nJETtf t"3t...' iU iswir ^M. 1/ <-' x\ '^-. ®0wll Hmrraitj phmg 4-iZJ'3 Cornell University Library arV15570 Astronomy for students and general reade 3 1924 031 322 054 olin,anx The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031322054 THE AMEEICAN SCIENCE SEEIES FOE HIGH SCHOOLS AND COLLEGES. Tlie principal objects of tliis series are to supply tlie lack — in some subjects very great — of authoritative books whose principles are, so far as practicable, illustrated by familiar American facts, and also to supply the other lack that the ad- vance of Science perennially creates, of text-books which at least do not contradict the latest generalizations. The volumes are large 13mo. The books arranged for are as follows. They systematically outline the field of Science, as the term is usually employed with reference to general education. Those marked * had been published up to July 1, 1880. J. Physics. By Alfred M. Maykk, Pro- fessor in the Stevens Institute of Technology, and ARTHtm W. Weight, Professor in Yale College. II. Clietnistry. By Samuel W. Johnson and William G. Mixtbr, Professors in Yale College. III. Astronomy.* By Simon Newcomb, Supt. American Nautical Almanac, and Edward S. Holden, Professor in the United States Naval Observatory. $3.50. IV. Geology. By Raphael Pumpblly, late Professor in Harvard University. V. Botany. * By C. E. Bessey, Professor in the Iowa Agricultural Col- lege and late Lecturer in the University of Ca'.ifomia. VI. Zoology.* By A. S. Packard, Jr., Professor of Zoology and Ge- ology in Brown University, Editor of the Ame/riean Nat- uralist. $3. VII. The Human Body. By H. Newbll Martin, Proiessor in the Johns Hop- kins University. VIII. Psychology. By William James, Pro- fessor in Harvard University. IX. Political Economy. By Francis A. Walker, Professor in Yale College. X. Government. By Edwin L. Godkin, Ed- itor of the Nation, HENRY HOLT & CO.. Publishers, NEW YORK. THE PLANET JUPITER. As Been with the 26-mch telescope at Washington, 1875, June 24. O AMERIOAN SOIENOE 8ERIE8, 3 ASTEQN«MT STUDENTS AND GENERAL READERS BY SIMON J^EWCOMB, LL.D., SUPERINTENDENT AMERICAN EPHEMERIS AND NAUTICAL ALMANAC, AND EDWAED S. HOLDEN, M.A., FKOFESSOR IN THE U. S. NAVAL OBSERVATORY, SECOND EDITION, REVISED NEW YORK HENKY HOLT AND COMPANY 1880 s^.i7^3 Copyright, 1879, BY Henry HoiT & Co. 0,E. The John A. Gray Press, AND Steam Type-Setting Office, Cor. Frankfort and Jacob Sts., NEW YORK. PREFACE, The following work is designed principally for the use of those who desire to pursue the study of Astronomy as a hranch of liberal education. To facilitate its use by stu- dents of different grades, the subject-matter is divided into two classes, distinguished by the size of the type, and the volume is thus made to contain two courses. The portions in large type form a complete course for the use of those who desire only such a genera,! knowledge of the subject as can be acquired without the application of advanced mathematics. It is believed that this course can be mastered by persons having at command only those geometrical ideas which are familiar to most intelligent students in our advanced schools ; though sometimes, es- pecially in the earlier chapters, a knowledge of elementary trigonometry and physics will be found conducive to a full understanding of a few details. The portions in small type comprise additions for the use of those students who either desire a more detailed and precise knowledge of the subject, or who intend to ' make astronomy a special study. In this, as in the ele- mentary course, the rule has been never to use more ad- vanced mathematical methods than are necessary to the development of the subject, but in some cases a knowl- edge of Analytic Geometry, in others of the Differential Calculus, and in others of Elementary Mechanics, is neees- vi PBEFAUE. sarily presupposed. The object aimed at has been to lay a broad foundation for further study rather than to at- tempt the detailed presentation of any special branch. As some students, especially in seminaries, may not de- sire so extended a knowledge of the subject as that em- braced in the course in large type, the following hints are added for their benefit : Chapter I., on the relation of the earth to the heavens, Chapter III. , on the motion of the earth, and the chapter on Chronology should, so far as pos- sible, be mastered by all. The remaining parts of the course may be left to the selection of the teacher or student. Most persons will desire to know something of the tele- scope (Chapter II.), of the arrangement of the solar system (Chapter IV. , §§ 1-2, and Part II. , Chapter II.), of ecUpses, of the phases of the moon, of the physical constitution of the sun (Part II., Chapter II.), and of the constellations (Part III., Chapter I.). It is to be expected that all wiU be interested in the subjects of the planets, comets, and meteors, treated in Part II., the study of which involves no difficulty. An acknowledgment is due to the managers of the Clarendon Press, Oxford, who have allowed the use of a number of electrotypes from Chambers's Desoriptme Astronomy. Messrs. Fatjth & Co., instrument-makers, of . Washington, have also lent electrotypes of instruments, and a few electrotypes have been kindly furnished by the editors of the American Journal of Science and of the Popular Science Monthly. The greater part of the illus- trations have, however, been prepared expressly for the work. CONTENTS. PAKT I. PAGE Introduction 1 CHAPTER I. THE RELATION OF THE EARTH TO THE HEAVENS. The Earth — The Diurnal Motion and the Celestial Sphere — Corre- spondence of the Terrestrial and Celestial Spheres — The Diurnal Motion in different Latitudes — Relation of Time to the Sphere — Determination of Terrestrial Loncritudes — Mathe- matical Theory of the Celestial Sphere — Determination of Latitudes on the Earth by Astronomical Observations — Parallax and Semidiameter 9 CHAPTER II. ASTRONOMICAL INSTRUMENTS. The Refractinif Telescope — Reflecting Telescopes — Chronometers and Clocks — The Transit Instrument — Graduated Circles — The Meridian Circle — The Equatorial — The Zenith Telescope —The Sextant 53 CHAPTER III. MOTION OF THE EARTH. Ancient Ideas of the Planets — Annual Revolution of the Earth — The Sun's apparent Path — Obliquity of the Ecliptic — The Seasons 96 CHAPTER IV. THE PLANETARY MOTIONS. Apparent and Real Motions of the Planets— Gravitation in the Heavens— Kepler's Laws of Planetary Motion Ill PAOE viii CONTENTS. CHAPTER V. UNIVERSAL GRAVITATION. Newton's Laws of Motion — Problems of Gravitation — Results of Gravitation — Remarks on the Theory of Gravitation 131 CHAPTER VI. THE MOTION AND ATTRACTION OF THE MOON. The Moon's Motion and Phases — The Sun's disturbing Force — Motion of the Moon's Nodes — Motion of the Perigee — Rotation of the Moon— The Tides 152 CHAPTER VII. ECLIPSES OP THE SUN AND MOON. The Earth's Shadow and Penumbra — Eclipses of the Moon — Eclipses of the Sun — The Recurrence of Eclipses — Character of Eclipses r . 168 CHAPTER VIII. THE EARTH. Mass and Density of the Earth — Laws of Terrestrial Gravitation — Figure and Magnitude of the Eartli — Change of Gravity with the Latitude — Motion of the Earth's Axis, or Precession of the Equinoxes 188 CHAPTER IX. CELESTIAL MEASimEMENTS OF MASS AITD DISTANCE. The Celestial Scale of Measurement — Measures of the Solar Parallax — Relative Masses of the Sun and Planets 213 CHAPTER X. THE REFRACTION AND ABERRATION OF LIGHT. Atmospheric Refraction — Aberration and the Motion of Light 231 CHAPTER XL CHRONOLOGY. Astronomical Measures of Time — Formation of Calendars Division of the Day — Remarks on improving the Calendar The Astronomical Ephemeris or Nautical Almanac 345 CONTENTS. PART II. THE SOLAR SYSTEM IN DETAIL. CHAPTER I. bTBTJCTDKE OF THE SOIiAR SYSTEM. 267 CHAPTER 11. THE SUN. General Summary — The Photosphere — Sun-Spots and Faculae — The Sun's Chromosphere and Corona — Sources of the Sun's Heat 278 CHAPTER HI. THE INFERIOR PLANETS. Motions and Aspects — Aspect and Rotation of Mercury — The Aspect and supposed Rotation of Venus — Transits of Mercury and Venus — Supposed intramercurlal Planets 310 CHAPTER IV. The Moon 326 CHAPTER V. THE PLANET MABB. The Description of the Planet — Satellites of Mars 334 CHAPTER VI. The Minor Planets 340 CHAPTER VII. JUPITER AND HIS SATELLITES. The Planet Jupiter— The Satellites of Jupiter 343 CHAPTER VIII. SATURN AND HIS SYSTEM. General Description — The Rings of Saturn— Satellites of Saturn. . 353 X CONTENTS. CHAPTER IX. PAGE The Planet Ukanus — Satellites of Uranus 36;i CHAPTER X. The Planet Neptune — Satellite of Neptune 305 CHAPTER XI. The Physical Constitution of the Planets 370 CHAPTER XII. METEORS. Phenomena and Causes of Meteors — Meteoric Showers 375 CHAPTER XIII. comets. Aspect of Comets — The Vaporous Envelopes — The Physical Con- stitution of Comets — Motion of Comets — Origin of Comets — Remarkable Comets 888 PAET III. THE UNIVERSE AT LARGE. Introduction 411 CHAPTER I. the constellations. Oeneral Aspect of the Heavens — Magnitude of the Stars — The Constellations and Names of the Stars — Description of Con- stellations — Numbering and Cataloguing the Stars 415 CHAPTER n. variable and temporary stabs. stars Regularly Variable— Temporary or New Stars— Theoiy of Variable Stars 440 CONTENTS. XI CHAPTER III. MrLTIPLB STABS. PAOB Character of Donble and Multiple Stars— Orbits of Binary Stars. . 448 CHAPTER IV. NEBULA AND CLTISTERS. Discovery of Nebulae — Classification of Nebulae and Clusters — Star Clusters — Spectra of Nebulae and Clusters — Distribution of Nebulae and Clusters on the Surface of the Celestial Sphere 457 CHAPTER V. SPKCTBA OP FIXED STARS. Characters of Stellar Spectra — Motion of Stars in the Line of Sight. 468 CHAPTER VI. MOTIONS AND DISTANCES OP THE STABS. Proper Motions — Proper Motion of the Sun — Distances of the Fixed Stars 473 CHAPTER VII. Construction op the Heavens 478 CHAPTER VIII. Cosmogony 492 Index 503 ASTRONOMY, INTRODUCTION. Astronomy (asrrjp — a star, and vdjjLoi — a law) is the science which has to do with the heavenly bodies, their appearances, their nature, and the laws governing their real and their apparent motions. In approachiag the study of this, the most ancient of the sciences depending upon observation, it must be borne in mind that its progress is most intimately connected with that of the race, it having always been the basis of geog- raphy and navigation, and the soul of chronology. Some of the chief advances and discoveries in abstract mathe- matics have been made in its service, and the methods both of observation and analysis once peculiar to its prac- tice now furnish the firm bases upon which rest that great group of exact sciences which we call physics. It is more important to the student that he should be- come penetrated with the spirit of the methods of astron- omy than that he should recollect its minutise, and it is most important that the knowledge which he may gain from this or other books should be referred by him to its true sources. For example, it will often be necessary to speak of certain planes or circles, the ecliptic, the equa- tor, the meridian, etc., and of the relation of the appa- rent positions of stars and planets to them ; but his labor will be useless if it has not succeeded in giving him a precise notion of these circles and planes as they exist in 2 ASTRONOMY. the sky, and not merely in the figures of his text -book. Above all, the study of this science, in which not a single step could have been taken without careful and painstak- ing observation of the heavens, should lead its student himself to attentively regard the phenomena daily and hourly presented to him by the heavens. Does the sun set daily in the same point of the hori- zon ? Does a change of his own station affect this and other aspects of the sky ? At what time does the full moon rise ? "Which way are the horns of the young moon pointed ? Tliese and a thousand other questions are already answered by the observant eyes of the an- cients, who discovered not only the existence, but the motions, of the various planets, and gave special names to no less than fourscore stars. The modern pupil is more richly equipped for observation than the ancient philoso- pher. If one could have put a mere opera-glass in the hands of Hipparchus the world need not have waited two thousand years to know the nature of that early mystery, the Milky Way, nor would it have required a Galileo to discover the phases of Venus and the spots on the sun. From the earliest times the science has steadily progress- ed by means of faithful observation and sound reasoning upon the data which observation gives. The advances in our special knowledge of this science have made it con- venient to regard it as divided into certain portions, which it is often convenient to consider separately, although the boundaries cannot be precisely fixed. Spherioal and Practical Astronomy — First in logical order we have the instruments and methods by which the positions of the heavenly bodies are determined from obser- vation, and by which geographical positions are also fixed. The branch which treats of these is called spherical and practical astronomy. Spherical astronomy provides tlie mathematical theory, and practical astronomy (which is almost as mucli an art as a science) treats of the applica- tion of this theory. DIVISIONS OF THE SUBJECT. 3 Theoretical Astronomy deals with the laws of motion of the celestial bodies as determined by repeated observations of their positions, and by the laws according to which they ought to move under the influence of their mutual gravi- tation. The purely mathematical part of the science, by which the laws of the celestial motions are deduced from the theory of gravitation alone, is also called Celestial Mechanics, a term first applied by La Place in the title of his great work Mecanique Celeste. Cosm.ical Physics.— A third branch which has received its greatest developments in quite recent times may be called Cosmieal Physics. Physical astronomy might be a better appellation, were it not sometimes ajDplied to celestial mechanics. This branch treats of the physical constitution and aspects of the heavenly bodies as investi- gated with the telescope, the spectroscope, etc. We thus have three great branches which run into each other by insensible gradations, but under which a large part of the astronomical research of the present day may be included. In a work like the present, however, it will not be advisable to follow strictly this order of sub- jects ; we shall rather strive to present the whole subject in the order in which it can best be understood. This order will be somewhat like that in which the knowl- edge has been actually acquired by the astronomers of different ages. Owing to the freqiiency with which we have to use terms expressing angular measure, or referring to circles on a sphere, it may be admissible, at the outset, to give an idea of these terms, and to recapitulate some prop- erties of the sphere. Angular Measures. — The unit of angular measure most used for considerable angles, is the degree, 360 of which extend round the circle. The reader knows that it is 90° from the horizon to the zenith, and that two objects 180° apart are diainetrically opposite. An idea of distances of 4 ASTUONOMY. a few degrees may be obtained by looking at the two stars which form the pointers in the constellation Ursa Major (the Dipper), soon to be described. These stars are 5° apart. The angular diameters of the sun and moon are each a little more than half a degree, or 30'. An object subtending an angle of only one minute ap- pears as a point rather than a disk, but is still plainly vis- ible to the ordinary eye. Helmholtz finds that if two minute points are nearer together than about 1' 12", no eye can any longer distinguish them as two. If the ob- jects are not plainly visible — if they are small stars, for instance, they may have to be separated 3', 5', or even 10', to be seen as separate objects. Near the star a Lyrce are a pair of stars 3|^' apart, which can be separated only by very good eyes. If the object be not a point, but a long line, it may be Been by a good eye when its breadth subtends an angle of only a fraction of a minute ; the limit probably ranges from 10' to 15". If the object be much brighter than the background on which it is seen, there is no limit below which it is neces- sarily invisible. Its visibility then depends solely on the quantity of light wliich it sends to the eye. It is not likely that the brightest stars subtend an angle of ^ra ^^ a second. So long as the angle subtended by an object is small, we may regard it as varying directly as the linear magnitude of the body, and inversely as its distance from the ob- server. A line seen perpendicularly subtends an angle of 1" when it is a little less than 60 times its length dis- tant from the observer (more exactly when it is 57-3 lengths distant) ; an angle of 1' when it is 3438 lengths distant, and of 1" when it is 206265 lengths distant. These numbers are obtained by dividing the number of degrees, minutes, and seconds, respectively, in the cir- cumference, by 2 X 3 •14159265, the ratio of the circum- ference of a circle to the radius. CIRCLES OF THE SPHERE. 5 Great Circles of the Sphere.— In Fig. 1 let the outline represent that of a sphere, around which are described the two great circles A EB F and C E D F. These cir- cles are the lines in which two planes passing through the centre of the sphere intersect the latter. We may con- sider them as representing the planes. The points P and P' , each of which is 90° distant from every point of the circle A E B F, are called the Fig. 1. — SECTIONS of a spheke by planes. poles of that circle. The poles are the points in which a line passing through the centre O perpendicular to the j)lane of the circle meets the sphere. They may be con- sidered as representing this line. The angle B P, or A C, equal to the greatest distance of the two circles, is the same as the angle which the planes of the circles make with each other. The dis- tance between the poles P Q ov P' Q' is equal to the same angle. There are therefore three equivalent representa- tives for what may be considered the same element ; namely : (1) the inclination of the planes of two circles ; (2) the angle between their poles ; and (3) the greatest angles, A C or B P. between the circles on the celestial sphere. ® or SYMBOLS AND ABBREVIATIONS. SIGNS OF THE PLANETS, ETC. The Sun. The Moon. Mercury. Venus. The Earth. i Mars. n Jupiter. ^ Saturn. $ Uranus. W Neptune The asteroids are distinguished by a circle inclosing a number, which number indicates the order of discovery, or by their names, or by both, aal^j; Hecate SIGNS OF THE ZODIAC. Spring signs, Summer eigns. "1^ f Aries. S Taurus. n Gemini. © Cancer. SI Leo. VS. Virgo. Autumn signs. Winter signs. 7. 8. 9. (10. (12. :3= Libra. TU Scorpius. # Sagittariua V3 Capricornus. Zf Aquarius. H Pisces. ASPECTS. i Conjunction, or Laving the same longitude or right ascension. D Quadrature, or differing 90° in " " " a Opposition, or differing 180° in " " " ASTRONOMICAL SYMBOLS. MISCELLANEOUS SYMBOLS. fl Ascending node. J5 Descending node. N. North. S. South. E. East. W. West. " Degrees. ' Minutes of arc. ' Seconds of arc. ■■ Hours. '" Minutes of time. " Seconds of time, i, Mean longitude of a body. g. Mean anomaly. /, True anomaly. n. Mean sidereal motion in a of time. r, Radius vector. ^, Angle of eccentricity. IT, Longitude of perihelion parallax). p. Earth R.A. or a, Right ascension. Dee. or 6, Declination. C, True zenith distance. f, Apparent zenith distance. A Distance from the eartli. I, Helioceniric longitude. h, Heliocentric latitude. J., Geocentric longitude. /?, Geocentric latitude. 6 or U, Longitude of ascending node. i. Inclination of orhit to the eclip- tic. unit ! a. Angular distance from perihe- lion to node. i u. Distance from node, or argu- 1 ment for latitude, (also ' a. Altitude. ' A, Azimutli. 's Equatorial radius. The Greek alphabet is here inserted to aid those who are not already familiar with it in reading the parts of the text in which its letters Letters. Names. A a Alpha B p 6 Beta T yr Gamma A d Delta E e Epsilon z ?^ Zeta H V Eta e-96 Theta 1 1 lota Kk Kappa. A?i Lambda U/i Mu LetterB. Names. N V Nu S ? Xi O Omicron J] ar Pi V pg Rho S as Sigma T r7 Tau Tv Upsilon *^ Phi Xx Chi ■iij) Psi fiu Omega THE METRIC SYSTEM. The metric system of weights and measures being employed in tliis volume; the following relations between the units of this system most used and those of our ordinary one will be found convenient for reference : MEASURES OF LENGTH. 1 kilometre = 1000 metres = 0-63137 mile. 1 metre = tlie unit = 39-37 inches. 1 millimetre = ygVu of a meire = 0-03937 inch. MEASDRES OP WEIGHT. 1 niillier or tonneau = 1,000,000 grammes = 2201-G pounds. 1 kilogramme = lOUO grammes = 2-2046 pounds. 1 gramme = tUe unit = 15-432 grains. 1 milligramme = roW "f a gramme = 0-01513 grain. The following rough approximations may be memorized : The kilometre is a little more tbau if^ of a mile, but less than f of a mile. The mile is 1-,% kilometres. 'I lie kilogramme is 2^ pounds. The pound is less than half a kilogramme. CHAPTER I. THE EELATION OF THE EARTH TO THE HEAVENS. § 1. THE EAETH. In considering the relation of the earth to the heavens, we necessarily begin witli the earth itself ; not simply because we now know it to be one of the heavenly bodies, but because it is from its surface that all observations of the heavens have to be made. A consideratioji of well-known facts will show that this earth upon which we live is, at least approximately, a globe whose dimensions are gigantic when compared to our ordinary and daily ideas of size. Its shape is in several ways known to be nearly that of a sphere. I. It has been repeatedly circum- navigated in various directions. II. Portions of its sin-face, visi- ble from elevated positions in the midst of extensive plains or at sea, appear to be bounded by circles. This appearance at all points of the niusu-ating the fact that the .« !!• ,.1 portions of the eartti visible suiiace of a body is a geometrical from elevated positions, s, s', . , ., , J. T 1 1 J- 1 S", etc. , are bounded by circles. attnbute oi a globular form only. III. Further than this we know that careful measure- ments of portions of the globe by the various national geodetic surveys have agreed with this general conclusion. Pig. 3. 10 ASTRONOMY. More precise reasons will be apparent later, bnt these will be sufficient to base our general considerations upon. Of the size of the earth we may form a rough idea by the time required to travel completely around it, which is now about three months. We find next that this globe is completely isolated in space. It neither rests on any thing else, nor is it in contact with any surrounding body. The most obvious proof of this which presents itself is, that mankind have ^'isited nearly every part of its surface without finding any such connection, and that the heavenly bodies seem to perform complete circuits around it and under it with- out meeting with any obstacles. The sun which rose to- day is the same body as the setting sun of yesterday, but it has been seen to move (apparently) about the earth from east to west during the day, and it regularly reap- pears each morning. Moreover, if attentively watched, it will be found to rise and set at different parts of the' horizon of any place at different times of the year, which negatives the ancient belief that its nocturnal journey was made through a huge subterranean tunnel. § 2. THE DIURlf AL MOTION ATTD THE CELESTIAL SPHERE. Passing now from the earth to the heavens, and viewing the sun by day, or the stars by night, the first phenomenon which claims our attention is that of the diurnal motion. We must here caution the reader to carefuUy distin- guish between apparent and real motions. For example, when the phenomena of the diurnal motion are set forth as real visible motions, he must be prepared to learn sub- sequently that this appearance, which is obvious to all is yet a consequence of a real motion only to be detected'by reason. We shall first describe the diurnal motion as it appears, and show that all the appearances to a spectator at any one place may be represented by supposing the earth to remain fixed in space, and the whole concave of THE DIUliNAL MOIION. 11 the heavens to turn about it, and finally it will be shown that we have reason to believe that the solid earth itself is in constant rotation while the heavens remain immov- able, presenting different portions in turn to the observer. The motion in question is most obvious in the case of the ■ sun, which appears to make a daily circuit in the heavens, rising in the east, passing over toward the south, setting in the west, and moving around under the earth until it reaches the eastern horizon again. Observations of the stars made through any one evening sliOM' that they also appear to perform a similar circuit. Whatever stars we see near the eastern horizon will be found constantly rising higher, and moving toward the south, while those in the west will be constantly setting. If we watch a star which is rising at the same point of the horizon where the sun rises, we shall find it to pursue nearly the same course in the heavens through the night that tlie sun follows through the day. Continued observations will show, however, that there are some stars which do not set at all — namely, those in the north. Instead of rising and setting, they appear to perform a daily revolution around a point in the heavens which in our latitudes is nearly half way between the zenith and the northern horizon. This cen- tral point is called the pole of the heavens. Near it is situated Polaris, or the pole star. It may be recog- nized by the Pointers, two stars in the coustellation Ursa Major, familiarly known as The Dipper. These stars are shown in Fig. 3. If we watch any star be- tween the pole and the north horizon, we shall find that instead of mo\'ing from east to west, as the stars generally appear to move, it really appears to move toward the east ; but instead of continuing its naotion and setting in the east, we shall find that it gradually curves its course upward. If we could follow it for twenty-four hours we should see it move upwards in the north-east, and then pass over toward the west between the zenith and the pole, then sink down in tlie north-west ; and on the 12 ASTRON^OMY. following night curve its couree once more toward the east. Tlie arc which it appears to describe is a perfect circle, having the pole in its centre. The farther from the pole we go, the larger the circle which eacli star seems to describe ; and when we get to a distance equal to that between the pole and the horizon, each star in its appa^ rent passage below the pole just grazes the horizon. ^G. 3. — THE APPARENT DIURNAI, MOTION. As a result of this apparent motion, each individual constellation changes its configuration with respect to the horizon, that part which is liighest when the constellation is above the polo being lowest when below it. This is shown in Figure 4, which represents a supposed constel- lation at five different times of the night. Going fartlier still from tlie pole, the stars will dip be- THE DIURSAL MOTION. 13 low the horizon during a portion of their course, and the fraction of the circle which is below the pole will be con- tinually increasing. Looking yet farther south we shall find one half of the circle to be above and one half below the horizon. Farther yet, we shall find the stars describing sliorter arcs while above the horizon, and therefore longer ones below it. Near the south horizon, each star rises for only a short time a httle to the east of south, and soon sets a little to the west of it. NORTH Fio. 4. If we carefully study this motion, we shall find that it does not arise from each star pursuing an independent course, for not only do all the stars perform this ap- parent revolution in the very same time, but they also preserve unchanged their relati^^e distances from each other, with the exception of five, called planets or wan- dering stars. The thousands of otliers which are visible to the naked eye preserve tlioir relative positions with sucii exactness that the ordinary observer could perceive no change even after the lapse of centuries. This fact naturally suggested to the ancients the idea that there must be some material connection between the stars. An 14 ASTRONOMY. apparent explanation, both ©f this and of the phenomena of the diurnal motion, was offered by the conception of the celestial sphere. The salient phenomena of the heavens, from whatever point of the earth's surface they might be viewed, were represented by supposing that the globe of the earth was situated centrally within an im- mensely larger hollow sphere of the heavens. The vis- ible portion, or upper half of this hollow sphere, as seen from any point, constituted the celestial vault, and the whole sphere, with the stars which studded it, was called the lirmainent. The stars were set in its interior surface, or the firmament might be supposed to be of a perfectly transparent crystal, and the stars might be situated in any portion of its thickness. About one half of the sphere could he seen from any point of the earth's surface, the view of the other half being necessarily cut off by the earth itself. This sphere was conceived to make a diurnal revolution around an axis, necessarily a purely mathemat- ical line, passing centrally through it and through the earth. The ends of this axis were the poles. Tlie situa- tion of the north end, or north pole, was visible in north- ern latitudes, while the south pole was invisible, being below the horizon. A navigator sailing south woukl so change his horizon, ov^ing to the sjihoricity of the earth, that the location of the north pole would sink out of sight, while that of the south pole would come into view. It was clearly seen, even by the ancients, that the diur- nal motion could be as well represented by supposing the celestial sphere to be at rest, and the earth to revolve ill ound tliis axis, as by supposing the sphere to revolve. Tiiis doctrine of the earth's rotation was iuaintained by several of the ancient astronomers, notably bv Akis'iae- c'HUs and Timocharis. The opjjosite view, however, was maintained by Ptolemy, who could not conceive that the earth could be endowed with such a rapid rotation with- out disturbing the motion of bodies at its surface. We now know that Ptolkmv was wrong, and his opponents TEE CELESTIAL SPHEliE. 15 right. Still, so far as the apparent diurnal motion is con- cerned, it is indifferent whether we conceive the earth or the heavens to be in motion. Sometimes the one concep- tion, and sometimes the other, will make the phenomena the more clear. As a matter of fact, astronomers speak of the snn rising and setting, jnst as others do, although it is in reality the earth whicli turns. This is a form of language which, being designed only to represent the ap- pearances, need not lead us into error. The celestial sphere which we have described has long ceased to tigure in astronomy as a reality, We now know that the celestial spaces are practically perfectly void ; that some of the heavenly bodies, which appear to be on the surface of the celestial sphere at equal distances from the earth as a centre, are thousands, or even millions of times farther from the earth than others ; that there is no material connection between them, and that the celestial sphere itself is only a result of optical perspective. But the language and the conception are still retained, because they afford the most clear and definite method of repre- senting the directions of the heavenly bodies from the observer, wherever he may be situated. In this respect it serves the same purpose that the geometric sphere does in spherical trigonometry. The student of this sci- ence knows that there is really no need of supposing a sphere or a spherical triangle, because every spherical arc is only the representative of an angle between two lines which emanate from the centre, one to each end of the arc, while the angles of the triangle are only those of the planes containing the three lines which are drawn to each angle from the centre. Spherical trigonometry is, therefore, in reality, only the trigonometry of solid angles ; and the purpose of the sphere is only to afford a convenient method of conceiving of such angles. In the same way, although the celestial sphere has no real ex- istence, yet by conceiving of it as a reality, and supposing certain lines of reference drawn upon it, we are enabled to 16 ASTRONOMY. form an idea of tlie relative directions of the heavenly bodies. We may conceive of it in two ways : firstly, as having an infinite radius, in which case the centre of the earth, or any point of its surface, may equally be supposed to be in the centre of the celestial sphere ; or, secondly, we may suppose it to be finite, the observer carrying the cen- FlG. 5. — STARS SEEN ON THE CELESTIAL SPHERE. tre with him wherever he goes. The first assumption will probably be the one which it is best to adopt. The object attained by each mode of representation is that of having the observer always in the centre of the supposed sphere. Fig. 5 will give the reader an idea of its application. He is supposed to be stationed in the centre, 0, and to have around him the bodies pqrst, etc. The sphere itself being supposed at an immense distance, outside of all these bodies, we may suppose lines to be drawn from each of them directly away from the centre until they reach the sphere. The points PQRST, etc. , in which THE CELESTIAL SPHERE. Vt these lines intersect the sphere, will represent the appa- rent positions of the heavenly bodies as seen by the ob- server at 0. If several of them, as those marked t tt, are in the same direction from the observer, they will ap- pear to be projected on the same point of the sphere. Thus positions on the sphere represent simply the direc- tions in which the bodies are seen, but have no direct re- lations to the distance. It was seen by the ancients that the earth was only a point in comparison with the apparent sphere of the fixed stars. This was shown by the uniformity of the diurnal motion ; if the earth had any sensible magnitude in com- parison with the sphere of the heavens, the sun, or a star, would seem to be nearer to the observer when it passed the meridian, or any point near his zenith, than it would when it was below the horizon, or nearly under his feet, by a quantity equal to the diameter of the earth. Being nearer to him, it would seem to move more rapidly when above the horizon than when below, and its apparent angular dimensions would be greater in the zenith than in the horizon. As a matter of fact, however, the most refined observations do not show the shghtest variation from perfect uniformity, no matter what the point at which the observer may stand. Therefore, observers all over the earth are apparently equally near the stars at every point of their apparent diurnal paths ; whence their distance must be so great that in proportion to them the diameter of the earth entirely vanishes. This argument holds equally true whether we suppose the earth or the heavens to reyolve, because the observer, carried around by the rotating earth, will be brought nearer to those stars which are over his head, and carried farther from them when he is on the opposite side of the circle in which he moves. Suppose the earth to be at 0, and the celestial sphere of the fixed stars to be represented in the figure by the circle N Z Q Sn, etc. Suppose N E S W U} represent the plane of the horizon of some IS ASTRONOMY. observer on the earth's surface. Fig. 6. He will then see every thing above this plane, and nothing below it. If N E 8 \& his eastern horizon, stars will appear to rise at various points, g, E, d, a, etc., and will appear to describe circles until they attain their highest points at h, Q, e, &, etc., sinking into the western horizon at Ic, W, f, c, etc. These are facts of observa^ tion. The common axis of these circles is P p, and stars about P (the pole) never set. The appa- rent diurnal arc I m, for instance, represents the apparent orbit of a eircumpola/r star. 3. COBRESPONDENCE OP THE TERBESTSIAL AND CELESTIAL SPHERES. "We have said that the direction of a heavenly body from an observer, or, which is the same thing, its ap- parent position, is defined by the point of the celestial sphere on which it seems to be. This point is that in which the straight line drawn from the observer to the body, and continued forward indefinitely, meets the celes- tial sphere. Its position is fixed by reference to certain fundamental circles supposed to be drawn on the sphere, on the same plan by which longitude and latitude on the earth are fixed. The system of thus defining terrestrial positions by reference to the earth's equator, and to some prime meridian from which we reckon the longitudes, is one with which the reader may be supposed familiar. We shall therefore commence with those circles of the celestial sphere which correspond to the meridians, parallels, etc. , on the earth. First, we remark that if we consider the earth to be at rest for a moment, every point on its surface is at the end of a radiu^ which, if extended, would touch a correspond- THE CELESTIAL AND TERRESTRIAL SPHERES. 19 ■ ing point upon the celestial sphere. This point is called the zenith of the point on the earth. In other words, the zenith is defined by a line passing through the centre of the earth to the observer, and continuing directly up- ward until it meets the celestial sphere. To the observer this line necessarily appears vertical, because, wherever he may be, he understands by a vertical line one passing from where he stands toward the centre of the earth. As the earth revolves, the direction of this line in relation to any fixed diameter of the celestial sphere necessarily varies, and therefore the point in which it cuts the celestial sphei'e or the zenith of the observer varies also in space. Let lis suppose first that the observer is on the earth's equator. Then he will see both the north and the south pole in the horizon directly opposite each other. Looking upward he will see his zenith half way between the poles. Then, as the earth revolves on its axis, his zenith will describe a great circle around the celestial sphere, every point of which will be equally distant from the two poles. If we imagine an infinitely long pencil reaching from any point of the earth's equator vertically up to the stars, we may conceive that its point marks out an equator among them. A complete revolution of the earth brings it back to the place from which it started, and thus completes the circle. The imaginary circle thus described in the heavens is called the celestial equator. The relation which it bears to the terrestrial equator is that every point of it is above a corresponding point of the latter. The two equators lie in the same plane, passing through the centre of the earth, which plane is called the plane of the equatw, and belongs to both the celestial and terrestrial spheres. Now suppose that the observer passes from the equator to 45° of north latitude. His horizon having changed by 45°, the north pole wiU ■ now be 45° above the horizon, and 45° from the zenith. Then, by the revolution of the earth, his zenith wiU describe a circle on the celestial 20 ASTRONOMY. sphere which will be everywhere 45° distant from the celestial equator. This circle will thus correspond to the parallel of 45° north upon the earth. If he goes to lati- tude 60° north, he will sec the pole at an elevation of 60°, and his zenith will in the same way describe a circle which will be everywhere 60° from the celestial equator, and 30"^' from the pole. If he passes to the pole, the latter will be directly over his head, and his zenith will not move at Fig. 7. — terrestrial and celestial spheres. all. The celestial pole is simply the point in which the earth's axis of rotation, if continued out in a straight line of infinite length, would meet the celestial sphere. "We thus have a series of circles on the celestial sphere corre- sponding to the parallels of latitude upon the earth. Unfortunately the celestial element corresponding to latitude on the earth is not called by that name, but by that of decimation. The declmaiion of a star is its distance north or south from the celestial equator, pre- CELESTIAL AND TEURESTRIAL MERIDIANS. 21 cisely as latitude on the earth is distance from the eartli's equator. Let i be a place on the earth, P^ju Q, Pp being the earth's axis, and E Q its equator. Z is the zenith and H R the horizon of L. L Q\s the latitude ot L accord- ing to ordinary geographical de- finitions : i.e., it is its angular dis- tance from the equator. Prolong P indefinitely to P*, and draw L P' parallel to it. To an observer at L the elevated pole of the heavens will be seen along the line L P\ because at an in- finite distance the distance P P' will appear like a point. H L Z=. POe and Z£P'=Z OP', hence P"Zi7=Z0e— that is, the eleva- tion of the pole above the celestial horizon is equal to the latitude of the place. Referring to Fig. 9, it can at once be seen that the latitude of a pkux on the earth'' s surface is equal to the declination of the zenith of that place, since the declination of the zenith is equal to the altitude of the elevated pole. "We have next to consider the correspondence between the celestial and terrestrial meridians. A terrestrial me- ridian is an imaginary line drawn along the earth's surface in a north and south direction from one pole to the other. These meridians diverge from one pole in every direc- tion, and meet at the other pole. Sometimes they are called by the names of places they pass through, as the meridian of Greenwich, or the meridian of Washington. Each meridian may be considered as the intersection witli the earth's surface of a plane passing through the axis of the earth, and therefore through both poles. Such a plane will cut the earth into two equal hemispheres, and will of course be vertical with the earth's surface along every part of its line of intersection. This plane is called the plane of the meridian ; and by continuing it out to the celestial sphere, we should have a celestial meridian corresponding to each terrestrial one, precisely as we have 22 ASTRONOMY. circles of declination corresponding to parallels of latitude on the earth. But owing to the rotation of the earth, the circle in which the plane of the meridian of any place in- tersects the celestial sphere will be continually moving atnong the stars, so that tliere is no such permanent cor- respondence as in the case of the declinations. This does not prevent us from conceiving imaginary meridians passing from one pole of the heavens to the other pre- cisely as the meridians on the earth do, only these me- ridians will be apparently in motion, owing to the rotation of the earth. We may,, in fact, conceive of two sets of meridians — one really at rest among the stars, but appa- rently moving from east to west around the pole as the stars do, and the other the terrestrial meridians continued to the celestial sphere, apparently at rest, but really in motion from west to east. The relations of these me- ridians will be best understood when we explain the in- struments and methods by which they are fixed, and by which the positions of the stars in tlie heavens are deter- mined. At present we will condne ourselves to the con- sideration of the celestial meridians. The reader will understand that these meridians pass from one pole of the celestial sphere to tlie other, pre- cisely as on the globe terrestrial meridians pass from one pole to the otlier, and that being fixed among the stars, they appear to turn around the pole as the stars appear to do. As on the earth differences of longitude between different places are fixed by the differences between the meridians of the two places, so in the heavens what cor- responds to longitude is fixed by the difference between the celestial meridians. This co-ordinate is, however, in the heavens not called longitude, but right ascension. Let the student very thoroughly impress upon his mind this term — right ascension — which is longitude on the celestial sphere, and also the term we have before spoken of — declination — which is latitude on the celestial sphere. In order to fix the right ascension of a heavenly body RIGHT ASCENSION. 33 we must have a fii'st ineridian to count from, precisely as on the earth we count longitudes from the meridian of Greenwich or of Washington. It is indifferent what me- ridian we take as the first one ; but it is customary to adopt the meridian of the vernal equinox. What the ver- nal equinox is will he described hereafter : for our pres- ent purposes, nothing more is necessary than to under- stand that a certain meridian is arbitrarily taken. If now we wish to fix the right ascension of a star, we have only to imagine a meridian passing through it, and to deter- mine the angle which this meridian makes with the meri- dian of the vernal equinox, as measured from west to east on the equator. Tliat angle will be the right ascension of a star. As already indicated, the declination of a star will be its angular distance from the equator measured on this meridian. Thus, the right ascension and declination of a star fix its apparent position on the celestial sphere, precisely as latitude and longitude fix the position of a ])oint on the surface of the earth. To give precision to tlie ideas, we present a brief con- densation of this subject, with additional definitions. Let /^Z^iV^ represent the celestial sphere of an ob- server in the northern hemisphere, being the position of the earth. P p is the axis of the celestial spfiere, or the line about which the apparent diurnal orbits of the stars and the actual revolution of the earth are performed. The zenith, Z, is the point immediately above, the nadir n, the point immediately below the observer. The direction Zn h defined in practice by the position freely assumed by the plumb line. The celestial horizon is the plane perpendicular to the line joining the zenith and nadir N E S W ; or it is the terrestrial horizon continued till it meets the celestial sphere. The celestial horizon intersects the earth in the rational horizon, which passes through the earth's centre, and which is so called in distinction to the sensible horizon, which is the plane tangent to the earth's surface at any 24 ASTRONOMT. poiut. But, since tlie earth itself is considered as Imt a point in comparison with the celestial sphere, the rational and sensible horizons are considered as one and the same circle on this sphere. The celestial poles are the extremities of the axis of the celestial sphere P p, the north pole being that one which is above the horizon in the latitude of New York, in the northern hemisphere. The circles apparently described by the stars in their diurnal orbits are caReA parallels of declination, KN ; Fig. 9. — cniCLES op the sphere. that one whose plane passes through the centre of the sphere being tlie celestial equator, or the equinoctial C WD. The celestial equator is then that parallel of declination which is a great circle of the celestial sphere. Tlie figure illustrates the phenomena which appear in the heavens to an observer upon the earth. The stars which lie in the equator have their diurnal paths bisecte 1 by the horizon, and are as long above the horizon as below CIRCLES OF THE SPHERE. 25 it ; those whose distances from the pole {polar-distances) are greater than 90° will be a shorter time above the ho- rizon ; those whose polar-distances are less than 90° a longer time. The circle 2V K drawn aronnd the pole P as a centre so as to graze the horizon is called the circle of perpetual apparition, because stars situated within it never set. The corresponding circle 8 It round the south pole is called the ciTcle of perpetual disappearance, because stars within it never rise above our horizon. The great circle passing through the zenith and the pole is the celestial meridian, N P Z 8. The meridiam, intersects the horizon in the meridian line, and the points iVand 8 axe the north and south points. The prime vertical, E ZW,\& pei-pendicular to the m-eri- dian line and to the horizon : its extremities in the hori- zon are the east and west points. The meridian plane is perpendicular to the equator and to the hxyrizon, and therefore to their intersection. Hence this intersection is the east and west line, which is tlius determined by the intersection of the planes of the equator and of the horizon. The altitude of a heavenly body is its apparent distance above the horizon, expressed in degrees, minutes, and seconds of arc. In the zenith the altitude is 90°, which is the greatest possible altitude. If A be any heavenly body, the angle ZP A which the circle P A drawn from the pole to the body makes with the meridian is called the hour angle of the body. The hour angle is the angle through which the earth has ro- tated on its axis since the body was on the meridian. It is so called because it measures the time which has elapsed since the passage of the body over the meri, dian. That diameter of the earth which is coincident with the constant direction of the axis of the celestial sphere is its axis, and intersects the earth in its north and south poles. 2G ASTRONOMY. S 4 THE PTTTRTSTAT. MOTION IN" DrPPBBEWT LATI- TUDES. As we have seen, the celestial horizon of an observer -will change its place on the celestial sphere as the observer travels from place to place on the surface of the earth. If he moves directly toward the north his zenith wiU ap- proach the north pole, but as the zenith is not a visible point, the motion will be naturally attributed to the pole, which will seem to approach the point overhead. The new apparent position of the pole will change the aspect of the observer's sky, as the liigher the pole appears above the horizon the greater the circle of perpetual apparition, and therefore the greater the number of stars, which never set. Fig. 10. — THE PARALLEL SPHERE. If the observer is at the north pole his zenith and the pole itseK will coincide : half of the stars only will be vis- ible, and these will never rise or set, but appear to move around in circles parallel to the horizon. The horizon and equator will coincide. The meridian will be indeter- minate since Zand P coincide ; there will be no east and west line, and no direction but south, The sphere in this case is called a parallel ipfiere. DIURNAL MOTION IN DIFFERENT LATITUDES. 27 If instead of travelling to tlie north the observer should go toward the er[uator, the north pole would seem to ap- proach his horizon. "When he reached the equator both poles would be in the horizon, one north and tlie other south. All the stars in succession would then be visible, and each would be an equal time above and below the horizon. Fig. 11. — THE EIGHT SPHERE. The sphere in this ease is called a right sphere, because the diurnal motion is at right angles to the horizon. If now the observer travels southward from the equator, the south pole will become elevated above his horizon, and in the southern hemisphere appearances will be reproduced which we have already described for the northern, except that the direction of the motion will, in one respect, be different. The heavenly bodies will still rise in the east and set in the west, but those near the equator will pass north of the zenith instead of south of it, as in our lati- tudes. The sun, instead of moving from left to right, there moves from right to left. The bounding line be- tween the two directions of motion is the equator, where the sun culminates north of the zenith from March till September, and south of it from September till March. If the observer travels west or east of his first sta- tion, his zenith will still remain at the same angular 28 ASTRONOMY. distance from the north pole as before, and as the phe- nomena caused by the earth's diurnal motion at any place depend only upon the altitude of the elevated pole at that place, these will not be changed except as to the times of their occurrence. A star which appears to pass through the zenith of his first station will also appear to pass through the zenith of the second (since each star re- mains at a constant angular distance from the pole), but later in time, since it has to pass through the zenith of every place between the two stations. The horizons of the two stations will intercept different portions of the celestial sphere at any one instant, but the earth's rotation will present the same portions successively, and in the same order, at both. § 5. EELATION OF TIME TO THE SPHEBE. Different Kinds of Time.— We have seen (p. 17) that the earth rotates imiformly on its axis — that is, it turns through equal angles in equal intervals of time. This ro- tation can be used to measure intervals of time when once a unit of time is agreed upon. The most natural unit is a day. A sidereal day is the interval of time required for the earth to make one complete revolution on its axis. Or, what is the same thing, it is the interval of time between two consecutive transits of a star over the same meridian. The sidereal day is divided into 24 sidereal hours ; each hour is divided into 60 minutes ; each minute into 60 seconds. In mating one revolution, the earth turns through 360°, so that 34 hours =; 360° ; also 1 hour = 15° ; 1° = 4 minute.s ; 1 minute = 15' ; 1' = 4 seconds ; 1 second = 15' ; 1" = 0-066 . . . sec. The hour-angle of any star on the meridian of a place is zero (by definition p. 25). It is then at its transit or culmination. SIDEREAL TIME. 29 As the earth rotates, the meridian moves away (east- wardly) from this star, whose hour-angle continually in- creases from 0° to 360°, or from hours to 24 hours. Sidereal time can then be directly measured by the hoiir- angle of any star in the heavens which is on the meridian at an instant we agree to call sidereal hours. When this star has an hour-angle of 90°, the sidereal time is 6 hours ; when the star has an hour-angle of 180° (and is again on the meridian, but invisible unless it is a eircumpolar star) it is 12 hours ; when its hour-angle is 270° the sidereal time is 18 hours, and, finally, when the star reaches the upper meridian again, it is 24 hours or hours. See Fig. 9 where E G TI^Z^ is the apparent diurnal path of a star in the equator. It is on the meridian at C. Instead of choosiug a star as the determining point ■whose transit marks sidereal hours, it is found more con- venient to select that point in the sky from which the right ascensions of stars are counted — the vernal equinox — the point V in the figure. The fundamental theorem of si- dereal time is, the hour-angle of the vernal equinox or the sidereal time is equal to the right ascension of the meri- dian, that is T^= Y C. To avoid continual reference to the stars, we set a clock so that its hands shall mark hours minutes seconds at the transit of the vernal equinox, and regulate it so that its hour-hand revolves once in 24 sidereal hours. Such a clock is called a sidereal clock. Time measured by the hour-angle of the sun is called true or apparent solar time. An apparent solar day is the interval of time between two consecutive transits of the sun over the upper meridian. The instant of the transit of the sun over the meridian of any place is the apparent noon of that place, or local apparent noon. When the sun's hour-angle is 12 hours or 180°, it is local apparent midnight. The ordinary sun-dial marks apparent solar time. As a matter of fact, apparent solar days are not equal. The 30 ASTBO'SOMT. reason for this is fully explained later (p. 258). Hence our clocks are not made to keep this kind of time, for if once set right they would sometimes lose and sometimes gain on such time. A modified kind of solar time is therefore used, called viean solar time. This is the time kept by ordinary watches and clocks. It is sometimes called civil time. Mean solar time is measured by the hour-angle of the mean sun, a fictitious body which is imagined to move uniformly in the heavens. The law according to which the mean sun is supposed to move enables us to compute its exact position in the heavens at any instant, and to define this position by the two co-ordinates right ascension and declination. Thus we know the position of this imaginary body just as we know the position of a star whose co-ordinates are given, and we may speak of its transit as if it were a bright ma- terial point in the sky. A mean solar day is the interval of time between two consecutive transits of the mean sun over the upper meritiian. Mean noon at any place on the earth is the instant of the mean sun's transit over the meri- dian of that place. Twelve hours after local mean noon is local mean midnight. The mean solar day is divided into 24 hours of 60 minutes each. Each minute of mean time contains 60 mean solar seconds. We liave thus three kinds of time. They are alike in one point. Each is measured by the hour-angle of some body, real or assumed. The body chosen determines the kind of time, and the absolute length of the unit — the day. The simplest unit is that determined by the uniformly rotating earth — the sidereal day ; the most natural unit is that determined by the sun itself — the apparent solar day, which, howsver, is a variable unit ; the most convenient unit is the mean Bolar day. Comparative Lengths of the Mean Solar and Sidereal Day.— As a fact of observation, it is found that the sun appears to move from west to east among the stars, about 1° daily, making a complete revolution around the sphere in a year. The reason of this will be explained later (p. 101). SIDEREAL TIME. 31 Hence an apparent solar day will be longer than a sidereal day. For suppose the sun to be at the vernal equinox exactly at sidereal noon (0 hours) of "Washington time on March 21st — that is, the vernal equinox and the sun are both on the meridian of Washington at the same instant. In 2i sidereal hours the vernal equinox will again be on the same meridian, but the sun Vi'ill have moved eastwardly by about a degree, and the earth will have to turn through this angle and a little more in order that the sun shall again be on the Washington meridian, or in order that it may be apparent noon on March 22d. For the meridian to overtake the sun requires about 4 minutes of sidereal time. The true sun does not move, as we have said, uniformly. The mean sun is supposed to move uniformly, but to make the circuit of the heavens in the same time as the real sun. Hence a mean solar day will also be longer than a sidereal day, for the same reason that the apparent solar day is longer. The exact rela- tion is : 1 sidereal day = 0-997 mean solar day, 2'l: sidereal hours = SS"" 56"" 4' ■ 091 mean solar time, 1 mean solar day = 1-003 sidereal days, 34 mean solar hours = 24'' 3"^ 56^-555 sidereal time, and 36G-24323 sidereal days = 360-24333 mean solar days. Local Time.— When the mean sun is on the meridian of a place, as Boston, it is mean noon at Boston. When the mean sun is on the meridian of St. Louis, it is mean noon at St. Louis. St. Louis being west of Boston, and the earth rotating from west to east, the local noon of Boston occurs before the local noon at St. Louis. In the same way the local sidereal time at Boston at any given instant is expressed by a larger number than the local sidereal time of St. Louis at that instant. The sidereal time of our common noon is given in the astronomical ephemeris for every day of the year. It can be found within ten or twelve -minutes at any time by re- membering that on March 21stit is sidereal hours about 32 ASTRONOMY. noon, on April 21st it is about 2 hours sidereal time at noon, and so on through the year. - Thus, by adding two hours for each month, and four minutes for each day after the 21st day last preceding, we have the sidereal time at the noon we require. Adding to it the number of hours since noon, and one minute more for every fourth of a day on account of the constant gain of the clock, we have the sidereal time at any moment. Example. — Find the sidereal time on July ith, 1881, at 4 o'clock A.M. We have : h m June 21st, 3 months after March 21st ; to be X 2, 6 July 3d, 12 days after June 21st ; X 4, 48 4 A.M., 16 hours after noon, nearly | of a day, 16 3 22 51 Tliis result is within a minute of the truth. Ilelation of Time and Longitude. — Considering our civil time which depends ou the sun, it will be seen that it is noon at any and every place on the earth when the sun crosses the meridian of that place, or, to speak with more precision, Avhen the meridian of the places passes under the sun. In the lapse of 24 hours, the rotation of the earth ou its axis brings all its meridians uader the sun in succession, or, which is the same thing, the sun appears to pass in succession all the meridians of the earth. Hence, noon continually travels westward at the rate of 15° in an hour, making the circuit of the earth in 24 hours. The difference between the time of day, or local time as it is called, at any two places, wiU be in proportion to the differ- ence of longitude, amounting to one hour for every 15 degrees of longitude, four minutes for every degree, and so on. Vice versa, if at the same real moment of time we can determine the local times at two different places, the difference of these times, multiplied by 16, will give the difference of longitude. CHANOE OF DA T. 33 The longitudes of places are determined astronoinicallj on this principle. Astronomers are, however, in the habit, of expressing tlie longitude of places on the earth like the right ascensions of the heavenly bodies, not in degrees, but in hours. For instance, instead of saying that Washington is 77° 3' west of Greenwich, we com- monly say that it is 5 hours 8 minutes 12 seconds west, meaning that when it is noon at Washington it is 6 hours 8 minutes 12 seconds after noon at Greenwich. This course is adopted to prevent the trouble and confusion which might arise from constantly having to change hours into degrees, and the reverse. A. question frequently asked in this connection is, Wliere does the day change ? It is, we will suppose, Sun- day noon at Washington. That noon travels all the way round the earth, and when it gets back to Washington again it is Monday. Where or when did it change from Sunday to Monday ? We answer, wherever people choose to make the change. Navigators make the change occur in longitude 180° from Greenwich. As this meri- dian lies in the Pacific Ocean, and scarcely meets any land through its course, it is very convenient for this purpose. If its use were universal, the day in question would be Sunday to all the inhabita,nts east of this line, and Mon- day to every one west of it. But in practice there have been some deviations. As a general rule, on those islands of the Pacific which are settled by men travelling east, the day would at first be called Monday, even though they might cross the meridian of 180°. Indeed the Eus- sian settlers carried their count into Alaska, so that when our people took possession of that territory they found that the inhabitants called the day Monday when they themselves called it Sunday. These deviations have, how- ever, almost entirely disappeared, and with few exceptions the day is changed by common consent in longitude 180° from Greenwich. 34 ASTRONOMY. § 6. DETERMINATIONS OF TERRESTRIAL LONGI- TUDES. "We have remarked that, owing to the rotation of the earth, there is no such fixed correspondence between meridians on the earth and among the stars as there is between latitude on the earth and declination in the heavens. The observer can always determine his latitude by finding the declination of his zenith, but he cannot find his longitude from the right ascension of his zenith with the same facility, be- cause that right ascension is constantly changing. To deter- mine the longitude of a place, the element of time as mea- sured by the diumal motion of the earth necessarily comes in. Let us once more consider the plane of the meridian of a place extended out to the celestial sphere so as to mark out on the latter the celestial meridian of the place. Consider two such places, Washington and San Francisco for example ; then there will be two such celestial meri- dians cutting the celestial sphere so as to make an angle of about forty-five degrees with each other in this case. Let the observer imagine himself at San Francisco. Then he may conceive the meridian of Washington to be visible on the celestial sphere, and to extend from the pole over toward his south-east horizon so as to pass at a distance of about forty-five degrees east of his own meridian. It would appear to him to be at rest, although really both his own meridian and that of Washington are movino- in consequence of the earth's rotation. Apparently the stars in their course will first pass the meridian of Washington, and about three hours later will pass his own meridian. ISTow it is evident that if he can determine the interval which the star requires to pass from the meridian of Wash- ington to that of his own place, he will at once have the difference of longitude of the two places by simply turn- ing the interval in time into degrees at the rate of fifteen degrees to each hour. Essentially the same idea may perhaps be more readily grasped by considering the star as apparently passing over LONGITUDE. 35 the successive terrestrial meridians on the surface of the earth, the earth being now supposed for a moment to be at rest. If we imagine a straight line drawn from the centre of the earth to a star, this line will in the course of twenty-four sidereal hours apparently make a complete revolution, passing in succession the meridians of all the places on the earth at the rate of fifteen degrees in an hour of sidereal time. If, then, V\rashington and San Francisco are forty-five degrees apart, any one star, no matter what its declination, will require three sidereal hours to pass from the meridian of Washington to that of San Francisco, and the sun will require three solar Jiours for the same passage. Whichever idea we adopt, the result will be the same : difference of longitude is measured by the time required for a star to apparently pass from the meridian of one place to that of another. There is yet another way of defining what is in effect the same thing. The sidereal time of any place at any instant being the same with the right ascension of its meridian at that instant, it follows that at any instant the sidereal times of the two places will differ by the amount of the difference of longitude. For instance : suppose that a star in hours right ascension is crossing the meridian of Washington. Then it is hoLirs of local sidereal time at Washington. Three hours later the star will have reached the meridian of San Francisco. Then it will be hours local sidereal time at San Fran- cisco. Hence the difference of longitude of two places is •" measured by the difference of their sidereal times at the same instant of absolute time. Instead of sidereal times, we may equally well take mean times as measured by the sun. It being noon when the sun crosses the meridian of any place, and the sun requiring three hours to pass from the meridian of Washington to that of San Francisco, it follows that when it is noon at San Francisco it is three o'clock in the afternoon at Wasliington.* * The difference of longituiie thus depends upon the nnc/ular dis- tance of terrestrial rfieridCaas, and not upon the motion of a celestial body, 36 ASTRONOMY. T]ie whole problem of the determination of terrestrial longitudes is thus reduced to one of these two : either to find the moment of Greenwich or "Washington time corresponding to some moment of time at the place which is to he determined, or to find the time required for the sun or a star to move from the meridian of Green- wich or Washington to that of the place. If it were possible to fire a gun every day at Washington noon which could be heard in an instant all over the earth, tlien observers everywhere, with instniments to deter- mine their local time by the sun or by the stars, would be able at once to fix their longitudes by noting the hour, minute, and second of local time at which the gun was heard. As a matter of fact, the time of Washington noon is daily sent by telegraph to many telegraph stations, and an observer at any such station who knows his local time can get a very close value of his longitude by observing the local time of the arrival of this signal. Human ingenuity has for several centuries been exercised in the effort to in- vent some practical way of accomplishing the equivalent of such a signal which could be used anywhere on the earth. The British Government long had a standing offer of a reward of ten thousand pounds to any person who would discover a practical method of determining the lon- gitude at sea with the necessary accuracy. This reward was at length divided between a mathematician who con- structed improved tables of the moon's motion and a mechanician who invented an improved chronometer. Before the invention of the telegraph the motion of the moon and the transportation of chronometers afforded almost the only practicable and widely extended methods of solving the problem in question. The invention of the telegraph offered a third, far more perfect in its appli- and hence the longitude of a place is the same whether expressed as a difference of two sidereal times or of two solar times. The lonn-itude of Washington west from Greenwich is 5'' 8"' or 77°, and this is, in fact, the ratio of the angular distance of the meridian of Washington froni that of Greenwich to 360" or 34''. It is tlius plain that the lonsitude is the difference of flie simultaneous local times, whether solar or sidereal. LONGITUDE B Y VHRONOMETERS. 37 cation, but necessarily limited to places in telegraphic communication with each other. Longitude by Motion of the Moon. — When we de- scribe the motion of the moon, we shall see that it moves eastward among the stars at the rate of about thirteen de- grees per day, more or less. In other words, its right as- cension is constantly increasing at the rate of a degree in something less than two hours. If, then, its right ascension can be predicted in advance for each hour of Greenwich or Washington time, an observer at any point of the earth, by noting the local time at his station, when the moon has any given right ascension, can thence determine the corresponding moment of Greenwich time ; and hence, from the difference of the local times, the longitude of his place. The moon will thus serve the purpose of a sort of clock running on Greenwich time, upon the face of which any observer with the proper appliances can read the Greenwich hour. This method of determining longitudes has its difficulties and drawbacks. The motion of the moon is so slow that a very small change in its right ascen- sion will produce a comparatively large one in the Green- wich time deduced from it — about 27 times as great an error in the deduced longitudes as exists in the determi- nation of the moon's right ascension. With such instru- ments as an observer can easily carry from place to place, it is hardly possible to determine the moon's right ascen- sion within five seconds of arc ; and an error of tliis amount will produce an error of nine seconds in the Greenwich time, and hence of two miles or more in his deduced longitude. Besides, the mathematical processes of deducing from an observed right-ascension of the moon the corresponding Greenwich time are, under ordinary circumstances, too troublesome and laborious to make this method of value to the navigator. Transportation of Chronometers. — The transportation of chronometers affords a simple and convenient method of obtaining the time of the standard meridian at any moment. The observer sets his chronometer as nearly as 38 ASTEOIS^OMT. possible on Greenwicli or Washington time, and deter- mines its correction and rate. This he can do at any sta- tion of which the longitude is correctly known, and at which the local time can be determined. Then, wherever he travels, he can read the time of his standard meridian from the face of his chronometer at any moment, and compare it with the local time determined with his transit instrument or sextant. The principal error to which this method is subject arises from the necessary uncertainty in the rate of even the best chronometers. This is the method almost universally used at sea where the object is simply to get an approximate knowledge of the ship's position. The accuracy can, however, be increased by carrying a large number of chronometers, or by repeating the de- termination a number of times, and this method is often employed for fixing the longitudes of seaports, etc. Between the years 1848 and 1855, great numbers of chro- nometers were transported on the Cunard steamers plying between Boston and Liverpool, to determine the difference of longitude between Greenwich and the Cambridge Ob- servatory, Massachusetts. At Liverpool the chronometers were carefuUy compared with Greenwich time at a locjal observatory — that is, the astronomer at Liverpool found the error of the chronometer on its arrival in the ship, and then again when the ship was about to sail. "When the chronometer reached Boston, in like manner its error on Cambridge time was determined, and the determination was repeated when the ship was about to return. Having a number of such determinations made alternately on the two sides of the Atlantic, the rates of the chronometers could be determined for each double voyage, and thus the error on Greenwich time could be calculated for the mo- ment of each Cambridge comparison, and the moment of Cambridge time for each Greenwich comparison. Longitude by the Electric Telegraph. — As soon as the electric telegraph was introduced it was seen by American LONGITUDE BY TELEGRAPH. 39 astronomers that we here laad a method of determining longitudes which for rapidity and convenience would supersede all others. The first application of this method was made in ISM between Washington and Baltimore, under the direction of the late Admiral Charles Wilkes, U. S. N. During the next two years the method was intro- duced into the Coast Survey, and the difEercnce of longitude between New York, Philadelphia, and Washington was thus determined, and since that time this method has had wide extension not only in the United States, but between America and Europe, in Europe itself, in the East and West Indies, and South America. The principle of the method is extremely simple. Each place, of which the difference of time (or longitude) is to be determined, is furnished with a transit instrument, a clock and a chronograph ; instruments described in the next chapter. Each clock is placed in galvanic communication not only with its own chronograph, but if necessary is so connected with the telegraph wires that it can record its own beat upon a chronograph at the other station. The observer, looking into the telescope and noting the crossing of the stars over the meridian, can, by his signals, record the instant of transit both on his own chronograph and on that of the other station. The plan of making a determination between Philadelphia and Washington, for instance, was essentially this : When some previously selected star reached the meridian at Phil- adelphia, the observer pointed his transit upon it, and as it crossed the wires, recorded the signal of time not only on his own chronograph, but on that at Washington. About eight minutes afterward the star reached the meridian at Washington, and there the observer recorded its transit both on his own chronograph and on that at Philadelphia. The interval between the transit over the two places, as measured by either sidereal clock, at once gave the difference of longitude. If the record was in- stantaneous at the two stations, this interval ought to be the same, whether read off the Pliiladelphia or the Wash- 40 ASTRONOMY. ington clironograph. It was found, however, that there was a difference of a small fraction of a second, arising from the fact that electricity required an interval of tiiiie, minute but yet appreciable, to pass between the two cities. The Philadelphia record was a little too late in being recorded at Washington, and the "Washington one a little too late in being recorded at^ Philadelphia. "We may illustrate this by an example as follows : Suppose E to be a station one degree of longitude east of another station, "W ; and that at each station there is a clock exactly regulated to the time of its own place, in which case the clock at E will be of course four minutes fast of the clock at "W ; let us also suppose that a signal takes a quarter of a second to pass from one station to the other : Then if the observer at E sends a signal to W at exactly noon by his clock 12'> 0" O'.OO It will be received at W at 11'' 56"" 0«.25 Showing an apparent difference of time of 3"" 59*. 75 Then if the observer at W sends a signal at noon by his clock la"" O™ O'.OO It will be received at E at lai- i" 0".25 Showing an apparent difference of time of 4" 0'.25 One half the sum of these difEerences is four minutes, which is exactly the difference of time, or one degree of longitude ; and one half their difference is twenty -five hundredths of a second, the time taken by the electric im- pulse to traverse the wire and telegraph instruments. This is technically called the " wave and armature time." We have seen that if a signal could be made at Wash- ington noon, and observed by an observer anywhere sit- uated who knew the local time of his station, his longi- tude would thus become known. This principle is often employed in methods of determining longitude other than those named. For example, the instant of the beginning THEORY OF THE SPHERE. 41 and ending of an eclipse of the sun (by the moon) is a perfectly definite phenomenon. If this is observed by two observers, and these instants noted by each in the local time of his station, then the difference of these local times (subject to small corrections, due to parallax, etc.) will be the difference of longitude of the two sta- tions. The satellites of Jupiter suffer eclipses frequently, and the Greenwich and "Washington times of these ph3nomena are computed and set down in the Nautical Almanac. Ob- servations of these at any station will also give the differ- ence of longitude between this station and Greenwich or Washington. As, however, they require a larger tele- scope and a higher magnifying power than can be used at sea, this method is not a practical one for navigators. § 7. MATHEMATICAL THEORY OP THE CELESTIAL SPHERE. In this explanation of tlie mathematical theory of the relations of the heavenly bodies to circles on the sphere, an acquaintance with spherical trigonometry on tlie part of the reader is necessarily pre- supposed. The general method by which tlie position of a point on the sphere is referred to fixed points or circles is as follows : A fundamental great circle E V Q„ Fig. 13 is taken as a basis, and the first co-ordinate * of the body is its angular distance from this circle. When the earth's equator is taken as the fundamental circle, this distance is on the earth's surface called Xa tan d, and the arc during which the star is above the horizon is 2 h. From this formula may be deduced at once many of the results given in the preceding sections. (I). At the poles = 90°, tan = infinity, and therefore cos h = infinity. But the cosine of an angle can never be greater than unity; there is therefore no value of h which fulfils the condition. Hence, a star at the pole can neither rise nor set. (2). At the earth's equator * = 0°, tan ^ = 0, whence cos A = 0, h = 90°, and 2 A = 180°, whatever be 6. This being a semicircum- ference all the heavenly bodies are half the time above the horizon to an observer on the equator. (3). If (5 = 0° (that is, if the star is on the celestial equator), then tan (! = 0, and cos A = 0, A = 90°, 2 ^ = 180°, so that all stars on the equator are half the time above the horizon, whatever be the lati- tude of the observer. Here we except the pole, where, in this case, tan 9 tan (5 = a X 0, an indeterminate quantity. In fact, a star on the celestial equator will, at the pole of the earth, seem to move round in the horizon. (4). The above value of cos h may be expressed in the form : Fig. 15. — upper and lower Drmi- NAL ARCS. cos A = — tan <5 cot l/) tan 6 tan (90° — This shows that when S lies outside the limits + (90° — 0) and _ (90° — ^\ cos Ti will lie without the limits — 1 and + 1, and there will be no value of h to correspond. Hence, in this case, the stars neither rise nor set. These limits correspond to those of per- petual apparition and perpetual disappearance. (5). In the northern hemisphere ^ and tan. ^ are positive.^ w^l' when <5 is positive, cos h is negative, and li > 90°, 2 A > 180". With 46 ABTRONOMT. negative i, cos A is positive, h < 90°, 2 A. < 180°. Hence, in north- ern latitudes, a northern star is more than half of the time ahove the horizon, and a southern star less. In the southern hemisphere, ^ and tan (p are negative, and the case is reversed. (6). If, in the preceding case, the declination of a body is supposed constant and north, then the greater we make (p the greater the nega- tive value of cos A and the greater A itself will be. Considering, in succession, the cases of north and south decli lation and north and south latitude, we leadily see that the farther we go to the north on the earth, the longer bodies of north declination remain above the horizon, and the more quickly those of south declination set. In the southern hemisphere the reverse is true. Thus, in the month of June, when the sun is north of the equal or, the days are shortest near the south pole, and continually increase in length as we go north. Examples. (1). On April 9, 1879, at Washington, the altitude of Eigel above the west horizon was observed to be 12° 25'. Its position -was: Right ascen-^ion = 5' 8" 44' -27 = a. Declination = — 8° 20' 86 " = sm 6 = sin a = 8 959560 091109 215020 Bin a — sin sin i = Ig cos = Ig cos (5 = 9 9 306129 891151 995379 Ig cos ^ cos c5 = Ig (sin a — sin sin t!, =: 9 9 886530 485905 Ig cos A = A = A ^ 15 = a = sidereal time = 9 66 4 5 9 599375 ° 34' 33" ' 26"" 18'.20 ' 8'°44'.27 '■35"' 2».47 (3). Had the star been observed at the same altitude in the east, what would have been the sidereal time ? Ans. a - A = O' 42-" 26'.07. DUrERMINAriON OF LATITUDE. 47 (3). At what sidereal time does Rigel rise, and at what sidereal time does it set in the latitude of Washington ? — tg (/i = — 9-906728 tg(i = — 9- 166301 cos A = — 9 . 073029 h = S3' 12' 19' 15 := a = 51- 32"' 49^27 5L 8"'44'.27 rises 23'' 35" sets 10'' 41" 55". 00 33'.54 (4). What is the greatest altitude of Rigel above the horizon of Washington, and what is its greatest depression below it V Ans. AItitude=42= 45' 45" ; (lepression=59' 26' 57'. (5). What is the greatest altitude of a star on the equator in the meridian of Washington ? Ans. 51" 6' 21" (6). The declination of the pointer in the Great Bear which is nearest the pole is 63' 30' N., at what altitude does it pass above the pole at Washmgton, and at what altitude does it pass below it V Ans. 66" 23' 39" above the pole, and 11" 23' 39' when below it. (7). If the dechnation of a star is 50° N., what length of sidereal time is it above the horizon of Washington and what length below it during its apparent diurnal circuit ? Ans. Above, 21'' 52'" ; below, 2'' 8"'- g 8. DETERMINATION OF LATITUDES ON THE EARTH BY ASTRONOMICAI. OBSERVATIONS. Latitude from clreumpolar »tars. — In Fig. 16 let iJ represent the zenith of the place of observation, Pthe pole, and HPZ R the me- ridian, the observer being at the centre of the sphere. Suppose /Sand S' to be the two pomts at which a circumpolar star crosses the meridian in tiie de- scription of its apparent diurnal orbit. Then, since P is midway between S and S', zs+zs „„ ■■ Z+ Z' = 90° - < If, then, we can measure the dis- tances Z and Z\ we have ;90° Z+ Z' Fig. which serves to determine f. The distances .Zand Z' can be meas- 48 ASTRONOMY. ured by the meridian circle or the sextant — both of which instru- ments are described in the next chapter — and the latitude is then known. Z and Z' must be freed from the effects of refraction. In this method no previous knowledge of the star's declination is re- (juired, provided it remains constant between the upper and lower transit, which is the case for fixed stars. Latitude by Circum-zenith Observations.— If two stars S and 8', whose declinations i and A' are known, cross the meridian, one north and the other south of the zenith, at zenith distances Z 8 and Z8\ which call Z and Z', and if we have measured Z and Z', we can from such measures find the latitude ; for (f ^=d -\- Z and = i' — Z', whence from a mcas- or, PAMALLAX. 49 ured altitude of a body of known declination. The last method is that commonly used at sea, the altitude being measured by the sextant. The student can deduce the formula for a north zenith-distance. § 9. PARALLAX AND SEMIDIAMETEE. An observation of the apparent position of a heavenly body can give onJy the directimi in which it Ues from the station occupied by the observer without any direct indi- cation of the distance. It is evident that two observers stationed in different parts of the earth will not see such a body in the same direction. In Fig. 18, let S' be a sta- Fia 18. — ^PARALLAX. tion on the earth, P a planet, Z' the zenith of S', and the outer arc a part of the celestial sphere. An observation of the apparent right ascension and declination of P taken from the station S' will give us an apparent position P'. A similar observation at S" will give an apparent position P", while if seen from the centre of the earth the appar- ent position would be P,. The angles P' P P, and P" P P^, which represent the differences of direction, are called parallaxes. It is clear that the parallax of a body depends upon its distance from the earth, being greater the nearer it is to the earth. The word parallax having several distinct applications, we shall give them in order, commencing with the most general signification. 50 ASTRONOMY. (1.) In its most general acceptation, parallax is the differ- ence between the directions of a body as seen from two different standpoints. This difference is evidently equal to the angle made between two lines, one drawn from each point of observation to the body. Thus in Fig. 18 the difference between the direction of the body P as seen from C and from S' is equal to the angle P' P P^, and this again is equal to its opposite angle S P C. This angle is, however, the angle between the two points G and S' as seen from P : we may therefore refer this most general definition of parallax to the body itself, and define parallax as the angle subtended by the line between two stations as seen from a heavenly body. (2.) In a more restricted sense, one of the two stations is supposed to be some centre of position from which we imagine the body to be viewed, and the parallax is the difference between the direction of the body from this centre and its direction from some other point. Thus the parallax of which we have just spoken is the differ- ence between the direction of the body as seen from the centre of the earth C and from a point on its surface as S'. If the observer at any station on the earth determines the exact direction of a body, the parallax of which we speak is the correction to be applied to that direction in order to reduce it to what it would have been had the ob- servation been made at the centre of the earth. Obser- vations made at different points on the earth's surface are compared by reducing them all to the centre of the earth. We may also suppose the point C to be the sun and the circle S' S" to be the earth's orbit around it. The paral- lax will then be the difference between the directions of the body as seen from the earth and from the sun. This is termed the annual parallax, because, owing to the an- nual revolution of the earth, it goes through its period in a year, always supposing the body observed to be at rest. (3.) A yet more restricted parallax is the horlscnital PAIiAZLAX. 51 pa/rallax of a heavenly body. The parallax first described in the last paragraph varies with the position of the ob- server on the surface of the earth, and has its greatest value when the body is seen in the horizon of the ob- server, as may be seen by an inspection of Fig. 19, in which the angle OPS attains its maximum when the line P S is tangent to the earth's surface, in which case P will appear in the horizon of the observer at S. Fig. 19. — ^HORIZONTAL PABAWiAX. The horizontal parallax depends upon the distance of a body in the following manner: In the triangle GPS, right-angled at S, we have CS= CP sin GPS. If, then, we put p, the radius of the earth els' ; r, the distance of the body P from the centre of the earth ; TT, the angle SPG, or the horizontal parallax, we shall have, P p ^:zi r sm n \ r = — ; sin n Since the earth is not perfectly spherical, the quantity p is not absolutely constant for all parts of the earth, and its greatest value is usually taken as that to which the hori- zontal value shall be referred. This greatest value is, as we shall hereafter see, the radius of the equator, and the 52 ASTRONOMY. corresponding value of the parallax is therefore called the equatorial horizontal i)arallax. When the distance r of the body is known, the equa- torial horizontal parallax can be found by the first of the above equations ; when the parallax can be observed, the distance r is found from the second equation. How this is done will be described in treating the subject of celes- tial measurement. It is easily seen that the equatorial horizontal parallax, or the angle C P 8, is the same as the angular semi- diameter of the earth seen from the object P. In fact, if we draw the line P 8' tangent to the earth at 8', the angle 8 P 8' will be the apparent angular diameter of the earth as seen from P, and will also be double the angle C P 8. The apparent semi-diameter of a heavenly body is therefore given by the same formulae as the parallax, its own radius being substituted for that of the earth. If we put, p, the radius of the body in linear measure ; r, the distance of its centre from the observer, expressed in the same measure.; 8, its angular semi-diameter, as seen by the observer ; we shall have, P sm s = — . r If we measure the semi-diameter s, and know the dis- tance, r, the radius of the body will be p = r sin s. Generally the angular semi-diameters of the heavenly bodies are so small that they may be considered the same as their sines. We may therefore say that the apparent angular diameter of a heavenly body varies inversely as its distance. CHAPTER II. ASTEONOMIOAL INSTEUMBNTS. § 1. THE REFRACTrNG TELESCOPE. In explaining the theory and use of the refracting tele- scope, we shall assume that the reader is acquainted with the fundamental principles of the refraction and disper- sion of Hght, so that the simple enumeration of them will recall them to his mind. These principles, so far as we have occasion to refer to them, are, that when a ray of light passing through a vacuum enters a trans- parent medium, it is refracted or bent from its course in a direction toward a line perpendicular to the sur- face at the point where the ray enters ; that this bend- ing follows a certain law known as the law of sines ; that when a pencil of rays emanating from a luminous point falls nearly perpendicularly upon a convex lene, the rays, after passing through it, all converge toward a point on the other side called a focus ; that light is com- pounded of rays of various degrees of refrangibility, so that, when thus refracted, the component rays pursue slightly different courses, and in passing through a lens come to shghtly different foci ; and finally, that the ap- parent angular magnitude subtended by an object when viewed from any point is inversely proportional to its distance. * * More exactly, in the case of a globe, the sine of the angle is in- versely as the distance of the object, as shown on the preceding page. 54 ASTRONOMT. We shall first describe the telescope in its simplest ^■I^^H form, showing the principles upon which H^^^l its action depends, leaving out of considera- l^^^l tion the defects of aberration which require special devices in order to avoid them. In the simplest form in which we can conceive g of a telescope, it consists of two lenses of I unequal focal lengths. The purpose of one ° of these lenses (called the objectime, or object % glass) is to bring the rays of light from a g distant object at which the telescope is ^ pointed, to a focus and there to form an fe image of the object. The purpose of the s other lens (called the eye-piec&) is to view ■& this object, or, more precisely, to form an- g other enlarged image of it on the retina of the eye. The figure gives a representation of the course of one pencil of the rays which go to fonn the image A T of an object / B after " passing through the objective 0' . The > pencil chosen is that composed of all the rays emanating from / which can possibly fall on the objective O 0'. All these are, by the action of the objective, concentrated at the point I'. In the same way each point of the image out of the optical axis A B emits an obhque pencil of diverging rays which are made to converge to some point ^^■^^1'^. of the image by the lens. The image of ^^Hlj^H (^ the point B of the object is the point A of ^^Hnl^B the image. We must conceive the image of ^^^H^H any object in the focus of any lens (or I^^^H^H mirror) to be formed by separate bundles ^^H||^| of rays as in the figure. The image thus ^HBh^B formed becomes, in its tuni, an object to be viewed by the eye-piece. After the rays meet to form o I o g p: o C o H o < MAGNIFYING POWER OF TELESCOPE. 55 the imago of an object, as "at T, they continue on their course, diverging from T as if the latter were a material object reflecting the light. There is, however, this excep- tion : that the rays, instead of diverging in every direction, only form a small cone having its vertex at T, and having its angle equal to T 0'. The reason of this is that only those rays which pass through the objective can form the image, and they must continue on their course in straight lines after fonning the image. This image can now be viewed by a lens, or even by the unassisted eye, if the observer places himself behind it in the direction A, so that the pencil of rays shall enter his eye. For the pres- ent we may consider the eye-piece as a simple lens of short focus like a common hand-magnifier, a more com- plete description being given later. Magnifying Power.— To understand the manner in which the telescope magnifies, we remark that if an eye at the object-glass could view the image, it would appear of the same size as the actual object, the image and the object subtending the same angle, but lying in opposite direc- tions. This angular magnitude being the same, whatever the focal distance at which the image is fonned, it follows that the size of the image varies directly as the focal length of the object-glass. But when we view an object with a lens of small focal distance, its apparent magnitude is the same as if it were seen at that focal distance. Consequently the apparent angular magnitude will be inversely as the focal distance of the lens. Hence the focal image as seen with the eye-piece will appear larger than it would when viewed from the objective, in the ratio of the focal distance of the objective to that of the eye-piece. But we have said that, seen through the objective, the image and the real object subtend the same angle. Hence the angu- lar njagnifying power is equal to tlie focal distance of the objective, divided by that of the eye piece. If we simply turn the telescope end for end, the objective becomes the eye-piece and the latter the objective. The ratio is in- 56 ASTRONOMY. verted, and the object is dimmished in size in the same ratio that it is increased when viewed in the ordinary waj. If we should form a telescope of two lenses of equal focal length, by placing them at double their focal distance, it would not magnify at all. The image formed by a convex lens, being upside down, and appearing in the same position when viewed with the eye-piece, it follows that the telescope, Avhen constructed in the simplest manner, shows all objects in- verted, or upside down, and right side left. This is the case with all refracting telescopes made for astronomical uses. Light-gathering Power.— It is not merely by magnify- ing that the telescope assists the vision, but also by in- creasing the quantity of light which reaches the eye from the object at which we look. Indeed, should we view an object through an instrument which magnified, but did not increase the amount of hght received by the eye, it is evident that the brilliancy would be diminished in propor- tion as the surface of the object was enlarged, since a con- stant amount of hght would be spread over an increased surface ; and thus, unless the Hght were faint, the object might become so darkened as to be less plainly seen than with the naked eye. How the telescope increases the quantity of hght will be seen by considering that when the unaided eye looks at any object, the retina can only re- ceive so many rays as fall upon the pupil of the eye. By the use of the telescope, it is evident that as many rays can be brought to the retina as fall on the entire object- glass. The pupil of the human eye, iu its normal state, has a diameter of about one fifth of an inch ; and by the use of the telescope it is virtually increased in surface in the ratio of the square of the diameter of the objective to the square of one fifth of an inch. Thus, with a two-inch aperture to our telescope, the number of rays collected is one hundred times as great as the number collected with the naked eye. POWER OF TELESCOPE. 57 With a 5-inch object-glass, the ratio is 625 to 1 " 10 " '"' " " " 2,500 to 1 " 16 " " " " " 5,625 to 1 " 20 " " " " " 10,000 to 1 " 26 " " " " " 16,900 to 1 AVhen a minute object, like a star, is viewed, it is necessary that a certain number of rays should fall on the retina in order that the star may be visible at all. It is therefore plain that the use of the telescope enables an observer to see much fainter stars than he could detect with the naked eye, and also to see faint objects much better than by unaided vision alone. Thus, with a 26- inch telescope we may see stars so minute that it would require many thousands to be visible to the unaided eye. An important remark is, however, to be made here. Inspecting Fig. 20 we see that the cone of rays passing through the object-glass converges to a focus, then diverges at the same angle in order to pass through the eye-piece. After this passage the rays emerge from the eye-piece parallel, as shown in Fig. 22. It is evident that the diameter of this cyhnder of parallel rays, or " emergent pencil," as it is called, is less than the diameter of the object-glass, in the same ratio that the focal length of the eye-piece is less than that of the object-glass. For the central ray II' is the common axis of two cones, A I and I 0', having the same angle, and equal in length to the respective focal distances of the glasses. But this ratio is also the magnifying power. Hence the diameter of the emergent pencil of rays is foimd by dividing the diameter of the object-glass by the magnifying power. Now it is clear that if the magnifying power is so small that this emergent pencil is larger than the pupil of the eye, all the hght which falls on the object-glass cannot enter the pupil. This will be the case whenever the magnifying power is less than five for every inch of aperture of the glass. If, for example, the observer should 58 ASTRONOMY. look througli a twelve-inch telescope with an eye-piece so large that the magnifying power was only 30, the emergent pencil would be two fifths of an inch in diam- eter, and only so much of the Ught could enter the pupil as fell on the central six inches of the object-glass. Practically, therefore, the observer would only be using a six-inch telescope, all the light which fell outside of the six-inch circle being lost. In order, therefore, that he may get the advantage of all Us object-glass, he must use a magnifying power at least five times the diameter of his objective in inches. When the magnifying power is carried beyond this limit, the action of a telescope will depend partly on the nature of the object one is looldng r.t. Viewing a star, the increase of power will give no increase of light, and therefore no increase in the apparent brightness of the star. If one is looking at an object having a sensible surface, as the moon, or a planet, the light coming from a given portion of the surface A^ill be spread over a larger portion of the I'etina, as the magnifying power is increased. xUl magnifj-ing must then be gained at the expense of the apparent illumination of the surface. Whether this loss of illumination is important or not wiU depend entirely on how much light is to spare. In a general way we may say that the moon and all the plan- ets nearer than Saturn are so brilliantly iRuminated by the sun that the magnifying power can be carried many times above the hmit without any loss in the distinctness of vision. Tlie Telescope in Measurement. — A telescope is gen- erally thought of only as an instrument to assist the eye by its magnifying and light-gathering power in the man- ner we have described. But it has a very important additional function in astronomical measurements by en- abling the astronomer to point at a celestial object with a certainty and accuracy otherwise unattainable. This func- tion of the telescope was not recognized for more than ITHE OP TELESCOPE. 59 half a century after its invention, and after a long and rather acrimonious contest between two schools of astron- omers. Until the middle of the seventeenth century, when an astronomer wished to determine the altitude of a celestial ol^ject, or to measure the angular distance be- tween two stars, he was obliged to point his quadrant or other measuring instrument at the object by means of ' ' pinnules. ' ' These served the same purpose as the sights on a rifle. In using them, however, a difiiculty arose. It was impossible for the observer to have distinct vision both of the object and of the pinnules at the same time, because when the eye Avas focused on either pinnule, or on the object, it was necessarily out of focus for the others. The only way to diminish this difficulty was to lengthen the arm on which the pinnules were fastened so that the latter should be as far apart as possible. Thus Tycho Beahe, before the year 1600, had measuring in- struments very much larger than any in use at the pres- ent time. But this plan only diminished the difficulty and could not entirely obviate it, because to be manageable the instrument must not be very large. About 1670 the English and French astronomers found that by simply inserting line threads or wires exactly in the focus of the telescope, and then pointing it at the ob- ject, the image of that object formed in the focus could be made to coincide with the threads, so that the observer could see the two exactly superimposed upon each other. "When thus brought into coincidence, it was known that the point of the object on which the wires were set was in a straight line passing through the wires, and through the centre of the object-glass. So exactly could such a point- ing be made, that if the telescope did not magnify at all (the eye-piece and object-glass being of equal focal length), a very important advance would still be made in the ac- curacy of astronomical measurements. This line, passing centrally through the telescope, we call the line of col- limation of the telescope, ^ ^ in Fig. 20. If we have 60 ASTRONOMY. any way of determmiug it we at once realize the idea ex- pressed in the opening chapter of this book, of a pencil ex- tended in a definite direction from the earth to the heav- ens. If the observer simply sets his telescope in a fixed position, looks through it and notices what stars pass along the threads in tlie eye-piece, he knows that those stars all lie in the line of coUimation of his telescope at that instant. By the diurnal motion, a pencil-mark, as it were, is thus being made in the heavens, the direction of which can be determined with far greater precision than by any meas- urements with the unaided eye. The direction of this line of coUimation can be determined by methods which we need not now describe in detail. The Achromatic Telescope. — The simple form of tele- scope which we have described is rather a geometrical conception than an actual instrument. Only the earli- est instruments of this class ware made with so few as two lenses. Galileo's telescope was not made in the form which we have described, for instead of two convex lenses having a common focus, the eye-piece was concave, and was placed at the proper distance inside of the focus of the objective. This form of instrument is still tised in opera- glasses, but is objectionable in large instruments, owing to the smallness of the field of view. The use of two con- vex lenses was, we believe, first proposed by Kepler. Although telescopes of this simple form were wonderful instrnmsnts in their day, yet they would not now be re- garded as serving any of the purposes of such an instru- ment, owing to the aberrations with which a single lens is affected. We know that when ordinary light passes through a simple lens it is partially decomposed, the differ- ent rays coming to a focas at difEerent distances. The focus for red rays is most distant from the object-glass, and that for violet rays the nearest to it. Thus arises the chromatio aberration of a lens. But this is not all. Even if the light is but of a single degree of refrangi- bility, if the surfaces of our lens are spherical, the rays ACHHOMATIG OBJECT-GLASS. CI which pass near the edge will come to a shorter focus than those which pass near the centre. Tims arises spherical aberration. This aberration might be avoided if lenses could be ground with a proper gradation of curvature from the centre to the circumference. Prac- tically, however, this is impossible ; the deviation from uniform sphericity, which an optician can produce, is too small to neutralize the defect. Of these two defects, the chromatic aberration is much the more serious ; and no way of avoiding it was known until the latter part of the last century. The fact had, indeed, been recognized })y mathematicians and physicists, that if two glasses could bo found having very different ratios of refractive to dispersive powers,* the defect could be cured by combining lenses made of these different kinds of glass. But this idea was not realized imtil the time of DoLLOND, an English optician who lived during the last century. This artist found that a concave lens of flint glass could be combined with a convex lens of crown of double the curvature in such a manner that the dispersive powers of the two lenses should neutrahze each other, being equal and acting in opposite di- rections. But the crown glass having the greater refractive power, owng to its greater cur- vature, the rays would be brought to a focus without dispersion. Such is the construction of the Fig. 31.— section of object- achromatic objective. As now glass. made, the outer or crown glass lens is double convex ; the inner or flint one is generally nearly plano-concave. Fig. 21 shows the section of such an objective as made by Alvan Clark & Soxs, the inner curves of the crown and flint being nearly equal. *By the refractiix powev of a glass is meant its power of bending the rays out of tlxeir course, so as to bring tliem to a focus. By its disfper- sive power is meant its power of separating the colors so as to form a spectnim, or to produce chromalic aberration. C2 ASTRONOMY. A great advantage of the aeliromatic objective is that it may be made to correct the spherical as well as the chro- matic aberration. This is effected by giving the proper curvatui'e to the various surfaces, and by making such slight deviations from perfect sphericity that rays passing through all parts of the glass shall come to the same focus. The Secondary Spectrum. — It is now known that the chromatic aberration of an objective cannot be perfectly corrected with any combination of glasses yet discovered. In the best telescopes the lu-ightest rays of the spectrum, which are the yellow and green ones, are all brought to the same focus, but the red and blue ones reach a focus a little farther from the objective, and the violet ones a focus still farther. Hence, if we look at a bright star through a large telescope, it will be seen surrounded by a blue or violet light. If we push the eye-piece in a little the enlarged image of the star will be yellow in the centre and purple around the border. This separation of colors by a pair of lenses is called a secondary spectrum. Eye-Piece. — In tlie skeleton form of telescope before described the eye-piece as well as the objective was con- sidered as consisting of but a single lens. But with such an eye-piece vision is imperfect, except in the centre of the field, from the fact that the image does not throw rays in every direction, but only in straight lines away from the objective. Hence, the rays from near the edges of the focal image fall on or near the edge of the eye- piece, whence arises distortion of the image formed on the retina, and loss of light. To remedy this difficulty a lens is inserted at or very near the place where the focal image is formed, for the purpose of throwing the different pencils of rays which emanate from the several parts of the image toward the axis of the telescope, so that they shall all pass nearly through the centre of the eye lens pro- per. Tliese two lenses are together called the eye-piece. There are some small differences of detail in the con- struction of eye-pieces, but tiie general principle is the THEORY OP OBJEGT-GLASS. 63 same in all. The two recognized classes are tlie posi- tive and negative, the former being those in which the image is formed before the light reaches the field lens ; the negative those in which it is formed between the lenses. The figure shows the positive eye-piece drawn accurately to scale. / is one of the converging pencils from the object-glass which forms one point (J) of the focal image la. This image is viewed by the field lens F oi the eye-piece as a real object, and the shaded pencil between F and E shows the course of these rays after de- viation by F. If there were no eye-lens E an eye properly placed beyond F would see an image at /' a'. The eye-lens E receives the pencil of rays, and deviates it to the observer's eye placed at such a point that the whole incident pencil will pass through the pupil and fall on the retina, and thus be effective. As we saw in the SECTION OF A POSITTVE KTE-PIKCE. figure of the refracting telescope, every point of the object produces a pencil similar to /, and the whole surfaces of the lenses F and E are covered with rays. All of these pencils passing through the pupil go to make up the retinal image. This image is referred by the mind to the distance of distinct vision (about ten inches), and the image A I" represents the dimension of the final image relative to the image a F as formed by the objective and — =- is evidently the magnifying power of this particular eye-piece used in combination with this particular objective. More Exact Theory of the Objective.— For the benefit of the reader who wishes a more precise knowledge of the optical princi- ples on which the action of the objective or other system of lenses depends, we present the following geometrical theory of the sub- ject. This theory is not rigidly exact, but is sufficiently so for all ordinary computations of the focal lengths and sizes of image in the usual combinations of lenses. 64 ASTRONOMY. Centres of Convergence and Divergence.— Suppose A B, Fig. 23, to be a lens or combination of lenses on which the light falls from the left hand and passes through to the right. Suppose rays parallel to i? P to fall on every part of the first surface of the glass. After passing through it they are all supposed to converge nearly or ex- actly to the same point R. Among all these rays there is one, and one "only, the course of vchich, after emerging from the glass at Q, will be parallel to its original direction RF. het R P Q R" he this central ray, which will bo completely determined by the direction from which it comes. Next, let us take a ray coming from another direction as S P. Among all the rays parallel to 8 P, let us take that one which, after emerging from the glass at T, moves in a line parallel to its original direction. Continuing the process, let us suppose isolated rays coming from all parts of a distant object sub- ject to the single condition that the course of each, after passing through the glass or system of glasses, shall be parallel to its original course. These rays we may call central rays. They have this re- markable property, pointed out by Gauss : that they all converge Fig. 33. toward a single point, P, in coming to the glass, and diverge from another point, P", after passing through the last lens. These points were termed by Gauss " Hauptpunkte, " or principal points. But they will probably be better understood if we call the first one the centre of convergence, and the second the centre of divergence. It must not be understood that the central rays necessarily pass through these centres. If one of them lies outside the first or last refracting surface, then the central rays must actually pass through it. But if they lie between the surfaces, they will be fixed by the continuation of the straight line in which the rays move, the latter being refracted out of their course by passing through the surface, and thus avoiding the points in question. If the lens or system of lenses be turned around, or if the light passes through them in an opposite direction, the centre of convergence in the first case be- comes the centre of divergence in the second, and nice term. The necessity of this will be clearly seen by reflecting that a return ray of light will always keep on the course of the original ray in the opposite direction. THEORY OF OBJECT-GLASS. 65 The figure represents a plano-convex lens with light falling on the convex side. In this case the centre of convergence will be on the convex surface, and that of divergence inside the glass about one third or two fifths of the way from the convex to the plane surface, the positions varying with the refractive index of the glass. In a double convex lens, both points will lie inside the glass, while if a glass is concave on one side and convex on the other, one of the points will be outside the glass on the concave side. It must be remembered that the positions of these centres of conver- gence and divergence depend solely on the form and size of the lenses and their refractive indices, and do not refer in any way to the distances of the objects whose images they form. The principal properties of a lens or objective, by which the size of images are determined, are as follows : Since the angle S' P S made by the diverging rays is equal to R P S, made by the con- verging ones, it follows, that if a lens form the image of an object, the size of the image will be to that of the object as their respec- tive distances from the centres of convergence and divergence. In other words, the object seen from the centre of convergence P will be of the same angular magnitude as the image seen from the centre of divergence P'- Bj' conjugate foci of a lens or system of lenses we mean a pair of points such that if rays diverge from the one, they will converge to the other. Hence if an object is in one of a pau- of such foci, the image will be formed in the other. By the refractive power of a lens or combination of lenses, we mean its influence in refracting parallel rays to a focus which we may measure by the reciprocal of its focal distance or 1 -=-/. Thus, the power of a piece of plain glass is 0, because it cannot bring rays to a focus at all. The power of a convex lens is positive, while that of a concave lens is negative. In the latter case, it will be remembered by the student of optics that the virtual focus is on the same side of the lens from which the rays proceed. It is to be noted that when we speak of the focal distance of a lens, we mean the distance from the centre of divergence to the focus for parallel rays. In astronomical language this focus is called the stellar focus, being that for celestial objects, all of which we may regard as infinitely distant. If, now, we put p, the power of the lens ; /, its stellar focal distance ; f, the distance of an object from the centre of convergence ; /', the distance of its image from the centre of divergence ; theu the equation which determines/ will be f + f'-f -P! or, f / / fi ^ ' ^J J-f+f'^-' f-f Prom these equations may be found the focal length, having the distance at which the image of an object is formed, or vice versa. 66 ASTJiONOMT. I 2. REFLECTING TELESCOPES. As we have seen, the most essential part of a refracting telescope is the objective, which brings all the incident rays from an object to one focus, forming there an image of thdu object. In reilecting telescopes (reflectors) the objective is a mirror of speculum metal or silvered glass ground to the shape of a paraboloid. The figure shows the action of such a mirroi on a bundle of parallel rays, which, after impinging on it, are brought by reflection to one focus ^. The image formed at this focus may be viewed with an eye-piece, as in the case of the refracting telescope. The eye- pieces used with such a mirror are of the kinds already described. In the flgure the eye-piece would Pig. 24. concave mirror forming an image. have to be placed to the right of the point F, and the observer's head would thus interfere with the incident light. Various devices have been proposed' to remedy this inconvenience, of which we will describe the two most common. The Newtonian Telescope. — In this form the rays of light reflected from the mirror are made to fall on a small plane mirror placed diagonally just before they reach the principal focus. The rays are tlius reflected out laterally through an opening in the telescope tube, and are there brought to a focus, and the image formed at the point marked by a heavy white line in Fig. 25, instead of at the point inside the telescope marked by a dotted line. REFLECTING TELESCOPES. 67 This focal image is then examined by means of an or- dinary eye-piece, the head of the observer being outside of the telescope tube. This device is the invention of Sir Isaac I^ewton. ^^-j^ % Pig. 25. newtonian tblescope. Pig. 26. cassegkainian telescope. The Cassegrainian Telescope. — In this form a second- ary convex mirror is placed in the tube of the telescope 68 ASTRONOMY. about three fourths of the way from the large speculum to the focus. The rays, after being reflected from the large speculum, fall on this mirror before reaching the focus, and are reflected back again to the speculum ; an opening is made in the centre of the latter to let the rays pass through. The position and cm-vature of the secondary mirror are adjusted so that the focus shall be formed just after passing through the opening in the speculum. In this telescope the observer stands behind or under the speculum, and, with the eye-piece, looks through the opening in the centre, in the direction of the object. This form of reflector is much more convenient in use than the Newtonian, in using which the observer has to be near the top of the tube. This form was devised by Cassegeain in 1672. The advantages of reflectors are found in their cheap- ness, and in the fact that, supposing the mirrors perfect in figure, all the rays of the spectrum are brought to one focus. Thus the reflector is suitable for spectroscopic or photographic researches Mdthout any change from its or- dinary form. This is not true of the refractor, since the rays by which we now photograph (the blue and violet rays) are, in that instrument, owing to the secondary spectrum, brought to a focus slightly dijEferent from that of the yellow and adjacent rays by means of which we see. Reflectors have been made as large as six feet in aper- ture, the greatest being that of Lord Eosse, but those which have been most successful have hardly ever been larger than two or three feet. The smallest satellite of Saturn {Mimas) was discovered by Sir William Heeschel with a four-foot speculum, but all the other satellites dis- covered by him were seen with mirrors of about eighteen inches in aperture. "With these the vast majority of his faint nebulae were also discovered. The satellites of JVeptune and Uranus were discovered by Lassell with a two-foot speculum, and much of the REFLECriNO TELESCOPES. 69 work of Lord Kosse has been done with his three-foot mirror, instead of his celebrated six-foot one. From the time of Newton till qnite recently it was nsual to make the lai-gc mirror or objective out of specu- Inm metal, a brilliant alloy liable to tarnish. When the jnirror was once tarnished through exposure to the weather, it coxild be renewed oidy by a process of polish- ing almost equivalent to figuring and polishing the mirror anew. Consequently, in such a speculum, after the cor- rect form and polish were attained, there was great diffi- culty in preserving them. In recent years this difficulty has been largely overcome in two ways : first, by im- provements in the composition of the alloy, by which its liability to tarnish under exposure is greatly diminished, and, secondly, by a plan proposed by FoccArLT, which consists in making, once for all, a mirror of glass which will always retain its good figure, and depositing upon it a thin film of silver which may be removed and restored with little labor as often as it becomes tarnished. In this way, one important defect in the reflector has been avoided. Another great defect has been less success- fully treated. It is not a process of exceeding difficulty to give to the reflecting surface of either metal or glass the correct pai'abolic shape by which the incident rays are brought accurately to one focus. But to maintain this shape constantly when the mirror is mounted in a tube, and when this tube is directed in succession to various parts of the sky, is a mechanical problem of extreme diffi- culty. However the mirror may bo supported, all the tmsupported points tend by their weight to sag away from the proper position. "When the mirror is pointed near the horizon, this effect of flexure is quite different from what it is when pointed near the zenith. As long as the mirror is small (not greater than eight to twelve inches in diameter), it is found easy to support it so that these variations in the strains of flexure have little practical effect. As we increase its diameter up to 4S or 70 ASTRONOMY. 1'2i inches, the effect of flexure rapidly increases, and special devices have to be used to counterbalance the injury done to the shape of the mirror. § 3. CHEOIfOMETEES AU"D CLOCKS. In Chapter I., § .5, we described how the right ascen- sions of the heavenly bodies are measured by the times of their transits over the meridian, this quantity increas- ing by a minute of arc in four seconds of time. In order to determine it with all required accuracy, it is necessary that the time-pieces with which it is measured shall go with the greatest possible precision. There is no great difficulty in making astronomical measures to a second of arc, and a star, by its diurnal motion, passes over this space in one fifteenth of a second of time. It is there- fore desirable that the astronomical clock shall not vary from a uniform rate more than a few hundredths of a second in the course of a day. It is not, however, necessary that it should be perfectly correct ; it may go too fast or too slow without detracting from its char- acter for accuracy, if the intervals of time which it tells off — hours, minutes, or seconds — are always of ex- actly the same length, or, in other words, if it gains or loses exactly the same amount every hour and every day. The time-pieces used in astronomical observation are the chronometer and the clock. The chronometer is merely a very perfect time-piece with a balance-wheel so constructed that changes of tem- perature have the least possible effect upon the time o"f its oscillation. Such a balance is called a compensation bal- ance. The ordinary house clock goes faster in cold than in warm weather, because the pendulum rod shortens under the influence of cold. This effect is such that the clock will gain about one second a day for every fall of 3° Cent. (5°. 4 Fahr.) in the temperature, supposing the pendulum THE ASTRONOMICAL CLOCK. 71 rod to be of iron. S-ucli cliauges of rate would be entirelj inadmissible in a clock used for astronomical purposes. The astronomical clock is therefore provided with a com- pensation pendulum, by which the disturbing effects oi changes of temperature are avoided. • There are two forms now in use, the Harrison {grid- iron) and the mercurial. In the gridiron pendulmn the rod is composed in part of a number of parallel bars of steel and brass, so connected together that while the expansion of the steel bars produced by an increase of temperature tends to depress the lob of the pendulum, the greater expansion of the brass bars tends to raise it. When the total lengths of the steel and brass bars have been properly adjusted a nearly perfect compensation occurs, and the centre of oscillation remains at a con- stant distance from tlie point of sus- pension. The rate of the clock, so far as it depends on the length of the pendulum, will therefore be constant. In the mercurial pendulum the weight which forms the bob is a cylindric glass vessel nearly filled with mercury. With an increase of temperature the steel suspension rod lengthens, thus throwing the centre of oscillation away from the point of suspension ; at the same time the expanding mercury rises in the cylinder, and tends therefore to raise the centre of oscillation. When the length of the rod and the dimensions of the cylinder of mercury are properly proportioned, the centre of oscillation is kept at a constant distance from the point of suspension. Other methods of making this compensa- tion have been used, but these are the two in most gen- eral use for astronomical clocks. Fig. 27. — gridiron pendulum:. 72 ASTRONOMY. The correction of a chronometer (or clock) is the quantity of time (expressed in hours, minutes, seconds, and decimals of a second) which it is necessary to add iilgebiaically to the indication of the hands, in order that the sum may be the correct time. Thus, if at sidereal 0'', May 18, at New Yoik, a sidereal clock or chronometer indicates 23" 58'" SC-T, its coir.ction is + 1'" 39'-3; if atO'' (sidereal noon), of May 17, its correction was + 1'" 38' -3, its daily rate or the change of its correction in a sidereal day is + I'O: in other words, this clock is loning 1'' daily. For clock slow the sign of the correction is -|- ; " " fant " " " " " is—; " " gaining" " " " rate is — ; " losing " '' '• " " is + . A clock or chronometer may be well compensated for temperature, and yet its rate may be gaining or losing on the time; it is intended to keep : it is not even necessary that the rate should be small (ex- cept that a small rate is practically convenient), provided only that it is constant. It is continually necessary to compute the clock cor- rection at a given time from its known correction at some other time, and its known rate. If for some definite instant we denote the time as shown by the clock (technically "the clock-face") by T, the truo time by T' and the clock correction by a T, we have r = T + aT. and A r = T - T. In all observatories and at sea observations are made daily to de- termine A T. At the instant of the observation the time T \& noted by the clock ; from the data of the observation the time T' is com- puted. If these agree, the clock is correct. If they differ, A T is found from the above equations. If by observation we have found ATo = the clock correction at a clock-time jf'o, A T = the clock correction at a clock-time T, (jy = the clock rate in a unit of time, we have A T= AT„ + dT{T— Tu) where T — To must be expressed in days, hours, etc., according as c5 T is the rate in one daj', one hour, etc. When, therefore, the clock correction A To and rate 6T have been determined for a certain instant, To, we can deduce the true time fi om the clock-face 2' at any other instant by the equation T' = T + A.T-, + AT{T— 1\). If the clock correction has been deter- mined at two different times, ?'u and T to be A To and A T, the rate is inferred from the equation AT-ATo ^1 =-j,--y,- • THE ASTliONOMWAL CLOCK. Ti These equations apply onl}' so long as we can regard the rate as constant. As observations can be made only in clear weather, it is plain that during periods of overcast sky we must depend on these equations for our knowledge of T — i.e., the true time at a clock- time T. The intervals between the determination of the clock correction should be small, since even with the best clocks and chronometeis too much dependence must not be placed upon the rate. The follow- ing example from Chauvenet's Astronomy will illustrate the practi cal processes : " Exnmple. — At sidereal noon, May 5, the correction of a sidereal clock is— Ifl'" 47-0; at sidereal noon, May 12, it is — 16"' IS'-oO; what is the sidereal time on May 25, when the clock-face is 11'' 13'" 12' -6, supposing the rate to be uniform ? May 5, correction = — 16'»47".30 " 12, '• = — 16'" 13". 50 7 days' rate = + SS'-dO 6T= + 4'-829. Taking then as our starting-point T« = May 12, 0'', we have for the interval to 7'= May 25. 1 !'■ 13"' 12'- 6, T — r„ = 13'' 11'' 13'" 12' -6 = 13''-467. Hence we have A 2"„ = - 16'" 18' -50 dT(T- r„) = + 1'" 5' -03 A r= 15'" 8'-4r T= 11'' 13'" 12'-60 r = J0''~58'"^^"i3 But in this example the rate is obtained for one true sidereal day, while the unit of the interval 13'' -467 is a sidereal day as shown by the clock. The proper interval with which to compute the rate in this case is 13'' 10'' 58"' 4''- 18 = 13''-457, with which we tind A r„ = — 16" 13' dT X 13-457= + 1'" 4« A r = — 15'>' 8' 2' = 11'' 13'" 12" T' = IC- 58'" 4' 50 98 52 60 08 This repetition will be rendered unnecessary by always giving the rate in a. unit of the clock. Thus, suppose that on June 3, at 4" 11"' 12'- -35 by the clock, we have found the correction -I- 2"i 10' -14; and on June 4, at 14'' 17"' 49'. 82 we have found the correction + 2"" 19'-89; the rate in one hour of tie dock will be ''^=^04=+°'-^«^«-" U ASTJiOA'OMY. % 4. THE TRAIfSIT nsrSTHUMEIfT. The meridian trcmait instrument, or Lriefly tlie " tran- sit," is used to observe the transits of the heavenly bodies. Fig. 28. — a transit rNSTKUMENT. and from the times of these tra:nsits as read from the clock to determine either the corrections of the clock or the right ascension of the observed body, as explained in Chapter I., §5. THE TRANSIT INSTRUMENT. 75 It lias two general forms, one (Fig. 28) for use in lixel observatories and one (Fi^. 29) fornse in tbe field. It consists essentially of a telescope T T (Fig. 28) mounted on an axis V Fat right angles to it. —I jJ aJjEYl Fig. 29.— portable transit instrument. The ends of this axis terminate in accurately cylindrical steel pivots which rest in metallic bearings V F, in shape like the letter Y, and hence called the Ys. 76 A8TU0N0MT. These are fastened to two pillars of stone, brick, or iron. Two counterpoises W W are connected with the axis as in the plate, so as to take a large portion of the weight of the axis and telescope from the Ys, and thus to diminish the friction upon these and to render the rota- tion about V V more easy and regular. In the ordinary use of the transit, the line V Y\s placed accurately level and perpendicular to the meridian, or in the east and west hne. To effect this " adjustment," there are two sets of adjusting screws, by which the ends of F T^ in the Ys may be moved either up and down or north and south. The plate gives the form of transit used in permanent observa- tories, and shows the observing chair C, the reversing car- riage R, and the level L. The anns of the latter have Y's, which can be placed over the pivots V V. The line of collimation of the transit telescope is the line drawn through the centre of the objective perpendic- ular to the rotation axis V V. The reticle is a network of fine spider lines placed in the focus of the objective. In Fig. 30 the circle represents the field of view of a transit as seen thi'ough the eye-piece. The seven ver- tical lines, I, II, III, lY, Y, YI, YIl, are seven fine spider lines tightly stretched across a metal plate or diaphragm, and so adjusted as to be perpendicular to the direction of a star's apparent diurnal motion. This metal plate can be moved right and left by five screws. The hori- zontal wires, guide-wires, a and h, mark the centre of the field. The field is illuminated at night by a lamp at the end of the axis which shines through the hollow interior of the lat- ter, and causes the field to appear bright. The wires are dark against a bright ground. The line of sight is a line joining the centre of the objective and the central one, lY, of the seven vertical wires. THE TRANSIT INSTRUMENT. 11 The whole transit is in adjustment when, first, the axis F F is horizontal ; second, when it lies east and west ; and third, when the line of sight and the line of collima- tion coincide. When these conditions are fulfilled the line of sight intersects the celestial sphere in the meridian of the place, and when T Tis. rotated about V V the line of sight marlis out the meridian on the sphere. In practice the three adjustments are not exactly made, since it is impossible to effect them with mathematical precision. The errors of each of them are first made as small as is convenient, and are then determined and allowed for. To find the error of level, we place on the pivots a fine level (shown in position in the figure of the portable transit), and determine how much higher one pivot Is than the other in terms of the divisions marked on the level tube. Such a level is shown in Fig. 4 of plate 36, page 86. The value of one of these divisions in seconds of arc can be determined by knowing the length I of the wliole level and the number n of divisions through which the bubble will run when one end is raised one hundredth of an inch. If I is the length of the level in inches or the radius of the circle in which either end of the level moves when it is raised, then as the radius of any circle is equal to 57° -396, 3437'- 7.5 or 206, 264" -8, we have in this particular circle one inch = 206,264" -8 -i- Z ; 0-01 inch = 206,264-8 -^ 100 I = a, certain arc in seconds, say a". That is, n divisions =: a", or one division d = a" -r- n. The error of collimation can be found by pointing the telescope at a distant mark whose image is brought to the middle wire. The telescope (with the axis) is then lifted bodily from the Ys and re- placed so that the axis V Fis reversed end for end. The telescope is again pointed to the distant mark. If this is still on the middle thread the line of sight and the line of collimation coincide. If not, the reticle must be moved bodily west or east until these conditions are fulfilled after repeated reversals. To find the error of azimuth or the departure of the direction of FFfrom an east and west line, we must observe the transits of two stars of different declinations i and S' and right ascensions a and or'. Suppose the clock to be running correctly— that is, with no rate — and the sidereal times of transit of the two stars over the mid- dle thread to be f) and i9'. If — (/ = a — a', then the middle wire is in the meridian and the azimuth is zero. For if the azimuth was not zero, but the west end of the axis was too far south, for example, the line of sight would fall east of the meridian for a south star, and further and further east the further south the star was. Hence if the two stars have widely different declinations 6 and (5', then the star furthest south would come proportionately sooner to the middle wire than the other, and 9—8' would be different from a — a'. The amount of this difference gives a 7S ASTRONOMY. means of deducing the deviation oi A A from an east and west tine. In a similar way the effect of a given error of level on the time of the transit of a star of declination 6 is found. Methods of Observing with the Transit Instrument.— We liave so far assumed that the time of a star's transit over the middle thread was known, or could be noted. It is necessary to speak more in detail of how it is noted. "When the telescope is pointed to any star the earth's diurnal motion will carry the image of the star slowly across the field of view of the telescope (which is kept fixed), as before explained. As it crosses each of the threads, the time at which it is exactly on the thread is noted from the clock, which must be near the transit. The mean of these times gives the time at which this star was on the middle thread, the threads being at equal intervals ; or on the " mean thread," if, as is the case in practice, they are at imequal intervals. If it were possible for an astronomer to note the exact instant of the transit of a star over a thread, it is plain that one thread would be suflScient ; but, as all estima- tions of this time are, from the very nature of the case, but aj)proximations, several threads are inserted in order that the accidental errors of estimations may be eliminated as far as possible. Five, or at most seven, threads are suflicient for this purpose. In the figure of the reticle of a transit instru- ment the star (the planet Vemos in this case) may enter on the right hand in the figure, and may be supposed to cross each of the wires, the time of its tran- sit over each of them, or over a sufii- P gj cient number, being noted. The method of noting this time may be best understood by referring to the next figure. Suppose that the line in the middle of Fig. 32 is one of the transit- threads, and that the star is passing from the right hand of the figure toward the left ; if it is on this wire at an Fig. 32. THE TRANSIT INSTliVMENT. 70 exact second by tlie clock (which is always near the ob- server, beating seconds audibly), this second must be writ- ten down as the time of the transit over this thread. As a rule, however, the transit cannot occur on the exact beat of the clock, but at the seventeenth second (for exam- ple) the star may be on the right of the wire, say at a ; while at the eighteenth second it will have passed this wire and may be at h. If the distance of a from the wire is six tenths of the distance ah, then the time of transit is to be recorded as — hours — minutes (to be taken from the clock-face), and seven- teen and six tenths seconds ; and in this way the transit over each wire is observed. This is the method of " eye- and-ear" observation, the basis of such work as we have described, and it is so called from the part which both the eye and the ear play in the appreciation of intervals of time. The ear catches the beat of the clock, the eye fixes the place of the star at a ; at the next beat of the clock, the eye fixes the star at 5, and subdivides the space a h into tenths, at the same time appreciating the ratio which the distance from the thread to a bears to the distance a h. This is recorded as above. This method, which is still used in many observatories, was introduced by the celebrated Bradley, astronomer royal of England in 1750, and per- fected by Maskelyne, his successor. A practiced observer can note the time within a tenth of a second in three cases out of four. There is yet another method now in common use, which it is necessary to understand. This is called the American or chronographic method, and consists, in the present practice, in the use of a sheet of a paper wound about and fastened to a horizontal cylindrical barrel, which is caused to revolve by machinery once in one min- ute of time. A pen of glass which will make a continu- 80 ASTBONOMT. Oils line is allowed to rest on the paper, and to this pen a continuous motion of translation in the direction of the length of the cylinder is given. Now, if the pen is allow- ed to mark, it is evident that it will trace on the paper an endless spii-al line. An electric current is caused to run through the observing clock, through a key which is held in the observer's hand and through an electro-magnet connected with the pen. A simple device enables the clock every second to give a slight lateral motion to the pen, which lasts about a thirtieth of a second. Thus every second is automatically marked by the clock on the chronograph paper. The ob- server also has the power to make a signal by his key (easily distinguished from the clock-signal by its different length), which is likewise permanently registered on the sheet. In this way, after the chronograph is in motion, the observer has merely to notice the instant at which the star is on the thread, and to press the key at that moment. At any subsequent time, he must mark some hour, min- ute, and second, taken from the clock, on the sheet at its appropriate place, and the translation of the spaces on the sheet into times may be done at leisure. § 5. GBADUATED CIRCLES. Nearly every datum in practical astronomy depends either directly or indirectly upon the measure of an angle. To make the necessary measures, it is customary to em- ploy what are called graduated or divided circles. These are made of metal, as light and yet as rigid as possible, and they have at their circumferences a narrow flat band of silver, gold, or platinum on which fine radial lines called "divisions" are cut by a "dividing engine" at regular and equal intervals. These intervals may be of 10', 5', or 2', according to the size of the circle and the degree of accuracy desired. The narrow band is called the divided htnb, and the circle is said to be di- THE VERNIER. 81 vided to 10', 5', 2'. The separate divisions are numbered consecutively from 0° to 360° or from 0° to 90°, etc. The graduated circle has an axis at its centre, and to this may be attached the telescope by which to vievir the points whose angular distance is to be determined. To this centre is also attached an arm wliich revolves with it, and by its motion past a certain number of divi- sions on the circle, determines the angle through which the centre has been rotated. This arm is called the index arm, and it xisually carries a vernier on its extremity, by means of which the spaces on the graduated circle are subdivided. The reading of the circle when the index arm is in any position is the number of degrees, minutes, and seconds which correspond to that po- sition ; when the index arm is in an- other position there is a different reading, and the differences of the two readings S^ — 5", S" — S\ S* — S"' are the angles through which the index arm has turned. The process of measuring the angle between the objects by means of a divided circle consists then of pointing the telescope at the first object and reading the position of the index arm, and then turning the telescope (the index arm turning with it) until it points at the second object, and again reading the position of the index arm. The difference of these readings is the angle sought. To facilitate the determination of the exact reading of the circle, we have to employ special devices, as the ■vernier and the reading microscope. The Vernier. — In Fig. 34, M JV is, a, portion of the divided limb of a graduated circle ; C D is, the index arm which revolves with the telescope about the centre of the circle. The end a 5 of C D is also a part of a circle con- centric with M W, and it is divided into n parts or divi- sions. The length of these n parts is so chosen that it is 83 ASTRONOMY. the same as that of {n — 1) parts on the divided limb M N or the reverse. The first stroke a is the zero of the vernier, and the reading is always determined by the position of this zero or pointer. If this has revolved past exactly twenty di- visions of the circle, then the angle to be measured is 20 X <^, <^ being the value of one division on the limb {N M) in arc. Fig. 34.— the vernikb. Call the angular value of one division on the vernier d'; fi 1 1 (n — l)d = n-d', ord' = -d, and d—d'^ -d ; d — d' is called the least count of the vtriiier which is one ji"" part of a circle division. If the zero a does not fall exactly on a division on the circle, but is at some other point (as in the figure), for ex- ample between two divisions whose numbers are P and \P + 1), the wliole reading of the circle in this position is P X d + the fraction of a division from P to a. If the m"" division of the vernier is in the prolongation of a division on the limb, then this fraction Pa is m TUK MERIDIAN CIRCLE. 83 {d — d') — - -d. In the figure n = 10, and as the Itli division is almost exactly in coincidence, m = 4, so that 4 the whole reading of the circle \?, P X d + ^y d. li d is 10', for example, and if the division P is numbered 297° 40', then this reading would be 297° 44', the least count being 1', and so in other cases. If the zero had started from the reading 280° 20', it must have moved past 17° 24', and this is the angle which has been measured. § 6. THE MERIDIAN CIRCLE. The meridian circle is a combination of the transit in- strument with a graduated circle fastened to its axis and moving with it. The meridian circle made by Repsold for the United States Naval Academy at Annapolis is shown in the figure. It has two circles, c c and c' c', finely divided on their sides. The graduation of each circle is viewed by four microscopes, two of which, P P, are shown in the cut. The microscopes are 90° apart. The cnt shows also the hanging -level Z Z, by which the error of level of the axis ^ ^ is found. The instrument can be used as a transit to determine right ascensions, as before described. It can be also used to measure declinations in the following way. If the tele- scope is pointed to the nadir, a certain division of the cir- cles, as JY, is under the first microscope. If it- is pointed to the iJole, the reading will change by the angular distance between the nadir and the pole, or by 90° + being the latitude of the place (supposed to be known). The polar reading P is thus known when the nadir reading i\7"is found. If the telescope is then pointed to various stars of unknown polar distances, p', p",p"', etc., as they successively cross the meridian, and if the circle readings for these stars are P', P", P'" , etc. , it follows that p' = P'-P;p" = P" - P; p'" = P" - P, etc. 84 AUTltUNOMr. Fig. 35. — the mertdiatt cibci.b. THE MERIDIAN CIRCLE. 85 To determine tlie reading j P, P', P", etc., we use the micro- scopes R, R, etc. The observer, after haviug set the telescope so that one of the stars shall cross tlie field of view exactly at its cen- tre (whicli may be liere marked by a single horizontal thread in the reticle), goes to each of the microscopes in succession and places his eye at A (see Fig. 1, page 80). Ho sees in the field of the microscope the imago of the divisions of the graduated scale (Fig. 3) formed at D (Fig. 1), the common focus of the lenses A and C. Just at that focus is placed a notched scale (Fig. 2) and two crossed spider lines. These lines are fixed to a sliding frame a a, which can be moved by turning the graduated head F. This head is divided usually into sixty parts, each of which is 1" of arc on the circle, one whole revolution of the head serving to move the sliding frame a a, and its crossed wires through 60" or V on the graduated circle. The notched scale is not movable, but serves to count the number of complete revolutions made by the screw, there being one notch for each revolution. The index i (Fig. 2) is fixed, and serves to count the number of jjarts of F which are carried past it by the revolution of this head. If on setting the crossed threads at the centre of the motion of F, and looking into the microscope, a division on the circle coin- cides with the cross, the reading of the circle P is the exact num- ber of degrees and minutes corresponding to that particular divi- sion on the divided circle. Usually, however, the cross has been apparently carried past one of the exact divisions of the circle by a certain quantity, which is now to be measured and added to the reading corresponding to this adjacent division. This measure can be made by turning the screw back say four revolutions (measured on the notched scale) plus ST S pavts (measured by the index i). If the division of the circle in question was 179' 30', for example, the complete reading would be in this ease 179° 50' + 4' 37''-3 or 179° 54' 37"-3. Such a reading is made by each microscope, and the mean of the min- utes and seconds from all four taken as the circle reading. We now know how to obtain the readings of our circle when directed to any point. "We require some zero of reference, as the nadir reading (iV), the polar reading (P), the equator reading, (Q), or the zenith reading (Z). Any one of these being known, the circle readings for any stars as P, P", P'', etc., can be turned into polar distances p', p", p'", etc. The nadir reading (iV) is the zero commonly employed. It can be determined by pointing the telescope vertically downward at a basin of mercury placed immediately beneath the instrument, and turning the whole instrument about the axis until the middle wire of the reticle seen directly exactly coincides with the image of this wire seen by reflection from the surface of the qiiicksilver. When this is the case, tlie telescope is vertical, as can be easily seen, and the nadir reading may be found from the circles. The meridian circle thus serves to determine both tlie right ascen- sion and declination of a given star at the same culmination. Zone observations are made with it by clamping tlie telescope in one 86 ASTRONOMY. nil. K|.2. 1 : a rTi _i -mi r ~ ": - i Fig. 3. *E W r.j.4. Fig. 3C. — reading microscope, micrometer axd level. THE EQUATORIAL. 87 direction, and observing successively the stars which pass through its field of view. It is by this rapid method of observing that the largest catalogues of stars have been formed. § 7. THE EQUATORIAL. To complete the enumeration and description of the principal instruments of astronomy, we require an account of the equatorial. This term, properly speaking, refers to a form of mounting, but it is commonly used to in- clude both mounting and telescope. In this class of instruments the object to be attained is in general the easy finding and following of any celestial object whose apparent place in the heavens is known by its right as- cension and declination. The equatorial mounting con- sists essentially of a pair of axes at right angles to each other. One of these S JV (the polar axis) is directed to- ward the elevated pole of the heavens, and it therefore makes an angle with the horizon equal to the latitude of the place (p. 2] ). This axis can be turned about its own axial line. On one extremity it carries another axis J, D (the declination axis), which is fixed at right angles to it, but which can again be rotated about its axial line. To this last axis a telescope is attached, which may either be a reflector or a refractor. It is plain that such a telescope may be directed to any point of the heavens ; for we can rotate the declination axis until the telescope points to any given polar distance or declination. Then, keeping the telescope fixed in respect to the declination axis, we can rotate the whole instrument as one mass about the polar axis until the telescope points to any por- tion of the parallel of declination defined by the given right ascension or hour-angle. Fig. 37 is an equatorial of six-inch aperture which can be moved from place to place. If we point such a telescope to a star when it is rising (doing this by rotating the telescope first about its decli- nation axis, and then about the polar axis), and fix the telescope in this position, we can, by simply rotating the 88 ASIliONOMT. Fig. 37. — EQUAToniAL telescope pointed towabd the pole. THE MICROMETER. 89 whole apparatus on the polar axis, cause the telescope to trace out on the celestial sphere the apparent diiimal path which this star will appear to follow from rising to set- ting. In such telescopes a driving-clock is so arranged that it can turn the telescope round the polar axis at the same rate at which the earth itself turns about its own axis of rotation, but in a contrary direction. Hence such a telescope once pointed at a star will continue to point at it as long as the driving-clock is in operation, thus enabling the astronomer to observe it at his leisure. Fig. 38. — measueembnt of position-angle. Every equatorial telescope intended for making exact measures has a Jfllar micrometer, which is precisely the same in principle as the reading microscope in Fig. 3, page 86, except that its two wires are parallel. A figure of this instrument is given in Fig. 3, page 86. One of the wires is fixed and the other is movable by the screw. To measure the distance apart, of two objects A and B, wire 1 (the fixed wire) is placed on A and wire 2 (movable by the screw) is placed on B. The number of revolutions and parts of a revolution of the screw is noted, say t0'-367 ; then wires I and 2 are placed in coincidence, and this eero-reading noted, say 5' -143. The dis- tance A Bis equal to S'- 124. Placing wires 1 and 2 a known num- ber of revolutions apart, we may observe the transits of a star in the equator over them ; and from the interval of time required for this star to move over say fifty revolutions, the value of one revolution 90 ASTRONOMY. is known, and can always be used to turn distances measured in revolutions to distances in time or arc. By the filar micrometer we can determine the distance apart in seconds of arc of any two stars A and B. To completely fix the relative position of A and B, we require not only this distance, but also the angle which the line A B makes with some fixed direction in space. We assume as the fixed direction that of the meridian passing through A. Suppose in Fig. 38 A and B to be two stars visible in the field of the equatorial. The clock-work is detached, and by the diurnal motion of the earth the two stars will cross the field slowly in the direction of the fo/rallel of declination passing through A, or in the direction of the arrow in the figure from E. to W., east to west. The filar micrometer is con* structed so that it can be rotated bodily about the axis of the tele- scope, and a graduated circle measures the amount of this rotation. The micrometer is then rotated until the star A will pass along one of its wires. This wire marks the direction of the parallel. The wire perpendicular to this is then in the meridian of the star. The 'position angle of B with respect to A is then the angle which A B makes with the meridian A N passing through A toward the north. It is zero when B is north of ^, 90° when B is east, 180 when B is south, and 270° when B is west of A. Knowing p, the position angle (iV^ 5 in the figure), and « {A B) the distance of B, we can find the difference of right ascension (A or), and the differ- ence of declination ( A<5) of J5 from A by the formulae, Aor = « sin p ; Ad = s cos f. Conversely knowing Aar and AcS, we can deduce s and 'p from these formulae. The angle p is measured while the clock-work keeps the star A in the centre of the field. § 8. THE ZElflTH TELESCOPE. The accompanying figure gives a view of the zenith telescope in the form in which it is used by the United States Coast Survey. It consists of a vertical pillar which supports two Ya. In these rests the horizontal axis of the instrument which carries the tele- scope at one end, and a counterpoise at the other. The whole in- strument can revolve 180° in azimuth about this pillar. The tele- scope has a micrometer at its eye-end, and it also carries a divided circle provided with a fine level. A second level is provided, whose use is to make the rotation axis horizontal. The peculiar features of the zenith telescope are the divided circle and its at- tached level. The level is, as shown in the cut, in the plane of motion of the telescope (usually the plane of the meridian), and it can be independently rotated on the axis of the divided circle, and set by means of it to any a^gle with the optical axis of the telescope. The circle is divided from zero (0°) at its lowest point to 90° in each direction, and is firmly attached to the telescope tube, and moves with it. By setting the vernier or index-arm of the circle to any degree and minute as «, and clamping it there (the level moving with it). TEE ZENITH TELEHGOPE. 91 Fig. 39. — the zenith telescope. 9a ASTRONOMY. and then rotating the telescope and the whole system about the horizontal axis until the bubble of the level is in the centre of the level-tube, the axis of the telescopes will be directed to the zenith distance a. The filar micrometer is so adjusted that a motion of its screw measures differences of zenith distance. The use of the ze- nith telescope is for determining the latitude by TaI/Cott's method. The theory of this operation has been already given on page 48. A description of the actual process of observation will illustrate the excellences of this method. Two stars, A and B, are selected beforehand (from Star Cata- logues), which culminate, A south of the zenith of the place of ob- servation, B north of it. They are chosen at nearly equal zenith dis- tances I* and I", and so that i'' — 4° is less than the breadth of the field of view. Their right ascensions are also chosen so as to be about the same. The circle is then set to the mean zenith distance of the two stars, and the telescope is pointed so that the bubble is nearly in the middle of the level. Suppose the nght ascension of A is the smaller, it will then culminate first. The telescope is then turned to the south. As A passes near the centre of the field its distance from the centre is measured by the micrometer. The level and micrometer are read, the whole instrument is revolved 180°, and star B is observed in the same way. By these operations we have determined the difference of the zenith distances of two stars whose declinations i^ and i^ are known. But ^ being the latitude, ^ = d-' + 4^ and ^ = (5° — f, whence ^ = J (d^ -K)") -f- J {t - f ). The first term of this is known ; the second is measured ; so that each pair of stars so observed gives a value of the latitude which depends on the measure of a very small arc with the micrometer, and as this arc can be measured with great precision, tlie exactness of the determination of the latitude is equally great. § 9. THE SEXTANT. The sextant is a portable instrument by which the altitudes of celestial bodies or the angular distances between them may be measured. It is used chiefly by navigators for determining the latitude and the local time of the position of the ship. Knowing the local time, and comparing it with a chronometer regulated on Greenwich time, the longitude becomes known and the ship's place is fixed. It consists of the arc of a divided circle usually 60° in extent, whence the name. This arc is in fact divided into 120 equal parts, each marked as a degree, and these are again divided into smaller spaces, so that by means of the vernier at the end of the index-arm Jfir 5 an arc of 10" (usually) may be read. The indej.-arm M 8 carries the index-glass M, which is a silvered plane mirror set perpendicular to the plane of the divided arc. The 'IHE SEXTANT. 93 horisoTi-glass m is also a plane mirror fixed perpendicular to the plane of the divided circle. This last glass is fixed in position, while the first revolves with the index-arm. The horizon-glass is divided into two parts, of which the lower one is silvered, the upper half being transparent. ^ is a telescope of low power pointed toward the horizon-glass. By it any object to which it is directed can be seen through the un- sil^ered half of the horizon-glass. Any other object in the same plane can be brought into the same field by rotating the index-arm Fig. 40. — the sextant. (and the index-glass with it), so that a beam of light from this second object shall strike the index-glass at the proper angle, there to be reflected to the horizon-glass, and again reflected down the telescope M Thus the images of any two objects in the plane of the sextant may be brought together in the telescope by viewing one directly, and the other by reflection. The principle upon which the sextant depends is the following, which is proved in optical works. The angle tetween the first and the last direction of a ray which has suffered two reflections in the same 94 ASTRONOMY. plane is equal to twice the angle which the two reflecting sutfaees make with each other. In the figure 5 ^ is the ray incident upon A, and this ray is by reflection brought to the direction B E. The theorem declares that the angle B E S is equal to twice D C B, or twice the angle of Fig. 41. the mirrors, since B G and D G are perpendicular to B and A. To measure the altitude of a star (or the sun) at sea, the sextant is held in the hand, and the telescope is pointed to the sea-horizon, which appears like a definite line. The index-arm is then moved until the reflected image of the sun or of the star coincides with the Fig. 42. — artificial horizon. image of the sea-horizon seen directly. "When this occurs the time is to be noted from a chronometer. If a star is observed, the read- ing of the divided limb gives the altitude directly ; -if it is the sun or moon which has been observed, the lower limb of these is brought to coincide with the horizon, and the altitude of the centre THE SEXTANT. SS is found by applying tlie semi-diameter as found in the Nautical Almanac to the observed altitude of the limb. The angular distance apart of a star and the moon can be meas- ured by pointing the telescope at the star, revolving the whole sex- tant about the sight-line of the telescope until the plane of the di- vided arc passes through both star and moon, and then by moving the index-arm until the reflected moon is just in contact with the star's image seen directly. On shore the horizon is broken up by buildings, trees, etc., and the observer is therefore obliged to have recourse to an artificial hoi-izon, which consists usually of the reflecting surface of some liquid, as mercury, contained in a small vessel A, whose upper surface is necessarily parallel to the horizon BAG. A ray of light S A, from a star at S, incident on the mercury at A, will be reflected in the direction A E, making the angle S A G — C A S' (A S' be- ing E A produced), and the reflected image of the star will appear to an eye at E as far below the horizon as the real star is above it. With a sextant whose index and horizon -glasses are at /and H, the angle 8 E S' may be measured \ h\xt 8 E 8' = S A 8 — A 8 E, and if ^ Eis exceedingly small as compared with A 8, as it is for all celestial bodies, the angle A 8 E may be neglected, and 8 E 8' will equal -i ASTRONOMY. the latter would seem to move more rapidly when nearest the earth than when farther from it. It was not until the time of Keplek that the eccentric was shown to be in- capable of accounting for the real motion ; and it is his discoveries which we are next to describe. § 2. KEPLER'S LAWS OP PLANET AB.T MOTIOlf. The direction of the sun, or its longitude, can be deter- mined from day to day by direct observation. If we could also observe its distance on each day, we should, by laying down the distances and directions on a large piece of paper, through a whole year, be able to trace the curve which the earth describes in its annual course, this course being, as already shown, the counterpart of the apparent one of the sun. A rough determination of the rela- tive distances of the sun at different times of the year may be made by measuring the sun's apparent angular diame- ter, because this diameter varies inversely as the distance of the object observed. Such measures would show that the diameter was at a maxunum of 32' 36" on January 1st, and at a minimum of 31' 32" on July 1st of every year. The difference, 64", is, in round numbers, -^^ the mean diameter — that is, the earth is nearer the sun on January 1st than on July 1st by about ^. We may consider it as ^ greater than the mean on the one date, and -^ less on the other. This is therefore the actual displacement of the sun from the centre of the earth's orbit. Again, observations of the apparent daily motion of the sun among the stars, corresponding to the real daily motion of the earth round the sun, show this motion to be least about July 1st, when it amounts to 57' 12" = 3432", and greatest about January 1st, when it amounts to 61' 11" = 3671". The difference, 239", is, in round num- bers, -jij^ the mean motion, so that the range of variation is, in proportion to the mean, double what it is in the case pf the distances. If the actual velocity of the earth in its KEPLER'S LAWS. 123 orbit were uniform, the ajiparent angular motion round the sun would be inversely as its distance from the sun. Actually, however, the angular motion, as given above, is inversely as the square of the distance from the sun, be- cause (1 -)- -^y = 1 4- tV "^^^y nearly. The actual ve- locity of the earth is therefore greater the nearer it is to the sun. On the ancient theory of the eccentric circle, as pro- pounded by HippAECHUs, the actual motion of the earth was supposed to be uniform, and it was necessary to sup- pose the displacement of the sun (or, on the ancient theo- ry, of the earth) from the centre to be -^ its mean distance, in order to account for the observed changes in the motion in longitude. We now know that, in round numbers, one half the inequality of the apparent motion of the sun in longitude arises from the variations in the distance of the earth from it, and one half from the earth's actually mov- ing with a greater velocity as it comes nearer the sun. By attributing the whole inequality to a variation of distance, the ancient astronomers made the eccentricity of the or- bit — that is, the distance of the sun from the geometrical centre of the orbit (or, as they supposed, the distance of the earth from the centre of the sun's orbit) — twice as great as it really was. An immediate consequence of these facts of observa- tion is Keflee's second law of planetary motion, that the radii vectores drawn from the sun to a planet revolving round it, sweep over equal areas in equal times. Sup- pose, in Fig. 51, that S represents the position of the sun, and that the earth, or a planet, in a unit of time, say a day or a week, moves from P^ to ^3. At another part of its orbit it moves from P to P^ in the same time, and at a third part from P, to P„. Then the areas SP,P,, SPP„ SP,P, will all be equal. A little geometrical consideration will, in fact, make it clear that the areas of the triangles are equal when the angles at IS are inversely as the square of the radii vectores, 8P, etc., 124 ASTRONOMY. since the expression for the area of a triangle in which the angle at S is very small is ^ angle S X S P'.* Fig. 51. — law of areas. In the time of Kepleb the means of measuring the sun's angular diameter were so imperfect that the preced- ing method of determining the path of the earth around the sun could not be applied. It was by a study of the motions of the planet Mars, as observed by Ttcho Beahe, that Keplee was led to his celebrated laws of planetary motion. He foimd that no possible motion of Mars in a truly circular orbit, however eccentric, would represent the observations. After long and laborious calculations, and the trial and rejection of a great number of hypotheses, he was led to the conclusion that the planet Mars m.oved in an ellipse, having the sun in one focus. As the analo- gies of nature led to the inference that all the planets, the earth included, moved in curves of the same class, and according to the same law, he was led to enunciate the first two of his celebrated laws of planetary motion, which were as follow : * More exactly If we consider the arc PPi as a straight line, the area of the triangle PPi ;SwUl be equal to i5Px SPx xsin angle S. But in considering only very small angles we may suppose SP= SPi and the sine of the angle S equal to the angle itself. This supposition will give the area mentioned above. KEPLER'S LAWS. 125 I. Each planet moves around the sun in an ellipse, hav- ing the stm iti one of its foci. II. The radius vector joining each planet with the sun, moves over equal areas in equal times. To these he afterward added another showing the rela- tion between the times of revolution of the separate planets. III. The square of the time of revolution of each flanet is proportional to the cube of its mean distance from the sun. These three laws comprise a complete theory of plan- etary motion, so far as the main features of the motion are concerned. There are, indeed, small variations from these laws of Keplee, hnt the laws are so nearly correct that they are always employed by astronomers as the basis of their theories. Mathematical Theory of the Elliptic Motion. — The laws of Keplee lead to problems of such mathematical elegance that we give a brief synopsis of the most impor- tant elements of the theory. A knowledge of the ele- ments of analytic geometry is necessary to understand it. Let us put : ffi, the semi-major axis of the ellipse in which the planet moves. In the figure, if C is the centre of the el- lipse, and 8 the focus in which the sun is situated, then a = ^ = C t. oa e, the eccentricity of the ellipse = — : TT, the longitude of the perihelion, rep- resented by the angle n 8 E, E being the direction of the vernal equinox from which longitudes are counted. n, the mean angular motion of the planet round the sun in a unit ot time. The actual motion being variable, the mean motion is found by dividing the circumference = 360° by the time of revolution. T, the time of revolution. r, the distance of the planet from the sun, or its radius vector, a variable quantity. I. The first remark we have to make is that the eUipticitus of the 12(i ASTRONOMY. planetary orbits — that is, the proportions in which the orbits are flat- tened — is much less than their eccentricities. By the properties of the ellipse we have : 8 B ^ semi-major axis = a, B C = semi-minor axis = aVl — e'', or, B G =a{\ — \ e^) nearly, when e is very small. The most eccentric of the orbits of the eight major planets is that of Mercury, for which e = 0.3. Hence for Mercury BC = a{l--,\) very nearly, so that flattening of the orbit is only about j^ or .03 of the major axis. The next most eccentric orbit is that of Mars for which e = .093 ; B O = a {1 — .0043), so that the flattening of the orbit is only about ^^. We see from this that the hypothesis of the eccentric circle makes a very close approximation to the true form of the planetary orbits. It is only necessary to suppose the sun removed from the centre of the orbit by a quantity equal to the product of the eccentricity into the radius of the orbit to have a nearly true representation of the orbit and of the position of the sun. II. The least distance of the planet from the sun is 3n = a (1 — e), and the greatest distance is AS = a{l + e). III. The angular velocity of the planet around the sun at any point of the orbit, which we may call 8, is, by the second law of Kepler : "5 = ^' C being a constant to be determined. To determine it we remark that 8 is the angle through which the planet moves in a unit of time. If we suppose this unit to be very small, the quantity S r" is double the area of the very small triangle swept over by the radius vector during such unit. This area is called the areolar veloaity of the planet, and is a constant, by Kepler's second law. Therefore, in the last equation, G = S r^ represents the double of the areolar velocity of the planet. When the planet completes an entire revo- lution, the radius vector has swept over the whole area of the ellipse which is n a' V 1 — e''.* The time required to do this be- "^ In this formula v represents the ratio of the circumference of the circle to its diameter. KEPLER- 8 LAWS 127 ing called T, the area swept over with the areolar velocity iC7 is also \0 T. Therefore I C r = ir a" Vl — e' ; 'ina'Vl — e' The quantity 3 t here represents 360^, or the whole circumference, which, being divided by T, the time of describing it will give the mean angular velocity of the planet around the sun which we have called n. Therefore 3 TT and C=a''n Vl — e'. This value of C being substituted in the expression f or iS, we have a' n VI —~7 S=- IV. By Kepler's third law "P is proportioned to a" ; that is, — J- is a constant for all the planets. The numerical value of this constant will depend upon the quantities which we adopt as the units of time and distance. If we take the year as the unit of time and the mean distance of the earth from the sun as that of distance, T and a for the earth will both be unity, and the ratio — ■ will there- fore be unity for all the planets. Therefore a' = r^ ; a= T*. Therefore if ice sqmre the period of reeolution of any planet in years, and extract the eube root of the square, we shall home its mean distance from, the sun in units of the earth'' s distance. It is thus that the mean distances of the planets are determined in practice, because, by a long series of observations, the times of revolution of the planets have been determined with very great pre- cision. ^^^ V. To find the position of a planet we must know the epoch at which it passed its perihelion, or some equivalent quantity. To find its position at any other time let t be the time which has elapsed since passing the perihelion. Then, by the law of areas, if P be the position of the planet at this time we shall have Area of sector P 5 TT t Area of whole ellipse ~ T 128 ASTRONOMY. 'I'lie times r and T beiug both given, the problem is reduced to that of cutting a given area of the ellipse by a line drawn from the focus to some point of its circumference to be found. This is known as Kepler's problem, and may be solved by analytic geom- PlG. 53. etry as follows : Let A Bhe the major axis of the ellipse, P the position of the planet, and Sthat of the focus in which the sun is situated. On A B a,s s. diameter describe a circle, and through P draw the right line P' P D perpendicular to A B. TJie area of the elliptic sector SPB, over which the radius vector of the planet has swept since the planet passed the perihelion at B, is equal to the sector O P B minus the triangle GPS. Since an ellipse is formed from a circle by shortening all the ordinates in the same ratio (namely, the ratio of the minor axis li to the major axis a), it follows that the elliptic sector G P B may be formed from the circular sector C P" Bhy shortening all the ordinates in the ratio oi D PX,o D P', or of a to 6. Hence, Area GPB : area CP'B = b : a. But area C P' B = angle P' C B x i a", taking the unit radius as the unit of angular measure. Hence, putting u for the angle POBwehave Area GPB=- area CP B = \alu (2)- Again, the area of the triangle GP 8 is equal to i base C 8 x al- titude PD. Also PJ5 = - P'D, and P* D = CP* sin M = asintt. a Wherefore, PZ) = ?/ sin u. (3). KEPLER'S LAWS. 129 By the first principles of conic sections, C S, the base of the tnangle, is equal to a e. Hence Area CP8= ^abesinu, and, from (2) and (3), Area 8PB = ^ah{u — esinw). Substituting in equation (1) this value of the sector area, and TT a 6 for the area of the ellipse, we have 2n- ~ T' or, M — e sin M = 2 5r —- . T From this equation the unknown angle u is to be found. The equation being a transcendental one, this cannot be done directly, but it may be rapidly done by successive approximation, or the value of u may be developed in an infinite series. Next we wish to express the position of the planet, which is given by its radius vector S P and the angle £ 8 P which this radius vector makes with the major axis of the orbit. Let us put r, the radius vector SP, y, the angle B SP, called the true anomaly. Then r sin/ = PD = h!i\nu (Equation 3), r cos/= 8D = C D — CS= C P' cosu — ae = a (cos u — e), from which r and /can both be determined. By taking the square root of the sums of the squares, they give, by suitable reduction and putting i' = a" (1 — e'^), r=a{l—e cos u), and, by dividing the first by the second. 5 sin u Vl tan/: a (cos u — e) cos v — e Putting, as before, t for the longitude of the perihelion, the true longitude of the planet in its orbit will be/ + t. VI. To find the position of the planet relatively to the ecliptic, 130 ASTRONOMY. the inclination of the orbit to the ecliptic has to be taken into ac- count. The orbits of the several large planets do not lie in the same plane, but are inclined to each other, and to the ecliptic, by various small angles. A table giving the values of these angles will be given hereafter, from which it will be seen that the orbit of Mercury has the greatest inclination, amounting to 7°, and that of Uranus the least, being only 46'. The reduction of the position of the planet to the ecliptic is a problem of spherical trigonometry, the solution of which need not be discussed here. CHAPTER V. UNIVEKSAL GRAVITATION. § 1. NEWTON'S LAWS OF MOTION. The establishment of the theory of universal gravitation furnishes one of the best examples of scientific method which is to be found. We shall describe its leading features, less for the purpose of making known to the reader the technical nature of the process than for illus- trating the true theory of scientific investigation, and showing that such investigation has for its object the dis- covery of what we may call generalized facts. The real test of progress is found in our constantly increased ability to foresee either the course of nature or the effects of any accidental or artificial combination of causes. So long as prediction is not possible, the desires of the inves- tigator remain unsatisfied. When certainty of prediction is once attained, and the laws on which the prediction is founded are stated in their simplest form, the work of science is complete. The whole process of scientific generalization consists in grouping facts, new and old, under such general laws that they are seen to be the result of those laws, combined with those relations in space and time which we may suppose to exist among the material objects investigated. It is essen- tial to such generaHzation that a single law shall suffice for grouping and predicting several distinct facts. A law invented simply to account for an isolated fact, however 1.32 ASTRONOMY. general, cannot be regarded in science as a law of nature. It may, indeed, be true, but its truth cannot be proved until it is shown that several distinct facts can be accounted for by it better than by any other law. The reader will call to mind the old fable which represented the earth as supported on the back of a tortoise, but totally forgot that the support of the tortoise needed to be accounted for as much as that of the earth. Te the pre-Newtonian astronomers, the phenomena of the geometrical laws of planetary motion, which we have just described, formed a group of facts having no connection with any thing on the earth. The epicycles of Hippaechus and Ptolkmy were.a truly scientific conception, in that they explained the seemingly erratic motions of the planets by a single simple law. In the heliocentric theory of Copee- Nicus this law was still further simplified by dispensing in great part with the epicycle, and replacing the latter by a motion of the earth around the sun, of the same nature with the motions of the planets. But CoPEENions had no way of accounting for, or even of describing with rigor- ous accuracy, the small deviations in the motions of the planets around the sun. In this respect he made no real advance upon the ideas of the ancients. Keplee, in his discoveries, made a great advance in representing the motions of all the planets by a single set of simple and easily understood geometrical laws. Had the planets followed his laws exactly, the theory of planetary motion would have been substantially complete. Still, furth'er progress was desired for two reasons. In the first place, the laws of Kepler did not perfectly represent all the planetary motions. "When ob- servations of the greatest accuracy were made, it was f omid that the planets deviated by small amounts from the ellipse of Kepler. Some small emendations to the motions com- puted on the elliptic theory were therefore necessar}'. Had this requirement been fulfilled, still another step would have been desirable — namely, that of connecting the LA W8 OP MOTION. 133 motions of the planets with motion upon the earth, and I'educing them to the same laws. Notwithstanding the great step which Kepler made in describing the celestial motions, he unveiled none of the great mystery in which they were enshrouded. This mys- tery was then, to all appearance, impenetrable, because not the slightest likeness could be perceived between the celestial motions and motions on the surface of the earth. The difficulty was recognized by the older philosophers in the division of motions into "forced" and "natural." The latter, they conceived, went on perpetually from the very nature of things, wliile the former always tended to cease. So when Kepler said that observation showed the law of planetary motion to be that around the circum- ference of an ellipse, as asserted in his law, he said all that it seemed possible to learn, supposing the statement per- fectly exact. And it was all that could be learned from the mere study of the planetary motions. In order to connect these motions with those on the earth, the next step was to study the laws of force and motion here around us. Sin- gular though it may appear, the ideas of the ancients on this subject were far more erroneous than their concep- tions of the motions of the planets. We might almost say that before the time of Galileo scarcely a single correct idea of the laws of motion was generally entertained by men of learning. There were, indeed, one or two who in this respect were far ahead of their age. Leonardo da Vmci, the celebrated painter, was noted in this respect. But the correct ideas entertained by him did not seem to make any headway in the world until the early part of the seventeenth century. Among those who, before the time of jSTewton, prepared the way for the theory in question, Galileo, Hutghens, and Hooke are entitled to especial mention. As, however, we cannot develop the history of this subject, we must pass at once to the gen- eral laws of motion laid down by Newton. These were three in number. 134 ASTRONOMY. Law First : Every body preserves its state of rest or of uniform motion in a right line, unless it is com,pelled to change that state hy forces impressed thereon. It was formerly supposed that a body acted on by no force tended to come to rest. Here lay one of the great- est difficulties which the predecessors of J^ewton found, in accounting for the motion of the planets. The idea that the sun in some way caused these motions was enter- tained from the earliest times. Even Ptolemy" had a vague idea of a force which was always directed toward the centre of the earth, or, which was to him the same thing, toward the centre of the universe, and which not only caused heavy bodies to fall, but bound the whole uni- verse together. Kepi.ee, again, distinctly affirms the ex- istence of a gravitating force by which the sun acts on the planets ; but he supposed that the sun must also exercise an impulsive forward force to keep the planets in motion. The reason of this incorrect idea was, of course, that all bodies in motion on the surface of the earth had practically come to rest. But what was not clearly seen before the time of Newton, or at least before Galileo, was, that this arose from the inevitable resisting forces which act upon all moving bodies around us. Law Second : The alteration of motion *s ever propor- tional to the moving force impressed, and is made in the direction of the right line in which that force acts. The first law might be considered as a particular case of this second one arising when the force is supposed to van- ish. The accuracy of both laws can be proved only by very carefully conducted experiments. They are now considered as mathematically proved. Law Third : To every action there is always opposed an equal reaction ; or the mutual actions of two bodies upon each otfisr are always equal, and in opposite directions. That is, if a body A acts in any way upon a body B, B will exert a force exactly equal on A in the opposite direction. ORAVITATION OF TUB PLANETS. 135 These laws once established, it became possible to calcu- late the motion of any body or system of bodies when once the forces which act on them were known, and, vice versa, to define what forces were requisite to produce any given motion. The question which presented itself to the mind of Newton and his contemporaries was this : Under what law of force will planets move round the sun in accord- ance with Keplee's laws ? The laws of central forces had been discovered by Huy- GHENs some time before ^^ewton commenced his re- searches, and there was one result of them which, taken in connection with Keplee's third law of motion, was so obvious that no mathematician could have had much diffi- culty in perceiving it. Supposing a body to move around in a circle, and putting B the radius of the circle, 1' the period of revolution, Hutghens showed that the centrifugal force of the body, or, which is the same thing, the attract- ive force toward the centre which would keep it in the circle, was proportional to =j. But by Keplee's third law jT" is proportional to H^. Therefore this centripetal R 1 force is proportional to -pj, that is, to -^. Thus it fol- lowed immediately from Keplee's third law, that the central force which would keep the planets in their or- bits was inversely as the square of the distance from the sun, supposing each orbit to be circular. The first law of motion once completely understood, it was evident that the planet needed no force impelling it forward to keep up its motion, but that, once started, it would keep on forever. The next step was to solve the problem, what law of force will make a planet describe an ellipse around the sun, having the latter in one of its foci ? Or, supposing a planet to move round the sun, the latter attracting it with a force inversely as the square of the distance ; what will be the form of the orbit of the planet if it is not cir- 136 ASTRONOMY. eular ? A solution of either of these problems was beyond the power of mathematicians before the time of Newton ; and it thus remained uncertain whether the planets mov- ing under the influence of the sun's gravitation would or would not describe ellipses. Unable, at first, to reach a satisfactory solution, Newton attacked the problem in another direction, starting from the gravitation, not of the sun, but of the earth, as explained in the following section. § 2. GRAVITATION IN THE HEAVENS. The reader is probably familiar with the story of New- ton and the falling apple. Although it has no authorita- tive foundation, it is strikingly illustrative of the method by which Newton first reached a solution of the problem. The course of reasoning by which he ascended from grav- itation on the earth to the celestial motions was as follows : We see that there is a force acting all over the earth by which all bodies are di'awn toward its centre. This force is familiar to every one from his infancy, and is properly called gravitation. It extends without sensible diminution to the tops not only of the highest buildings, but of tlie highest mountains. How much higher does it extend ? Why should it not extend to the moon ? If it does, the moon would tend to drop toward the earth, just as a stone thrown from the hand drops. As the moon moves round the earth in her monthly course, there must be some force drawing her toward the earth ; else, by the first law of motion, she would fly entirely away in a straight line. Why should not the force which makes the apple fall be the same force which keeps her in her orbit ? To answer this question, it was not only necessary to calculate the intensity of the force which would keep the moon herself in her orbit, but to compare it with the intensity of gravity at the earth's surface. It had long been known that the distance of the moon was about sixty radii of the earth. If this GRAVITATION OF THE PLANETS. 137 force diminislied as the inverse square of the distance, then, at the moon, it would be only ^^Vir ^® great as at the surface of the earth. On the earth a body falls six- teen feet in a second. If, then, the theory of gravitation were correct, the moon ought to fall toward the earth "JTiVtr oi ^^^^ amount, or about J-^ of an inch in a second. The moon being in motion, if we imagine it moving in a s raight line at the beginning of any second, it ought to be drawn away from that line ^V of ^-n inch at the end of the second. When the calculation was made with the connect distance of the inoon, it was found to agree ex- actly with this result of theory. Thus it was shown that the force which holds the moon in her orbit is the same which makes the stone fall, only diminished as the inverse sqnai-e of the distance from the centre of the earth.* As it appeared that the central forces, both toward the sun and toward the earth, varied inversely as the squares of the distances, Newton proceeded to attack the mathe- matical problems involved in a more systematic way than any of his predecessors had done. Keplee's second law showed that the line drawn from the planet to the sun will describe equal areas in equal times. Newton showed that this could not be true, unless the force which held the planet was directed toward the sun. We have already stated that the third law showed that the force was in- versely as the square of the distance, and thus agreed ex- actly with the theory of gravitation. It only remained to * It is a remarkable fact in the history of science that Newton would have reached this result twenty years sooner than he did, had he not been misled by adopting an erroneous value of the earth's diame- ter. His first attempt to compute the earth's gravitation at the distance of the moon was made in 1665, when he was only twenty-three years of age. At that time he supposed that a degree on the earth's surface was sixty statute miles, and was in consequence led to erroneous results by supposing the earth to be smaller and the moon nearer than they really were. He therefore did not make public his ideas ; but twenty years later he learned from the measures of PrcARD in France what the true diameter of the earth was, when he repeated his calculation with entire success. 138 ASTRONOMY. consider the results of the first law, that of the elliptic motion. After long and laborious efforts, Newton was enabled to demonstrate rigorously that this law also re- sulted from the law of the inverse square, and could result from no other. Thus all mystery disappeared from the celestial motions ; and planets were shown to be simply heavy bodies moving according to the same laws that were acting here around us, only under very different circum- stances. All three of Keplee's laws were embraced in the single law of gravitation toward the sun. The sun attracts the planets as the earth attracts bodies here around us. Mutual Action of the Planets. —It remained to extend and prove the theory by considering the attractions of the planets themselves. By K^ewton's third law of motion, each planet must attract the sun with a force equal to that which the sun exerts upon the planet. The moon also must attract the earth as much as the earth attracts the moon. Such being the case, it must be highly probable that the planets attract each other. K so, Keplee's laws can only be an approximation to the truth. The sun, being immensely more massive than any of the planets, overpowers their attraction upon each other, and makes the law of elliptic motion very nearly true. But still the comparatively small attraction of the planets must cause some deviations. Now, deviations from the pure elliptic motion were known to exist in the case of several of the planets, notably in that of the moon, which, if gravitation were universal, must move under the influence of the com- bined attraction of the earth and of the sun. Newton, therefore, attacked the complicated problem of the deter- mination of the motion of the moon under the combined action of these two forces. He showed in a general way that its deviations would be of the same nature as those shown by observation. But the complete solution of the problem, which, required the answer to be expressed in numbers, was beyond his power. ATTRACTION OF GRAVITATION. 139 Gravitation Resides in each Particle of Matter. — Still another question arose. Were these mutuallj attractive forces resident in the centres of the several bodies attracted, or in each particle of the matter composing them ? New- ton showed that the latter must be the ease, because the smallest bodies, as well as the largest, tended to fall toward the earth, thus showing an equal gravitation in every separate part. The question then arose : what would be the action of the earth upon a body if the body was attracted — not toward the centre of the earth alone, but toward every particle of matter in the earth ? It was shown by a quite simple mathematical demonstra- tion that if a planet were on the surface of the earth or outside of it, it would be attracted with the same force as if the whole mass of the earth were concentrated in its centre. Putting together the various results thus arrived at, Newton was able to formulate his great law of uni- versal gravitation in these comprehensive words : ' ' Every particle of matter in the universe attracts every other jparticle with a force directly as the masses of the two particles, and inversely as the square of the distance which separates them.'''' To show the nature of the attractive forces among these various particles, let us represent by m and m' the masses of two attracting bodies. We may conceive the body m to be composed of m particles, and the other body to be composed of ml particles. Let us conceive that each particle of the one body attracts each particle of the other with a force -^ . Then every particle of m will be attracted by each of the ml particles of the other, and therefore the total attractive force on each of these m pai-- nvy tides will be —5. Each of the m particles being equally subject to this attraction, the total attractive force between the two bodies will be — ^-. When a given force acts 140 ASTRONOMY. upon a body, it will produce less motion the larger the body is, the accelerating force being proportional to the total attracting force divided by the mass of the body moved. Therefore the accelerating force which acts on the body m', and which determines the amount of motion, will be —J ; and conversely the accelerating force acting on the body m will be represented by the fraction —^. § 3. PROBIiKMS OP GRAVITATION. The problem solved by Newton, considered in its great- est generality, was this : Two bodies of which the masses are given are projected into space, in certain directions, and with certain velocities. What will be their motion under the influence of their mutual gravitation ? If their rela- tive motion does not exceed a certain definite amount, they will each revolve around their common centre of gi-avity in an ellipse, as in the case of planetary motions. If, how- ever, the relative velocity exceeds a certain limit, the two bodies wiU separate forever, each describing around the common centre of gravity a curve having infinite branches. These curves are found to be parabolas in the case where the velocity is exactly at the limit, and hyperbolas when the velocity exceeds it. Whatever curves may be de- scribed, the common centra of gravity of the two bodies will be in the focus of the curve. Thus, when restricted to two bodies, the problem admits of a perfectly rigorous mathematical solution. Having succeeded in solving the problem of planetary motion for the ease of two bodies, Newton and his con- temporaries very naturally desired to effect a similar solu- tion for the case of three bodies. The problem of motion in our solar system is that of the mutual action of a great number of bodies ; and having succeeded in the case of two bodies, it was necessary next to try that of three. PROBLEMS OF OBAVITATION. 141 Thus arose the celebrated problem of three bodies. It is found that no rigorous and general solution of this problem is possible. The curves described by the several bodies would, in general, be so complex as to defy mathematical definition. But in the special case of motions in the solar system, the problem admits of being solved by approxima- tion with any required degree of accuracy. The princi- ples involved in this system of approximation may be com- pared to those involved in extracting the square root of any number which is not an exact square ; 2 for instance. The square root of 2 cannot be exactly expressed either by a decimal or vulgar fraction ; but by increasing the number of figures it can be expressed to any required limit of approximation. Thus, the vulgar fractions -|, -f |, |-J|-, etc. , are fractions which approach more and more to the required quantity ; and by using larger numbers the errors of such fraction may be made as small as we please. So, in using decimals, we diminish the en-or t^jjne tenth for eve- ry decimal we add, but never reduce it to zero. A process of the same nature, but immensely more complicated, has to be used in computing the motions of the planets from their mutual gravitation. The possibility of such an ap- proximation arises from the fact that the planetary orbits are nearly circular, and that their masses are very small compared with that of the sun. The first approximation is that of motion in an ellipse. In this way the motion of a planet through several revolutions can neaj'ly always be predicted within a small fraction of a degree, though it may wander widely in the course of centuries. Then sup- pose each planet to move in a known ellipse ; their mutual attraction at each point of their respective orbits can be expressed by algebraic formulae. In constructing these formula, the orbits are first supposed to be circular ; and afterward account is taken by several successive steps of the eccentricity. Having thus found approximately their action on each other, the deviations from the pure elliptic motion produced by this action may bo approximately cal- 142 ASTRONOMY. culated. This being done, tlie motions will be more exact- ly determined, and the mutual action can be more exactly calculated. Thus, the process can be carried on step by step to any degree of precision ; but an enormous aniount of calculation is necessary to satisfy the requirements of modern times with respect to precision." As a general iTile, every successive step in the approximation is much more laborious than all the preceding ones. To understand the principle of astronomical investiga- tion into the motion of the planets, the distinction be- tween observed and theoretical motions must be borne in mind. When the astronomer with his meridian circle de- termines the position of a planet on the celestial sphere, that position is an observed one. When he calcxdates it, for the same instant, from theory, or from tables founded on the theory, the result will be a calciilated or theoretical position. The two are to be regarded as separate, no mat- ter if they should be exactly the same in reality, because they have an entirely different origin. But it must be re- membered that no position can be calculated from theory alone independent of observation, because all soimd theory requires some data to start with, which observation alone can furnish. In the case of planetary motions, these data are the elements of the planetary orbit already described, or, which amounts to the same tiling, the velocity and di- rection of the motion of the planet as well as its mass at some given time. If these quantities were once given with mathematical precision, it would be possible, from the theory of gravitation alone, without recourse to observa- tion, to predict the motions of the planets day by day and generation after generation with any required degree of precision, always supposing that they are subjected to no influence except their mutual gravitation according to the law of Xewton. But it is impossible to determine the elements or the velocities without recourse to observation ; * In the works of the great mathematicians on this subject, algebraic formulse extending through many pages are sometimes given. PROBLEMS OF GRAVITATION. 143 and however correctly they may seemingly be determined for the time being, subsequent observations always show them to have been more or less in error. The reader must understand that no astronomical observation can be mathematically exact. Both the instruments and the observer are subjected to influences which prevent more than an approximation being attained from any one observation. The great art of the astronomer consists in so treating and combining his observations as to eliminate their errors, and give a result as near the truth as possible. When, by thus combining his observations, the astrono- mer has obtained the elements of the planet's motion which he considers to be near the truth, he calculates from them a series of positions of the planet from day to day in the future, to be compared with subsequent observations. If he desires his work to be more permanent in its nature, he may construct tables by which the position can be de- termined at any future time. Having thus a series of the- oretical or calculated places of the planet, he, or others, Avill compare his predictions with observation, and from the differences deduce coiTections to his elements. We may say in a rough way that if a planet has been observed through a certain number of years, it is possible to calculate its place for an equal number of years in advance with some approach to precision. Accurate observations arc commonly supposed to commence with Beadlet, Astron- omer Royal of England in 1750. A century and a quarter having elapsed since that time, it is now possible to con- struct tables of the planets, which we may expect to bt; tolerably accurate, until the year 2000. But this is a possibility rather than a reality. The amount of calcu- lation required for such work is so immense as to be en- tirely beyond the power of any one person, and hence it is only when a mathematician is able to command the ser- vices of others, or when several mathematicians in some way combine for an object, that the best astronomical tables can hereafter be constructed. 144 ASTIiONOMT. § 4. RESULTS OP GRAVrTATION. From what we have said, it will be seen that the problem of the motions of the planets under the influence of grav- itation has called forth all the skill of the mathematicians who have attacked it. They actually find themselves able to reach a solution, which, so far as the mathematics of the subject are concerned, may be true for many centuries, but not a solution which shall be true for all time. Among those who have brought the solution so near to perfec- tion, La Place is entitled to the first rank, although there are others, especially La Geange, who are fully worthy to be named along with him. It will be of interest to state the general results reached by these and other mathema- ticians. We call to mind that but for the attraction of the planets upon each other, every planet would move around the sun in an invariable ellipse, according to Kepler's laws. The deviations from this elliptic motion produced by their mutual attraction are czH^eA. periurbatioTis. When they were investigated, it was found that they were of two classes, which were denominated respectively periodic 'perturbations and secular v.ariations. The periodic perturbations consist of oscillations depend- ent upon the mutual positions of the planets, and there- fore of comparatively short period. Whenever, after a number of revolutions, two planets return to the same position in their orbits, the periodic perturbations are of the same amount so far as these two planets are concerned. They may therefore be algebraically expressed as depend- ent upon the longitude of the two planets, the disturbing one and the disturbed one. For instance, the perturba- tions of the earth produced by the action of Mercury depend on the longitude of the earth and on that of Mer- cury. Those produced by the attraction of Yenus de- pend upon the longitude of the earth and on that of Venus, and so on. RESULTS OF GRAVITATION: 1-45 Tlie secular peHurhations, or secular variations as they are -commonly called, consist of slow changes in the forms and positions of the several orbits. It is found that the perihelia of all the orbits are slowly changing their ap- parent directions from the sun ; that the eccentricities of some are increasing and of others diminishing ; and that the positions of the orbits are also changing. One of the first questions which arose in reference to these secular variations was, will they go on indefinitely .' If they should, they would evidently end in the subversion of the solar system and the destruction of all life upon the earth. The orbits of the earth and planets would, in the course of ages, become so eccentric, that, approaching near the sun at one time and receding far away from it at another, the variations of temperature would be destruc- tive to life. This problem was first solved by La Grange. He showed that the changes could not go on forever, but that each eccentricity would always be confined between two quite narrow limits. His results may be expressed by a very simplo geometrical construction. Let S repre- sent the sun situated in the focus of the ellipse in which Fig. 54. the planet moves, and let C be the centre of the ellipse. Let a straight line SB emanate from the sun to B, another line pass from B to D, and so on ; the number of these lines being equal to that of the planets, and the last one terminating in O, the centre of the ellipse. Then the line S B will be moving around the sun with a very slow motion ; B D will move around B with a slow motion somewhat different, and so each one will revolve in the 146 ASTIiOyOMY. same maimer until we reacli the line which carries on its end the centre of the ellipse. These motions are so slow that some of them require tens of thousands, and others hundreds of thousands of years to perform the revolution. By the combined motion of them all, the centre of the elhpse describes a somewhat irregular curve. It is evi- dent, however, that the distance of the centre from the sun can never be greater than the sum of these revolving hnes. Now this distance shows the eccentricity of the ellipse, which is equal to half the difference between the greatest and least distances of the planet from the sun. The perihelion being in the direction C S, on the opposite side of the sun from C, it is evident that the motion of C will carry the perihelion with it. It is found in this way that the eccentricity of the earth's orbit has been diminishing for about eighteen thousand years, and will continue to diminish for twenty-five thousand years to come, when it will be more nearly circular than any orbit of our system now i.^. But before becoming quite circu- lar, the eeeentrioity will begin to increase again, and so go on oscillating inilefinitely. Secular Acceleration of the Moon. — Another remark- able result reached by mathematical research is that of the acceleration of the moon's motion. More than a century ago it was found, by comparing the ancient and modern observations of the moon, that the latter moved around the earth at a slightly greater rate than she did in ancient times. The existence of this acceleration was a source of great perplexity to La Geange and La Place, because they thought that they had demonstrated mathematically that the attraction could not have accelerated or retarded the mean motion of the moon. Biit on continuing his in- vestigation, La Place found that there was one cause which he omitted to take account of — -namely, the secular diminution in the eccentricity of the earth's orbit, of which we have just spoken. He found that this change in the eccentricity would slightly alter the action of the ACCELERATION OF THE MOON. 147 snn upon the moon, and that this alteration of action would be such that so long as the eccentricity grew smaller, the motion of the moon would continue to be ac- celerated. Computing the moon's acceleration, he found it to be equal to ten seconds into the square of the number of centuries, the law being the same as that for the motion of a falUng body. That is, while in one century she would be ten seconds ahead of the place she would have occupied had her mean motion been uniform, she would, in two centuries, be forty seconds ahead, in three centuries ninety seconds, and so on ; and during the two thousand years which have elapsed since the observations of Hippaechus, the acceleration would be more than a degree. It has re- cently been found that La. Place's calculation was not com- plete, and that with the more exact methods of recent times the real acceleration computed from the theory of gravita- tion is only about six seconds. The observations of ancient eclipses, however, compared with our modern tables, show an acceleration greater than this ; but owing to the rude and doubtful character of nearly all the ancient data, there is some doubt about the exact amount. From the most celebrated total eclipses of the sun, an acceleration of about twelve seconds is deduced, while the observations of Ptolemy and the Arabian astronomers indicate only eight or nine seconds. There is thus an apparent discrepancy between theory and observation, the latter giving a larger value to the acceleration. This difference is now accounted for by supposing that the motion of the earth on its axis is retarded— that is, that the day is gradually growing longer. From the modem theory of friction, it is found that the motion of the ocean under the influence of the moon's attraction which causes the tides, must be accom- ' panied with some friction, and that this friction must re- tard the earth's rotation. There is, however, no way of determining the amount of this retardation unless we assume that it causes the observed discrepancy between the theoretical and observed accelerations of the moon. H8 ASTMUNOMY. TI ow this effect is produced will be seen by reflecting that if the day is continually growing longer without our know- ing it, our observations of the moon, wliich we may suppose to be made at noon, for example, will be constantly made a little later, because the interval from one noon to another will be continually growing a little longer. The moon con- tinually moving forward, the observation will place her fur- ther and further ahead than she would have been observed had there been no retardation of the time of noon. If in the course of ages our noon-dials get to be an hour too late, we should find the moon ahead of her calculated place by one hour's motion, or about a degree. The present theory of acceleration is, therefore, that the moon is really accelerated about, six seconds in a century, and that the motion of the earth on its axis is gradiially diminishing at such a rate as to produce an apparent additional ac- celeration which may range from two to six seconds. § 5. REMAEKS ON THE THEORY OP GRAVTTA- TION. The real nature of the great discover}' of Newton is so frequently misunderstood that a little attention may be given to its elucidation. Gravitation is frequently spoken of as if it were a theory of Newton's, and very generally received by astronomers, but still liable to be iJtimately rejected as a great many other theories have been. Not infrequently people of greater or less intelligence are found making great efforts to prove it erroneous. Every prominent scientific institution in the world frequently receives essays having this object in view. Now, the fact is that Newton did not discover any new force, but only showed that the motions of the heavens could be accounted for by a force which we all know to exist. Gravitation (Latin gravitas — weight, heaviness) is, properly speaking, tlie force which makes all bodies here at the surface of the earth tend to fall downward ; and if any one wishes to REALITY OF GRAVITATION: 149 subvert the theory of gravitation, he must begin by prov- ing that this force does not exist. This no one would think of doing. "What Newton did was to show that this force, which, before his time, had been recognized only as acting on the surface of the earth, really extended to the heavens, and that it resided not only in the earth itself, but in the heavenly bodies also, and in each particle , of matter, however situated. To put the matter in a terse form, what Newton discovered was not gravitation, but the universality of gravitation. It may be inquired, is the induction which supposes gravitation universal so complete as to be entirely beyond doubt ? We reply that within the solar system' it certainly is. The laws of motion as established by observation and experiment at the surface of the earth must be considered as mathematically certain. Now, it is an observed fact that the planets in their motions deviate from straight lines in a certain way. By the first law of motion, such deviation can be produced only by a force ; and the direc- tion and intensity of this force admit of being calculated once that the motion is determined. When thus calculated, it is found to be exactly represented by one great force constantly directed toward the sun, and smaller subsidiary forces directed toward the several planets. Therefore, no fact in nature is more firmly established than is that of universal gravitation, as laid down by Newton, at least within the solar system. We shall find, in describing double stars, that gravita- tion is also found to act between the components of a great number of such stars. It is certain, therefore, that at least some stars gravitate toward each other, as the bodies of the solar system do ; but the distance which separates most of the stars from each other and from our sun is so immense that no evidence of gravitation between them has yet been given by observation. Still, that they do gravitate according to Newton's law can hardly be serri- ously doubted by any one who understands the subject. 150 ASTRONOMY. The reader may now be supposed to see the absurdity of supposing that the theory of gravitation can ever be sub- verted. It is not, however, absurd to suppose that it may yet be shown to be the result of some more general law. Attempts to do this are made from time to time by men of a philosophic spirit ; but thus far no theory of the sub- ject having the slightest probability in its favor has been propounded. Perhaps one of the most celebrated of these theories is that of George Lewis Le Sage, a Swiss physicist of the last century. He supposed an infinite number of ultra- mundane corpuscles, of transcendent minuteness and veloc- ity, traversing space in straight lines in all directions. A single body placed in the midst of such an ocean of mov- ing corpuscles would remain at rest, since it would be equal- ly impelled in every direction. But two bodies would ad- vance toward each other, because each of them would screen the other from these corpuscles moving in the straight line joining their centres, and there would be a slight excess of corpuscles acting on that side of each body which was turned away from the other.* One of the commonest conceptions to accoimt for grav- itation is that of a fluid, or ether, extending through all space, which is supposed to be animated by certain vibra- tions, and forms a vehicle, as it were, for the transmission of gravitation. This and all other theories of the kind are subject to the fatal objection of proposing complicated systems to account for the most simple and elementary ' facts. If, indeed, such systems were otherwise known to exist, and if it could be shown that they really would produce the effect of gravitation, they would be entitled to reception. But since they have been imagined only to account for gravitation itself, and since there is no proof of their existence except that of accounting for it, they * Reference may be made to an article on the kinetic theories of gravitation by William B. Taylor, in the Smithsonian Report for 1S76. CAUSE OF aUAVITATION. 151 are not entitled to any weight whatever. In the present state of science, we are justified in regarding gravitation as an ultimate principle of matter, incapable of alteration by any transformation to which matter can he subjected. The most careful experiments show that no chemical pro- cess to which matter can be subjected either increases or diminishes its gravitating principles in the slightest degree. We cannot therefore see how this principle can ever be referred to any more general cause. ^ CHAPTER VI. THE MOTIONS AND ATTRACTION OP THE MOON. Each of the planets, except Mercury and Yenus, is at- tended by one or more satellites, or moons as they are some- tiroes familiarly called. These objects revolve aroimd their several planets in nearly circular orbits, accompanying them in their revolutions around the sun. Their distances from their planets are very small compared with the distances of the latter from each other and from the sun. Their magnitudes also are very small compared with those of the planets around which they revolve. Where there are several satellites revolving around a planet, the whole of these bodies forms a small system similar to the solar sys- tem in arrangement. Considering each system by itself, the satellites revolve around their central planets or " primaries," in nearly circular orbits, much as the planets revolve around the sun. Biit each system is carried around the sun vdthoiit any serious derangement of the motion of its several bodies among themselves. Our earth has a single satellite accompanying it in this way, the familiar moon. It revolves around the earth in a little less than a month. The nature, causes and con- sequences of this motion form the subject of the present chapter. § 1. THE MOON'S MOTIONS AND PHASES. That the moon performs a monthly circuit in the heav- ens is a fact with which we are all familiar from child- hood. At certain times we see her newly emerged from MOTION OF THE MOON. 153 the snn's rays in the western twilight, and then we call her the new moon. On each succeeding evening, we see her fnrther to the east, so that in two weeks she is oppo- site the snn, rising in the east as he sets in the west. Continuing lier course two weeks more, she has approaclied the snn on the other side, or from the west, and is once more lost in his rays. At the end of twenty-nine or thirty days, we see her again emerging as new moon, and her cir- cuit is complete. It is, however, to be remembered that the sun lias been apparently moving toward the east among the stars during the whole month, so that during the interval from one new moon to the next the moon has to make a complete circuit relatively to the stars, and move forward some 30° further to overtake the sun. The revolution of the moon among the stars is performed in about 27^ days,* so that if we observe when the moon is very near some star, we shall find her in the same position relative to the star at the end of this interval. The motion of the moon in this circuit differs from the apparent motions of the planets in being always forward. We have seen that the planets, though, on the whole, mov- ing directly, or toward the east, are affected with an ap- parent retrograde motion at certain intervals, owing to the motion of the earth around the sun. But the eai'th is the real centre of the moon's motion, and carries the moon along with it in its annual revolution around the sun. To form a correct idea of the real motion of these three bodies, we must imagine the earth performing its circuit around the sun in one year, and carrying witli it the moon, which makes a revolution around it in 27 days, at a distance only about -^ that of the sun. In Fig. 55 suppose S to represent the sun, the large circle to represent the orbit of the earth around it, E to be some position of the earth, and the dotted circle to rep- resent the orbit of the moon around the earth. We must * More exactly, 2" •33166'*, 154 ASTBONOMT. Fig. 55. imagine the latter to carry this circle with it in its an- nual course around the sun. Suppose that when the earth is at E the moon is at M. Then if the earth move, to E^ in 27^ days, the moon win have made a complete revolution relative to the stars — that is, it will be at M„ the line E^ M^ being par- allel to EM. Bat new moon will not have arrived again because the sun is not m the same direction as be- fore. The moon must move through the additional arc J/, EM,, and a little more, owing to the continual ad- vance of the earth, before it will again be new moon. Phases of the Moon. — The moon being a non-luminous body shines only by reflecting the light falling on her from some other body. The principal source of light is the sun. Since the moon is spherical in shape, the sun can illuminate one half her surface. The appearance of the moon varies according to the amount of her illumi- nated hemisphere which is turned toward the earth, as can be seen by studying Fig. 56. Here the central globe is the earth ; the circle around it represents the orbit of the moon. The rays of the sun fall on both earth and moon from the right, the distance of the sun being, on the scale of the figure, some 30 feet. Eight positions of the moon are shown around the orbit at A, E, C, etc., and the right-hand hemisphere of the moon is illuminated in each position. Outside these eight positions are eight others showing how the moon looks as seen from the earth in each position. At A it is "new moon," the moon being nearly between the earth and the sun. Its dark hemisphere PBAtfU'S OF THE MOON. IJJ is then turned toward the earth, so that it is entirely invisible. At E the observer on the earth sees about a fourth of the illuminated hemisphere, which looks like a crescent, as shown in the outside figure. In this position a great deal of light is reflected from the earth to the moon, ren- dering the dark part of the latter visible by a gray light. Pig. 56. This appearance is sometimes called the "old moon in the new moon's arms." At Cthe moon is said to be in her " first quarter," and one half her illuminated hemisphere is visible. At G three fourths of the illuminated hemisphere is visible, and at B the whole of it. The latter position, when the moon is opposite the sun, is called " full moon." After this, at H, D, F, the same appearances are re- peated in the reversed order, the position £> being called the "last quarter." 156 ASTRONOMY. The four principal phases of the moon are, ' ' New- moon," " First quarter," " Full moon," " Last quarter," which occur in regular and unending succession, at inter- vals of between 7 and 8 days. §2. THE SUN'S DISTUEBINa FORCE. The distances of the sun and planets being so immensely gi'eat compared with that of the moon, their attraction upon tlie earth and the moon is at all times very nearly equal. Now it is an elementary princijjle of mechanics that if two bodies are acted upon by equal and parallel forces, no matter how great these forces may be, tJie bodies will move relatively to each other as if those forces did not act at all, though of course the absolute motion of each will be different from what it otherwise would be. If we calculate the absolute attraction of the sun u])on the moon, we shall find it to be about twice as great us that of the earth, because, although it is situated at 400 times the distance, its mass is about 330,000 times as groat as that of the earth, and if we divide this mass by the square of the distance 400 we have 2 as the quotient. To those unacquainted with mechanics, the difficulty often suggests itself that the sun ought to draw the moon away from the earth entirely. But we are to remember that the sun attracts the earth in the same way that it at- tracts the moon, so that the difference between the sun's attraction on the moon and on the earth is only a small fraction of the attraction between the earth and the moon* As a consequence of these forces, the moon moves around the earth nearly as if neither of them were attracted by * In lUis comparisoa of tUe attractive forces of the sun upon the moon and upon the earth, the reader will remember that we are speak- ing not of the absolute force, but of what is called the accelerating force, ■which is properly the ratio of the absolute force to the mass of llie body attracted. Th'i earth having 80 times the mass of the moon, the sun must of course attract it with 80 times the absolute force in order to produce the same motion, or the same accelerating force. SUN'IS ATTRACTION ON MOON. 157 the sun — that is, nearly in an ellipse, having the earth in its focns. But there is always a small difference between the attractive forces of the sun upon the moon and upon the earth, and this difference constitutes a disturbing force which makes the moon deviate from the elliptic orbit which it would otherwise describe, and, in fact, keeps the ellipse which it approximately describes in a state of con- stant change. A more precise idea of the manner in which the sun disturbs the motion of the moon around the earth ma,y be gathered from Fig. 57. Here 8 represents the sun, aiid the circle F Q M N repre- sents the orbit of the moon. First suppose the moon at N, the posi- tion corresponding to new moon. Then the moon, being nearer to the sun than the earth is, will be attracted more powerfully by it than the earth is. It will therefore be drawn away from the earth, or the a,ction of the sun will tend to separate the two bodies. Fig. 57. Next suppose the moon at F the position corresponding to full moon. Here the action of the sun upon the earth will be more powerful than upon the moon, and the earth will in conscijuence I)e drawn away from the moon. In this position also the effect of the disturbing force is to separate the two bodies. If, on the other hand, the moon is near the first quarter or near Q, the sun will exert a nearly equal attraction on both bodies ; and ince the lines of at- traction B S and Q S then converge toward S, it follows that there will be a tendency to bring the two bodies together. The same will evidently be true at the third quarter. Hence the influence of the disturbing force changes back and forth twice in the course of each lunar month. The disturbing force in question may be constructed for any po- sition of the moon in its orbit in the following way, which is be- lieved to be due to Mr. R. A. Pkoctok : Let Mhe the position of the moon ; let us represent the sun's attraction upon it by the line jy S, and let us investigate what line will represent the sun's attrac- tion U])un the earth on the same scale. From i/drop the perpen- 158 ASTRONOMY. dicular iWPupon the line £?