,000 miles in diameter, and even very near its nucleus, with hardly any perceptible diminution of their lustre. In such cases the writer has noticed that the image of a star is rendered a little indistinct ; and recent observations of several astronomers have shown a very small apparent displacement of the star, such as might be ascribed to a slight refraction produced by the gaseous matter of the comet. Students often find difficulty in conceiving how bodies of so infinitesimal density as comets can move in orbits like solid masses, and with such enormous velocities. They forget that in a vacuum a feather falls as freely and as swiftly as a stone. Interplanetary space is a vacuum far more per- fect than any air-pump could produce, and in it the rarest and most tenuous bodies move as freely as the densest. 408 COMETS. 721. The reader,- however, must bear in mind that, although the mean density of a comet (that is, the quantity of matter in a cubic mile) is small, the density of the constituent particles of a comet need not necessarily be so. The comet may be composed of small, heavy bodies, widely separated, and there is some reason for thinking that this is the case ; that, in fact, the head of a comet is a swarm of meteoric stones ; though whether these stones are many feet in diameter, or only a few inches, or only a few thousandths of an inch, like particles of dust, no one can say. In fact, it now seems quite likely that the greatest portion of a comet's mass is made up of such particles of solid matter, carrying with them a certain quantity of enveloping gas. 722. Light of Comets. — There has been much discussion whether these bodies shine by light reflected or intrinsic. The fact that they become less brilliant as they recede from the sun, and finally dis- appear while they are in full sight simply on account of faintness, 1 and not by becoming too small to be seen, shows that their light is in some way derived from the sun. The further fact that the light shows traces of polarization also indicates the presence of reflected sunlight. But while the light of a comet is thus in some way attribu- table to the sun's action, the spectroscope shows that it does not consist, to any considerable extent, of mere reflected sunlight, like that of the moon or a planet. 723. If a comet shone by mere reflected light, or by any light the intensity of which is proportional inversely to the square of the sun's distance (as would naturally be the case if the light were excited directly by the sun's radiation, and proportional to it), we should have its apparent brightness at any time equal to the quantity — - — , in which D and A are the comet's distances from the sun and from the earth respectively. The brightness of a comet does, in fact, generally follow this law roughly, but with many and striking exceptions. The light of a comet often varies greatly and almost capriciously, shining out for a few hours with a splendor seven or eight fold multiplied, and then falling back to the normal state or even below it. The Pons-Brooks comet in 1883 furnished several remark- able instances of this sort (Clerke, p. 418). 724. The Spectra of Comets. — The spectrum of most comets consists of a more or less faint continuous spectrum (which mav be 1 If a comet shone with its own independent light, like a star or a nebula, then, so long as it continued to show a. disc of sensible diameter, the intrinsic brightness of this disc would remain unchanged : it Would only grow smaller as it receded from the earth, not fainter. METALLIC LINES IN SPECTRUM. 409 due to reflected sunlight; though it is usually too faiut to show the Frauuhot'er lines) overlaid by three bright bands, — one in the yellow, one in the green, and the third in the blue. These bands are sharply defined on the lower, or less refrangible, edge, and fade out towards the blue end of the spectrum. A fourth band is sometimes visible in the violet. The green band, which is much the brightest of the three, in some cases is crossed by a number of fine, bright lines, and there are traces of similar lines in the yellow and blue bands. This spectrum is absolutely identical with that given by the blue base of an ordinary gas or candle flame ; or better, by the bhie flame of a Bun- sen burner consuming ordinary illuminating gas. Almost beyond question it indicates the presence in the comet of some gaseous hydro- carbon, which in some way is made to shine ; either by a general heating of the whole body to the point of luminosity (which is hardly probable) , or by electric discharges within it, or by local heatings due to collisions between the solid masses disseminated through the gas- eous envelope ; or possibly to phosphorescence due to the action of sunlight : or none of these surmises may be correct, and we may have to seek some other explanation not yet suggested. It is not at all certain that the temperature of the comet, considered as a whole, is very much elevated. Nor will it do to suppose that because the spectrum reveals the presence in the comet of gaseous hydrocarbon, this substance, therefore, composes the greater part of the comet's mass. The probability is that the gaseous portion of the comet is only a small percentage of the whole quantity of matter contained in it. 725. metallic Lines in Spectrum. — When a comet approaches very near to the sun, as did Wells's comet in 1882, and a few weeks later the great .comet of that year, the spectrum shows bright metal- lic lines in addition to the hydrocarbon bands. The lines of sodium and magnesium are most easily and certainly recognizable. As for the other lines — a multitude of which were seen by Eicco (of Palermo) for a few hours, in the spectrum of the great comet of 1882 — they are probably due to iron ; though that is not certain, for they were not seen long enough to be studied thoroughly. t 726. Anomalous Spectra. — While most comets show the hydro- carbon spectrum, occasionally a different spectrum of bands appears. Fig. 198 shows the spectra of three comets compared with the solar spectrum and with that of hydrocarbon gas. 410 COMETS. The first, the spectrum of Tebbutt's comet of J881, is the usual one. The other two are unique. Brorsen's comet, at its later returns, showed the ordinary comet spectrum, and it might perhaps be considered possible that an error was made in fixing the position of the bands at the first observa- tion. But the peculiar spectrum of comet C, 1877, hardly permits such an explanation. It was observed at Dunecht on the same night, by the same observers and with the same spectroscope, as another comet which Fig. 198. — Comet Spectra. (For convenience in engraving, the dark lines of the solar spectrum in the lowest strip of the figure are represented as bright.) gave the usual spectrum ; so that in this case it hardly seems possible that the anomalous result can be a mistake, though the spectrum itself as yet remains unidentified and unexplained. Indeed, from some points of view, it is strange that comets, coming as they do from such widely separated regions of space, do not show an almost infinite variety of spectra ; a priori we should expect differences rather than uniformity. It is maintained by Mr. Lockyer that the spectrum of a comet changes as it varies its distance from the sun, the bands altering in appearance and shifting their position. But the evidence of this is not yet conclusive. 727. Development of Jets and Envelopes. — When a comet is first seen at a great distance from the sun it is ordinarily a mere roundish, hazy patch of faint nebulosity, a little brighter near the centre. DEVELOPMENT OF JETS AND ENVELOPES. 411 As the comet draws near the sun it brightens, and the central con- densation becomes more conspicuous and sharply defined, or star-like. Fig. 199. — Head of Donati's Comet, Oct. 5, 1858. (Bond.) Then, on the side next the sun, the newly formed nucleus begins to emit jets and streamers of light, or to throw off more or less sym- metrical envelopes, which fol- low each other concentrically at intervals of some hours, ex- panding and growing fainter as they ascend, until they are lost in the general nebulosity which forms the head. Dur- ing these processes the nucleus continually changes in bril- liancy and magnitude, usually growing smaller and brighter just before the liberation of each envelope. When jets are thrown off, the nucleus seems to oscillate, moving slightly from side to side ; but no evidences of a continuous rotation have ever been discovered. The two figures, 199 and 200, represent the Fiq. 200. — Tebbutt's Comet, 1881. (Common.) 412 COMETS. heads cf two comets which behaved quite differently. Fig. 199 is the head of Donati's comet as seen on Oct. 5, 1858. This comet was characterized by the quiet, orderly vigor of its action. It did very little that was anomalous or erratic, but behaved in all respects with perfect propriety. The system of envelopes in the head of this comet was probably the most symmetrical and beautiful ever seen. Fi\ years. Fig. 211. — Orbits of Meteoric Swarms which are known to be associated with Comets. The meteoric shower occurred in 1866 as predicted, and was repeated in 1867, the meteor-swarm being stretched out along its orbit for such a distance that the procession is nearly three years in passing any given point. 1 In a.d. 902 (the "year of the stars" in the old Arab chronicles), the date was what would be Oct. 19, in our '' new style " reckoning. In 1202 the shower occurred five days later, and in 1833 the date was Nov. 12. IDENTIFICATION OF COMETARY AND METEORIC ORBITS. 445 786. Identification of Cometary and Meteoric Orbits. — The re- searches of Newton and Adams had awakened lively interest in the subject, and Schiaparelli, of Milan, a few weeks after the Leonid shower, published a paper upon the Perseids, or August meteors, in which he brought out the remarkable fact that they were moving in the same path as that of the bright comet of 1862, knoivn as Tuttle's Comet. Shortly after this Leverrier published his orbit of the Leonid meteors, derived from the observed position of the radiant in connection with the periodic time assigned by Adams ; and almost simultaneously, but without any idea of a connection between them, Oppolzer published his orbit of Tempel's comet of 1866 ; and the two orbits were at once seen to be practically identical. Now a single case of such a coincidence as that pointed out by Schiaparelli, might u=place of Uranus l$6 A. D. v\ / / I 1 I 1 1 / ' '' / / • / s / - "£' J M // /& v /#* \ ' ff — ^ / \ / & v? \ ■'' n\ ' \ y ( ' \ V r \ \ V — -~^k 1 ■ \ \ \ / \ \ \ / \ N \ / \ ^^ \ / \ \^^\ / Fig. 212. — Transformation of the Orbit of the Leonids by the Encounter -with Uranns, a.d. 126. possibly be accidental, but hardly two. Then five years later, in 1872, came the meteoric shower of the Andromedes, following in the track of Biela's comet ; and among the more than one hundred dis- tinct meteor-swarms now recognized, Professor Alexander Herschel finds four or five others which have a " comet annexed," so to speak. Fig. 211 represents the orbits of four of the meteoric swarms which are known to be associated with comets. 446 METEORS. 787. In the cases of the Leonids and Andromedes the meteor-swarm follows the comet. Many believe, however, that the comet itself is simply the thickest part of the swarm. Kirkwood and Schiaparelli have both pointed out that a body constituted as a comet is supposed to be, must almost necessarily break up in consequence of the " tide-producing " pertur- bations of the sun, independent of any repulsive action such as is supposed to be the cause of a comet's tail. They hold that these meteor-swarms are therefore merely the product of a comet's disintegration. The longer the comet has been in the system, the more widely scattered will be its particles. The Perseids are supposed, therefore, to be old inhab- itants of the solar system, while the Leonids and Andromedes are compara- tively new-comers. Leverrier has shown that in the year a.d. 126 Tempel's comet must have been very near to Uranus, and a natural inference is that it was introduced into the solar system at that time. Fig. 212 illustrates his hypothesis. However these things may be, it is now certain that the connec- tion between comets and meteors is a very close one, though it can hardly be considered certain as yet that every scattered group of meteors is the result of cometary disintegration. We are not sure that when a cometary mass first enters the solar system from outer space, it comes in as a close- packed swarm. THE STARS : THEIR NATURE AND NUMBER. 447 CHAPTER XIX. THE STARS: THEIR NATURE AND NUMBER. -V THE CONSTEL- LATIONS. — STAR-CATALOGUES. — DESIGNATION. AND NOMEN- CLATURE. — PROPER MOTIONS AND THE MOTION OE THE SUN IN SPACE. — STELLAR PARALLAX AND DISTANCE. 788. We enter now upon a vaster subject. Leaving the confines of the- solar system we cross the void that makes an island 1 of the sun's domains, and enter the universe of the stars. The nearest star, so far as we have yet been able to ascertain, is one whose distance is more than 200,000 times the radius of the earth's annual orbit ; so remote that, seen from that star, the sun itself would appear only about as bright as the pole star, and from it no telescope ever yet constructed could render visible a single one of all the retinue of planets and comets that make up the solar system. 789. Nature of the Stars. — As shown by their spectra the stars are suns; that is, they are bodies comparable in magnitude and in physical condition with our own sun, shining by their own light as the sun does, and emitting a radiance which in many cases could not \>e distinguished from sunlight by any of its spectroscopic characteristics. Some of them are vastly larger and hotter than our sun, others smaller and cooler, for, as we shall see, they differ enormously among them- selves. 790. Number of the Stars. — The impression on a dark night is of absolute countlessness ; but, in fact, the number visible to the naked eye is very limited, as one can easily discover by taking some definite area in the sky, say the " bowl of the dipper," and counting the stars which he can see within it. He will find that the number which he 1 That the solar system is thus isolated by a surrounding void is proved by the almost undisturbed movements of Uranus and Neptune ; for their perturbations would betray the presence of any body, at all comparable with the sun in magni- tude, within a distance a thousand times as great as that between the earth and sun. 448 THE STAliS, can fairly count is surprisingly small, though by averted vision he will get uncertain glimpses of many more. In the whole celestial sphere the number bright enough to be visible to the naked eye is only from 6000 to 7000 in a clear, moonless sky. A little haze or moonlight cuts out fully half of them, and of course there is a great difference in eyes. But the sharpest eyes could probably never fairly see more than 2000 or 3000 at one time, since near the horizon the smaller stars are invisible, and they are immensely the most numerous, fully half of the whole number being those which are just on the verge of visi- bility. The total number that can be seen well enough for observation ivith such instruments as were used before the invention of the telescope is not quite 1100. With even a small telescope the number is enormously increased. A mere opera-glass an incli and a half in diameter brings out at least 100,000. The telescope with which Argelander made his Durchmus- terung of more than 300,000 stars — all north of the celestial equator — had a diameter of only two inches and a half. The number visible in the great Lick * telescope of three feet diameter is probably nearly 100,000000. ^v 791. Constellations. — In ancient times the stars were grouped by "constellations," or " asterisms," partly as a matter of convenient reference and partly as superstition. Many of the constellations now recognized, — all of those in the zodiac and those about the northern pole, — are of prehistoric antiquity. To these groups were given fanci- ful names, mostly of persons or objects conspicuous in the mythological records of antiquity ; a great number of them are connected in some way or other with the Argonautic expedition. In some cases the eye can trace in the arrangement of the stars a vague resemblance to the object which gives name to the constellation ; but gener- ally no reason can be assigned why the constellation should be so named or so bounded. Of the sixty-seven constellations now usually recognized on celestial globes, forty-eight have come down from Ptolemy. The others have been formed by Hevelius, Bayer, Royer, and one or two other astron- omers, to embrace stars not included in Ptolemy's constellations, and espe- cially to furnish a nomenclature for the stars never seen by Ptolemy on account of their nearness to the southern pole. A considerable number of 1 Neglecting the loss of light in the lenses, the Lick telescope ought, theoreti- cally, to show stars so faint that it would take more than 30,000 of them to make a star equal to the faintest that can he seen with the naked eye. (See Art. 38.) LIST OF CONSTELLATIONS. 449 other constellations, which have been tentatively established at various times, and are sometimes found on globes and star-maps, have been given up as useless and impertinent. 792. We present a list of the constellations, omitting, however, some of the modern ones which are now not usually recognized by astronomers. The constellations are arranged both vertically and horizontally. The order in the vertical columns is determined by right ascension, as indicated by the Roman numbers at the left. Horizontally the arrangement is according to distance from the north pole, as shown by the headings of the columns. The number appended to each constellation gives the number of stars it contains, down to and including the 6th magnitude. The zodiacal constel- lations are italicized, and the modern constellations are marked by an asterisk. The different groups of constellations are found near the meridian at half-past eight o'clock, p.m., on the dates indicated below. Group (I., II.), Dec. 1. These constellations contain no first-magnitude stars, but Cassiopeia, Andromeda, Aries, and Cetus include enough stars of the second and third magnitude to be fairly conspicuous. Group (III., IV.), Jan. 1. Perseus north of the zenith, and the Pleiades and Aldebaran in Taurus, are characteristic. Group (V., VI.), Feb. 1. On the whole_. this is the most brilliant region of the sky and Orion the finest constellation. Group (VII., VIII.), March 1. Characterized by Procyon and Sirius, the latter incomparably the brightest of all the fixed stars. Group (IX., X.), April 1. Leo is the only conspicuous constellation. Group (XI., XII.), May 1. A barren region, except for Ursa Major north of the zenith. Group (XIII., XIV.), June 1. Marked by Arcturus, the brightest of the northern stars, with the paler Spica south of the equator. Group (XV., XVI.), July 1. The Northern Crown and Hercules are the most characteristic configurations. Group (XVIL, XVIII.), Aug. 1. Vega is nearly overhead, and the red Antares low down in the south, with Altair near the equator, just east of Ophiuchus. Group (XIX., XX,), Sept. 1. Cygnus is in the zenith, and Sagittarius low down, while the brightest part of the Milky Way lies athwart the meridian. Group (XXI., XXII.), Oct. 1. A barren region, relieved only by the bright star Fomalhaut of the Southern Fish near the southern horizon. Group (XXIII., XXIV.), Nov. 1. This region also is rather barren, though the " great square " of Pegasus is a notable configuration of stars. 450 THE STABS. 00 r-1 i-H COtH ft o to .2 3* eg « a, p * § CO =1 < . o ■a s ho 3 o " -S3 -5 Jug 3 _B '5 k- 5 * 3 bo „ B co cd 3 •t, ft * * g -S m t. » --3 3 cs 3 a) OS g 2 « i 8^ ft* 3 MM * * o.2 ° * * * faOQ * * B 3 S "> k- 5 * * HPhO * * "So * * 3 _H C CJ HO * * CI CO /-^ to i— i CO CO -* CO T-H •o l^. CO T-H o CO o .to 1 c kr CO 1 & -o CO ^ CO O _"3 .Boo c . 3 B a T3 of e 3 J? o 3 Lc 3 C3 3 of s c o 3 < CO CO 1 8 ft be 1 ft u of u 3 * H O O * < O 3 1-1 o kc o O s O * CM ~f 00 r- (M O CO OS 00 O CO -+ O CO o CO o OT T-H -# CO CM co ■-# co .¥ 3 CD ^ .~ cj c S £ M CCS 5 o o * 1 C ft CD 02 E"3s rt to" "3 g U .- ^^ r-J 'q; >fi * CD* a* a* CrT 3 CO CO bo ■ 01 ■ Ph CO lO o UO 00 >o iO »o C- '-O 00 t^ CO 1— 1 "* CO CM I— 1 i— i co I-H O T-H CO rH f,-, „ «. 1 +j 1 CN a- .2 c t*> t, E cS (J + O © go* to of .5? a tH o a o 1 >.- co aj CD -S 3 co o «8" eg k- &1 of k- CO 3 B bo ecT a> CJ 1 + £'£ a> 3 ^> CD 03 O o» >■. c*> efl 1 | s i o 3 cu 1 '! 1 O) cc a cJ rt CJ J + to (fi 03 ft rt rt fi ^- ai a * P t3 fi O ! ft / ~ i— < i—i > > K! i— l > > l-H 1— 1 > X rH hH X X A r- 1 l-H i> 1— 1 M h- ( HH rH l-H > k- 1 X HH H M a DESIGNATION OP BRIGHT STARS. 451 793. A thorough knowledge of these artificial groups, and of the names and locations of the stars in them, is not at all essential, even to an accomplished astronomer ; but it is a matter of very great con- venience to know the principal constellations, and perhaps a hundred of the brightest stars, well enough to be able to recognize them read- ily and to use them as points of reference. This amount of knowl- edge is easily acquired by three or four evenings' study of the sky in connection with a good star-map or celestial globe, taking care to observe on evenings at different seasons of the year, so as to com- mand the whole sky. At present the best star-atlas for reference is probably that of Mr. Proctor. The maps of Argelander's "Uranometria Nova" and Heis's atlas (both in German) are handsomer, and for some purposes more convenient. There are many others, also, which are excellent. The smaller maps which are found in the text-books on astronomy are not on a scale sufficiently large to be of much scientific use (as, for instance, in the observation of meteors), though they answer well enough the purpose for which they were designed, of introducing the student to the principal star-groups. 794. Designation of Bright Stars. — (a) Names. Some fifty or sixty of the brighter stars have names of their own in common use. A majority of the names belonging to stars of the first magnitude are of Greek or Latin origin, and significant, as, for instance, Arcturus, Sirius, Procyon, Regulus, etc. Some of the brightest stars, however, have Arabic names, as Aldebaran, Vega, and Betelguese, and the names of most of the smaller stars are Arabic, when they have names at all. (6) Place in Constellation. Spica is the star in the handful of wheat carried by Virgo ; Cynosure signifies the star at the end of the Dog's Tail (in ancient times the constellation we now call Ursa Minor seems to have been a dog) ; Capella is the goat which Auriga, the charioteer, carries in his arms. Hipparchus, Ptolemy, and, in fact, all the older astronomers, including Tycho Brahe, used this clumsy method almost entirely in designating particular stars; speaking, for instance, of the star in the " head of Hercules," or in the " right knee of Bootes," and so on. (c) Constellation and Letters. In 1603 Bayer, in publishing a new star-map, adopted the excellent plan, ever since in vogue, of designat- ing the stars in the different constellations by the letters of the Greek alphabet, assigned Usually in order of brightness. Thus Aldebaran is 452 THE STABS. a Tauri, the next brightest star in the constellation is /? Tauri, and so on, as long as the Greek letters hold out ; then the Roman letters are used as long as they last ; and finally, whenever it is found neces- sary, we use the numbers which Flamsteed assigned a century later. At present every naked-eye star can be referred to and identified by some letter or number in the constellation to which it belongs. (d) Current Number in a Star -Catalogue. Of course all the above methods fail for the hundreds of thousands of smaller stars. In their case it is usual to refer to them as number so-and-so of some well- known star-catalogue; as, for instance, 22,500 LI. (Lalande), or 2573 B. A. C. (British Association Catalogue) . At present our various star-catalogues contain from 600,000 to 800,000 stars, so that, except in the Milky Way, almost any star visible in a telescope of two or three inches' aperture can be identified and referred to by means of some star-catalogue or other. Synonyms. Of course all the brighter stars which have names have also letters, and are sure to be included in every star-catalogue which covers their part of the sky. A given star, therefore, has often a large number of aliases, and in dealing with the smaller stars great pains must be taken to avoid mistakes arising from this cause. STAR-CATALOGUES. 795. These are lists of stars arranged in regular order (at present usually in order of right ascension), and giving the places of the stars at some given epoch, either by means of their right ascensions and declinations, or by their (celestial) latitudes and longitudes. The so-called "magnitude," or brightness of the star, is also ordinarily indicated. The first of these star-catalogues was that of Hipparchus, containing 1080 stars (all that are easily visible and measurable by naked-eye instruments) , and giving their longitudes and latitudes for the epoch of 125 b.c. This catalogue has been preserved for us by Ptolemy in the Almagest, and from it he formed his own catalogue, reducing the positions of the stars (i.e., correcting for precession the positions given by Hipparchus) to his own epoch, about 150 a.d. The next of the old catalogues of any value is that of Ulugh Beigh made at Samarcand about 1450 a.d. This appears to have been formed from independent observations. It was followed in 1580 by the catalogue of Tycho Brahe containing 1005 stars, the last which was constructed before the invention of the telescope. The modern catalogues are numerous. Some give the places of a great number of stars rather roughly, merely as a means of identifying them when STAR-CATALOGUES. 453 used for cometary observations or other similar purposes. To this class belongs Argelander's Durchmusterung of the northern heavens, which con- . tains over 324,000 stars, — the largest number in any one catalogue thus far published. Then there are the " catalogues of precision," like the Pulkowa and Greenwich catalogues, which give the places of a few hundred stars as accurately as possible in order to furnish " fundamental stars," or reference points in the sky. The so-called " Zones " of Bessel, Argelander, and many others, are catalogues covering limited portions of the heavens, containing stars arranged in zones about a degree wide iu declination, and running some hours in right ascension. To the practical astronomer the most useful catalogue is likely to be the one which is now in process of formation by the co-operation of various observatories under the auspices of the Astronomische Gesellschaft (an International Astronomical Society, with its headquarters in Germany). This catalogue will contain accurate places of all stars above the ninth magnitude in the northern sky. Most of the necessary observa- tions have already been made. 796. Determination of Star-Places. — The observations from which a star-catalogue is constructed are usually made with the meridian circle (Art. 63). For the catalogues of precision, comparatively few stars are observed, but all with the utmost care and during several years, taking all possible means to eliminate instrumental and obser- vational errors of every sort. In the more extensive catalogues most of the stars are observed only once or twice, and everything is made to depend upon the accuracy of the places of the fundamental stars, which are assumed as correct. The instrument in this case is used only " differentially " to measure the comparatively small differences between the right ascension and declination of the fundamental stars and those of the stars to be cata- logued. 797. Method of using a Catalogue. — The catalogue contains the mean right ascension and declination of its stars for the beginning of some given year; i.e., the right ascension and declination the star would have at that time if there were no aberration of light and no irregular motion in the celestial pole to affect the position of the equator and equinox. To determine the actual apparent right ascen- sion and declination of a star for a given date (which is what we want in practice) , the catalogue place must be ' ' reduced " to the date in question ; i.e., it must be corrected for precession, nutation, and aberration. The operation with modern tables and formula is not a very tedious one, involving perhaps five minutes' work, but without it the catalogue places are 454 THE STABS. Fie. 2JS. - The Photographic Telescope of the Henry Brothers, Paris. STAR-CHARTS. 455 ageless for most purposes. Vice versa, the observations of a fixed star with the meridian circle do not give its mean right ascension and declination ready to go into the catalogue, but the observations must be reduced from apparent place to mean before they can be tabulated. t798. Star-Charts; — For many purposes charts of the stars are more convenient than a catalogue, as, for instance, in searching for new planets. The old-fashioned way of making such charts was by plotting the results of zone observations. The modern way, intro- duced within the last few years, is to do it by photography. The plan decided upon at the Paris Astronomical Congress in 1887* contemplates the photographing of the whole sky upon glass plates about six inches square, each covering au area of 2° square (four square degrees) , showing all stars down to the fourteenth magnitude, — a project which is entirely feasible, and can be accomplished in five or six years by the co-operation of about a dozen different observ- atories in the northern and southern hemispheres. The instruments are now (1888) in process of construction. The figure (Fig. 213) is a representation of the Paris instrument of the Henry Brothers, which was adopted as the typical instrument for the opera- tion. It has an aperture of about fourteen inches, and a length of about eleven feet, the object-glass being specially corrected for the photographic rays. A 9-inch visual telescope is enclosed in the same tube so that the ob- server can watch the position of the instrument during the whole operation. It was originally planned to give each plate 20 minutes' exposure, but improvements in the photographic plates since the meeting of the Congress now make it possible to cut down the time very materially. It will require about 11,000 plates of the size named to cover the whole sky, and as each star is to appear on two plates at least, the whole number of plates, allowing for overlaps, will be about 22,000. As every plate will contain upon it a number of well-determined catalogue stars, it will furnish the means of determining accurately, whenever needed, the place of any other star which appears upon the same plate. STAR MOTIONS. 799. The stars are ordinarily called "fixed," in distinction from the planets or "wanderers," because as compared with the sun and moon and planets they have no evident motion, but keep their relative positions and configurations unchanged. Observations made at suffi- ciently wide intervals of time, and observations with the spectroscope, show, however, that they are really moving, and that with velocities which are comparable to the motion of the earth in her orbit- 456 THE STARS. If we compare the right ascension and declination of a star de- termined to-day with that determined a hundred years ago, they will be found different. The difference is mainly due to precession and nutation, which are not motions of the stars at all, but simply changes in the position of the reference circles used, and due to alterations in the direction of the earth's axis (Arts. 205 and 214). Aberration also comes in, and this also is not a real motion of the stars, but only an apparent one. 800. Proper Motions. — But after allowing for all these apparent and common motions, which depend upon the stars' places in the sky, and are sensibly the same for all stars in the same telescopic field of view, whatever may be their real distance from us, we find that most of the larger stars have a '■'■proper motion " of their own, ("proper" as opposed to " common,") which displaces them slightly with reference to the stars about them. There are only a few stars for which this proper motion amounts to as much as 1" a year ; per- haps 150 such stars are now known, but the number is constantly in- creasing, as more and more of the smaller stars come to be accurately observed. The maximum proper motion known is that of the seventh magnitude star 1830 Groombridge {i.e., No. 1830 in Groombridge's catalogue of circumpolar stars) , which has an apparent drift of 7" annually, — enough to carry it completely around the heavens in 185,000 years. The largest known proper motions are the following : — ■ 1830, Groombridge, 7 tli mag. , 7".0 e indi, 5th mag. 4".5 9352, Lacaille, 7th " 6".9 Lalande 21,258 8th " 4".4 32,416, Gould, 9th " 6".2 o 2 Eridani 6th " 4'M 61 Cygni, 6th " 5".2 H Cassiopeise, 5th " 3".8 Lalande 21,185, 7th " 4".7 a Centauri, 1st " 3".7 The proper motions of Arcturus (2".l), and of Sirius (1".2), are considered "large," but are exceeded by a considerable num- ber of stars besides those given above. Since the time of Ptolemy, Arcturus has moved more than a degree, and Sirius about half as much. These motions were first detected by Halley in 1718. It is found, as might be expected, that the brighter stars, which as a class are presumably nearer than the fainter ones, have on the average a greater proper motion ; on the average only, however, as is evident from the list given above. Many smaller stars have larger proper motions than any bright one, for there are more of them. REAL MOTIONS OF STABS. 457 801. Real Motions of Stars.— The average proper motion of the first- magnitude stars appears to be about \" annually, and that of a sixth- magnitude star, — the smallest visible to the eye, — is about JJ'. The proper motion of a star gives comparatively little informa- tion as to its real motion until we know the distance of the star and the true direction of the motion, since the proper motion as deter- *— ® To tfie E art h .__ E mined from the star-catalogues is p ^\. only the angular value of that part - ; _isJ or component of the star's whole 6 -° motion which is perpendicular to . Components of a Star's Proper Motion. the line of sight, as is clear from the figure. When the star really moves from A to B (Fig. 214), it will appear, as seen from the earth, to have moved from A to b. The angular value of Ab as seen from the earth is the proper mo- tion (usually denoted by p), as determined from the comparison of star-catalogues. Expressed in seconds of arc, we have Ab \ distance; A body moving directly towards or from the earth has, therefore, no (angular) proper motion at all, — none that can be obtained from the comparison of star-catalogues. Since Ab in miles _ /x" x distance _ 206 265 ' these motions cannot be translated into miles without a knowledge of the star's, distance ; and this knowledge, as we shall see, is at present exceedingly limited ; nor can the true motion AB be found until we also know either the angle BAE or else the line Aa. But since AB is necessarily greater than Ab, it is possible in some cases to determine a minor limit of velocity, which must certainly be exceeded by the star. In the case of 1830, Groombridge, for instance, we have certain knowledge that its distance is not less than 2,000000 times the earth's distance from the sun. It may be vastly greater; but it cannot be less. Now at that distance the observed proper motion of 7" a year would corre- spond to an actual velocity along the line A b of more than 200 miles a second, and the star may be moving many times more swiftly. This star has some- times been called the " runaway star." 458 THE STARS. +802. Motion in the Line of Sight. — Although the comparison of star-catalogues gives us no information of the body's motion towards or from us along the line of sight, the velocity Aa in the figure, yet it is possible, when the star is reasonably bright, to determine some- what roughly its rate of approach or recession by means of the spec- troscope. If the star is approaching us, the lines in the spectroscope, according to Doppler's principle (Art. 321, note), will be shifted towards the blue ; and vice versa, towards the red, if it is receding from us. Dr. Huggins was the first actually to use this method of investigating the star movements in 1868, and by means of it he arrived at some very interesting results, which, however, must be admitted to be somewhat uncertain as, regards their quantitative value. He found, for instance, that Sirius was receding from us at the rate of nineteen miles a second, and that Arcturus was rush- ing towards us at the rate of nearly sixty miles a second ; and re- sults of a similar character were found for a considerable number of other stars. Of late the investigations of this class have been carried on mainly at the Greenwich Observatory, usually hy comparing the stellar spectra with hydrogen and sodium. The observations, however, are extremely difficult to make, for the. displacements of the lines are very small, and in most star spectra the lines are broad and hazy and not well adapted for accurate measurements. The results of dif- ferent days' observations, therefore, for a single star are sometimes mournfully discrepant. Vogel has recently taken up the work pho- Blue Red , . .. .„ r mp- - jj tograplucauy at roisdam, with very encourag- P*-<^-- -- - -- --■ - i- - - q ing results. Fig. 215 is from one of his Spectrum of Rigd P lates ( a ne 9<*tive), showing the displacement pjg. 215- towards the red of the H y line in the spec- Displacement of By Line in the Spec- trum of . P Orionis, or Rigel, and indicating tram p Orionie. a recession of the star at a very rapid rate. Much is hoped in this line from the photo- graphic spectrum-work of the Draper Memorial at Cambridge, of which more will be said a few pages farther on. 803. Star-Groups. — Star-atlases have been constructed by Proc- tor and Flammarion, which show by arrows the direction and rate of the angular proper motion of the stars as far as now known. A moment's inspection shows that in many cases stars in the same neighborhood have a proper motion nearly the same iu direction and in amount. THE sun's way. 459 Thus, Flammarion has pointed out that the stars in the " dipper " of Ursa Major have such a community of motion, except a and y\, — the brighter of the pointers and the star in the end of the handle, — which are moving in entirely different directions, and refuse to be counted as belonging to the same group. Fig. 216 shows the proper motions of the stare which compose „.-"aS* \ . — ■" Fie. 216. — Common Proper Motions of Stare in the " Dipper " of Ursa Major. this group. The same thing appears when their motion is tested- by the spectroscope. Huggins found that the five associated stars are rapidly receding from the earth, while a is approaching us, and t\, though receding, has a widely different rate of motion from the others. The brighter stars of the Pleiades are found in the same way to have a common motion. In fact, it appears to be the rule rather than the exception that stars apparently near each other are really connected as comrades, travelling together in groups of twos and threes, dozens or hundreds. They show, as Miss Clerke graphically expresses it, a distinctly "gregarious tendency." t804. The "Sun's Way." — The proper motions of the stars are due partly to their own real motion, and partly also to the motion of our sun, which is moving swiftly through space, taking with it the earth and the planets. Sir William Herschel was the first to investi- gate and determine the direction of this motion a little more than 100 years ago. The principle involved is this : that the apparent motion of each star is made up of its own motion combined with the motion of the sun reversed (Art. 492) . The effect must be that on the whole, the stars in that part of the sky towards which the sun is moving are separating from each other, — the intervals between them widening out, — while in the opposite part of the heavens they are closing up; and in the intermediate part of the sky the general drift must be backward with reference to the sun's (and earth's) real motion. Just as one walking in a park filled with people moving indiscriminately in differ- ent directions, would, on the whole, find that those in front of him 460 THE STAKS. appeared to grow larger, 1 and the spaces between them to open out, while at the sides they would drift backwards, and in the rear close up. The spectroscope, moreover, ought to indicate this motion and undoubt- edly will do so when the apparent motion in the line of sight has been accu- rately determined for a considerable number of stars. In fact one or two attempts have already been made to determine the solar motion in this way; in the quarter of the sky towards which the sun is moving, the star spectra should, on the whole, show displacement of their lines indicating approach, and vice versa in the opposite quarter ; and these observations will have the advantage of showing directly the sun's rate of motion in miles, a result which is not given by investigations founded upon the angular proper motions of the stars. From Vogel's spectroscopic observations (Table VH), the velocity of the solar system comes out about eleven miles a second. 805. About twenty different determinations of the point in the sky towards which this motion of the sun is directed have been worked out by various astronomers, using in their discussions the angular proper motions of from twenty to twent3 - -five hundred stars. All the investigations present a reasonable accordance of results, differing from each other only by a few degrees, and show that the sun is now moving towards a point in the constellation of Hercules, having a right ascension of about 267° and a declination of about + 31°. This point is known as the " apex of the sun's way." 806. Velocity of the Sun's Movement. — This also is determined by the discussion, and comes out to be such as would carry the sun and its system about 5" in 100 years as seen from the average sixth magnitude star (the sixth magnitude is the smallest easily visible to the naked eye) . If we knew with any certainty the distance of this average sixth magnitude star we could translate this motion into miles; but at present this indispensable datum can be little more than guessed at. On the reasonable assumption adopted by Ludwig Struve (who has made the most recent and extensive of all the investigations upon the motion of the solar system), that this distance is about 20,000000 times the astronomical unit, the velocity of the sun's motion in space comes out about five units per year, that is, about five-sixths of the earth's orbital velocity, or nearly sixteen miles per second ; but this result must be considered as still very uncertain. 1 Theoretically, of course, the stars towards which we are moving must appear jo grow brighter as well as to drift apart; but this change of brightness, though real, is entirely imperceptible within a human lifetime. THE CENTRAL SUN. 461 It is to be noted that this swift motion of the solar system, while of course it affects the real motion of the planets in space, converting them into a sort of corkscrew spiral like the figure (Fig. 217), does not in the least affect the relative motion of sun and plan- ets, as some paradoxers have supposed it must. tf% .* f / J ' %\ Si •.•si \a.T %' -V ?/ /06}|/'\ '\55 c I \ / J7 ^^/ S/ Fie. 217. The Earth's Motion in Space as affected by the Sun's Drift. 807. The Central Sun. — We men- tion this subject simply to say that there is no real foundation for the be- lief in the existence of such a body. The idea that the motion of our sun and of the other stars is a revolution around some great central sun is a very fascinating one to certain minds, and one that has been frequently sug- gested. It was seriously advocated some fifty years ago by Madler, who placed this centre of the stellar uni- verse at Alcyone, the principal star in the Pleiades. It is certainly within bounds to deny that any such motion has been demonstrated, and it is still less probable that the star Alcyone is the centre of such a motion, if the motion exists. So far as we can judge at present it is most likely that the stars are moving, not in regular closed orbits around any centre whatever, but rather as bees do in a swarm, each for itself, under the action of the predominant attraction of its nearest neighbors. The solar system is an absolute monarchy with the sun supreme. The great stellar system appears to be a republic, without any such central, unique, and dominant authority. THE PARALLAX AND DISTANCE OE THE STARS. 808. When we speak of the " parallax " of the moon, the sun, or a planet, we always mean the diurnal parallax, i.e., the angular semi-diameter of the earth as seen from the body in question. In the case of the stars, this kind of parallax is hopelessly insensible, never reaching an amount of ao ^ 00 of a second of arc. The expression ' ' parallax of a star " always means its annual par- allax, that is, the semi-diameter of the earth's orbit as seen from the star. Even this in the case of all stars but a very few is a mere frac- tion of a second of arc, too small to be measured. In a few instances 462 THE STARS. it rises to about half a second, and in the one case of our nearest neighbor (so far as known at present) , the star a Centauri, it appears to be about 0".9, according to the earlier observers, or about 0".75, according to the latest determination of Gill and Elkin. In Fig. 218 the angle at the star is the star's parallax. In accordance with the principle of relative motion (Art. 492) , every star has, superposed upon its own motion and combined with it, an apparent motion equal to that of the earth but reversed. If the star is really at rest it must seem to travel around each year in a little orbit 186,000000 miles in diameter, the precise counterpart of the earth's orbit in size and form, and having its plane parallel to the ecliptic. Fig. 218. — The Annual Parallax of a Star. If the star is near the pole of the ecliptic this apparent " parallactic " orbit will be viewed perpendicularly and appear as a circle ; if the star is on the ecliptic it will be seen edgewise as a short, straight line, while in interme- diate latitudes the parallactic orbit will appear as an ellipse. In this respect it is just like the "aberrational" orbit of a star (Art. 226); but while the aberrational orbit is of the same size for every star, having always a semi-major axis of 20". 492, the size of the parallactic orbit depends upon the distance of the star. Moreover, in the parallactic orbit the star is always opposite to the earth, while in the aberrational orbit it keeps just 90° ahead of her. 809. If we can find a way of measuring this parallactic orbit, the star's distance is at once determined. It equals 206 265 x R p" in which p" is the parallax in seconds of arc (the apparent semi- major axis of the parallactic orbit), and R is the earth's distance from the sun. The determination of stellar parallax had been attempted over and over again from the days of Tycho down, but without success until Bessel, in 1838, succeeded in demonstrating and measuring the par- allax of the star 61 Cygni ; and the next year Henderson, of the Cape of Good Hope, brought out that of a Centauri. It will METHODS OF DETERMINING STELLAK PARALLAX. 463 be remembered that it was mainly on account of his failure to detect stellar parallax that Tycho rejected the Coperuican theory and sub- stituted his own (Art. 504). Eoemer of Copenhagen, in 1690, thought that he had detected the effect of stellar parallax in his observations of the difference of right ascension between Sirius and Vega at different times of the year. A few years later, Horrebow, his successor, from his own discussion of Koemer's observations, made out the amount to be nearly four seconds of time or 1', and published his pre- mature exultation in a book entitled " Copernicus Triumphans." The discov- ery of aberration by Bradley explained many abnormal results of the early astronomers which had been thought to arise from stellar parallax, and proved that the parallax must be extremely small. About the beginning of the present century, Brinkley of Dublin and Pond, the Astronomer Royal, had a lively controversy over their observations of u Lyrse (Vega). Brink- ley considered that his observations indicated a parallax of nearly 3". Pond, on the other hand, from his observations deduced a minute negative parallax, which, as some one has expressed it, would put the star " somewhere on the other side of nowhere." In fact, as it turns out, Pond was nearer right than Brinkley, the actual parallax as deduced from the latest observations being only about 0".2. The negative parallax, like the much too large result of Brinkley, simply indicates the uncertainties and errors incident to the instruments and methods of observation then used. The periodical changes of temperature and air pressure continually lead to fallacious results, except under the most extreme precautions. 810. Methods of determining Parallax. — The operation of meas- uring a stellar parallax is, on the whole, the most delicate in the whole range of practical astronomy. Two methods have been suc- cessfully employed so far — the absolute and the differential. (a) The first method consists in making meridian observations of the right ascension and declination of the star in question at differ- ent seasons of the year, applying all known corrections for preces- sion, nutation, aberration, and proper motion, and then studying the resulting star-places. If the star is without parallax, the places should be identical after the corrections have been duly applied. If it has parallax, the star will be found to change its right ascension and declination systematically, though slightly, through the year. But the changes of the seasons so disturb the constants of the instrument that the method is treacherous and uncertain. There is no possibil- ity of getting rid of these temperature effects (in producing changes of refraction and varying expansions of the instrument itself) by merely multiplying observations and taking averages, since the 464 THE STABS. changes of temperature are themselves annually periodic, just as is the parallax itself. Still, in a few cases the method has proved successful. Different observers at different places with different instruments have found for a few stars fairly accordant results ; as, for instance, in the case of a Centauri, already mentioned as our nearest neighbor. 811. (6) The Differential Method. — This consists in measuring the change of position of the star whose parallax we are seeking (which is supposed to be comparatively near to us), with reference to other small stars, which are in the same telescopic field of view, but are supposed to be so far beyond the principal star as to have no sensible parallax of their own. If the comparison stars are near the large one (say within two or three minutes of arc) , the ordinary wire micrometer answers very well for the necessary measures ; but if they are farther away, the heliometer (Art. 677) represents special and very great advantages. It was with this instrument that Bessel, in the case of 61 Cygni, obtained the first success in this line of research. The great advantage of the differential method is that it avoids entirely the difficulties which arise from the uncertainties as to the exact amount of precession, etc. ; and in great measure, though not entirely, those arising from the effect of the seasons upon refraction and the condition of the instruments. On the other hand, however, it gives as the final result, not the absolute parallax of the star, but only the difference between its parallax and that of the comparison star. If the work is accurate the parallax deduced cannot be too great; but it may be sensibly too small, and so may make the star apparently too remote. This is because the parallax of the comparison star can never be quite zero : if the comparison star happens to have a parallax of its own as large as that of the principal star, there will be no relative parallax at all ; if larger, the parallax sought will come out negative. 812. Determination of Parallax by Photography. — Recently it lias been attempted to press photography into the service, and Professor Pritchard has obtained apparently excellent results from an extensive series of photographs of 61 Cygni and its neighboring stars, made at Oxford during 1886. Now and then a plate was found in which the sensitive film appeared to have slipped a little during the development of the picture, but the meas- SELECTION OP STARS. 465 urements at once showed up the faulty plates, so that they could be confi- dently rejected. The measurements made upon the other plates appeared to be just as trustworthy as measurements made upon the actual objects in the sky ; and of course the measurement of a photographic plate at leisure in one's laboratory is a vastly more comfortable operation than that of making micrometric settings by night. 813. Selection of Stars. — It is important to select for investigations of this kind those stars which may reasonably be supposed to be near, and, therefore, to have a sensible parallax. The most important indication of proximity is a large proper motion , and brightness is, of course, confirm- atory. At the same time, while it is probable that a bright star with large proper motion is comparatively near, it is not certain. The small stars are so much more numerous than the large ones that it will be nothing sur- prising if we should find among them one or more neighbors nearer than a Centauri itself. 814. Unit of Stellar Distance. — The Light- Year. The ordinary "astronomical unit," or distance of the sun from the earth', is not sufficiently large to be convenient in expressing the distances of the stars. It is found more satisfactory to take as a unit the distance that light travels in a year, which is about 63,000 times the distance of the earth from the sun. A star with a parallax of 1" is at a dis- tance of 3.26 " light-years " so that the distance of any star in " light- years " is expressed by the formula 815. Table IV. in the Appendix, based upon that given in Houzeau's " Vade Mecum de FAstronome," but brought down to date by some additions and changes, gives the parallaxes, and the distances in light-years, of those stars whose parallaxes may be considered as now fairly determined. The student will, of course, see that the tabulated distance in the case of a remote star is liable to an enormous percentage of error. Considering the amount of discordance between the results of different observers, it is extremely charitable to assume that any of the parallaxes are certain within -fa of a second of arc : but in the case of a star like the pole-star, which appears to have a parallax of less than 0".08, this -fa of a second is \ of the whole amount ; so that the distance of that star is uncertain by at least twenty-five per cent. (^ of a second is the angle subtended by T '^ of an inch at the distance of ten miles.) 466 THE STARS. A vigorous campaign has lately been organized for the purpose of obtain- ing within a reasonable time the parallaxes of a considerable number of stars (perhaps one or two hundred), — enough to enable us to deduce some general laws by statistical methods. The instruments to* be used are heli- ometers of six or seven inches' aperture, one of which is at the Cape of Good Hope, another has just been erected at Bamberg in Germany, and a third is at New Haven in this country, under the charge of Dr. Elkin. As regards the distance of stars, the parallax of which has not yet been measured, very little can be said with certainty. It is probable that the remoter ones are so far away that light in making its journey occupies a thousand and perhaps many thousand years. 815*. Since the above was written Dr. Elkin has published the result of his observations upon ten first-magnitude stars, as follows: u. Tauri (Aldebaran), 0".116±.029; a Aurigse (Capella), 0".107±.047; a Ononis' (Betelgueze), - 0".009 ± .049 ; a Canis Minoris (Procyon), 0".266 ± .047 ; /? Gem- inorum (Pollux), 0".068 + .047; u. Leonis (Regulus), 0".093±.048; a Bootis (Arcturus), 0".018± .022; a Lyrse (Vega), 0".034±.045; ^ Aquilte (Allair), 0".199 ± .047 ; a Cygni (Deneb), - 0".042 ± .047. Of course the two negative results simply indicate that the parallax of the large star was less than that of the comparison stars employed. The very small results for Vega and Arcturus are also rather surprising. See Table IV. of Appendix. ADDENDA TO CHAPTER XIX. 466 A 798 and 861. A number of large photographic telescopes have been recently erected. At Cambridge (U.S.) the Bruce telescope has a four-lens objective of 24 inches diameter, constructed like that of a camera. Its focal length was made only 11 feet, in order to secure a wide field of view, and great brilliance of image. It is provided with two objective prisms (Art. 861). The photographic telescope at Greenwich is also 24 inches in diameter, and an instrument of the same size, with a visual 'finder' of 18 inches' diameter is being constructed for the Cape of Good Hope observ- atory. The enormous instrument at Meudon has also two telescopes combined, — a visual telescope of 32 inches' aperture, and a photo- graphic of 25 inches', each of 55 feet focal length. 802 and 804. Vogel prosecuted his work to complete success so far as relates to all the brighter stars with which his instrument was able to deal, and his results are given in Table VII on page 536. THE LIGHT OF THE STABS. 467 CHAPTER XX. THE LIGHT OF THE STABS. — STAB MAGNITUDES AND PHO- TOMETBY. — VABIABLE STABS. — STELLAB SPECTEA. — SCIN- TILLATION OF STABS. 816. Star Magnitudes. — The term " magnitude," as applied to a star, refers simply to its brightness. It has nothing to do with its apparent angular diameter. Hipparchus and Ptolemy arbitrarily graded the visible stars, according to their brightness, into six classes, the stars of the sixth magnitude being the smallest visible to the eye, while the first class comprises about twenty of the brightest. There is no assignable reason why six classes should have been constituted, rather than eight or ten. After the invention of the telescope the same system was extended to the smaller stars, but without any general agreement or concert, so that the magnitudes assigned by different observers to telescopic stars vary enormously. Sir William Herschel, especially, used very high numbers : his twentieth magnitude being about the same as the fourteenth on the scale now generally used, which more nearly corresponds with that of the elder Struve. 817. Fractional Magnitudes. — Of course, the stars classed to- gether under one magnitude are not exactly alike in brightness, but shade from the brighter to the fainter, so that exactness requires the use of fractional magnitudes. It is now usual to employ decimals giving the brightness of a star to the nearest tenth of a magnitude. Thus, a star of 4.3 magnitude is a shade brighter than one of 4.4, and so on. A peculiar notation was employed by Ptolemy, and used by Argelander in his " Uranometria x Nova." It recognizes thirds of a magnitude as the smallest subdivision. Thus, 2, 2,3, 3,2, and 3 express the gradations between second and third magnitude, 2,3 being applied to a star whose brightness is a little inferior to the second, and 3,2 to one a little brighter than the third magnitude. 1 The term " Uranometria" has come to mean a catalogue of naked-eye stars: like the catalogues of Hipparchus, Ptolemy, and Ulugh Beigh. 468 STARS VISIBLE TO THE NAKED EYE. 818. Stars Visible to the Naked Eye. — Heis enumerates the stars clearly visible to the naked eye in the part of the sky north of 35° south declination, as follows : — 1st magnitude, . . . . 14 2d " 48 3d " 152 4th magnitude, 313 5th " ... '.854 6th " .... 2010 Total 3391 According to Newcomb, the number of stars of each magnitude is such that united they would give, roughly speaking, somewhere nearly the same amount of light as that received from the aggregate of those of the next brighter magnitude. But the relation is very far from exact, and seems to fail entirely for the fainter magnitudes below the tenth or eleventh, the smaller stars beiug less numerous than they should be. In fact, if the law held out perfectly, and if light was transmitted through space without loss, the whole sky would be a blaze of light like the surface of the sun. 819. Light-Ratio and Absolute Scale of Star Magnitudes. — It was found by Sir John Herschel, about fifty years ago, that the light given by the average star of the first magnitude is just about one hundred times as great as that received from one of the sixth, and that a corresponding ratio has. been pretty nearly maintained through- out the scale of magnitudes, the stars of each magnitude being about 2^ times ( VlOO) brighter than those of the next inferior magnitude. The number which expresses the ratio of the light of a star to that of another one magnitude fainter is called the light-ratio. In the star magnitudes of the maps by Argelander, Heis, and others, which are most used at present, the divergence from a strict uniformity of light-ratio is, however, sometimes serious. Some forty years ago it was proposed by Pogson to reform the system, by adopting a scale with the uniform light-ratio of a/T00, adjusting the first six magnitudes to correspond as nearly as possible with the magnitudes hitherto assigned by leading authorities, and then carrying forward the scale indefinitely among the telescopic stars. Until recently this "' absolute scale of magnitude" as it has been called, has not been much used ; but in the New TJranometrias made at Cambridge (U.S.) and Oxford it has been adopted, and astronomers generally now endeavor to conform to it. 820. Relative Brightness of Different Star Magnitudes. — In this scale the light-ratio between successive magnitudes is made exactly NEGATIVE MAGNITUDES. 469 Vl 00, or the number whose logarithm is 0.4000, viz., 2.512. Its recip- rocal is the number whose logarithm. is 9.6000, viz., 0.3981. If 6 X is the brightness of a standard first-magnitude star, expressed either iu candle-power or other convenient unit, and 6„ be the brightness of a star of tbe nth magnitude on this scale, we shall therefore have log6„ = log6 1 -^ r (»-l) ; (n— 1) in this equation being the number of magnitudes between the star of the first magnitude and the star of the nth magnitude ; i.e., for a star of the sixth magnitude (n — 1) is 5 ; so that for a star of the sixth magnitude the equation reads, log& 6 = log6 1 --^ r x 5 = log6!-2. With this light-ratio, every difference of five magnitudes corresponds to a multiplication or division of the star's light by 100 ; i.e., to make one star as bright as the standard star of the first magnitude it would require 100 of the sixth, 10,000 of the eleventh, 1,000000 of the six- teenth, and 100,000000 of the twenty-first magnitude. As nearly standard stars of the first magnitude on this scale we have a Aquilse and Aldebaran (a Tauri). The other stars usually counted as of first magnitude are some of them sensibly brighter, and others fainter than these. The pole-star and the two "pointers " are very nearly standard stars of the second magnitude. 821. Negative Magnitudes. — According to this scale, stars that are one magnitude brighter than those of the standard first would be of the zero magnitude (as is the case with Arcturus), and those that are brighter yet would be of a negative magnitude; e.g., the magnitude of Sirius is —1.43; and Jupiter at opposition, in conformity to this system, is described as a star of nearly —2d magnitude, which means that it is nearly 2.51 3 , or about 16 times brighter than a star of the + 1st magnitude like Aldebaran. Accord- ing to Seidel, Jupiter at opposition is about 8£ times as bright as "Vega, which would make its " magnitude " —2.09, Vega being of magnitude 0.2. 822. Relation of Size of Telescope to the Magnitude of the Smallest Star Visible with it. — If a telescope just shows a star of a given magnitude, then to show stars one magnitude smaller w e requ ire an instr ument having its aperture larger in the ratio of V2.512 (or ty'lOO) to 1; i.e., as 1.59:1. Every tenfold increase in the diameter of the object-glass will therefore carry the power of vision just Jive magnitudes lower. 470 MEASUREMENT OF STAB, MAGNITUDES. Assuming what seems to be very nearly true for normal eyes and good telescopes, that the minimum visible for a one-inch aperture is a star of the ninth magnitude, we obtain the following little table of apertures required to show stars of a given magnitude. Star Magnitude . . 7 in .40 8 in .63 9 l in .00 10 l in .59 11 2 in .51 12 3 in .98 Star Magnitude . . 13 14 6 ta .31 10 in .00 15 15 in .90 16 25 iu .10 17 39 in .80 18 63 in .10 But on account of the increased thickness necessary in the lenses of large telescopes, they never quite equal their theoretical capacity as compared with smaller ones. The smallest stars visible by the Lick telescope (thirty-six inches aperture), after allowing for all the advantage of the site, are not quite a whole magnitude below the smallest visible with the Washington telescope ; but the number visible will be at least double ; since the smaller stars are vastly the more numerous. 823. Measurement of Star Magnitudes and Brightness. — Until recently all such measurements were mere eye-estimates, and even yet all photometric measurements depend ultimately on the judgment of the eye. But it is possible by the help of instruments to aid this judgment very much by limiting the point to be decided, to the question whether two lights as seen are, or are not, exactly equal, or else making the decision depend on the visibility or non-visibility of some appearance. 824. Method of Sequences. — For some purposes the unassisted eye is quite as good as any photometric instrument. It judges directly with great precision of the order of brightness in which a number of objects stand. In the method of " sequences," as it is called, the observer merely arranges the stars he is comparing, say to the number of fifty or so, in the order of their brightness, taking care that the stars in each sequence list are nearly at the same altitude, and seen under equally favorable circumstances. Then he makes a second sequence, taking care to include in it some of the stars that were in the first; and so on. Finally, from the whole set of sequences, a list can be formed, including all the stars contained in any of them, arranged in the order of brightness. This process gives, however, no determination of the light-ratio, nor of the number of times by which the light of the brightest exceeds that of the faintest. INSTRUMENTAL METHODS. 471 Variable stars are still often observed in this way, the stars with which they are compared, being such as have their magnitudes already well deter- mined. 825. 2. Instrumental Methods. —These are based on two different principles : — a. The measurement is made by causing the star to disappear by diminishing its light in some measurable way. This is usually referred to as the " method of extinctions." b. The measurement is effected by causing the light of the star to appear just equal to some other standard light, by decreasing the brightness of the star or of the standard in some known ratio until they are perfectly equalized. Under the first head come the photometers which act upon the principle of "limiting apertures." The telescope is fitted with some arrangement, often a so-called " cat's-eye," by which the available aperture of the object- glass can be diminished at will, and the observation consists in determining with what area of object-glass the star is just visible. The method is em- barrassed by constant errors from the fact that the greater thickness of the glass in the middle of the lens comes into account, and, still worse, from the fact that the image of the star becomes large and diffuse on account of diffraction when the aperture is very much reduced. 826. The Wedge Photometer. — The method of producing the "extinction" by a "wedge" of dark, neutral- tinted glass is much better. The wedge is usually five or six inches long, by perhaps a quarter of an inch wide, and at the thick end cuts off light enough to extinguish the brightest stars that are to be observed. In the Pritchard form of the instrument this wedge is placed close to the eye at the eye-hole of the eye-piece ; in some other forms it is placed at the principal focus of the object-glass, where micrometer wires would be put. In observation the wedge is pushed along promptly until the star just disappears, and a graduation on the edge of the slider is read. The great simplicity of the instrument commends it, and if the wedge is a good one of really neutral glass (which is not easy to get), the results are remarkably accurate. But the observations are very trying to the eyes on account of the straining to keep in sight an object just as it is becoming invisible. The constant of the wedge must be carefully determined in the laboratory, i.e., what length of the wedge corresponds to a diminution of the light of a star by just one magnitude (cutting off 0.602 of its light). It is convenient to have the slider graduated into inches or millimeters on 472 STELLAR PHOTOMETRY. the one edge and magnitudes on the other. The " Uranoinetria Nova Oxo- niensis " is a catalogue of the magnitudes of the naked-eye stars to the number of 2784, between the pole and 10° south declination, observed with an instrument of this kind by Professor Pritchard, and published in 1885. 827. Polarization Photometers. — The instruments, however, with which most of the accurate photometric work upon the stars has been done, are such as compare the light of the star with some standard by means of an "equalizing apparatus" based on the application of the principles of double refraction and polarization. The light of either the observed star or the comparison star (real or artificial) is polarized by transmission through a Nicol prism, or else both pencils are sent through a double refracting prism. The images are viewed with a Nicol prism in the eye-piece ; and by turning this the polarized image or images can be varied in brightness at pleasure, and the amount of varia- tion determined by reading a small circle attached to it. In the photometers of Seidel and Zbllner, who observed comparatively few objects, but very accurately, the artificial star with which the real stars were compared was formed by light from a petroleum lamp, shining through a small aperture, and reflected to the eye by a plate of glass in the telescope tube. Professor Pickering, in his extensive work embodied in the " Harvard Photometry " (published in 1884, and giving the magnitudes of 4260 stars) used the pole-star as the standard, bringing it by an ingenious arrangement into the same field with the star observed. Photometric observations in many cases require large and some- what uncertain corrections, especially for the absorption of light by the atmosphere at different altitudes, and the final results of different observers naturally fail of absolute accordance. Still the agreement between the two catalogues of Pickering and Pritchard is remarkably close, generally within one or two tenths of a magnitude. 828. The Meridian Photometer. — This instrument, contrived and used by Professor Pickering in the observations of the Harvard Photometry, consists of a telescope with two object-glasses side by side. The telescope is pointed nearly east and west, and in front of each object-glass is placed a silvered glass mirror (M, and J/ a , Fig. 219) at an angle of 45°. One of the mirrors is so set as to bring the rays of the pole-star to one object-glass ; the other mirror is capable of being turned around the optical axis of the telescope, in such a way as to command a star at any part of the meridian, and bring its light into the other object-glass. At the eye-end is placed, first {i.e., next the object-glass), a double-image prism D, which separates any pencil of light falling upon it into two, polarized at right angles to each other. The « ordi- nary" rays come through nearly undeflected, but the "extraordinary" are PHOTOMETRY AND PHOTOGRAPHY. 473 bent out of their course, as indicated in the figure, where the pencil A, coming from object-glass No. 1, is divided into two pencils a„ and a„ and in the same way the pencil B, from the second objective, is divided into b and 6„. The angle of the double-image prism D is so chosen that (Miva) i i \ \ \ A i i i i i i i i \ \ \ \ \ \ \ i i i ,-— -v B / ---' \ S fiLyrce. P= ■-Wd.mn. n^ ^ C P Persei \ (Algol) \ I P=2 d. 20 ft. AS m. 65 s A. i \_J Fio. 221. — Light Curves of Variable Stars. that appear to be the result of some unknown disturbance. During most of the time the star remains of the second magnitude. At the time of obscuration it loses about five-sixths of its light, falling to the fourth magnitude in about four and one-half hours, remaining at the minimum for about twenty minutes, and then in three and one- half hours recovering its original condition. During the minimum the spectrum undergoes no considerable change, though there are suspicions of some slight variations in its lines. 484 VARIABLE STAES. Only nine stars are so far known belonging to this class, and among them are those of the shortest period. One in Cepheus, discovered by Ceraski in 1880, has a period of two days aud eleven hours ; anothei in Ophiuchus of only twenty hours. 849. Explanation of Variability. — Evidently no simple explana- tion will hold for all the different classes. For the gradually pro- gressive changes no explanation need be looked for ; on the contrary, it is surprising that such changes are no greater than they are, for the stars are all growing older. As for the irregular changes, no sure account can yet be given of them. Where the range of variation is small, as it is in most cases, one thinks of spots on the surface like those of our own sun, (but much more extensive and numerous) and running through a period just as our sun spots do. Let a star with such spots upon it revolve on its own axis, as of course it must do, and in the combination we have at least a possible explanation of a great proportion of all the known cases, both the irregular variables and the regularly periodic. 850. Collision Theory. — For the temporary stars, and those of the o Ceti type, Mr. Lockyer has recently (in connection with a much more extended subject) suggested a collision theory, illustrated by Fig. 222. The fun- damental idea that the phenomena of the tem- porary stars may be due to collisions is not new. Newton long ago brought it out, and to some extent discussed it ; but considering the probable diameters of the stars as compared with the distances be- tween them, it seems impossible that collis- ions could have been frequent enough to ac- Fio. 222. — The CollisiOD Theory of Variable Stars. ^ s count for the number of temporary stars actually observed. Mr. Lockyer, however, imagines that the temporary f tars, and also variable stars of the o Ceti class, NUMBER AND DESIGNATION OF VARIABLE STARS. 485 are, in their present stage of development, not compact bodies,' but only pretty dense swarm of meteorites of considerable extent, each such swarm being accompanied by another smaller one revolving around it in an eccentric orbit, just as comets revolve around the sun, or as the components of double stars revolve around each other. He supposes that the perihelion distance is so small that the swarms interpenetrate and pass through each other at the perihelion, which could happen without disturbing the general motion of either of the two meteoric flocks ; but while they are thus passing, the collisions are immensely increased in number and violence, with a corresponding increase in the evolution of light. There are many good points about this ingenious theory, but also serious objections to it — as, for instance, the great irregularity of the periods of stars of this class, an irregu- larity which seems hardly consistent with such an orbital revolution. t 851. Stellar Eclipses. — As to the Algol type, the natural explana- tion is by means of an eclipse of some sort; The interposition of a more or less opaque object between the observer and the star, — a dark companion revolving around it, — would produce just the effect observed. If, however, this is really the case, the mass of Algol must be absolutely enormous compared with that of our sun in order to produce so swift a revolution in the eclipsing body. But Ceraski's variable of this type in Cepheus is very refractory and exhibits changes of period and other phenomena that are extremely difficult to reconcile with the idea that its obscuration is due to an eclipse. 852. Number and Designation of Variables. — The "Catalogue of Variable Stars," of Mr. S. C. Chandler, published in 1888, con- tains 225 objects. One hundred and sixty are distinctly periodic stars : in 12 cases the periodicity is perhaps uncertain, while 14 are certainly irregular. The remainder includes the temporary stars and some twenty-five or thirty stars in respect to the variations of which very little is yet known. Table VI. in the Appendix presents the principal data for the naked-eye variables which are visible in the United States. When a star is discovered to be variable which previously had no special appellation of its own, it is customary to designate it by one of the last letters in the alphabet, beginning with R. Thus R Sagittarii is the first discovered variable in Sagittarius ; S Sagittarii is the second ; T Sagittarii, the third, and so on. 853. Range of Variation. — In many cases the whole range is only a fraction of a magnitude (especially among the more newly 486 VARIABLE STARS. discovered variables) , but in a great number it extends from four to eight magnitudes, the maximum brightness exceeding the mini- mum by from fifty to a thousand times ; and in a few cases the range is greater yet. Not unfrequently considerable changes of color accompany the changes of brightness ; the star as a rule being whiter at its maximum, and frequently showing bright lines in its spectrum. 854. Method of Observation. — There is no better way than that of comparing the star by the eye, or with the help of an opera-glass, with surrounding stars of about the same brightness at the time when its light is near the maximum or minimum ; noting to which of them it is just equal at the moment, and also those which are a shade brighter or fainter. It is possible for an amateur to do really valuable work in this way, by putting himself in relation with some observatory which is interested in the subject. The observations themselves require so much time that it is diffi- cult for the working force in a regular observatory to attend to the matter properly, and outside assistance is heartily welcomed in gathering the needed facts. The observations themselves are not specially difficult, require no very great labor or mathematical skill in their reduction, and, as has been said, can be made without instruments ; but they require patience, assiduity, and a keen eye. STAR SPECTRA. 855. In 1824 Fraunhofer, in connection with his study of the lines of the solar spectrum, investigated also the spectra of certain stars, using an apparatus essentially similar to that which is now employed at Cambridge. He placed a prism in front of the object- glass of a small telescope and looked at the stars through this, using a cylindrical lens in the eye-piece to widen the spectrum, which other- wise would be a mere line. He found that Sirius, Castor, and many other stars show very few dark lines in their spectrum, but strong ones ; while, on the other hand, the spectra of Pollux (/J Geminorum) and Capella resemble closely the spectrum of the sun, In all the spectra he recognized the D line, although it was not then known that it had anything to do with sodium. 856. Observations of Huggins and Secchi. — Almost as soon as the spectroscope had taken its place as a recognized instrument of science it was applied by Huggins to the study of the stars, and STAB SPECTRA. 487 "Secchi followed hard in his footsteps. The former studied the spectra of comparatively few stars, but with all the dispersive power he could obtain, and in detail; while Secchi, using a much less pow- erful instrument, examined several thousand star spectra, in a more general way, for purposes of classification. Huggins identified with considerable certainty in the spectra of aOrionis (Betelgueze) andaTauri (Aldebaran) a number of elements that are familiar on the earth, and are most of them prominent in the solar spectrum. He found in the former sodium, magnesium, cal- cium, iron, bismuth, and hydrogen ; and in a Tauri he reported in addition, tellurium, antimony, and mercury ; but these latter metals have not yet been Verified. 857. Classification of Stellar Spectra. — Secchi, in his spectro- scopic survey, found that the 4000 stars which he observed could all be reduced to four classes. The first class comprises the white or blue stars. To it belong Sirius and Vega, and, in fact, considerably more than half of all the stars examined. The spectrum is characterized by the great strength of the hydrogen lines, which are wide, hazy bands, much like the H and K lines in the solar spectrum. Other lines are extremely faint or entirely absent; the _BT line especially, which in the solar spectrum is especially prominent, in the spectra of most of these stars is hardly visible. The second class is also numerous, and is composed of stars with a spectrum substantially like that of our sun. The H and K lines are both strong. Capella and Pollux (/? Geminorum) are prominent examples of this class. There are certain stars which form a con- necting link between these two first classes, stars like Procyon and a Aquilse, which, while they show the hydrogen lines very strongly, also exhibit a great number of other lines between them. The first and second classes together embrace fully seven-eighths of all the stars he observed. The third class includes most of the red and variable stars, some 500 in number, and the spectrum is characterized by dark bands instead of lines (though lines are generally present also). These bands, which are probably due to carbon, shade from the blue towards the red ; that is, they are sharply defined and darkest at the more refrang- ible edge. Occasionally in spectra of this type some of the hydrogen lines are bright, a Hercules, a Orionis, and Mira (o Ceti) are fine examples of this third class. 488 STAR SPECTRA. The fourth class is composed of a very small number of stars, less than sixty so far as now known, mostly small red stars. This spec- trum is also a banded one ; but compared with the third class the bands are reversed, that is, faced the other way, and shaded towards the blue. These generally show also a number of bright lines. None of tke conspicuous stars belong to this class — none above the Fig. 223. — Secchi's Types of Stellar Spectra. fifth magnitude. The sixth magnitude star, 152 Schjellerup, may be taken as its finest example (a, 12 h 39 m ; 8, +40° 9', in the constel- lation of Canes Venatici). Fig. 223 exhibits the light-curves of these four types of spectrum. 1 858. Vogel has revised Secchi's classification of spectra as follows, making only three main classes, but with subdivisions : 1 It is difficult to represent spectra accurately by any process of engraving that can be readily reproduced in «. book like the present. The curve, on the other hand, is easily managed, and, though it does not please the eye like the spectrum itself, it is capable of conveying all the information that could be obtained from the most finished engraving. Dark lines are represented by lines running down- ward from the upper boundary line of the curve, and bright lines by lines running upward, while the bands and their shading are represented by variations in the contour of the curve. PHOTOGRAPHY OF STELLAR SPECTRA. I. (a) Same as Secchi's I. The white stars. (b) Nearly continuous ; all lines wanting or very faint. /3 On- onis is the type. (c) Showing the lines of hydrogen bright, and also the helium line D s (Art. 323). II. (a) Same as Secchi's II. (6) Like II. (a) , but showing in addition bright lines which are not the lines of hydrogen or helium. (This is rare.) III. (a) Same as Secchi's III. (6) Same as Secchi's IV. Vogel's classification is based in part on the very doubtful assumption that stars of Class I. are hottest and also youngest, while the other classes belong to stars which are either beginning to fail or are already far gone in decrepitude. But it is very far from certain that a red star is not just as likely to be younger than a white one, as to be older. It probably is now at a lower temperature, and possesses a more extensive envelope of gases ; but it may be increasing in temperature as well as decreasing. At any rate we have no certain knowledge about its age. 859. Photography of Stellar Spectra. — As early as 1863 Huggins attempted to photograph the spectrum of Vega, and succeeded in getting an impression of the spectrum, but without any of the lines. In 1872 Dr. Henry Draper of New York, working with the reflector which he had him- self constructed, succeeded in getting an impression of the spectrum of the same star, showing for the first time four of its hydrogen lines. The intro- duction of the more sensitive dry plates in 1876, induced Mr. Huggins to resume the subject (as did Dr. Draper soon after), and they soon succeeded in getting pictures showing many lines. The spectra were about half an inch long by ^ or T *y of an inch wide. After the lamented death of Dr. Draper in 1882, Professor Pickering took up the work at Cambridge (U. S.) ; and with such success that Mrs. Draper, who had intended to establish and to endow her husband's observatory as an establishment for astro-physical research, and a most fitting monument to his memory, concluded to transfer the instruments to Cambridge, and there establish the " Draper Memorial." 860. The Slitless Spectroscope. — Professor Pickering has attained his remarkable success by reverting to the "slitless spectroscope," arranged in the manner first used by Fraunhofer, and later revived by Secchi. The instrument consists of a telescope with the objective corrected not for the visual, but for the photographic rays, equatorially mounted and carrying in front of the object-glass one or more prisms, with a refracting angle of nearly 10°, and large enough to cover the whole lens. 490 STAR SPECTKA. I The refracting edge of the prism is placed east and west, so that the linear spectrum of a star formed on a plate at the focus of the object-glass runs north and south. If, now, the clock-work of the instrument is adjusted to follow the star exactly, the image {i.e., the spectrum) will be a mere line, broken here and there where the dark lines of the spectrum should appear. By merely retarding or accelerating the clock a" trifle, the linear spectrum will drift a little sidewise upon the plate, and so will form a spectrum having a width depending on the amount of this drift during the time of exposure. If the air is calm the lines of the spectrum thus formed are as clean and sharp as if a slit were used ; otherwise not. t 861. ] The most powerful instrument used in this work at Cambridge is Dr. Draper's eleven-inch photographic refractor, with four huge glass prisms in a box in front of the object-glass, arranged as indi- cated in Fig. 224. With this apparatus, photographic spectra of the brighter stars are now obtained having, before enlargement, a length of fully three inches from F in the blue of the spec- trum to the extremity of the ultra violet. It is a pity, of course, that the lower portions of the spec- ¥lg 224 trum below F cannot be Arrangement of the Prisms in the Slitless Spectroscope. * ' but no plates sufficiently sensitive to green, yellow, and red rays have yet been found. The exposure necessary to obtain the impression of even the most power- ful photographic rays is from half an hour to an hour. Fig. 225 is I li K H ffr Fig. 225. — Photographic Spectrum of Vega. Cambridge, 1887. enlarged about one-third from one of these photographs of the spectrum of Vega, which extends far into the ultra violet. These spectra bear tenfold enlargement perfectly, making them more than two feet long by two inches in width, and then in the spec- 1 See p. 466^. PHOTOGRAPHY OF STELLAR SPECTRA. 491 trum of such a star as Capella they show hundreds of lines. It is simply amazing that the feeble, twinkling light of a star can be made to produce such an autographic record of the substance and condition of the inconceivably distant luminary. 862. Peculiar Advantages of the Slitless Spectroscope. — The slitless spectroscope has three great advantages. First, that it utilizes all the light that comes from the star to the object-glass, much of which in the usual form of the instrument is lost in the jaws of the slit : Secondly, that by taking advantage of the length of a large telescope, it produces a very high dispersion with even a single prism x : Thirdly, and most important, it gives on the same plate and with a single exposure the spectra of all the many stars whose images fall upon it. "With the smaller eight-inch instrument made at Cambridge, and one prism, as many as 100 or 150 spectra are sometimes taken together ; as, for instance, in a spectrum photo- graph of the Pleiades. 863. Disadvantages of the Slitless Spectroscope. — Per contra, the giving up of the slit precludes all the usual methods of identify- ing the lines by actually confronting them with comparison spectra ; the comparison prism (Art. 315) cannot be used. This makes it extremely difficult to utilize these magnificent pictures for purposes of scientific measurement. If it turns out that any of the lines photographed in the spectrum are of atmospheric origin, the difficulty will be largely removed, as these atmospheric lines could be used as reference points. Professor Pickering has tried, by the interposition of various vapors in the path of the light within the telescope tube, to get lines that will answer the purpose, but thus far without any really satisfactory results. Until this difficulty is overcome it will be impossible to make these spectra yield us all of the information which they undoubtedly contain with respect to the motions of the stars in the line of sight. Vogel in his photographic work mentioned in connection with this subject (Art. 802) used an ordinary spectroscope with a slit. This arrangement required a very long exposure, and limited the dispersion it was possible to employ; but it permitted him to use a hydrogen Geissler tube placed within the telescope itself, to furnish a comparison spectrum. 864. Twinkling or Scintillation of the Stars. — This is a purely atmospheric effect, usually violent near the horizon and almost null at the zenith. It differs greatly on different nights according to the steadiness of the air. 492 SCINTILLATION OP THE STARS. If the spectrum' of a star near the eastern horizon be examined with a spectroscope so held as to make the spectrum vertical, it will appear to be continually traversed by dark bands running through the spectrum from the blue end towards the red. At the western horizon the bands move in the opposite direction, from red to blue ; on the meridian they merely oscillate back and forth. Cause of Scintillation. — Authorities differ as to the exact explanation of scintillation, but probably it is mainly due to two causes (optically speak- ing), both depending on the fact that the air is full of streaks of unequal density that are carried by the wind. (1) In the first place, light transmitted through such a medium is con- centrated in some places and turned away from others by simple refraction: so that, if the light of a star were strong enough, a white surface illumi- nated by it would look like the sandy bottom of a shallow, rippling pool of water illuminated by sunlight, with light and dark mottlings which move with the ripples on the surface. So, as we look towards the star, and the mottlings due to the irregularities of the air move by us, we see the star alternately bright and faint; in other words, it twinkles; and if we look at it in a telescope we shall see that it not only twinkles, but dances, i.e., it is slightly displaced back and forth by the refraction. (2) The other cause of twinkling is " interference." Pencils of light coming from the star (which optically is a mere point), and feebly refracted by the air in the way above explained, reach the observer by slightly different routes, and are just in a condition to interfere. The result of the interference is the temporary destruction of rays of certain wave-lengths, and the reinforcement of others. At a given moment the green rays, for instance, will be destroyed, while the red and blue will be abnormally intense ; hence the quivering dark bands in the spectrum. If the star is very near the horizon, the effects are often sufficient to produce marked changes of color. 865. Why Planets Twinkle less than Stars. — This is mainly because they have discs of sensible diameter, so that there is a general unchanging average of brightness for the sum total of all the lumi- nous points of which the disc is composed. When, for instance, point A of the disc becomes dark for a moment, point B, very near it, is just as likely to become bright; the interference conditions being different for the two points. The different points of the disc do not keep step, so to speak, in their twinkling. ADDENDA TO CHAPTER XX. 492 A 837 and 851. The eclipse theory of the change of brightness in certain variable stars has lately received a striking confirmation from the spectroscopic work of Vogel, who has found by the method indicated in Art. 802 that about twelve or eighteen hours before the obscuration, Algol is receding from us at the rate of nearly twenty- seven miles a second, while after the minimum it approaches us at the same rate. This is just what it ought to do, if it had a large, dark companion, and the two were revolving around their common centre of gravity in an orbit nearly edgewise to the earth. When the dark star is rushing forward to interpose itself between us and Algol, Algol itself must be moving backwards, and vice versa when the dark star is receding after the eclipse. Vogel's conclusions are, that the distance of the dark star from Algol is about 3,250000 miles; that their diameters are respectively about 840,000 and 1,060000 miles ; that their united mass is about two-thirds that of the sun; and their density about one-fifth that of the sun, — not much greater than that of cork. Furthermore, from the variations in the observed period of the star, alluded to in Art. 848, combined with certain minute irregu- larities in its ' proper motion,' Chandler has shown almost beyond doubt that this swiftly-moving pair is itself revolving around another distant and invisible star in an orbit aboiit as large as that of Uranus, with a period of about 130 years. There are other variables of the Algol type which show symp- toms of a similar effect, — Y Cygni and X Tauri especially. 842 and 845. I* 1 Jan - 1892 a twelfth "Nova" appeared in the constellation of Auriga, which at its brightest, about Feb. 5th, was a star of the 4£ magnitude. Its spectrum was very peculiar, show- ing a great number of bright lines, especially those of Hydrogen and with them also the dark lines of the same substances. The bright and dark lines were so displaced relatively to each other as to show that they were respectively due to at least two different masses of gas, in swift relative motion at the rate of something like 500 miles a second; — the "bright-line" mass receding from us, and the other approaching. In the autumn the star, which had sunk to the 11th magnitude, again brightened up to about the 9th, and now the spectrum was found to be almost, if not absolutely, identical with that of a planetary nebula, just as was the case with Nova Cygni, of 1876. 492 B ADDENDA TO CHAPTER XX. 848 and 852. A tenth variable of the Algol type, 12, or S Antliae, discovered by Paul of Washington in 1888, has a period of only 7 h 47 m - Since 1892 three more variables of the Algol type have been discovered. Mr. Chandler's second catalogue, published in 1893, contains 260 certain variables, and 90 suspected, several of which have since been confirmed. DOUBLE AND MULTIPLE STARS. 493 CHAPTER XXI. DOUBLE AND MULTIPLE STARS. — ■ ORBITS AND MASSES OP DOUBLE STABS. — CLUSTERS. NEBULAE. — THE MILKY WAY. — DISTRIBUTION OP STARS. — CONSTITUTION OP THE STELLAR UNIVERSE. — COSMOGONY AND THE NEBULAR HYPOTHESIS. 866, Double and Multiple Stars. — The telescope shows numerous instances in which two stars lie very near each other, in many eases so near that they can be seen separate only under a high II II Fig. 426. — Double and Multiple Stars. magnifying power. These are called "double stars." At present something over 10,000 such couples are known, and the number is continually increasing. In not a few instances we have three stars 494 DOUBLE AND MULTIPLE STABS. together, two of which are usually very close and the third farther away ; and there are several cases of quadruple stars, where there are two pairs of stars lying close together (as in e Lyrae), or a pair of stars with two single stars close by ; and there are some cases where more than four form a " multiple star." Fig. 226 represents a number of such double and multiple stars. 867. Distance, Magnitudes, and Colors. — The apparent distances usually range from 30" to \", few telescopes being able to separate double stars closer than \ u . In a very large proportion of cases (perhaps about one-third of all) the two stars are nearly equal ; in many others they are extremely unequal, a minute star near a large one being usually known as its " companion." Not infrequently the components of a double star present a fine contrast of color; never, however, in cases where they are nearly equal in magnitude. It is a remarkable fact, as yet wholly unexplained, that when we have such a contrast of color the tint of the smaller star always lies higher in the spectrum than that of the larger one. The larger one is reddish or yellowish, and the smaller one green or blue, without a single exception among the many hundreds of such tinted couples now known, y Andromedae and yS Cygni are fine examples for a small telescope. 868. Measurement of Double Stars. — Such meas- ures are generally made with a filar position-mi- crometer, essentially such as shown in Figs. 28 and 29 (Art. 73). The quan- tities to be determined are the distance and position- angle of the couple. By " distance " we mean simply the apparent distance, in seconds of arc between the centres of the two star discs. The posi- tion-angle of a double star is the angle made with the hour-circle Fig. 227. Measurement of Distance and Position-Angle of a Double Star. DISTINGUISHING DOUBLE STARS. 495 by the line drawn from the larger star to the smaller, reckoning around from the north through the east, as shown in Fig. 227. Photography may also be used, and promises to become a favorite method. The first photograph of such an object was by Bond in 1851. 869. Stars Optically and Physically Double. — Stars may be double in two different ways. They may be merely optically double, — that is, simply in line with each other, but one far beyond the other ; or they may be really very near together, in which case they are said to be "physically connected" because they are then under the influence of their mutual attraction, and move accordingly. 870. Criterion for distinguishing between Physically and Optically Double Stars. — This cannot be done off-hand. It requires a series of measurements long enough continued to determine whether the relative movement of the stars is in a curve or a straight line. If the stars are really close together their attraction will force them to describe curves around each other. If they are really at a great distance and only accidentally in line, then their proper motions, being sensibly uniform and rectilinear, will produce a relative motion of the same kind. Taking either star as fixed, the other star will appear to pass it in a straight line, and with a steady, uniform drift. 871. Relative Number of Stars Optically Double, and Physically connected. — Double-star observations practically begau with Sir William Herschel only a little more than a hundred years ago. When he took up the subject less than 100 such pairs had been recog- nized, such as had been accidentally encountered in making observa- tions of various kinds. The great majority of double stars have been discovered so recently that sufficient time has not yet elapsed to make the criterion above given effective with more than a small proportion of them. But it is already perfectly clear that the optically double stars are, as the theory of probability shows they ought to be, very few in number, while several hundred pairs have shown themselves to be physically connected, i.e., to be what are known as " binary " stars, or couples which revolve around their common centre of gravity. 1 872. Binary Stars. — Sir W. Herschel began his observations of double stars in the hope of ascertaining stellar parallax. He had supposed in the case of couples where one was large and the other small that the smaller one was usually a long way beyond the other 496 DOUBLE AND MULTIPLE STARS. (as sometimes is really the fact) . In this case there should be per- ceptible variations in the distance and position of the two stars dur- ing the course of the 3'ear ; precisely such variations as those by which, fifty years later, Bessel succeeded in getting the parallax of 61 Cygni (Art. 811). But Herschel, instead of finding the yearly oscillation of distance and position which he expected, found quite a different, and, at the time, a surprising thing, — a regular, progres- sive change, which showed that one of the stars was slowly describing a regular orbit around the other. To use his own expression, he " went out like Saul to seek his father's asses, and found a king- dom," — the dominion of gravitation 1 extended to the stars, unlimited by the bounds of the solar system, y Virginis, £ Ursae Majoris, and £ Herculis were among the most prominent of the systems which he pointed out. At present the number of pairs known to be binary is at least 200, and as many more begin to show signs of movement. (Up to the present time of course only the quicker moving ones are obvious.) About fifty have progressed so far, — having made at least one entire revolution or a great part of one, — that their orbits have been computed more or less satisfactorily. 873. Orbits of Binary Stars. — The real orbit described by each of such a pair of stars is always found to be an ellipse, and assum- ing the applicability of the law of gravitation, the common centre of gravity must be at the focus. The two ellipses are precisely similar, the one described by the smaller star being larger than the other in inverse proportion to the star's mass. So far as the relative motion of the two bodies goes, we may regard either of them (usually the larger is preferred) as being at rest, and the other as moving around it in a relative orbit of precisely the same shape as either of the two actual orbits which are described around the centre of gravity. But the relative orbit is larger, having for its semi-major axis the sum of the two semi-axes of the real orbits (Art. 427). Usually the relative . orbit is all that we can ascertain, as this alone can be deduced from the micrometer measures when they consist only of position-angles and distances measured between the two stars. 1 It is not yet fully demonstrated that the motions of binary stars are due to gravitation, though it is extremely probable, and the burden of proof seems to be shifted upon those who are disposed to doubt it. See, however, the foot- note to Article 901. CALCULATION OF THE ORBIT OP A BINARY STAR. 497 In a few cases where such measures have been made from small stars in the same field of view with the couple, but not belonging to the system, or when the couple has long been observed with the meridian circle, it becomes possible to work out separately the orbit of each star of the pair with refer- ence to their common centre of gravity ; then we can deduce their relative masses, as for instance, in the case of Sirius and its companion. 874. Calculation of the Orbit of a Binary Star. — If the observer is so placed as to view the orbit perpendicularly, he will see it in its true form and having the larger star in its focus, while the smaller moves around it, describing " equal areas in equal times." But if the observer is anywhere else, the orbit will be apparently more or less distorted. It will still be an ellipse (since every projection of a conic is also a conic) , but the large star will no longer occupy its focus, nor will the major and minor axes be apparently at right angles to each other ; nor will they even coincide with the longest and shortest diameters of the ellipse. In this distorted ellipse the smaller star will, however, still describe equal areas in equal times around the larger one. Theoretically five absolutely accurate observations of the position and distance are sufficient to determine the elements of the relative orbit, if we assume that the orbital motion is described under the law of gravitation. Practically a greater number are needed in most cases, because the motions are so slow and the stars so near each other that observation-errors of 0".l (which in most calculations are of small account) here become important. The work requires not only labor, but judgment and skill, and unless the pair has completed or nearly completed an entire revolution the result is apt to be seriously uncer- tain. So far, as has been said, about fifty such orbits are fairly well determined. Catalogues, more or less complete, will be found in Flammarion's book on "Double Stars," also in Grledhill's "Hand- book of Double Stars," and Houzeau's " Vade Mecum." See Table V. in the Appendix. 875. Sirius and Procyon. — The cases of these two stars are remark- able. Jn both instances the large stars have been found from meridian- circle observations to be slowly moving in little ellipses, although when this discovery was first made neither of them was known to be double. In 1862 the minute companion, of Sirius was discovered by Clark with the object- glass of the Chicago telescope, then just finished, and at that time the largest object-glass in the world. And this little companion was found to be precisely the object needed to account for the peculiar motion of Sirius itself. In the case of Procyon, the companion, if it exists, is yet to be discovered ; not, however, because it has not been carefully looked for with the most powerful instruments. 498 DOUBLE AND MULTIPLE STABS. 876. Periods. — The periods of binary stars, so far as at present known, vary from fourteen years (the period of 8 Equulei) to nearly 1600 years, as £ Aquarii. It is possible that one or two others may be found with periods even shorter than fourteen years, and it is practically certain that as time goes on, pairs of longer period than 1500 years will pre- sent themselves. Computed pe- riods much exceeding 200 years most, however, be received at present with much reserve. Fig. 228 shows the apparent orbits of several of the most in- teresting binaries. The figure for £ Cancri (copied from Gled- hill) is incorrect ; 1878 and 1827 should be interchanged, and the arrow reversed. 1878. *^4827 500 ± Years. 180" 58 Years. /? "-1878_ ■90° 1827 f Cancri. Ursoe Majoris 1880 1750 -90° 0° 61 Cygni Fig. 228. — Orbits of Binary Stars. 877. Size of the Orbits. — The angular semi-major axes of the orbits thus far computed range from about 0".3 for 8 Equulei, to 18" for a Centanri. The real dimensions are, of course, only to be obtained when we know the star's parallax and distance. 1 Fortu- nately several of the stars whose parallaxes have been determined are also binary stars. Assuming the data as to parallax and orbits given in- the tables in the Appendix (mainly taken from Houzeau's " Vade Mecum") we find the following short table of results : — Namb. AsBumed Parallax. Angular Semi-axis. Real Semi-axis. Period. Mass. 0=1. i) Cassiopeia; 0".15 8".64 57.6 195J\2 5.0 Sirius 0.38 8.58 22.6 52.0? 4.26? a Geminorum 0.20? 5.54 27.7? 266 0.30? a Centauri 0.75 17.50 23.3 77.0 2.14 70 Ophiuchi 0.16 4.79 29.9 94.5 3.0 61 Cygni 0.43 15.40? 35.8? 450.0? 0.23? 1 The real semi-axis of the orbit in astronomical units is simply the angular semi-axis divided by the parallax. . MASSES OF BINARY STARS. 499 It is obvious from the table that the double-star orbits are com- parable with the larger orbits of the solar system ; that of a Centauri being just about equal in size to that of Uranus, and that of „ Cassiopeia; not quite double the size of Neptune's orbit. Of course the many binary stars whose distance is so great as to make their parallax insensible while their apparent orbits are as large as those given in the list must have real orbits of still vaster dimensions. 878. Masses of Binary Stars. — When we know both the size of the orbit of a binary and its period, the mass, according to the law of gravitation, follows at once from the equation of Article 536, If * and a are given respectively in years and astronomical units of distance, then, by omitting the factor 4tt 2 , M+m comes out in terms of the sun's mass. The final column of the little table above gives the masses of the six pairs of stars as compared with the mass of the sun. But the student must bear in mind that the parallaxes of stars are so uncertain, that these results are to be accepted with a very large margin of error. 879. Relation between the "Mass-Brightness" of Binary Stars. — Monck of Dublin has recently called attention to a curious relation between the apparent brightness of a binary, its period and (angular) dis- tance on the one hand, and its " mass-brightness," or candle-power per ton, so to speak, on the other, — a relation which does not involve a knowledge of the parallax of the stars. He shows that if we let / be the apparent bright- ness of any double star, photometrically determined, a the semi-major axis of its orbit (in seconds of arc), and t its period in years, while k is its "mass-brightness," or candle-power per ton, — then we have 1 {The Observatory, February, 1887.) 1 This is strictly true only on the assumption that either the two components of the double star have the same mass-brightness, or else that the smaller one is so much smaller that its motion is practically the same as if it were a mere particle. It breaks down when the two stars of a pair do not differ much in mass, but do differ greatly in brightness — probably an unusual case. It involves, however, the doubtful assumption that different double stars are of the same 500 DOUBLK AND MTTLTTPLE STAES. He takes £ Ursse Majoris as a standard (i.e., its k=V), and finds for y Leonis k= 93 ; for a Geminorura, 38 ; for Sirius, 7; for £ Herculis, 4;. for 77 Cor. Bor., 1.5; for 70 Ophiuchi, 0.36; for 17 Cassiopeise, 0.22; and for 61 Cygni, 0.08. In other words, y Leonis is enormously brilliant in proportion to its mass, Sirius is medium, while 61 Cygni is extremely faint. Upon this scale the sun's mass-brightness would be about 2, judging from its mass and brightness compared with that of Sirius (Arts. 832 and 877). (We have given above only a few of Mr. Monck's numbers.) 880. Have the Stars Planets attending them? — It is a very natural supposition that the minute companions which attend some of the larger stars may be really planetary in their nature, shining more or less by reflected light. A.s to this we can only say, that while it is quite possible that other stars besides our sun may have their retinues of planets, it is quite certain that such planets could not be seen by us with any existing telescope. If our sun were viewed from a Centauri, Jupiter would be a star of less than the twenty-first mag- nitude, at a distance of only 5" from the sun, which itself would be a smallish first-magnitude star. 881. The statement can be verified as follows : Jupiter at opposition is certainly not equivalent in brightness to twenty stars like Vega (most photometric measurements make it from eight to fourteen). Assuming, however, that it is equal to twenty Vegas, its light received by the earth would be about Tswri^FSTSis °f the sun's. At opposition our distance from Jupiter is about four astronomical units, so that seen from the same distance as the sun, its light would be sixteen times that quantity, or (nearly) t^-oWto of the sun's. Now a ratio of 125,000000 between the light of two stars corresponds to a difference of 20 + magnitudes (log 125,000000 = 8.0969 ; but %M®L = 20.24 magnitudes. (Art. 820.)] Accordingly, if the observer were removed to such a distance that the sun would appear like a first-magnitude star (as would be the case from a Centauri), Jupiter would be a star of the twenty-first magnitude. Accord- ing to Art. 822, it would require a 25-inch telescope to show a star of the sixteenth magnitude ; it would therefore require an instrument with an aperture of 250 inches, or nearly 21 feet, to show a star five magnitudes fainter, even if there were no large star near to add to the difficulty. 882. Triple and Multiple Stars. — There are a considerable number of objects of this kind, and some of them constitute physical systems. In the case of £ Cancri the two larger stars revolve around their common STAR-CLUSTERS. 501 centre in a nearly circular orbit less than 2" in diameter, and with a period of about sixty years ; while the third star, smaller and more distant, mores around the closed pair in an orbit not yet well determined, but with a period that must be several hundred years ; and in its motion there is evidence of a peculiar perturbation, as yet unexplained. In e Lyrse we have two pairs, each making a very slow revolution, of periods not yet determined, but probably ranging from 300 to 500 years. And since the pairs have also a common proper motion it is practically certain that they also are physically connected, and revolve around their common centre of gravity in a period to be reckoned by millenniums — the motion during the last hundred years being barely perceptible. In other cases, as for instance, in the multiple star Oriouis, we have a number of stars not organized in pairs, but at more or less equal distances from each other : we are confronted by the problem of n bodies in its most general and unmanageable form. Nature challenges the mathematicians. 883. Clusters. — There are in the sky numerous groups of stars containing from one hundred to many thousand members. Some of them are made up of stars visible separately to the naked eye, as the Pleiades ; some of them require a small telescope to resolve them, as, for instance, the Praesepe in Cancer, and the group of stars in the sword-handle of Perseus ; while others yet, even in telescopes of some size, look simply like wisps or balls of shining cloud, and break up into stars only in the most powerful instruments. In a large instrument some of the telescopic clusters are magnificent ob- jects, composed of thousands of stellar sparks compressed into a ball which is dazzlingly bright at the centre and thinning out towards the edge. In some of them vividly colored stars add to the beauty of the group. In the northern hemisphere the finest cluster is that known as 13 M Hercules (a, 16 h 37" and 8, 36° 40') not very far from the " apex of the sun's way." 884. The Pleiades. — Of the naked-eye clusters the Pleiades is the most interesting and important. To an ordinary eye six stars are easily visible in it, the six largest ones indicated in the figure (Fig. 229). Eyes a little better see easily five more — those next in size in the figure (the two stars of Asterope being seen as one). A very small telescope (a mere opera-glass) increases the number to nearly a hundred ; and with large instruments more than 400 are catalogued in the group. A few of the stars, apparently in the cluster, are really only accidentally on the same line of vision, and are distin- guished by proper motions different from those of the rest of the group ; but the great majority have proper motions nearly the same 502 THE PLEIADES. in amount and direction ; they have also identical spectra, and there- fore undoubtedly constitute a single system. $£ Fletone Atlas • • Asterope y Taygeta Mai a if;-'/ * •CeUeno ' : *'-\J Alcyone/ Electro, Merope Fig. 229. — The Pleiades. The distances and positions of the principal stars with respect to the central star Alcyone have been carefully measured three or four times during the last fifty years. The relative motions during the period have not proved large enough to admit of satisfactory determination, but it is clear that such motions exist. A curious and interesting fact is the presence of nebulous matter in considerable quantity. A portion of this nebulosity hanging around Merope (the northeast star of the dipper-bowl in the figure) was discovered many years ago; but it was reserved to photography to detect very recently other wisps of nebulosity attached to other stars, espe- cially to Maia, and to show that the whole space is covered with streaks and streamers of it, emitting light of such a character as to impress the photo- DISTANCE OF STAK-CLUSTERS. 503 graphic plate much more strongly than the eye. The eye cannot see the nebula with the same telescope which is able to photograph it strongly. The figure shows roughly the outlines of some of the principal nebulous filaments. 885. Distance of Star-Clusters and Size of the Component Stars. The question at once arises whether clusters, such as the one men- tioned in Hercules, are composed of stars each comparable with our sun, and separated by distances corresponding to the distance between the sun and its neighboring stars, or whether the bodies which com- pose the swarm are really very small, — mere sparks of stellar matter : whether the distance of the mass from us is about the same as that of the stars among which it seems to be set, or whether it is far beyond them. Forty years ago the accepted view was that the stars composing the clusters are no smaller than ordinary stars, and that the distance of the star-clusters is immensely greater than that of the isolated stars. There are many eloquent passages in the writings of that period based upon the belief that these star-clusters are stellar universes, — "galaxies," like the group of stars to which the writers supposed the sun to belong, but so inconceivably remote that in appearance they shrank to these mere balls of shin- ing dust. It is now, however, quite certain that the other view is correct, — that star-clusters are among our stars and form part of our universe. Large and small stars are so associated in the "same group in many cases, as to leave us no choice of belief in the matter. It is true that as yet no parallax has been detected in any star-cluster ; but that is not strange, since a cluster is not a convenient object for observations of the kind necessary to the detection of parallax. t 886. The Nebulae. — There are also in the sky a multitude of faintly shining bodies, — shreds and balls of cloudy stuff that are known as "nebulce" (the word meaning strictly a "little cloud"). Some 8000 of these objects are already catalogued. Two or three of them are visible to the naked eye. The nebula in the girdle of Andromeda is the brightest of them, in which, it will be remembered, the temporary star of 1885 appeared. The next brightest is the wonderful nebula of Orion, which, in the beauty and variety of its details, in the interesting relations of the included stars, the delicate tint of the filmy light, and in its spectro- scopic interest, far exceeds the other, — indeed, all others. 504 THE NEBULA. It is so difficult to represent these delicate objects by any process avail- able to text-books, that we limit ourselves to cuts of two only, the great nebula of Andromeda and the " ring-nebula " in Lyra (both from original drawings), referring the reader to the elaborate engravings that can be found in the "Philosophical Transactions," the "Memoirs of the Royal Astronomical Society," and other similar works, for adequate representa- tions of other interest- ing objects of this kind. With a small tele- scope a nebula cannot be distinguished from a close star-cluster, and it is quite likely that the clusters and nebulae shade into each other by insensi- ble gradations. Forty years ago it was sup- posed that there was no distinction between them except that of mere remoteness, — that all nebulas could be resolved into stars by sufficient increase of telescopic power. When Lord Eosse's great telescope was first erected, it was for a time reported (and the state- ment is still often met with) that it had " resolved" the Orion nebula. This was a mistake however. No telescope ever has resolved that nebula into stars nor ever will, for we now know that it is not com- posed of stars. Fig. 230. —The Great Nebula in Andromeda. 887. Forms and Magnitudes of Nebulae. — The larger and brighter nebulae are, many of them, very irregular in form, stretching out sprays and streamers in all directions, and containing dark openings or " lanes." The so-called " fish-mouth" in the nebula of Orion, and the dark streaks in the nebula of Andromeda, are striking examples. Some of these bodies are of enormous volume. The nebula of Orion, with its outlying streamers, extends over several square degrees, and the nebula of Andromeda covers more than one. Now, as seen from even the nearest star, the apparent distance of Neptune from the sun is only 30", and the diameter of its orbit 1'. It is perfectly certain that neither of these nebulae is as near as a Centauri, and therefore the cross-section of the Oriou nebula, as seen from the THE SMALLER NEBULAE. 506 earth, must be at least many thousand times the area of Neptune's oi'bit. We do not know what is the real shape of either of these nebula , whether it is a thin, flat sheet, or a voluminous bulk ; but some things about these two nebulae and several others favor strongly the idea that their thickness does not correspond to their apparent area. 888. The Smaller Nebulae. — The smaller nebulae are for the most part elliptical in outline, some nearly circular, others more elongated, and some narrow, slender streaks. of light. Generally they are brighter at the centre, and in many cases the centre is occu- pied by a star. Indeed, there is a considerable number of so- called " nebulous stars," that is, stars with a hazy envelope around them. There are some nebulae which present nearly a uniform disc of light, and are known as '■'■plan- etary" nebulas, and there are some which are dark in the centre and are known as " an- » nular" or ring nebulae. The finest of these annular nebulae is the one in the constellation of Lyra, about half-way between the stars /3 and y ; it is shown in Fig. 231. There are also a number of double nebulae, and some of these exhibit a remarkable spiral structure when examined by telescopes of the largest aperture like the great reflector of Lord Eosse. The so-called " whirlpool " nebula in the constellation " Canes Venatici " is the most striking specimen. This spiral structure, however, is to be made out only in large telescopes ; in fact, very little of the real beauty of most of these objects is accessible to instruments of less than 12 inches aperture. 889. Variable Nebulas. — There are several nebulae which vary in their brightness from time to time ; one especially, near t Tauri, at times has been visible with a small telescope, while at other times it Fig. 231. — The Annular Nebula in Lyra. 506 THE NEBULA. is entirely invisible even with large ones. So far no regular periodi- city has been ascertained in such cases. t 890. Their Spectra. — One of the earliest and most remarkable achievements of the spectroscope was its demonstration of the fact that the light of many of the nebulae proceeds mainly from luminous gas. They give a spectrum of six or seven bright lines ' (Fig. 232), three of which are fairly conspicuous ; while the rest are very faint. Fig. 232. — Spectrum of the Gaseous Nebulae. Three of the lines, F, H y , and h, are due to Hydrogen. One, the faintest of all, recently discovered by Copeland, seems to be identical with the D 3 line of the solar chromosphere ; but the origin of the rest remains unknown. The brightest line of the whole number is in the green, X,5007, and was for a while referred to nitrogen ; but under closer examination the identification breaks down, though the student will find it still called " the nitrogen line " in many astronomical works. Mr. Lockyer identified it with a " fluting " in the low-temperature spec- trum of magnesium, which he has found in the spectrum of meteorites ; but this seems rather doubtful, as the nebula line is fine and sharp, and does not look at all like the relic of a band; nor has the coincidence been satisfac- torily established by actually confronting the nebula spectrum with that of magnesium. All the nebulae which give a gaseous spectrum at all, present this same spec- trum entire or in part. If the nebula is faint only the brightest line appears, while the H y line and the other fainter lines are seen only in the brightest nebulae and under favorable circumstances. 891. But not all the nebula? by any means give a gaseous spec- trum : those which do so — about half the whole number — are of a more or less distinct greenish tint, which is at once recognizable in the telescope. The white nebulae, the nebula of Andromeda at their head, give only a continuous and perfectly expressionless spectrum, 1 The wave-lengths of the lines are the following, in the order of brightness: (1)5007.05? (2)4959.02? (3) 4861.50, Hydrogen (F); (4) 4340.66, Hydrogen (7); (5) 4101.85, Hydrogen (h); (6) 5875.98, Helium (Z> 8 ); (7) 4472 ? CHANGES IN THE NEBULAE. 507 unmarked by any lines or bands, either bright or dark. This must not be interpreted as showing that these nebulae cannot be gaseous ; for a gas under pressure gives just such a spectrum ; but so also do masses of solid or liquid when heated to incandescence. The spectro- scope simply declines to testify in this case. The telescopic evidence as to the nature of the white nebulae is the same as for the green. They withstand all attempts at resolution, none more firmly than the Andromeda nebula itself, the brightest of them all. 892. Changes in the Nebula. — The question has been raised whether some of the nebulae have not sensibly changed, even within the few years since it has become possible to observe them in detail. It is quite certain that in important respects the early drawings differ seriously from those of recent observers ; but the appearance of a nebula depends so much upon the telescope and the circumstances under which it is used, the features are so delicate and indefinite, and the difficulty of representing them on paper is such, that very little reliance can be placed on discrepancies between drawings, unless sup- ported by the evidence of measures at some kind. Thus far, the best authenticated instance of such a change, according to Professor Holden, is in the so-called "trifid" nebula, in Sagittarius. In this object there is a peculiar three-legged area of darkness which divides the nebula into three lobes. A bright triple star, which in the early part of the century was described and figured by Herschel and other observers as in the middle of one of these dark lanes, is now certainly in the edge of the nebula itself. The star does not seem to have moved with reference to the neighboring stars, and it seems therefore necessary to suppose that the nebula itself has drifted and changed its form. As to the nebula of Orion, Professor Holden's conclusion is, that while the outlines of the different features have probably undergone but little change, their relative brightness and prominence have been continually fluc- tuating. This, however, can hardly be considered certain ; to settle the question will probably require another fifty years or so, and the compari- son, not of drawings but of photographs — for it is now possible to photo- graph the brighter nebulae. 1 893. Photographs of Nebulae. — The first success in photograph- ing an object of this sort was obtained in 1880 by Dr- Henry Draper in experiments upon the nebula of Orion ; in 1881 and 1882 he greatly improved upon his first essays ; and not long before his death in 1882 he obtained a really fine photograph of the beautiful object. Mr. Common, of England, with his three-foot silver-on-glass reflector, and 508 THE NEBULA. exposures ranging from half an hour upward, has since then obtained pictures of the same object, still finer, and that seem to leave little to be desired. They fail of perfection only because the stars which jewel the nebula so brilliantly as seen by the eye, become in a photo- graph mere blotches, encroaching upon and cutting out large patches of the most interesting portions of the nebula. A number of nebulae have also been photographed by others as well as by Common; but he still maintains the lead, and iU is hoped that when his new five-foot reflector is finished, he will procure for us an authentic record of the present appearance of all the most important objects of this class. 894. Nature of the Nebulae. — As to the constitution of these clouds we can only speculate. In the green nebulae we can say with confidence that hydrogen and some other gas or gases are certainly present, and that the gases emit most of the light that reaches us from such objects. But how much solid or liquid matter in the form of grains and drops may be included within the gaseous cloud we have no means of knowing. The idea of Mr. Lockyer (a part of his wide induction as to what we may call the "meteoritic constitution of the universe") is that they are clouds of " sparse meteorites, the collisions of which bring about a rise of temperature sufficient to render luminous one of their chief constituents, — magnesium." How far this theory will stand the test of time and future investigations remains to be seen. At first view it seems very doubtful whether the colli- sions in such a body could be frequent or violent enough to account for its luminosity, and one is tempted to look to other causes for the source of light. 895. Number and Distribution of Nebulae. — Sir William Her- schel was the first extensive investigator of these interesting objects, and left his unfinished work as a legacy to his son, Sir John Her- schel, who completed the survey of the heavens by a residence of several years at the Cape of Good Hope. His " General Catalogue" is the standard of reference for objects of this kind, and contains about 5000 of them. Between 2000 and 3000 more have been added since its publication, most of them extremely faint. It is hardly possible that any important ones remain to be found. As to their distribution, it is a curious and important fact that it is in contrast to the distribution of the stars. The stars, as we shall DISTANCE OF THE NEBULA. 509 Boon see, gather especially in and about the Milky Way, as do also the star-clusters ; but the nebulae specially crowd together in regions as far from the Milky Way as it is possible to get. As has been pointed out by more than one, this shows, however, not a want of relation between the stars and the nebulae, but some " relation of contrariety." Precisely what this is, and why the nebulae avoid the regions thickly starred, is not yet clear. Possibly the stars devour (hem, that is, gather in and appropriate surrounding nebulosity so that it disappears from their neighborhood. 896. Distance of the Nebulae. — On this point we have very little absolute knowledge. Attempts have been made to measure the par- allax of one or two, but so far unsuccessfully. Still it is probable, indeed almost certain, that they are at the same order of distance as the stars. The wisps of nebulosity which photography shows at- tached to the stars in the Pleiades (and a number of similar cases appear elsewhere), the nebulous stars of Herschel, and numerous nebulae which have a star exactly in the centre, — these compel us to believe that in such cases the nebulosity is really at the star. Then in the southern hemisphere there are two remarkable luminous clouds which look like detached portions of the Milky Way (though they are not near it),- and are known as the Nubeculae (major and minor). These are made np of stars and star-clusters, and of nebulae also, all swarming together, and so associated that it is not possible to question their real proximity to each other. 897. Fifty years ago a very different view prevailed. As has been said already, astronomers at that time very generally believed that there was no distinction between nebulae and star-clusters except in regard to distance, the nebulse being only clusters too remote to show the separate stars. They considered a nebula, therefore, as a " universe of stars," like our own " galac- tic cluster " to which the sun belongs, but as far beyond the " star-clusters " as these were believed to be beyond the isolated stars. In some respects this old belief strikes one as grander than the truth even. It made our vision penetrate more deeply into space than we now dare think it can. 898. The Galaxy, or Milky Way. — This is a luminous belt which surrounds the heavens nearly in a great circle. It varies much in width and brightness, and for about a third of its extent, from Cyg- nus to Scorpio, is divided into two nearly parallel streams. In several constellations, as in Cygnus, Sagittarius, and Argo Navis, it is crossed by dark straight-edged bars that look as if some 510 THE GALAXY. light cloud lay athwart it, and in the constellation of Centaurus there is a dark pear-shaped orifice, — the " coal sack," as it is called. The galaxy intersects the ecliptic at two opposite points near the solstices, making with it an angle of about 60°. The northern "galactic pole," as it is called, lies, according to Sir John Herschel, in declination + 27°, and right ascension 12 h 47 ra ; the southern " galactic pole " is of course at the opposite point in the southern hemisphere. As Herschel remarks, the "galactic plane" "is to sidereal what the ecliptic is to planetary astronomy, a plane of ultimate reference, the ground plan of the sidereal system." The Milky Way is made up almost wholly of small stars from the eighth magnitude down. It contains also a large number of star- clusters, but (as has been already mentioned) very few true nebulae. In some places the stars are too thickly packed for counting, es- pecially in the bright knots which abound here and there. (An excellent detailed description of its appearance and course may be found in Herschel's " Outlines of Astronomy.") 899. Distribution of Stars in the Sky : Star-Gauges. — It is obvious that the stars are not uniformly scattered over the heavens. They show a decided tendency to collect in groups here and there, and to form connected streams ; but besides this, an enumeration of the stars in the great star-catalogues shows that the number increases with considerable regularity from the galactic poles, where they are most sparse, towards the galactic circle, where they are most crowded. The "star-gauges" of the Herschels make this fact still more obvious. These gauges consisted merely in the counting of the number of stars visible in the field of view (15' in diameter) of the twenty-foot reflector. Sir William Herschel made 3400 of these gauges, directing the telescope to different parts of the sky ; and his son followed up the work at the Cape of Good Hope. Struve's discussion of these gauges in their relation to the galactic circle gives the following result : — Distance from Galaxy. Number of Stars iu Field 90° 4.15 75° 4.68 60° . 6.52 45° . 10.36 30° . 17.68 15° . 30.30 0° . 122.00 STRUCTURE OP THE HEAVENS. 511 900. Structure of the Heavens. — Our space does not permit a discussion of the untenable conclusions reached by Heischel and others by combining the unquestionable data derived from observa- tion, with the unfounded and untrue assumptions that the stars are substantially of a size and spaced at approximately equal distances. Many of those conclusions relating to the form and dimensions of the Milky Way, and of the stellar universe to which our sun belongs, have become almost classical ; but they are none the less incorrect. It is certain, however, that the faint stars as a class are smaller and darker and more remote than are the bright ones as a class : and accepting this, we can safely draw from the star-gauges a few general conclusions, as follows : — We present them substantially as given by Newcomb in his "Popular Astronomy," p. 491. 1. "The great mass of the stars which compose this (stellar) system are spread out on all sides in or near a widely extended plane, passing through the Milky Way. In other words, the large majority of the stars which we can see with the telescope are con- tained in a space having the form of a round, flat disc, the diameter of which is eight or ten times its thickness. 2. " Within this space the stars are not scattered uniformly, but are for the most part collected into irregular clusters or masses, with comparatively vacant spaces between them." They are " gregarious," to use Miss Clerke's expression. 3. Our sun is near the centre of this disc-like space. 4. The naked-eye stars "are scattered in this space with a near approach to uniformity," the exceptions being a few star-clusters and star-groups like the Pleiades and Coma Berenices. 5. " The disc described above does not represent the form of the stellar system, but only the limits within which it is mostly con- tained." The circumstances are such as to "prevent our assigning any more definite form to the system than we could assign to a cloud of dust." 6. " On each side of the galactic region the stars are more evenly and thinly scattered, but probably do not extend out to a distance at all approaching the extent of the galactic region," or if they do they are very few in number ; but it is impossible to set any definite boundaries. 7. On each side of the galactic and stellar region we have a nebular region, comparatively starless, but occupied by great numbers of nebulre. 512 STRUCTURE OF THE HEAVENS. As to the Milky Way itself, it is not yet certain whether the stars which compose it are distributed pretty equally near the galactic circle, or whether they form something like a ring with a compara- tively vacant space in the middle. The ring theory seems at present rather the more probable of the two. As to the distance of the remotest stars in the stellar system, it is impossible to say anything very definite, but it seems quite certain that it must be at least so great that light would occupy from 10,000 to 20,000 years in traversing it. If one asks what is beyond the stellar system, whether the star-filled space extends indefinitely or not, no certain answer can be given. 901. Do the Stars Form a System? — That is, dp they form an organized unit, in which, as in the solar system, each of the different members has its own function and permanently maintains its relation to the rest? Gravitation certainly operates, as the binary stars demonstrate, 1 and the stars are moving swiftly in various directions with enormous velocities, as shown by their proper motions, and by the spectroscope. The question is whether these motions are con- trolled by gravitation, and whether they carr3* the stars in orbits that can be known and predicted. That the stars are organized into a system or sj'stems of some sort can hardly be doubted, for this seems to be a necessary consequence of their mutual attraction. But that the system is one at all after the pattern of the solar system, in which the different members move in closed orbits, — orbits that are permanent except for the slow changes produced by perturbation, — this is almost certainly im- possible, as was said a few pages back. 902. Is there a Revolution of the Whole Mass of Stars? — A favorite idea has been that the mass of stars which constitutes our system 1 This is perhaps rather too strong an 'expression. It would be truer to say that none of the observed phenomena of the binary stars contravene the univer- sality of gravitation, and for the most part they are just what gravitation easily accounts for. But they do not demonstrate that the central force varies inversely with the square of the distance, because we do not know the angle made by the orbit-plane with the line of sight, except from calculations based on the assump- tion that the central force really varies in that way. The orbits, as directly observed, are consistent with several other laws of central force than the law of inverse squares. (See " Astron. Journal," Vol. VIII., article by Prof. A. Hall.) STRUCTURE OP THE HEAVENS. 513 has a slow rotation lite that of a body on its axis, the plane of this general revolution coinciding with the plane of the galaxy. Such a general motion is not in any way inconsistent with the independent motions of the indi- vidual stars, and there is perhaps a slight inherent probability in favor of such a movement; but thus far we have no evidence that it really exists indeed, there hardly could be any such evidence at present, because exact Astronomy is not yet old enough to have gathered the necessary data. 903. Central Sims. — A number of speculative astronomers, Madler perhaps most prominently, have held the belief that there is a " central sun,'' standing in some such relation to the stellar system as our sun does to the solar system. It is hardly necessary to say that the notion has not the slightest foundation, or even probability. Lambert supposed many such suns as the centres of subordinate stellar systems, and because we cannot see them, he imagined them to be dark. If we conceive of boundaries drawn around our stellar system, and count all the stars within the limits as members of it, leaving out of the account all that fall outside, then, of course, our system so limited has at any moment a perfectly definite centre of gravity. There is no reason why some particular star may not be very near that centre, and in that sense a " central sun " is possible ; but its central position would not give it any preeminence or rule over its neighbors, or put it in any such relation to the rest of the stars as the sun bears to the planets. 904. Orbits of Sun and Stars. — It is practically certain that the motions of the stars are not orbital in any strict sense. Excepting stars which are in clusters, all other stars are simultaneously acted upon by manj- forces drawing in various and opposite directions ; and these forces must in most cases be so nearly balanced that the resultant cannot be very large. The motions of the stars must consequently, as a rule, be nearly rectilinear. Still the balancing of the forces will seldom be exact, and accord- ingly the path of a star will almost always be slightly curved ; and since the amount and direction of the resultant force which acts on the star is continually changing, the curvature of its motion will alter correspondingly, and the result will be a path which does not lie in any one plane, but is bent about in all ways like a piece of crooked wire. It is hardly likely, however, that the curvature of a star's path would, in any ordinary case, be such as could be detected by the observations of a single century, or even of a thousand years. As has been said before, in connection with the proper motions of the stars, the probability is that the separate stars move nearly inde- 514 COSMOGONY. pendently, " like bees in a swarm." In the solar system the central power is supreme, and perturbations or deviations from the path which the central power prescribes are small and transient. In the stellar system, on the other hand, the central force, if it exists at all (as an attraction towards the centre of gravity of the whole mass of stars) is trifling. Perturbation prevails over regularity, and " indi- vidualism ''-is the method of the greater system of the stars, as solar despotism is that of the smaller system of the planets. 905. Cosmogony. — Unquestionably one of the most interesting, and also most baffling, topics of speculation is the problem of the way in which the present condition of the universe came about. By what processes have moons and earths and Jupiters and Saturns, come to their present state aud into their relation to the sun ? What has been their past history, and what has the future in store for them? How has the sun come to his present glory and dominion? and in the stellar universe, what is the meaning and mutual relation of the various orders of bodies we see, — of the nebulae, the star- clusters, and the stars themselves ? In a forest, to use a comparison long ago employed by the elder Herschel, we see around us trees in all stages of their life history. There are the seedlings just sprouting from the acorn, the slender saplings, the sturdy oaks in their full vigor, those also that are old and near decay, and the prostrate trunks of the dead. Can we apply the analogy to the heavens, and if wo can, which of the objects before us are to be regarded as in their infancy, and whicli of them as old and near dissolution ? 906. Fundamental Principles of a Rational Cosmogony. — In the present state of science many of the questions thus suggested, seem to be hopelessly beyond the reach of investigation, while others appear like problems which time and patient work will solve, and others yet have already received clear and decided answers. In a general way it may be said that the condensation and aggregation bf rarefied masses of matter under the force of gravitation; the conversion into heat of the (potential) " energy of position" destroyed by the process of condensation; the effect of this heat upon the contracting mass itself, and the radiation of energy into space and to surrounding bodies as tvaves of light and heat, — these principles contain' nearly all the explanations that can thus far be given of the present state of the heavenlv bodies. THE NEBULAR HYPOTHESIS. 515 907. The Planetary System. — We see that our planetary system is not a mere accidental aggregation of bodies. Masses of matter coming hap-hazard towards the sun would move, as comets do, in orbits, always conic sections to be sure, but of every degree of eccen- tricity and inclination. There are a multitude of relations actually observed in the planetary system which are wholly independent of gravitation and demand an explanation. 1. The orbits are all nearly circular. 2. They are all nearly in one plane (excepting the cases of some of the little asteroids). 3. The revolution of all is in the same direction. 4. There is a curiously regular progression of distances (expressed by Bode's law, which, however, breaks down at Neptune) . 5. There is a roughly regular progression of density, increasing both ways from Saturn, the least dense of all the planets in the system. As regards the planets themselves, we have 6. The plane of the planets' rotation nearly coinciding with that of the orbit (probably excepting Uranus) . 7. The direction of the rotation the same as that of the orbital revolution (excepting probably Uranus and Neptune). 8. The plane of orbital revolution of the satellites coinciding nearly with that of the planet's rotation. 9. The direction of the satellites' revolution also coinciding with that of the planet's rotation. 10. The largest planets rotate most swiftly. 908. Origin of the Nebular Hypothesis. — Now this is evidently a good arrangement for a planetary system, and therefore some have inferred that the Deity made it so, perfect from the first. But to one who considers the way in which other perfect works of nature usually come to their perfec- tion — their processes of growth and development — this explanation seems improbable. It appears far more likely that the planetary system grew than that it was built outright. Three different philosophers in the last century, Swedenborg, Kant, and La Place (only one of them an astronomer), independently proposed essen- tially the same hypothesis to account for the system as we now know it. La Place's theory, as might have been expected from his mathematical and scientific attainments, was the most carefully and reasonably worked out in detail. It was formulated before the discovery of the great principle of the " conservation of energy," and before the mechanical equivalence of heat with other forms of energy was known, so that in some respects it is defec- 516 COSMOGONY. tive, and even certainly wrong. In its main idea, however, that the solai system once existed as a nebulous mass and has reached its present state as the result of a series of purely physical processes, it seems certain to prove correct, and it forms the foundation of all the current speculations upon the subject. 909. La Place's Theory. — (a) He supposed that at some past time, which may be taken as the starting-point of our system's history (though it is not to be considered as the beginning of the existence of the substance of which our system is composed), the matter now col- lected in the sun and planets was in the form of a nebula. (6) This nebula was a cloud of intensely heated gas, perhaps hotter, as he supposed, "than the sun is now. (c) This nebula under the action of its own gravitation, assumed an approximately globular form with a rotation around an axis. As to this movement of rotation, it appears to be necessary to account for it by supposing that the different portions of the nebula, before the time which has been taken as the starting-point, had motions of their own. Then, unless these motions happened to be balanced in the most perfect and improbable manner, a motion of rotation would set in of itself as the nebula contracted, just as water whirls in a basin when drawn off by an orifice in the bottom. The velocity of this rota- tion would become continually swifter as the volume of the nebula diminished, the so-called " moment of momentum " remaining neces- sarily unchanged. 910. (d) In consequence of this rotation, the mass, instead of remaining spherical, would become much flattened at the poles, and as the rotation went on and the motion became accelerated, the time would come when the centrifugal force at the equator of the neb- ula would become equal to gravity, and " rings of nebulous matter" would be abandoned (not thrown off) , resembling the rings of Saturn, which, indeed, suggested this feature of the theory. (e) A ring would revolve for a while as a whole, but in time would break, and the material would collect into a single globe. La Place sup- posed that the ring would revolve as if it were solid, the outer edge, therefore, moving more swiftly than the inner. If this were so, the mass formed from the collection of the matter of the ruptured ring would necessarily rotate in the same direction as the ring had revolved. (/) The planet thus formed would continue to revolve around the central mass, and might itself in turn abandon rings which might break, and so furnish it with a retinue of satellites. MODIFICATIONS OP LA PLACE'S THEORY. 517 911. It is obvious that this theory meets completely most of the conditions of the problem. It explains every one of the facts just mentioned as demanding explanation in the solar system. Indeed, it explains them almost too well ; for as the theory stands it meets a most serious difficulty in the exceptional cases of the planetary system, such as the anomalous and retrograde revolutions of the satellites of Uranus and Neptune. Another difficulty lies in the swift revolution of Phobos (Art. 589), the inner satellite of Mars. According to the unmodified nebular hypothesis, no planet or satellite could have a time of revolution less than the time of rotation which the central body would have, if expanded until its radius becomes equal to the radius of the satellite's orbit ; still less could it have a period shorter than the central body now has. 912. Mecessary Modifications. — The principal modifications which seem essential to the theory in the light of our present knowledge, are- the following. (The small letters indicate the articles of the original theory to which reference is made.) (b) It is not probable that the original nebula could have been at a temperature even nearly as high as the present temperature of the sun. The process of condensation of a gaseous cloud from loss of heat by radiation, would cause jthe temperature to rise, according to the remarkable and almost paradoxical law of Lane (Art. 357), until the mass had begun to liquefy or solidify. And it appears prob- able that the original nebula, instead of being purely gaseous, was rather a cloud of dust than a "fire-mist" ; i.e., that it was made up of finely divided particles of solid or liquid matter, each particle envel- oped in a mantle of permanent gas. Such a nebula in condensing would rise in temperature at first as if purely gaseous, so that its cen- tral mass after a time would reach the solar stage of temperature, the solid and liquid p'articles melting and vaporizing as the mass grew hotter. At a subsequent stage, when yet more of the original energy of the mass had been dissipated by radiation, the temperature of the bodies which were formed from and within the nebula would fall again. And yet La Place may have been right in ascribing a high temperature to tiie original nebula. If that were really the case, the only difference would be that the nebula would be longer in reaching the condition of a solar system ; but it is not necessary, as he supposed, to assume that the original temperature was high, and that the matter was originally in a purely gaseous condition, in order to account for the present existence of such a group of bodies as the incandescent sun and its cool attendant planets. 518 COSMOGONY. 913. (d) As regards the manner in which the planetary bodies were probably liberated i'rom the parent mass, it seems to be very doubtful whether the matter accumulated at the equator of the rotat- ing mass would usually separate itself as a ring. If a plastic mass in swift rotation is not absolutely homogeneous and symmetrical, it is more likely to become distorted by a lump formed somewhere on its equator, which lump may be finally detached and circulate around its primary. The formation of a ring, though possible, would seem likely to be only a rare occurrence. La Place seems to have believed also that the outer rings must necessarily have been abandoned first, and the others in regular suc- cession, so that the outer planets are much the older. It seems, how- ever, quite possible, and even probable, that several of the planets may be of about the same age, more than one ring having been liberated at the same time ; or several planets having been formed from different zones of the same ring. (e) In the case where a ring was formed, it is practically certain that it could not have revolved as a solid sheet ; i.e., with the same angular velocity for all the particles, and with the outer portions, therefore, moving more swiftly than the inner. If, for instance, the matter which now constitutes the earth were ever distributed to form a ring occupying anything like half the distance from Venus to Mars, it must have been of a tenuity comparable only to that of a comet. The separate particles of such a ring could have had very little con- trol over each other, and must have moved substantially as independ- ent bodies ; the outer ones, like remoter planets, making their cir- cuits in longer periods and moving more slowly than those near the inner edge of the ring. 914. Explanation of the Anomalous Rotation of Uranus and Nep- tune. — When the matter of such a ring concentrates into a single mass, the direction of the rotation of the resultant planet depends upon the manner in which the matter was originally distributed in the ring. If the ring be nearly of the same density throughout, the resulting planet (which would be formed at about the middle of the ring's width) must have a retrograde rotation like Uranus and Neptune. But if the particles of the ring are more closely packed near its inner edge, so that the resultant planet would be formed much with- in the middle of its width, its axial rotation must be direct. In the first case, illustrated in Fig. 233 (a), the particles near the inner edge of the ring would control the rotation, having a greater moment of THE ROTATION OF THE PLANETS. 519 rotation with respect to M, where the planet is supposed to be formed, than those at the outer edge. The rotation, therefore, will be retrograde, on account of then- greater velocity. In the other case, Fig. 233 (6) , where the inner edge of the ring is densest, and the planet is formed as at iV, much nearer the inner than the outer edge of the ring, the aggregate moment of rotation Fig. 233. — Rotation of Planets formed from Rings according to the Nebular Hypothesis. with respect to JV is greater for the particles beyond W (because of their greater distance from it) than that of the swifter moving parti- cles within, and this determines a direct rotation. The fact that the satellites of Uranus and Neptune revolve back- wards is not, therefore, at all a bar to the acceptance of the nebular hypothesis, as sometimes represented. If a new planet should ever be discovered outside of Neptune it is altogether probable that its satellites would be found to retrograde. This is not the only way in which the retrograde rotation of the outer planets may be accounted for. There are a number of other possible assumptions as to the constitution of the mass detached from the parent nebula, and the manner in which its particles ultimately coalesce to form a planet, which lead to a similar result. 915. Faye has recently propounded a modification of the nebular hypoth- esis which makes the planets of the "terrestrial group" (Mercury, Venus, the 520 COSMOGONY. Earth, and Mars) older than the outer ones. He supposes that the planets were formed by local condensations (not by the formation of rings) within the revolving nebula. At first, before the nebula was much condensed at the centre, the inward attraction would be at any point directly proportional tt the distance of that point from the centre of gravity of the nebula; i.e., tha force could be expressed by the equation F=ar. After the condensation has gone so far that practically almost the whole of the matter is collected at the centre of the nebula, the force is inversely proportional to the square of the distance, — the ordinary law of gravitation, i.e., F=-. At any intermediate time, during the gradual condensation of the nebula, the intensity of the central force will, therefore, be given by an expression having the form F=ar+^ »• being the distance of the body acted upon from the centre of gravity of the nebula, while a and b are coefficients which depend upon its age ; a con- tinually decreasing as the nebula grows older, while b increases. The planets formed within the nebula when it was young, i.e., when a was large and b was small, would have direct rotation upon their axes, while those formed after a had sensibly vanished would have a retrograde rotation ; and this he supposes to be the case with Uranus and Neptune, which he considers younger than the inner planets. Faye's work " L' Origine du Monde," 1885, contains an excellent summary of the views and theories of the different astronomers who have speculated upon the cosmogony. 916. Tidal Evolution. — Within a few years Prof. G. H. Darwin (son of the great naturalist) has made some important investiga- tions upon the effect of tidal reaction between a central mass and a body revolving about it, both of them being supposed to be of such a nature (i.e., not absolutely rigid), that tides can be raised upon them by their mutual attraction. We have already alluded to the subject in connection with the tides (Art. 484). He finds in this reaction an explanation of many puzzling facts. It appears, for instance, that if a planet and its satellite have ever had their times of rotation of the same length as the time of their orbital revolution around their common centre of gravity, then, starting from that time, either of two things might happen, — the satellite might begin to recede from the planet, or it might fall back to the central mass. The condition is one of unstable equilibrium, and the slightest cause might determine the subsequent course of things in either of the TIDAL EVOLUTION. 521 two opposite directions. Whenever the time of rotation of the planet is shorter than the orbital period of the satellite (as it would naturally become by condensation continuing after the separation of the satellite), the tendency would be, as explained in Article 484, slightly to accelerate the satellite, and so to cause it continually to recede by an action the reverse of that produced by the hypothetical resist- ing medium which is supposed to disturb Encke's comet. This, it will be remembered, is thought to be the case with our moon. 917. But if by any means the rotation of the planet were retarded, so that its day should become longer than the period of the satellite! the tides produced by the satellite upon the planet will then retard the motion of the satellite like a resisting medium, and so will cause a continual shortening of its period, precisely as in the case of Encke's comet. If nothing intervenes, this action will in time bring down the satellite upon the planefs surface. Now in the case of Mars there is a known cause operating to retard its rotation (namely, the tides which are raised by the sun upon the planet), and those who accept the theory of tidal evolution suggest that this was the cause which first made the length of the planet's day to exceed the period of the satellite, and so enabled the planet to establish upon the satellite that retardation which has shortened its little month, and must ultimately bring it down upon the planet. Processes such as these of tidal evolution must necessarily be extremely slow. How long are the periods involved, no one can yet estimate with any precision, but it is certain that the years are to be counted by the million. (We have already referred the reader (Art. 484*) to the last chapter of Ball's " Story of the Heavens " as containing an excellent and easily under- stood explanation of this subject.) 918. Conclusions derived from the Theory of Heat. — As Profes- sor Newcomb has said, " Kaut and La Place seem to have arrived at the nebular hypothesis by reasoning forwards. Modern science ob- tains a similar result by reasoning backwards from actions which we now see going on before our eyes." We have abundant evidence that the earth was once at a much higher temperature than now. As we penetrate below the surface we find the temperature continually rising at a rate of about 1° F. for every fifty or sixty feet, thus indicating that at the depth of a few miles the temperature must be far above incandescence. Now, 522 COSMOGONY. since the surface temperature is so much lower, this implies one (or both) of two things, — either that heat-making processes are going on within the earth (which may be true to some extent) , or else that the earth has been much hotter than it now is, and is cooling off, — and this seems to be a most probable supposition. It is just as reasonable, as Sir W. Thomson puts it, to suppose that the earth has lately been intensely heated as to suppose that a warm stone that one picks up in the field has been lately somewhere in the fire. 919. Evidence derived from the Condition of the Moon and Planets. — In the case of the moon we find a body bearing upon its surface all the marks of past igneous action, but now in appearance intensely cold. The planets, so far as we can judge from what we can see through the telescope, corroborate the same conclusion. Their testimony is not very strong, but it is at least true that nothing in the aspect of any of them militates against the view that they also are bodies cooling like the earth ; and in the cases of Jupiter and Saturn many phenomena go to show that they are still (or at least now) at a high temperature, — as might be expected of bodies of such an enormous mass, which, necessarily, other things being equal, would cool much more slowly than smaller globes like the earth. The ratio of surface to mass is smaller as the diameter of a globe grows larger, and upon this ratio the rate of cooling of a body largely depends. In short, everything we can ascertain from the observation of the planets agrees completely with the idea that they have come to their present condition by cooling down from a molten or even gaseous state. 920. The Sun's Testimony. — In the sun we have a body steadily pouring forth an absolutely inconceivable amount of heat, without any visible source of supply. Thus far the only reasonable hypoth- esis to account for this, and for a multitude of other phenomena which it shows us, is the one which makes it a great cloud-mantled ball of incandescent gases, slowly shrinking under its own central gravity, converting continually a portion of its "potential energy of position" 1 into the kinetic-energy of heat, which at present is mainly radiated off into space. 1 By "potential energy of position " is meant the energy due to the separated condition of its particles from each other. As they fall together and towards the centre in the shrinkage of the sun, they " do work " in precisely the same way as any falling weight. CONSIDERATIONS FROM THE THEORY OF HEAT. 523 "We say mainly, because it is not impossible that the sun's temperature is even yet slowly rising, and that the maximum has not yet been reached. We are not sure whether all the heat produced by the sun's annual shrinkage is radiated into space, or whether a portion is retained within its mass, thus raising its temperature ; or whether, again, it radiates more than the amount thus generated, so that its temperature is slowly diminishing. 921- That the sun is really shrinking is admittedly only an inference, for the shrinkage must be far too slow for direct observation. Our case is like that of a man who, to use one of Professor Newcomb's illustrations, when he comes into a room and finds a clock in motion, concludes that the clock-weight is descending, even though its motion is too slow to be observed. Knowing the construction of the clock and the arrangement of its gearing, and the number of teeth in each of its different wheels, he states confidently just how many thousandths of an inch the weight sinks at each vibration of the pendulum ; and looking into the clock-case and measuring the length of the space in which the weight can move, and noting its present place, he proceeds to state how long ago the clock was wound up, and how long it has yet to run. We must not push the analogy too far, but it is in some such way that we conclude from our measurements of the sun's annual output of energy in the form of heat, how fast it is shrinking, and we find that its diameter must diminish not far from 250 feet in a year ; at least, the loss of potential energy corresponding to that amount of shrinkage would account for one year's running of the solar mechanism. 922. Age of the Solar System. — Looking backward, then, in imagination we see the sun growing continually larger through the reversed course of time, expanding and becoming ever less and less dense, until at some epoch in the past it filled all the space now included within the largest orbit of the solar system. How long ago that was no one can say with certainty. If we could assume that the amount of potential energy lost by contraction, con- verted into the actual energy of heat and radiated into space, has been the same each year through all the intervening ages, and moreover, that all the heat radiated has come from this source only, without sub- sidy from any original store of heat contained in an original "fire mist," or from energy derived from outside sources, then it is not dif- ficult to conclude that the sun's past history must cover some 15,000000 or 20,000000 years. But the assumption that the loss of heat has been even nearly uni- form is extremely improbable, considering how high the present tem- perature of the Sun must be as compared with that of the original nebula, and how the ratio of surface to solid content has increased with the lessening diameter. 524 COSMOGONY. Nor is it unlikely that the sun may have received energy from other sources than its own contraction. Altogether it would seem that we must consider the 15,000000 years to be the least possible value of a duration which may have been many times more extended. If the nebular hypothesis and the theory of the solar contraction be true, the sun must be as old as that, — how much older no one can tell. 923. Future Prospects. — Looking forward toward the future, it is easy to conclude also that at its present rate of radiation and con- traction the sun must, within 5,000000 or 10,000000 years, become so dense that the conditions of its constitution will be radically changed, and to such an extent that life on the earth, as we now know life, would probably be impossible. If nothing intervenes to reverse the course of things, the sun must at last solidify and become a dark, rigid globe, frozen and lifeless among its lifeless family of planets. At least, this is the necessary consequence of what now seems to science to be the true account of its present activity and the story of its life. 924. Stars, Star-Clusters, and the Nebulae. — It is obvious that the same nebular hypothesis applies satisfactorily to the explanation of the relation of these different classes of bodies to each other. In fact, Herschel, appealing only to the law of continuity, had concluded before La Place formulated his theory, that nebulae develop sometimes into clusters, sometimes into double or multiple stars, and sometimes into single ones. He showed the existence in the sky of all the interme- diate forms between the nebula and the finished star. For a time, about the middle of our century, while it was generally supposed that all nebulas were nothing but star-clusters, too remote to be resolved by existing telescopes, his views fell rather into abeyance ; but when the spectroscope demonstrated the substantial differences between the gaseous nebulas and the star-clusters, they regained acceptance in their essential features ; with perhaps the reservation, that many are disposed to believe that the rarest even of nebulous matter, instead of being purely gaseous, is full of solid and liquid particles like a cloud of fog or smoke. 925. The Present System not Eternal. — One lesson seems to stand out clearly, — that the present system of stars and worlds is not an eternal one. We have before us irrefragable evidence of continuous, uncompensated progress, inexorable «in one direction. The hot bodies are losing their heat, and distributing it to the cold ones, so that there is a steady, unremitting tendency towards a THE PRESENT SYSTEM NOT ETEKNAX,. 525 uniform (and therefore useless) temperature throughout the uni- verse : for heat does work, and is available as energy only when it can pass from hotter to cooler bodies, so that this warming up of cooler bodies at the expense of hotter ones always involves a loss, not of energy (for that is indestructible), but of available energy. To use the technical language now usually employed, energy is unceasingly "dissipated" by the processes which maintain the present life of the universe ; and this dissipation of energy can have but one ultimate result, — that of absolute stagnation when a uniform temperature has been everywhere attained. If we carry our imagination backwards we reach at last a "beginning of things," which has no intelligible antecedent : if forwards, an end of things in stagnation. That by some process or other this end of things will result in "new heavens and a new earth" we can hardly doubt, but science has as yet no word of explanation. 926. Mr. Loekyer's Meteoric Hypothesis. — The idea that the heavenly bodies in their present state may have been formed by the aggregation of meteoric matter, rather than by the condensation of a gaseous mass, is not new, and not original with Mr. Lockj-er, as he him- self points out. But his adoption and advocacy of the theory, and the support he brings to it from spectroscopic experiments on the light emitted by fragments of meteoric stones under different conditions, has given it such currency within the last two years that his name will always be justly associated with it. "We have already referred to it in several places (Arts. 850 and 894 especially) . He believes that he finds in the spectra of meteorites, under various conditions, an explanation 1 of the spectra of comets, nebulae, and all the different types of stars, as well as the spectra of the Aurora Borealis and the Zodiacal Light. Assuming this, he considers that nebulae are meteoric swarms in the initial stages of condensation, the separate individuals being still widely separated, and collisions comparatively infrequent. As aggregation goes on, the nebulae become stars, which run through a long life-history, the temperature first increasing slowly to a maxi- mum, and then falling to non-luminosity. During this life-history the stars pass through successive stages, each stage characterized by i In some cases the explanation appears to be at least doubtful, especially in instances where the presence of a sharply denned line in the spectrum of a heav- enly body is referred to the degradation of a band observed in the meteoric spectrum. 526 COSMOGONY. its own typical spectrum ; and according to these views he divides the stars into six spectroscopic classes, as follows : — I. Those which, like the nebulae, show bright lines or flutings in the spectrum, without dark lines or bands. Vogel's II. (6) and I. (c) (Art. 858) are included in this class, y Cassiopeiae and 152 Schjel- lerup are types. II. Those which show both dark lines and dark flutings in the spec- trum. Vogel's III. (a) : (aOrionis). III. Those which show the fine dark lines of metals. Vogel's II. (a) is included here in part, a Auriga? and the sun are typical. IV. Those whose spectra are characterized by the conspicuous hydro- gen lines, all other lines and markings being faint. These stars are at the summit of the temperature curve. Vogel's. I- (a) : Sirius and Vega. V. This class, on the descending branch of the temperature curve (stars past middle life) , should have sensibly the same spectrum as those of Class III. We cannot be sure in which of the two classes our sun, for instance, should be counted. VI. The red stars, their spectrum characterized by heavy absorp- tion bands. These are stars verging to extinction according to Mr. Lockyer's view. Vogel's III. (b). It is impossible to go into detail and give here Mr. Lockyer's in- genious applications of the theory to explain the phenomena of the different classes of variable stars. (See Art. 850, however.) This hypothesis has recently been much strengthened by a most interesting mathematical investigation of Prof. G. Darwin, who shows that, if we assume a meteoric swarm comparable in dimen- sions with our solar system, composed of individual masses such. as fall on the earth, and endowed with such velocities as meteors are known to have, such a swarm, seen from the distance of the stars, would behave like a mass composed of a continuous gas. This is not strange since, according to the kinetic theory of gases, a gas is simply a swarm of molecules, behaving in just the way the meteorites are supposed to act. But it follows that the meteoric theory of a nebula does not in the least invalidate, or even to any great extent modify, the reasoning of La Place in respect to the development of suns and systems from a gaseous nebula. The old hypothesis lias no quarrel with the new. While it would be premature to indorse this speculation of Mr. Lockyer's as an established discovery (since there remain in it many obscure and doubtful points), there can be little doubt that it marks an epoch in the history of the science. ADDENDA TO CHAPTER XXI. 526 A 872. Spectroscopic Binaries. — One of the most interesting of recent astronomical results is the detection by the spectroscope of several pairs of double stars so close that no telescope can separate them. In 1889 the brighter component of the well-known double star.Mizar (Zeta Ursse Majoris, Fig. 226) was found by Pickering to show the dark lines double in the photographs of its spectrum, at regular intervals of about fifty-two days. The obvious explanation is that this star is composed of two, which revolve around their common centre of gravity in an orbit which is turned nearly edge- wise to us. When the stars are at right angles to the line along which we view them, one of the two will be moving towards us, while the other is moving in an opposite direction ; and as a conse- quence, the lines in their spectra will be shifted opposite ways, according to Doppler's principle (Art. 321, note). Now since the two stars are so close that their spectra overlie each other, the result will be simply to make the lines in the compound spectrum appar- ently double. From the distance apart of the lines, the relative velocity of the stars can be found, and from this the size of the orbit and the mass of the stars. Thus it appears that in the case of Mizar the relative velocity of the two components is about 100 miles per second, the period about 104 days, and the distance between the two stars about the same as the diameter of the orbit of Mars; from which it follows that their united mass is about forty times that of the sun. This really makes Mizar a triple star, the larger of the two that are seen with a small telescope being the one that is thus spectro- scopically split. The lines in the spectrum of Beta Aurigse exhibit the same peculiarity, but the doubling occurs once in four days ; the velocity being about 150 miles a second, and the diameter of the orbit about 8,000000 miles, the united mass of the two stars comes out about two and a half times that of the sun. These observations of Pro- fessor Pickering's were made by photographing the spectrum with the slitless spectroscope (Art. 861), and are only possible where the stars which compose the binary are both of them reasonably bright. With his slit-spectroscope, Vogel, as has already been stated in the preceding note, has been able to detect a similar orbital motion in Algol, although the companion of the brighter star is itself invisible. More recently, in the case of the bright star Alpha Vir- ginis (Spica), he has found a result of the same kind. At first the photographic observations of the spectrum of this star appeared 526 B ADDENDA TO CHAPTER XXI. very discordant. Some days they indicated that the star was moving towards us quite rapidly, and then again from us ; but it is found that everything can be explained by the simple assumption that the star is double with a small companion, like that of Algol, not bright enough to show itself by its light, but heavy enough to make its partner swing around in an orbit about 6,000000 miles in diameter, 41°0 0138" 0E34S - Mr. Koberts's Photograph of the Nebula of Andromeda. once in four days. The orbit is not quite edgewise to the earth, so that the dark companion does not eclipse Spica, as Algol is eclipsed by its attendant. Eigel (Beta Orionis) also shows traces of a simi- lar periodic variation, though the observations have not yet been con- ADDENDA TO OHAPTER XXI. 526° tinued long enough to determine its period. These orbits, of course, are very much smaller than those of most of the telescopic binaries. 886 and 893. We present an engraving of the photograph of the nebula of Andromeda obtained by Mr. Eoberts of Liverpool, in December, 1888, -with a twenty-inch reflector. It appears that the " dark lanes " hitherto seen as straight, and quite inexplicable, are really curved ovals, like the divisions in Saturn's ring. The photo- graph brings out a distinctly annular structure pervading the whole nebula, though not satisfactorily seen by the eye with any existing telescope, and Mr. Roberts's photographs show a similar structure in several other nebulae. Huggins has suggested that the small nebulae seen on the right and left of the large one may be planets in process of formation. Eecent photographs of Orion made with instruments of short focus and with a long exposure, show that the whole constellation is enveloped in a nebulosity, which for the most part attaches itself to the principal stars, like the nebulosity in the Pleiades (Art. 884). The well-known nebula of Orion is only the brightest portion of this inconceivably enormous mass of stellar fog. 890. Eecent photographs of the spectrum of the nebula of Orion by Huggins and Lockyer show a very considerable number of bright lines in the violet and ultra-violet; and some of the lines appear also in the spectrum of the stars of the " trapezium," showing clearly that these stars are of the same material as the surrounding nebula, only more condensed. During the summer of 1890, Keeler at the Lick Observatory observed a number of the planetary nebulae with a spectroscope of high dispersive power, and was able to detect and to measure the motion of several of them in the line of sight. The velocity of their motion appears to be of the same order as that of the stars, the nebulae observed giving results ranging from zero up to nearly forty miles a second, — some approaching and some receding. The number of bright lines observed in the spectra of gaseous nebulae has been greatly increased. The list now includes more than thirty that are accurately determined ; among them are all the Hydrogen lines, from C to the extremity of the ultra-violet series ) the D s line of "Helium," and a problematical line, at A 4472, which seems to be identical with a bright line always present in the spectrum of the chromosphere. APPENDIX. oJ*, Phi. Z, £, Zeta. n,t, Xi. x ' X' Chi. H, rj, Eta. O, o, Omicron. *>*. Psi. ©, 6 0, Theta. II, IT US, Pi. fi, 0), LS. Omega. MISCELLANEOUS SYMBO: 6 , Conjunction. A.R., or a, Right Ascension. D , Quadrature. Decl., or 8, Declination. 8 , Opposition. A, Longitude (Celestial). SI , Ascending Node. j8, Latitude (Celestial) . 15 1 Descending Node. , Latitude (Terrestrial) . a>, Angle between line of nodes and line of apsides. Al obliquity of the eclipt: ic. DIMENSIONS OF THE TERRESTRIAL SPHEROID. (According to Clarke's Spheroid of 1878. Tor the spheroid of 1866, see Art. 145.) Equatorial semidiameter, — 20 926 202 feet = 3963.296 miles = 6 378 190 metres. Polar semidiameter, — 20 854 895 feet = 3949.790 miles = 6 356 456 metres. Ellipticity, or Polar Compression, „ *„ • Length (in metres) of 1° of meridian in lat. 4> = 111 132.09 — 556.05 cos 2<£ + 1.20 cos4<£. Length (in metres) of 1° of parallel, in lat. <£ = 111415.10cos — 94.54 cos 3 . 528 APPENDIX. 1° of lat. at pole = 111 699.3 metres = 69.407 miles. 1° of lat. at equator = 110 567.2 metres = 68.704 miles. These formulae correspond to the Clarke Spheroid of 1866, used by the U.S. Coast and Geodetic Survey. TIME CONSTANTS. The sidereal day = 23 h 56 m 4 8 .090 of mean solar time. The mean solar day = 24 h 3 m 56 s .556 of sidereal time. To reduce a time-interval expressed in units of mean solar time to units of sidereal time, multiply by 1.00273791 ; Log. of 0.00273791 = [7.4374191]. To reduce a time-interval expressed in units of sidereal time to units of mean solar time, multiply by 0.99726957 = (1 — 0.00273043) ; Log. 0.00273043= [7.4362316]. Tropical year (Leverricr, reduced to 1900), 365 d 5 h 48 m 45 s .51. Sidereal year " " " 365 6 9 8.97. Anomalistic year " " " 365 6 13 48.09.- Mean synodical month (new moon to new), 29 d 12 h 44 m 2 s . 684. Sidereal month, . . . . . . 27 7 43 11.545. Tropical month (equinox to equinox), .27 7 43 4.68. Anomalistic month (perigee to perigee), . 27 13 18 37.44. Nodical monthfnode to node) , . . . 27 5 5 55.81. ff ' - I _ Obliquity of the ecliptio(Leverrier), 23°,27' 08".0 — 0".4757(f — 1900), Constant of precession (Struve), 50".264 -f 0". 000227 (t— 1900) Constant of nutation (Peters), 9". 223. Constant of aberration (Nyren), 20".492. APPENDIX. 529 Major Ter'strial Planets. Planets. 1 Jupite Saturn Uranu Neptu O (6 Kg 1 B . a . . - 3* *< ! cr,' <6©»t- II "0 H SI aa 3 E9 MMOI-* .^ *- Cn 4- J- CO M K -t MCOCn^ II to to P" B" il-«W ft pmB ~«-o 3 to B-* fefe B 2 ~° P Wi If* "38* to OiW KS cn-^ SWg - Cjto to CO i_i CO OOOO OOOO o *- cn en cr- o 4- to to OiOOCO -* (6 ** Major Terrestrial Planets. Planets. •as I* g g 3 s O pi 2 a> pi A : P ! ^ *S©»^-fc! e a, e +0 «>: Symbol. CQ O ffi <9 d to M M o o • o>-B to oo g! E 'fe cn o )-» o 10 CD •1 tO M Mean Dist. Million of Mile CO b O CO £j in Ed to b ? ■ ^ Sider Peri nean day H ffl ffl lO M cn cn K^i h w ta o< o cn cn ifr. O CD 00 to cn sf " M k! *« fr rfi to b i-j ' M b i»i b S CO Cn rbit- ocity esper ond). OOOO O 3"M • aS' cn 4- -i en M cn -i It- o "? ? M O tO I-" O M O CO -T o to o 65 o B I -J O CO 00 -J H> O •3 o g to O O l-» O to O cn oo P 5' (-J M CO -7 h-t O O CO to 00 1-1 to fin Oi CO o cn 4- o to o CO o M CO CO CO ° 2 S-" mag, ' p CD cn #». co -* N H « h- ( O O £ o H l-t P3 H S S o o « o ^ o hH R o hH K W Ph pa © t- t* © t^ o t- © CO o 00 t» 1 o T-l CO l-H o ■* ^ CM CM UD ^ UD CO CO t~- O CO T-H © ■* CO CO *o a UD TP CM CM CM •o CO -* rH T-H CO rH o CM CM UD © ■* o c?q U5 rH o -* •HH (M rH l-H CM r-H rH T-H T-H rH IO t^ ■"* co rH 0-1 o o CO © © OS OS T-H UD rH IO rH UD O . o CO l-H IN CM CM T-H ^ rH PT3 cb (M © © CO ^H CO r~ UD CM CD © © o -h" CO r-l iO oq U3 CO TH o CO © OS >* ■* UD CO (M © © © t- -* ^n © CD O 00 UD CO CM I-H C>1 T-H CM CM CM <*-! ^ ° ^ UD T* HH CO o t^ t>- OS CO !>• 00 ^* UD IO t-H T-H ^ iO -* "* "* cm t~- © UD ^ T-H 00 T^ CO 00 o UD 00 CO uq o t-H *■# CM CM "* -tf w-c o 9 » CO © CO © © T-H 00 CS © CD CO © UD UD © HH t- ^tl o t-H o OS ■^ co l-H CO i— " CM N T-H CO r-* rH rH >» T-H i-H 1^ UD 00 © r^ t- ■* CD 00 I-H CO tH UD © r- c^ r- -* OS CO CO r-- o © t^ t~- i-H T-H cq © IO OS © CM t> UD CI UD © © UD IO CO © -* rH 1— fl tH UD UD © ~ CD CM CO (M CM T-H CM T-H CM O >> •4^> ft 0Q P. 0D P. au r4 Oh 0) 03 a 4 o O > O © © co IO '5 CO JO tH a J-* o +H CJ JO bo £ 03 Ph 3 tH .o tH IV JO ^3 as [3 as ■a 3 tH as H= tH as tH as JO as "3 O >iP3 O 3 *£H O "JH o o O O 2 tH o o o 2 o Q G « Ph o S Ph o Q fl E-i o P fl H ^ ■o lO o fN l& eq CO >a lO •a o CD d CO CS d d «3 -* CO IT- t- d c- CO i>- IT- CO d CO a iO i-H io- o io" io" d d 60 -tfl © CO CO -H cs lO Tt( i-H Tt< T« co TJH co CO ^f CD CN iC CN 3 1 fl oq o> CO -* CO GSJ © I-- o o i-H OS CO CO iO O O CO CO e as r- o r- T" CO CO CD Ir- Ir- CO d o O © © d © d d d d d d d d d d d d o 1> ■* t- CO © o o o t- OS 00 CO UD CO - 00 CO l-H CO ■* 1-1 CO U0 CO OJ ■* ^d CO ■* TH CO l-H CM d J3 CO CO »o CO It- CO l-H ^* t- U0 cs -* o CM o o o CO r- CM T-i tH rt *" l-H l-H rH l-H !N i-H i-H i-H PS ■4 LH CD o H H '53 "3 OS a o "3 u tH as tH o eq % a o o os 3 o 3 c 0) CS .2 Oh O tH O eq = o tH o ■r. 'S O o 00 O O 1 '3 tH 8 Oh O 'm as ■a as S o u T3 C < CD CO '3 O as tH o Ph S 3 O tH O e 3 tH o 3 B as 3 W ■^ a CO O a o tJ 3 £ <- o 1-1 O - CM CO ^n >o CO t^ CO OS O - oq CO T)( iO CO t- 00 OS o APPENDIX. 535 TABLE VI. — THE PRINCIPAL VARIABLE STARS. A selection from S. 0. Chandler's catalogue of 225 variables (Astronomical Journal, Sept. 1888), containing sucsh as are visible to the naked eye, have a range of variation exceeding half n magnitude, and can be seen in the United States. Name. Place, 1900. Range of Variation. Period (days). Remarks. & a S 1 TCeti . . . Oh 16m.7 -20° 37' 5.1 to 7 65 ± Irregular. 2 R AndromedsB 18.8 + 88 1 5.6 13 411.2 3 R Sculptoris . 1 22.4 -33 4 5.8 7.8 207 t Mira. Varia- 4 o Ceti . . . 2 14.3 - 3 26 1.7 9.5 331.3363 < tions in length 5 p Persei . . . 2 58.7 + 38 27 3.4 4.2 33 ( of period. 6 (3 Fersei . . . 3 1.6 + 40 34 2.3 3.5 2d 20h 48m 65S.43 \ Algol. Period I now shortening. 7 ATauri . . . 3 55.1 + 12 12 3.4 4.2 3d 22h 52m 12s ( Algol type, but 8 e Auriga . . 4 54.8 + 43 41 3 4.5 Irregular ( irregular. 9 a Orionis . . 5 49.7 + 7 23 1 1.6 196 ? Irregular. 10 7} G-eminorum . 6 8.8 + 22 32 3.2 4.2 229.1 11 £ Q-eminorum . 6 58.2 + 20 43 3.7 4.5 10d 3h 41m 30s 12 R Canis Maj. . 7 14.9 -16 12 5.9 6.7 Id 3tt 15m 55s Algol type. 13 R Leonis Min. 9 39.6 + 34 58 6 13 373.5 Period ehort'ing. 14 R Leonis . . 9 42.2 + 11 54 5.2 10 312.87 15 U Hydra? . . 10 32.6 -12 52 4.5 6.3 194.65 16 RUrsaeMaj. . 10 37.6 + 69 18 6.0 13.2 305.4 Period short'ing. 17 R Hydras . . 13 24.2 -22 46 3.6 9.7 496.91 Period short'ing. 18 S Virginis . . 13 27.8 - 6 41 5.7 12.5 376.0 19 R Bootis . . > 14 32.8 + 27 10 5.9 12.2 223.9 20 S Libras . . . 14 55.6 - 8 7 5.0 6.2 • 2d 7h 51m 22". 8 Algol type. 21 S Coronae . . 15 17.3 + 31 44 6.0 12.5 360.57 22 R Coronas . . 15 44.4 + 28 28 5.8 13 Irregular 23 R Serpentis . 15 46.1 + 15 26 5.6 13 357.6 24 aHerculis . . 17 10.1 + 14 30 3.1 3.9 Two or three mon ths, but very irreg. 25 U Ophiucbi . 17 11.5 + 1 19 6.0 6.7 20h 7m41s.6 26 X Sagittarii . 17 41.3 -27 48 4 6 7.01185 27 "W Sagittarii . 17 58.6 -29 35 5 6.5 7.59445 28 T Sagittarii . 18 15.5 -18 54 5.8 6.6 5.76900 29 RScuti . . . 18 42.1 - 5 49 4.7 9 71.10 ( Secondary mini- 30 p LyrsB . . . 18 46.4 + 33 15 3.4 4.5 12d 21h 46m 588.3 1 mum about mid- 81 RCygrd. . . 19 34.1 + 49 58 5.9 13 425.7 ( way. 32 X Cygni . . . 19 46.7 + 32 40 4.0 13.5 406.045 Period length'ing 33 Tt Aquilse . . 19 47.4 + 45 3.5 4.7 7d 4h 14m Os.O 34 35 S Sagi . ttse . . X Cygni . . . 19 20 51.4 39.5 + 16 + 35 22 13 5.6 6.4 6.4 7.7 8d 9h 11m 15d 14h 24m I Minimum not I constant. 36 T Vulpeculse . 20 47.2 + 27 52 5.6 6.5 4d 10h 29m 37 TCephei. . . 21 8.2 + 68 5 6.6 9.9 383.20 38 fj. Cephei . . 21 40.4 + 58 19 4 5 432? 39 $ Cephei . . 22 25.4 + 57 54 3.7 4.9 5d 8h 47m 398.97 40 Fegasi . . . 22 58.9 + 27 32 2.2 2.7 Irregular 41 R Aquarii . . 23 38.6 -15 50 5.8 11 387.16 42 R Cassiopeia . 23 53.3 + 50 50 4.8 12 429.00 536 APPENDIX. HiOOCOO©0«>OH!Ot*(»00:X(NONNHHHHC3 CT t-( i— I i— IHH t— ' T-H r-H i— I (N r-< CI I I I I I I I I I I I I + I++I I I I I I +++ eS c3 c3 c3 e3 h- 1 hH h- 1 ei (3 l— l h- l e3 e3 c3i— t^l— lj— l^ IceeS (Si— ii— ih- ( eSesi— ii— i is « cs i— t *" h- 1 1— i « IHh 1MI— II— IhHI— II— I M h- II— IHHHHMH I l-H OC?OMOWO HO OOOOMMO CO "<# CO CO ^ © CqciHtNc4iHiCTC>ic^Cq ^ "3 '3 8 8 "Ei 8 88 '-S "1 88 S § p, | J ? a 6, 8i ^ OOaigffmaiOOcoK^Ert'-St'S-gPSPbBilOSO ■D i :a o (v > ZT >>>> to a> a> iJlHlt>P>^t=PPpqtii-10aiW0i-l<500(iiP-ipH O CO w a J3 OOCqNMaHt»nOiOH«HlOrHiOCONCO>OH^Tl -O 0) CO 01 5 $ M $ 3 £ 5 J3 js $ r 5 5 'S "S 3 - c - a -g -s 3 g.a ~« » INDEX* [AH references, unless expressly stated to the contrary, are to articles and not to pages.] Aberration of light, annual, 99, 224-226; used to determine the solar parallax, 692; diurnal, 226*; spherical and chro- matic, 39. Aboul Wefa, discoverer of the lunar variation, 457. Absolute scale of stellar magnitude, 819. Acceleration of Encke's comet, 710; of Winnecke's comet, 711; of the sun's equator, 283-285; secular, of moon's meau motion, 459-461; secular, of moon's mean motion as affected by meteors, 778. Achromatic object-glasses for telescopes, 41. Actinometer of Violle, 341. Adams, J. C, the discovery of Neptune, 654; investigation of the orbit of the Leonids, 785. Adjustments of the transit instrument, 60. Aerolites. See Meteorites. Age, relative, of the planets, 913,915; of the solar system, 922 ; of the sun, 359. Air-currents at high elevations, 773, note. Aikt, G. B., density of the earth, 169. Albedo defined and determined, 546; of Jupiter, 614; of Mars, 583; of Mercury, 558; of the Moon, 259; of Neptune, 660; of Saturn, 636; of Uranus, 648; of Venus, 572. Algol, or Persei, 848. Almagest of Ptolemy, 500, 700, 795. Almucantar defined, 12. Altitude defined, 21; parallels of, 12; of pole equals latitude, 30; of sun, how measured with sextant, 77. Altitude and azimuth instrument, 71. Amplitude defined, 22. Andromeda, the nebula in, 886; tempo- rary star in the nebula of, 845. Andromedes, the, 780, 784, 786. Angle, position, of a double star, 868; of the vertical, 156. Angular and linear dimensions, 5 ; veloc- ity under central force, its law, 408, 409. Annual equation of the moon's motion, 458; motion of the sun, 172, 173. Annular eclipse, 382; nebula in Lyra, 888. Anomalistic month, the, 397, note ; revo- lution of the moon, 250; year defined, 216. Anomaly defined, mean and true, 189. Apertures, limiting, in photometry, 825. Apex of the sun's way, 805. Apparition, perpetual, circle of, 33. Apsides, line of, defined, 183; its revolu- tion in case of the earth's orbit, 199 ; its revolution in case of the moon's orbit, 454 ; its revolution in case of the planets' orbits, 527. Arc of meridian, how measured, 147. Areal or areolar velocity, law of, under central force, 402-406. Areas, equable description of, in earth's orbit, 186, 187. Argelander, his Durchmusterung and zones, 795, 833 ; his star magnitudes, 817, 833. Argus, v, 841. Ariel, the inner satellite of Uranus, 650. Aries, first of, 17. Aristarchus, method of determining the sun's distance, 666, 670. Artificial horizon, the, 78. Ashes of meteors, 775. Aspects of planets defined by diagram, 494. Asteroids, the, or minor planets, 592-601 , theories as to their origin, 600. Astrsea, the fifth asteroid, discovered by Hencke, 693. 538 INDEX. | All references, unless expressly stated to tlje contrary, are to articles, and not to pages.] Astro-Physios defined, 2. Atlases of the stars, 793. Atmosphere of tho moon, 255-257; of Venus, 573; height of the earth's, 98. Attraction, intensity of the solar, on the earth, 436; within a hollow sphere, 169 ; of universal gravitation, 161, 162. Axis of the earth, its direction, 14 ; of the earth, disturbed by precession, 206 ; of the sun, its direction, 282. Azimuth defined, 22 ; determination of, 127 ; method of reckoning, 22 ; of tran- sit instrument, its adjustment, 60. B. Baily, determination of the density of the earth, 166. Balance, common, used in determining the density of the earth, 170; torsion, used in determining the density of the earth, 165. Barometer, changes of, affecting atmos- pheric refraction, 91 ; effect on height of the tides, 480. Barometric error of a clock and its com- pensation, 52. Beginning of the day, 123; of the year, 222. Benzenberg, experiments on the devia- tion of falling bodies, 138. Bessel, the parallax of 61 Cygni, 809, 811 ; formation of comets' tails, 728; his "zones," 795. Biela's comet, 744-746. Bielids, the, 746, 780, 784, 786. Bielid meteorite, Mazapil, 784. Binary stars, 872-875 ; number known at present, 872; their masses, 877, 878; their "mass-brightness," 879; their orbits, 873-877. Bissextile year, explanation of term, 219. Black Drop, the, at a transit of Venus, 681. Bode's law, 488, 489. Bolides, or detonating meteors, 768. Bolometer, the, of Langley, 343. Bond, S. P., first photograph of a double star, 868. \ Bond, W. C, discovery of Hyperion, 643; of Saturn's dusky ring, 637. Boyle, law of, 360, note. Brahe, Tycho. See Tycho. Bredichin, his theory of comets' tails, 731, 732. Brightness of comets, 699 ; of planets in various positions, Mercury, 551 ; Ve- nus, 563, 568; Mars, 579; Asteroids, 596, 699; Jupiter, 610; Saturn, 632; Uranus, 647; Neptune, 660; of an ob- ject in the telescope, 38; of shooting stars, 773 ; of stars, causes of the dif- ference in this respect, 836; of stars its measurement, 823-831. C. Calendar, the, 217-223. Callisto, the outer satellite of Jupiter, 621, 627. Calories of different magnitude, 338, noie. Candle power, its mechanical equivalent, 776; power of sunlight, 332, 333. Candle standard, 333, note. Capture theory of comets, 740. Cardinal points defined, 20. Carlini, earth's density, 168. Carrington, law of the sun's rotation, 283, 284. Cassegrainian telescope, 48. Cassini, J. D., discovery of the division in Saturn's ring, 637; discovery of four satellites of Saturn, 643. Catalogues of stars, 795. Cavendish, the torsion balance, 165. Celestial latitude and longitude, 178, 179; sphere, conceptions of it, 4. Cenis, Mt., determination of the earth's density, 168. Central force, motion under it, 400-410; force, its measure in case of circular motion, 411. Central suns, 807, 903. Centrifugal force of the earth's rotation, 154. Ceres, discovery of, 592. Chandler, S. C, catalogue of variable stars, 852, Appendix, Table VI. Changes on the moon's surface, 268; in the nebulae, 892. Characteristics of different meteoric swarms, 783. Charts of the stars, 798. Chemical elements recognized in comets, 724,725; elements recognized in stars, 856; elements recognized in the sun, 315-317. Chromatic aberration of a lens, 39. Chromosphere, the, 291, 322, 363. Chronograph, the, 56. Chronometer, the, 54 ; longitude by, 121 [A]. Circle, the meridian, 63. Circles of perpetual apparition and oc- cultation, 33. Circular motion, central force in, 411. Clairaut's equation concerning the ellipticity of the earth, 155. INDEX. 539 [All references, unless expressly stated to the contrary, are to arUelea, and not to pages.] Clarke, Col., dimensions of the earth, 145 and Appendix. v Classification of stellar spectra, 857, 858. Clerke, Miss A. M., her history of as- tronomy, Preface, 570, 626, note, 723, 746, 900. Clocks, general remarks on, 50. Clock-breaks (electric), 57. Clock-error, or correction, and rate, 53; or correction determined by transit instrument, 59. Clusters of stars, 883-885. Coggia's Comet, 730. Collimating eye-piece, 67. Collimation of transit instrument, 60. Collimator, the, used with transit instru- ment, 60; of a spectroscope, 311. Collision theory of variable stars, 850. Colors of stars in photometry, 830; of double stars, 867. Colures defined, 25. Comet, Biela's, 744; Donati's, 727,(730, 747; Encke's, 710, 743; great, of 1882, 748-752; bailey's, 742; Winnecke's, 711. Comets, acceleration of Encke's and "Win- necke's, 710, 711; brightness of, 699, 723; capture theory of, 740; chemical elements in, 724, 725 ; constituent parts, 713; contraction of head when near the sun, 715 ; danger from, 753, 754 ; den- sity of, 720; designation of, 697; di- mensions of, 714, 717 ; ejection theory, 741 ; fall upon earth or sun, probable effect, 751; groups of, with similar orbits, 705; their light, 721; their masses, 718, 719; and meteors, their connection, 785-787; nature of, 737; their orbits, 700-709; origin of, 738- 7*1; perihelia, distribution of, 706; physical characteristics, 712; plane- tary families of, 739; their spectra, 724, 726 ; superstitions regarding them, 695 ; their tails or trains, 713, 717, 728- 736 ; variations in brightness, 723 ; vis- itors in the solar system, 709. Comparison of starlight with sunlight, 334, 832. Compensation pendulums, 51. Compensation of pendulum for barome- tric changes, 52. Components of the disturbing force, 445. Co-ordinates, astronomical, 20. Common, A. A., photographs of nebulae, 893. Conies, the, 422, 423. Connection between comets and meteors, 785-787. Constant of aberration, the, 225; the so- lar, 338-340. Constancy, secular, of the mean distances and periods of the planets, 526. Constellations, list of, 792; their origin, 791. Contact observations, transit of Venus, 679-682. Contraction theory of solar heat, 35S. Conversion of E. A. and Decl. to latitude and longitude, 180. Copernicus, his system, 503; "Trium- phans," 809. Corntj, determination of the earth's den- sity, 166 ; photometric observation of eclipses of Jupiter's satellites, 630. Corona, the solar, 291, 327-331, 364. Cosmogony, 905-917. Cotidal lines, 475. Craters on the moon, 265-267. Crew, H., spectroscopic observations of the sun's rotation, 285, note. Crust of meteorites, 761. Curvature of comet's tails, 729. Curvilinear motion the effect of force, 401. Cycle, the metonic, 218. Cyclones as proofs of the earth's rotation, 143. D. DALTON.his law of gaseous mixtures, 360, note. Danger from comets, 753, 754. Darkening of the sun's limb. 337. Darwin, G. H. . rigidity of the earth, 171 ; tidal evolution, 484, 916. Dawes, diameter of the spurious discs of stars, 43 ; nucleoli in sun spots, 293. Day, the civil and the astronomical, 117 ; effect of tidal friction upon its length, 461 ; changes in its length, 144 ; where it begins, 123. Declination defined, 23 ; parallels of, 23 ; determined with the meridian circle, 128. Degree of the meridian, how measured, 135, 147. Deimos, the outer satellite of Mars, 590, 591. Delisle, method of determining the so- lar parallax, 682. Denning, drawings of Jupiter's red spot, 618. Density of comets, 720; of the earth, de- terminations of it, 164-170; of the moon, 246; of a planet, how deter- mined, 540; of the sun, 279. Detonating meteors, or " Bolides," 768. 540 INDEX. [All reference!, unless expressly stated to the contrary, are to articles, cud not to pages.] Development of sun spots, 297. Dhurmsala meteorite, ice-coated, 765. Diameter (apparent) as related to dis- tance, 6 ; of a planet, how measured, 534. Differential method of determining a body's place, 129; method of deter- mining stellar parallax, 811. Diffraction of an object-glass, 43. Dione, fourth satellite of Saturn, 643, note. Dip of the horizon, 81. Disc, spurious, of stars in telescope, 43. Discovery of comets, 698. Dissipation of energy, 925. Distance of the moon, 239 ; of the nebu- lae, 896; and parallax, relation be- tween, 84; of a planet in astronomi- cal units, how determined, 515-518 ; of the stars, 808-815; of the sun, 274, 275, also Chap. XVI. Distinctness of telescopic image, its con- ditions, 39. Distribution of the nebulae, 895; of the stars, 899 ; of the sun spots, 301. Disturbing force, the, 439-444; force, diagram of, 441 ; force, its resolution into components, 445. Diurnal aberration, 226*; inequality of the tides, 471 ; parallax, 82, 86 ; phe- nomena in various latitudes, 191. Divisions of astronomy, 2. Doeefel proves that a comet moves in a parabola, 700. Donati's comet, 727, 730, 747. Doppler's principle, 321, note. Double stars, 866-879 ; their colors, 867 ; criterion for distinguishing between those optically and physically double, 870 ; method of measuring them, 868 ; optically and physically double, 869; having orbital motion, see Binary Stars. Draper, H., oxygen in the sun, 316 ; pho- tograph of the nebula in Orion, 893 ; photography of stellar spectra, 859; memorial, the, 859. Duration, future, of the sun, 358; of sun spots, 300. Earth, the, her annual motion proved by aberration and stellar parallax, 174 ; approximate dimensions, how meas- ured, 134, 135 ; constitution of its in- terior, 171; its dimensions, Appendix and 145 ; its dimensions determined ge- odetically, 147-149 ; form of, from pen- dulum experiments, 152-155; growth of, by accession of meteoric matter, 777 j mass compared with that of the sun, 278 ; its mass and density, 159- 170; its orbit, form of, determined, 182 ; principal facts relating to it, 132 ; proofs of its rotation, 138-143. Earth-shine on the moon, 254. Eccentricity of the earth's orbit, discov- ered by Hipparchus, 184 ; of the earth's orbit, how determined, 185; of the earth's orbit, secular change of, 198 ; of an ellipse defined, 183, 506. Eclipses, dnration of lunar, 373; dura- tion of solar, 385 ; number in a year, 391-393 ; recurrence of, the saros, 395 ; of Jupiter's satellites, 627-630; of the moon, 370-378; of the sun, 379-390; total, of the sun, as showing the solar atmosphere and corona, 319, 323. Ecliptic, the, defined, 175 ; obliquity of, 176; limits, lunar, 374, 375; limits, so- lar, 386. Effective temperature of the sun, 351. Ejection theory of comets and meteors, 741. Elbowed equatorial, the, 74. Electrical registration of observatious,56. Electro-dynamic theory of gravitation, 602. Elements, chemical, not truly elemen- tary, 318; chemical, recognized in comets, 724, 725 ; chemical, recognized in stars, 856 ; chemical, recognized in sun, 316, 317; of a planet's orbit, SOS- SOS. Elkin, stellar parallaxes, 808, 814, 815, and Appendix, Table IV. Ellipse defined, 183 ; described as a conic, 422, 423. Elliptic comets, their number, 702 ; their orbits, 703 ; recognition of, 704. Ellipticity or oblateness of a planet de- fined, 150. Elongation of moon or planet defined, 230. Enceladus, the second satellite of Sa'.urn, 643, note. Encke's comet, 710, 743. Encke, his reduction of the transits of Venus, 667. Energy, the dissipation of, 925 ; and work of solar radiation, 345. Enlargement, apparent, of bodies near horizon, 4, note, 88, 93. Envelopes in the head of a comet, 713, 727. Epoch of a planet's orbit defined, 508. Epsilon Lyrae, 653, 866, 882. Equal altitudes, determination of time, 115. INDEX. 541 [All references, unless expressly stated to the Equation, annual, of moon's motion, 458 ; of ftie centre, 189; of the equinoxes, 213; of light, by means of Jupiter's satellites, 628-630; of time explained, 201-204; expressing the relation be- tween the light of different stellar magnitudes, 820. Equator, the celestial, 16. Equatorial acceleration of the sun's ro- tation, 283-285 ; coude', Paris, 74; par- allax, 85 ; telescope, 72 ; telescope used to determine the place of a heavenly body, 12! I. Equinoctial, the, see Equator, celestial; points, or equinoxes, 17. Equinoxes, the, equation of, 213; preces- sion of, 205-212. Eratosthenes, his measure of the earth, 136. Erecting eye-piece for telescope, 45. Ekicsson, his solar engine, 345 ; experi- ment upon radiation of molten iron, 350. Eruptive prominences, 325. Escapement of clock, 50. Establishment of a port (harbor) de- fined, 463. Europa, the second satellite of Jupiter, 021 . Evection. the, 456. Evolution, tidal, 484, 916. Eye-pieces, telescopic, 44. Extinctions, the method of, in photome- try, 825. K. Faculae, solar, 292. Fall of a planet to the sun, time required, 413, 3 ; of a comet on the sun, proba- ble effect, 754. Falling bodies, eastward deviation, 138. Families (planetary) of comets, 739. Fate, H. A., his modification of the neb- ular hypothesis, 915; theory of sun spots, 304. Flattening, apparent, of the celestial sphere, 4, note. Ferce, evidenced not by motion, but by change of motion, 400; projectile, term carelessly used, 401 ; central, mo- tion under it, 400-410 ; repulsive, action on comets, 728-733. Form of the earth, 145-155. Formation of comets' tails, 728. Foucaui/t, the gyroscope, showing earth's rotation, 142; his pendulum experi- ment, showing earth's rotation, 139- 141 ; measures velocity of light, 690. Fourteen hundred and seventy-four line of the spectrum of the corona, 329. contrary, are to articles, and not to pages. \ Fkaunhofkh lines in the solar spectrum, 315, 855 ; observations on stellar spec- tra, 855. Free wa.ve, velocity of, 473. Frequency, relative, of solar and lunar eclipses, 394. Galaxy, the, 898. Galileo, discovery of Jupiter's sat ellites, 621; discovery of Saturn's rings, 637; discovery of phases of Venus, 567 ; use of pendulum in time- keeping, 50. Galle, optical discovery of Neptune, 654. Ganymede, the third satellite of- Jupiter, 621. Gas contracting by loss of heat, Lane's law, 357. Gauss, computes the orbit of Ceres, 592 ; determination of the elements of an orbit, 519 ; peculiar form of achroma- tic object-glass, 41. Gay Lussac, law of gaseous expansion, 360, note. Geocentric latitude, 156; place of. a heavenly body, 511. Geodetic determination of the earth's dimensions, 147, 149. Genesis of the solar system, 908-915 ; of star clusters and nebulae, 924. Georgium Sidus, the original name for Uranus, 645. Gill, solar parallax from observations of Mars, 676; stellar parallaxes, 808. Appendix, Table IV. Globe, celestial, rectification of, 33, note. Gnomon, determination of latitude with it, 107 ; determination of the obliquity of the ecliptic, 176. Golden number, the, 218. Gradual changes in the light of the stars, 839. Graduation errors of a circle, 69. Grating diffraction, 311, note. Gravitation, electro-dynamic, theory of, 602; law stated, 161; nature unknown, 161 ; law extending to the stars, 872, note, 873, 901, note ; Newton's verifi- cation of the law by means of the moon's motion, 419, 420. Gravitational astronomy defined, 2; methods of determining the solar par- allax, 687-689. Gravity, increase of, below the earth's surface, 169; variation of, between equator and pole, 152. 542 INDEX. [All references, unless expressly stated to the contrary, are to articles, and not to pages.] Gregorian calendar, the, and its adop- tion in England, 220, 221 ; telescope, 48. Groups, cometary, 705 ; of stars having common motion, 803. Growth of the earth by meteoric matter, 777. Gyroscope, Foucault's proof of earth's rotation, 142 ; illustrating the preces- sion of the equinoxes, 210, 211. H. Hall, A., discovery of the satellites of Mars, 590 ; on the question whether it is certain that gravitation extends through the stellar universe, 901, note. Hallby, his comet, 742 ; his computation of cofnetary orbits, 700; his method of determining the sun's parallax, 679, 680; the moon's secular acceleration, 459 ; proper motions of stars, 800. Hansen, correction of the solar parallax, 667; opinion on the form of the moon, 258. Harding discovers Juno, 5!)3. Harkness, observations on the light of meteors, 776 ; observation of the cor- ona spectrum, 329. Harmonic law, Kepler's, 412-417. Harton colliery, density of the earth, 169. Harvard photometry, the, 827, 828. Harvest and hunter's moons explained, 237. Heat and light of meteors explained, 765 ; of the moon, 260 ; of the suu, 338-358 ; received by the earth from meteors, 355, 779 ; from the stars, 834. Height of lunar mountains, 270. Heis, enumeration of naked-eye stars, 818. Heliocentric place of a planet, 512. Heliometer, the, 677 ; used in determin- ing solar parallax, 676, 683 ; used in determining stellar parallax, 811 , 815. Helioscopes, or solar eye-pieces, 286, 287. Helium, an unidentified metal in the so- lar chromosphere, 323. Helmholtz, contraction theory of solar heat, 356. Henckk, discovers Astroea, the fifth aste- roid, 593. Henderson, measures the parallax of a Centauri, 809, 810. Henry Brothers, astronomical pho- tography, 798. Henry, Prof. J., heat of sun spots, 310 ; at sun's limb, 348. Herschel, Sir John, astrometry, 819; illustration of the planetary system, Herschel, SraW , discovery of the sun's motion in space, 804 ; discovery>of two satellites of Saturn, 643 ; discovery of Uranus, 645; discovery of two satel- lites of Uranus, 650 ; star-gauges, 899 ; theory of sun spots, 302; his reflect- ing telescope, 48. Hevelius, his view of cometary orbits, 700. Hipparchus, discovers eccentricity of earth's orbit, 184 ; discovers precession, 205; his value of the solar parallax, 671 ; the first star-catalogue, 795. Holden, E. S., on changes in nebulae, 892. Horizon, apparent enlargement of bodies near it, 4, note, 88, 93 ; artificial, 78 ; rational and apparent defined, 10; dip of, 81 ; visible, defined, 11. Horizontal parallax, 83, 84 ; point of the meridian circle, 67. Hour-angle defined, 24. Hour-circle defined, 18. Huggins, W., attempts to photograph the solar corona without an eclipse, 328; attempted observation of stellar heat, 834; observations of stellar spectra, 856; photography of stellar spectra, 859; spectroscopic observations of T coronas, 844; star-motions in line of sight, 802. Humboldt, A. von, classification of the planets, 549. Hunt, Sterry, carbonic acid brought to earth by comets, 735. Huyghens, discovery of Saturn's rings, 637; discovery of Saturn's satellite, Titan, 643 ; invention of the pendulum clock, 50; his long telescope, 40. Hydrogen in the solar chromosphere and prominences, 323-325 ; bright lines of its spectrum in the nebulae, 890; bright lines of its spectrum in temporary stars, 844 ; bright lines of its spectrum in variable stars, 857. /" Hyperbola, the, described as a conic, 422. Hyperbolic comets, 702. Hyperion, the seventh and last discovered satellite of Saturn, 643, 644. Hypothesis, nebular. See Nebular hy- pothesis. Iapetus, the outermost satellite of Sat- urn, 643. Ibn Jounis, use of pendulum in observa- tion, 50. DTDEX. 543 [All references, unless expressly stated to the contrary, are to articles, and not to pages.] Ice, amount melted by solar radiation, 344-346, 364*. Illumination of the moon's disc during a lunar eclipse, 376. Image, telescopic, conditions of distinct- ness, 39. Inequality, diurnal, of the tides, 471. Inferences deducible from Kepler's laws, 418. Inferior planet, motion of, 497. Infinity velocity from, 429. Influences of the moon on the earth, 262. Intra-Mercurial planets, 602-606; plan- ets, supposed observations of, during solar eclipse, 605. Interior of the earth, its constitution, 171. Invariable plane of the solar system , 531. Invariability of the earth's rotation, 144. Io, the first satellite of Jupiter, 621. Iron meteorites, 758 ; in the sun, 315. Irradiation in micrometric measures, 256,534. Janssen, discovery of the spectroscopic method of observing the solar promi- nences, 323; solar photography, 289. Jets issuing from the nucleus of a comet, 713, 727. Jolly, observations of the earth's den- sity, 170. Julian calendar, 219. Juno, discovered by Harding, 593. Jupiter, the planet, 609-631; brightness as seen from a Centauri, 881 ; a semi- sun, 619. K. Kant, proposes the nebular hypothesis, 908. Keplek, his belief as to cometary orbits, ' 700; his three laws of planetary mo- tion, 412-418; his "problem," 188; his " regular solid " theory of the planet- ary distances, 592, note. Kirchhoff, his fundamental principles of spectrum analysis, 314. Langley, S. P., his Bolometer, 343; the color of the sun, 337 ; observations on lunar heat, 260, 261 ; on solar heat, 348 ; sun-spot drawings, 292; heat in sun spots, 293. Lane's law, rise of temperature conse- quent on the contraction of a gaseous mass, 357. La Place, his equations relating to the eccentricities and inclinations of the planetary orbits, 532; explanation of the moon's secular acceleration, 459, 460; the invariable plane of the solar system, 531; the nebular hypothesis, 901-911. LASSELL,'discovery of the^two inner sat- ellites of Uranus, 650; independent discovery of Hyperion, 643. Latitude (astronomical) of a place on the earth's surface, 30, 100, 156 ; astronom- ical, geodetic, and geocentric, dis- tinguished, 156; determination of, methods used, 101-107; at sea, its de- termination, 103; possible variations of it, 108 ; station errors, 158 ; celes- tial, defined, 178, 179; and longitude (celestial), conversion into o and 5, 180. Law of angular velocity under central force, 408; Bode's, 488,489; of Boyle or Mariotte, density of a gas, 360, note ; of Dalton, mixture of gases, 360, note ; of earth's orbital motion, 186, 187; of equal areas, 186, 402-406, 412 ; of Gay Lussac, gaseous expansion, 360, note ; of gravitation, 161, 162, 419, 872; Lane's, of temperature in gaseous con- traction, 357; of linear velocity in angular motion, 407. Laws of Kepler, 412-418 ; motion under a central force, 400-411. Leap year, rule for, 220. Length, of the day, possible changes in it, 144 ; of the year, its invariability, 526, 778. Leonids, the, 780, 786. Lesoaebault, supposed discovery of Vulcan, 603. Level adjustment of the transit instru- ment, 60. Leverrier, discovery of Neptune, 653, 654 ; on an intra-Mercurial planet, 603 ; motion of the perihelion of Mercury's orbit, 602 ; method of determining the solar parallax by planetary perturba- tions, 689. Lexell's comet, approach to Jupiter, 718; recognition of Uranus as a planet, 645. Librations of the moon, 249, 250, 251. Light of comets, 722; of the moon, 259; emitted by certain stars, 835 ; received by the earth from certain stars, 832; of the sun, 332-337 ; total, of the stars, 833; equation of, from Jupiter's satel- lites, 628-630; mechanical equivalent 544 INDEX. | All references, unless expressly stated to the contrary, are to articles, and not to pages.] of, Thomsen, 776; time occupied by, in coming from the sun, 275, 629; ve- locity of, 225, note, 668, 690. Light-curves of variable stars, 848. Light-gathering power of telescopes, 38. Light-ratio, the, in scale of star-magni- tudes, 819. Light-year, the, the unit of stellar dis- tance, 814. Limb of the sun, darkening of, 337 ; of the sun, diminution of heat, 348. Limiting apertures in stellar photometry, 825. Linear and angular dimensions, their re- lation, 5 ; velocity under central force, its law, 407, 409. Linne, lunar crater supposed to have changed, 269. Listing, dimensions of the earth, 145. Local and standard time, 122. Lockyer, J. X., discovery of the spec- troscopic method of observing the solar prominences, 323; his "collision theory" of variable stars, 850; views as to the compound nature of the so- called chemical "elements," 318; origin of the Fraunhofer lines, 320; theory of sun spots, 306; meteoric theory of nebula;, 894; meteoric hy- pothesis, 926. Loewy, peculiar method of determining the refraction, 95. Longitude, arcs of, to determine the earth's dimensions, 151 ; (terrestrial), determination of, 118-121 ; (celestial) , 178-180; of perihelion, 505, 506. Luminosity of bodies at low tempera- tures, 737, note. lunar distances, 120, B; eclipses, 370- 378; influences on the earth, 262; methods of determining the longitude, 120 ; perturbations, 448^61; perturba- tions used to determine the solar par- allax, G87. Lyra, u, see Vega; 0, variable star, 847; e, quadruple star, 653, 866, 882. Madler, speculations as to a central sun, 807, 903. Magnifying power of a telescope, 37; power, highest available, 43. Magnitudes of stars, 816-822. Magnitude of smallest star visible in a given telescope, 822. Magnesium in the nebulse, 890, 894. Maintenance of the solar heat, 353-356. Mars, the planet, 578-591; observed fot solar parallax, 673-677. Maskelyne, his mountain method of determining the earth's density, 164. Mass and weight, distinction between them, 159, 160; of comets, 718, 719; of the earth compared with the sun, 278; of the earth in terms of the sun as determining the solar parallax, 689; of the moon, its determination, 243 ; of a planet, how determined, 536- - 539; of the sun, compared with the earth, 278; probable, of shooting stars, 776. Mass-brightness of binary stars, 879. Masses of binary stars, 877, 878. Mayer, R., meteoric theory of the solar heat, 353. Maxwell, Clerk, meteoric theory of Saturn's rings, 641. Mazapil, the meteorite of, 784. Mechanical equivalent of light, 776. Mercury, the planet, 551^562. Mercury's orbit, motion of its perihelion, 602. Meridian, the celestial, defined, 19; arc of, how measured, 147. Meridian-circle, the, 63; used to deter- mine the place of a heavenly body, 128. Meridian photometer, the, 828. Meteors and shooting stars, 755-787; ashes of, 775; and comets, their connection, 785-787; daily number of, 771 ; detonat- ing, 768; effect on the earth's orbital motion, 778; effect upon the moon's motion, 778; effect upon the trans- parency of space, 779; explanation of their light and heat, 765; heat from them, 355, 779; magnitude of, 762; method of observing them, 764; their trains, 766. Meteoric growth of the earth, 777; show- ers, 780-786; shower of the Bielids, 1872, 1886, 746; shower of the Leonids, 1833, 1866-67, 781; swarms and rings, 783; swarms, special characteristics, 783; theory of Saturn's rings, 641; theory of the sun's heat, 353-355; theory of sun spots, 306. Meteorites, or uranoliths, or aerolites, 755-769; chemical elements in them, 760; crust, 761; fall of, 756; which have fallen in the United States, 759; iron, list of, 758; number of, 769; ques- tion of their origin, 767; their paths, 763. Metonic cycle, the, 218. INDEX. 545 [All references, unless expressly stated to the contrary, are to articles, and not to pages.] Michelson, determination of the veloc- ity of light, 225, note, GU8, 690. Micrometer, the filar, 73, 534, 867. Microscope, the reading, 64. Midnight sun, the, 191. Milky Way, or galaxy, 898. Mimas, the innermost satellite of Saturn, 643 and note. Minor planets, or asteroids, 592-601. Mira, Omicron Ceti, 846. Missing stars, 840. Mohammedan calendar, 217. Monck, the mass-brightness of binary stars, 879. Monocentric. eye-piece for telescope, 45. Moon, the, 227-272; her atmosphere, 255- 258; the, regarded as a clock, 120; distance of, etc., 240; her heat and temperature, 260, 261; her light as compared with sunlight, 259; influ- ences on the earth, 262; mass of, de- termined, 243, 244; her motion (ap- parent), 228; her motion relative to the sun, 241 ; her mountains, measure- ment of their elevation, 270; her orbit with reference to the earth, 238; her parallax determined, 239; perturba- tions, 448-461 ; her rotation and libra- tions, 248-251; her phases, 253; her surface character, 263-270; culmina- tions for longitude, 120, A. Month, the anomalistic, 397, note ; the nodical, 397, note ; the sidereal, 229, 232; the synodic, 229, 232; length of, increased by perturbation, 453; slight- ly shortened by the secular accelera- tion, 459. Motion, direct and retrograde, of the planets, 494; of the solar system in space, 804-807 ; in line of sight, effect on spectrum, 321; of stars in line of sight, spectroscopically observed, 802. Motions, proper, of the stars, 800-803. Mountains, lunar, their height, 270. Mountain method of determining the earth's density, 164. Multiple stars, 882. Mural circle, the, 70. N. Nadir, the, defined, 9; point of meridian circle, 67. Names of the constellations, 792 ; of Jup- iter's satellites, 621; of satellites of Mars, 590; of the planets, 487, 489; of Saturn's satellites, 643, note ; of the satellites of Uranus, 650; of stars, 794. Neap tide defined, 463. Nebula, the great, in Andromeda, 886; the annular, in Lyra, 888; of Orion, the, 886, 892, 893. Nebulse, the, 886-897; changes in, 892; their distance, 896; Lockyer'smeteorie theory, 894; their nature, 894; their number and distribution, 895; photo- graphs of, 893; planetary, 888; spiral, 888; their spectra, and chemical ele-, ments in them, S90, 891. Nebular hypothesis, the, 908-915; modi- fications of the original theory, 9J2f 913. Negative eye-pieces for the telescope, 44; shadow of the moon, 381 ; star magni- tudes, 821. Neptune, the planet, 653-661 ; anomalous retrograde rotation, in relation to the nebular hypothesis, 914; appearance of sun and solar system from it, 658; (actual) discovery by Galle, 654; theo- retical discovery by Leverrier and Adams, 653, 654; its discovery "no accident, ' ' 655 ; the computed elements erroneous, 655; its satellite, 661; spec- trum of, 660. Newcomb, S., conclusions as to sun's age and duration, 358, 359 ; observations on meteors, 776; on the moon's secular acceleration, 461 ; on the structure of the heavens, 900; his value of the so- lar parallax, 667; velocity of light, 225, note, 668, 690. Newton, Prof. H. A., daily number of meteors, 771 ; investigation of meteoric orbits, 767, 785; theory of the consti- tution of a comet, 737. Newton, Sir Isaac, discovery of gravi- tation, 161, 419; verification of the idea of gravitation by means of the moon's motion, 419, 420; discovery that planetary orbits must be conies, 421; computation of a cometary orbit, 700; his reflecting telescope, 48. Nitrogen, suspected in the nebulse, 890. Node of an orbit defined, 233, 506. Nodes of moon's orbit, their regression, 455; of the planetary orbits, their mo- tion, 527. Nodical month, the, 249, 397, note. Nordenskiold, meteoric ashes, 775. Nucleus of a comet, 713, 716. Number of eclipses in a year, 391; in a saros, 398 ; of meteors and meteorites, 759, 769, 771 ; and designation of vari- able stars, 852. Nutation of the earth's axis, 214, 215. 546 INDEX. [All references, unless expressly stated to the contrary, are to articles, and not to pages.] O. Oberon, the outer satellite of Uranus, 650. Object-glass, achromatic, 41; residual or secondary spectrum of, 42; designed for photography, 42. Oblateness, or ellipticity of a spheroid, 150. Oblique sphere, the, 33. Obliquity of the ecliptic defined and measured, 176; of the ecliptic, secular change of, 197. Occultation, circle of perpetual, 33; of stars, 399 ; of stars used for longitude determination, 120 C; of stars proving absence of lunar atmosphere, 256. Oxbers, discovers Pallas and Testa, 593. Olmsted, D., his researches on meteors, 785. Oppolzer, effect of meteors on the moon's motion, 778 ; orbit of Tempel's Comet, 786; motion of Winnecke's Comet, 711, note. Orbit of the earth, its form determined, 182; of the earth, effect of meteors upon it, 778; of the earth, perturba- tions, 197; of the moon, 238; of the moon, its perturbations, 448-461; of a planet, determined graphically, 428, •431, 432; planetary, its elements, 505- 510; planetary, its elements, determi- nation of, 519. Orbits of binary stars, 873; of comets, 700-709; of planets, diagram, 489; of suu and stars in the stellar system, 904. Origin of comets, 738-741; of meteorites or aerolites, 767. Orthogonal component of the disturbing force, 445, 455. Pallas, discovered by Olbers, 593. Palisa, discoverer of sixty-five asteroids, 593. Parabola, the, described as a conic, 422, 423. Parabolic comets, their number, 702; ve- locity, the, 429. Parallax (diurnal) , denned and discussed, 82, 83; of the moon, determined, 239; of the sun, classification of methods,. 669; of the sun, gravitational methods, 687-689; of the sun, history of inves- tigations, 666-668; of the sun, method of Aristarchus, 670; of the sun, meth- od of Hipparchus, 671 ; of the sun by observations on Mars, 673, 676; of the sun by transits of Venus, 678, 686; of the sun by the velocity of light, 690- 692; of the sun, Ptolemy's value, 671; of the stars (annual), 808-814; of o Centauri, Henderson, 809, 810; of 61 Cygni, Bessel, 809-811; of a Lyrae, negative, Pond, 809; stellar, absolute method, 810; stellar, differential meth- od, 811; stellar, table of, Appendix, Table IV. Parallactic inequality of the moon, 687; orbit of a star, 808. Parallel sphere, the, 32. Parallels of declination, 23. Petbce, B., heat from meteors, 355; on the mass of comets, 719; theory of sun spots, 306. Pendulum, compensation, 51; use in clocks, 50; used in determining form of the earth, 152-155; free, of Foucault, showing earth's rotation, 139-141. Penumbra of the earth's shadow, 368; of the moon's shadow, 383; of a sun spot, 295. Perigee and apogee denned, 238. Perihelia of comets, their distribution, 706. Perihelion of earth's orbit denned, 183; its motion, 199; of Mercury's orbit, its motion, 602. Period, sidereal and synodic, of the moon, 229-232; sidereal and synodic, of a planet, defined, 490; sidereal, of a planet, determined, 513, 514. Periodic comets, 703, 704, 738-740; table of comets of short period, Appendix, Table III. Periodicity of sun spots, 307-309. Persei, j8, or Algol, 848. Perseids, the, meteoric swarm, 780, 782, 783. Personal equation, 114, 120 A, 121 B. Perturbations, lunar, 448-461; planetary, 521-523; of Mais and Venus by the earth as determining the sun's par- allax, 689. Peters, C. H. F., discovers fifty-two asteroids, 593. Piazzi, discovery of Ceres, 592. Picard, measure of earth's diameter, 136. Pickering, E. C, his meridian photome- ter, 828; photography of stellar spec- tra, 860, 862 ; photometric observations of the eclipses of Jupiter's satellites, 630; the Harvard photometry, 827. Phases of Mercury, Venus, and Mars, 559, 567, 582; of the moon, 253. Phobos, the inner satellite of Ma^s- £89- Fhotographs of the moon, 272; of the nebulse, 893; of the solar corona, 328; of the sun's surface and spots, 289. INDEX. 547 f All references, unless expressly stated to the Photographic object-glasses, 42; obser- vations of eclipses of Jupiter's satel- lites, 630, and note; observations of transit of Venus, 684-686. Photography as a means of photometry, 829; solar, 289; spectroscopic, motion in line of sight, 802; applied to star- charting, 798; in determination of stel- lar parallax, 812; of stellar spectra, 859-863. Photometer, the meridian, 828; polariza- tion, 827; the wedge, 826. Photometry, Harvard, the, 827, 828; by means of photography, 829; by the spectroscope, 831; of sunlight, 332- 335; stellar, 823-831. Photosphere of the sun, its nature, 291, 292, 361. Photo-tachymetrical determination of the sun's parallax, 690-692. Physical characteristics of comets, 712; method of determining sun's parallax, 690-692. Planet, intra-Mercurial, 602-306; trans- Neptunian, 662. Planets attending certain stars, 880, 881; distances and periods, 489; enumer- ated, 486, 487; relative age, according to nebular hypothesis, 913,915; orbits, diagram of, 489; orbits, elements of, 505, 510. Planetoid. See Asteroid, 591. ^Planetary data, tables of, Appendix, Table I.; data, accuracy of, 663; neb- ulae, 888 ; system, facts suggesting the theory of its origin, 907; system, Sir J. Herschel's illustration of its dimen- sions, 664. Pleiades, the, 884. Pogson, his absolute scale of star-magni- tudes, 819. Pole of the earth, 28; (celestial), denned, 14; its altitude equal to the latitude, 30, 100; its place affected by preces- sion, 206, 207. Pole-star, ancient, a Draconis, 207; its position and recognition, 15. Polar distance, defined, 23; point of me- ridian circle, 66. Position-angle of a double star, 868. Position of a heavenly body, how deter- mined, 128, 129. Positive eye-pieces for telescopes, 44. Pouillet, his pyrheliometer, 340. Power, magnifying, of telescope, 37. Povnting, determination of the earth's density, 170. Practical astronomy defined, 2. contrary, are to articles, and not to pages.] Precession of the equinoxes, 205-212. Prime vertical defined, 19; vertical in- strument, 62, 106. Priming and lagging of the tides, 470. Pritchakd, Prof. C, determination of stellar parallax by means of photog- raphy, 812; stellar photometry, 826; Uranometria Oxoniensis, 826. Prttchett, C. W., discovery of the great red spot on Jupiter, 618. Problem of three bodies, 437-461; of two bodies, 424-433. Problems illustrating Kepler's third law, 413. Proctor, E. A., on the origin of comets, 741; determination of the rotation period of Mars, 584. Projectiles, deviation caused by earth's rotation, 143; their path near the earth, 435. Projectile force, careless use of the term, 401. Prominences, or protuberances, the so- lar, 291, 323-326, 363; quiescent and eruptive, 325, 326. Proper motions of the stars, 800, 803. Proximity of a star, indications of it, 813. Ptolemaic system, the, 500, 502. Ptolemy, his almagest, 500, TOO, 795. Pyrheliometer of Pouillet, 340. ft- Quantity of the solar radiation in calo- ries, 338-340; of sunlight in candle power, 332, 333. Quiescent prominences, 325. R. Kadial compound of the disturbing force, 446. Kadian, the, defined as angular unit, 5, note. Radiant, the, in meteoric showers, 780. Kadius of curvature of a meridian, 149. Eantard, peculiar theory of the repul- sive force operative in comets' tails, 733. Bate of a clock defined, 53. Beading microscope, the, 64. Recognition of elliptic comets, difficul- ties, 704. Bed spot of Jupiter, the, 618. Beduction of mean star places to appar- ent and vice versa, 797. Reflecting telescope, various forms, 47, 48; telescopes, large instruments, 48, 548 INDEX. [All references, unless expressly stated to the contrary, are to articles, and not to pages.] Refraction, atmospheric, its law, 89, 90; determination of its amount, 94, 95 ; effect of temperature and barometric pressure, 91 ; effect upon form and size of discs of sun and moon near the hori- zon, 93; effect upon time of sunrise and sunset, 92. Refracting telescope (simple), 36; tele- scope, achromatic, 41. Refractors and reflectors compared, 49. Reich, determination of the density of the earth, 166; experiments upon fall- ing bodies, 138. Relative motion, law of, 492; sizes of the planets, diagram, 550. Repulsive force acting on comets, 728, 731, 732, 734. Retardation of earth's rotation by the tides, 461, 483. Reticle used in telescope for pointing, 46. Retrograde revolution of the satellites of Uranus and Neptune, 652, 661, 914. Reversing layer of the solar atmosphere, 291, 319, 320, 362. Reversal of the spectrum, 314. Rhea, the fifth satellite of Saturn, 643, note. Rigidity of the earth, 171, 482. Right ascension defined, 25, 27 ; ascension determined by transit instrument, 59, 128, 129; sphere, the, 31. Rings of Saturn, 637-642. Rosse, Lord, observations of lunar heat, 260,261; his great telescope, 48; spiral nebulas, 888. Rotation, distinguished from revolution, 248, 248*; of the earth, affected by the tides, 461, 483; of the earth, proofs of, 138-143; of planets, how determined, 543; period of Jupiter, 615; period of Mars, 584; period of the moon, 248, 252; period of Saturn, 635; of the sun, 281, 283; period of Venus, 570; periods, see also Appendix, Table I. Saros, the, 395-398 ; number of eclipses in a saros, 398. Satellites of Jupiter, 621-631; of Mars, 590, 591; of Neptune, 661; of Savsrn, 643, 644; of Uranus, 650-652; general table of, Appendix, Table II. Satellite orbits, generally circular, 548. Saturn, the planet, 632-644. Schehallien, determination of the earth's mass, 164. Schiafarelli, connection between com- ets and meteors, 786; his map of Mars, 588. Schroter, the rotation of Mercury, 559; the rotation of Venus, 570. Schwabe, discovery of the periodicity of sun spots, 307. Scintillation of the stars, 864. Sea, ship's place at, 103, 120 B, 121 A, 124- 126. Seasons, the, explained, 190, 192, 193; difference between northern and southern hemispheres, 194, 195. Secchi, theories of sun spots, 303, 305; observations on stellar spectra, 856, 857. Seidel, his photometer, 827. Secular acceleration of the moon's mean motion, 459-461 ; changes in the earth's orbit, 196-200; perturbations in the planetary system, 525-529. Semi-diameter, augmentation of the moon's, 88; correction for, in sextant observations, 88. Semi-major axis of a planet's orbit, de- fined and discussed, 505, 506; axis of the planets' orbits, invariable, 526; axis as depending on planet's velocity, 428-430. Separating power of a telescope, Dawes, 43. Sequences, method of, in stellar photom- etry, 824. Sextant, the, described, 76; the, used to determine latitude, 103; the, used in finding a ship's place at sea, 103, 116, 125, 126; the t used in determining time, 115, 116. Shadow of the earth, its dimensions, 367; of the moon, 379, 380; of the moon, its velocity over the earth, 384. "Sheath" of the comet of 1882, 752. Ship's place at sea, determination of, 103, 120 B, 121 A, 124-126. Shooting Stars, 770, 787; ashes of, 775; brightness of, 773; comparative num- bers in morning and evening, 772; daily number of, 771; elevation of, 774; mass of, 776; materials of, 775; path of, 774; showers of, 780-786; spectra of 775- velocity of, 774. Short-period comets, 703; comets, table of, Appendix, Table III. Showers, meteoric, 780-786. Sidereal day defined, 26, 110; month, 229; tim°, 26. 110; year, its length, 216. Signals used in determining difference of longitude, 119.