THE GETTY CENTER LIBRARY ^ ^^^^ TREATISE ON OPTICS. BY SIR DAVID BREWSTER, LL.D.,F.R.S. L.&E. CORRESPONDING MEMBER OF THE INSTITUTE OF FRANCE, HONORARY MEMBER OF THE IMPERIAL ACADEMY OF PETERSBURG, AND OF THE ROYAL ACADEMY OF SCIENCES OF BERLIN, STOCK- HOLM, COPENHAGEN, GOTTINGEN, &C. &.C. A NEW EDITION. WITH AN APPENDIX, CONTAINING AN ELF.MENTARY VIEW OF THE APPLICA- TION OF ANALYSIS TO REFLEXION AND REFRACTION, BY A. D. BAG HE, A.M. PROFESSOR OF NAT. PHILOB. AND CHEMISTRY IN THE UNIVERSITY OF PENNSYLVANIA, ONE OF THE SECRETARIES OF AM. PHILOS. 80C-, MEM. ACAD. NAT. SC, HON. MEM. ASSOCIATE SOC. OF NEWPORT, R. 1. PHILADELPHIA: LEA AND BLANCHARD, SUCCESSORS TO CAREY & CO. 561- 1839. Entered according to the act of congress, in the year 1833, by Carey, Lea, & Blanchard, in the clerk's office of the district court of the eastern district of Pennsylvania. STEREOTYPED BY J. HOWE. THE GETTY CENTER LIBr NOTE BY THE AMERICAN EDITOR. My object in undertaking the revision of the Treatise on Optics by Dr. Brewster was, principally, to introduce an Appendix, containing such a discussion of the subjects of Reflexion and Refraction, as might adapt the work to use in those of our colleges in which considerable exten. sion is given to the course of Natural Philosophy. In this revision, I have thought it best, without specially calling the attention of the reader to them, to correct such errors as my comparatively limited knowledge of the subject assured me, would not have been passed over by the author in a second Edition. A. D. BACHE. Philadelphia, Jan., 1833. CONTENTS. Introduction Page 11 PART I. ON TIIE REFLEXION AND REFRACTION OF LIGHT. CATOPTRICS. CHAP. I. Reflexion by Specula and Mirrors 13 Reflexion of Rays from Plane Mirrors 15 Reflexion of Rays from Concave Mirrors 16 Reflexion of Rays from Convex Mirrors 20 CHAP. II. Images formed by Mirrors 22 DIOPTRICS. CHAP. III. Refraction 26 CHAP. IV. Refraction through Prisms and Lenses 31 On the total Reflexion of Light 34 Refraction of Light through Plane Glasses 36 Refraction of Light through Curved Surfaces 37 Refraction of Light through Spheres 38 Refraction of Light through Convex and Concave Surfaces . 40 Refraction of Light through Convex Lenses 41 Refraction of Light through Concave Lenses 44 Refraction of Light through Meniscuses and Concavo-convex Lenses 45 CHAP. V- On the Formation of Images by Lenses, and on their magnifying Power 46 CHAP. VI. Spherical Aberrat ion of Lenses and Mirrors 51 Spherical Aberration of Mirrors 57 On Caustic Curves formed by Reflexion and Refraction 68 A2 CONTENTS. PART II. PHYSICAL OPTICS. CHAP> VII. On the Colors of Light, and its Decomposition 63 Decomposition of Light by Absorption 67 CHAR VIII. On the Dispersion of Light 70 CHAP. IX. On the Principle of Achromatic Telescopes 74 CHAP. X. On the Physical Properties of the Spectrum 78 On the Existence of Fixed Lines in the Spectrum ib. On, the Illuminating Power of the Spectrum 80 On the Heating Power of the Spectrum 81 On the Chemical Influence of the Spectrum 82 On the Magnetizing Power of the Solar Rays 83 CHAP. XI. On the Inflexion or Diffraction of Light 86 CHAP. XII. On the Colors of Thin Plates 90 Table of the Colors of Thin Plates of Air, Water, and Glass 93 CHAP. XIII. On the Colors of Thick Plates 97 CHAP. XIV. On the Colors of Fibres and grooved Surfaces 101 CHAP. XV. On Fits of Refiexiun and Transmission, and on the Interference of Light Ill CHAP. XVI. On the Absorption of Light 120 CHAP. XVII. On the Double Refraction of Light 125 On Crystals with one Axis of Double Refraction 128 On the Law of Double Refraction in Crystals with one Nega- tive Axis 130 CONTENTS. 7 On the Law of Double Refraction in Crystals with one Posi- tive Axis 132 On Crystals with two Axes of Double Refraction 133 On Crystals with innumerable Axes of Double Refraction ... 134 On Bodies to which Double Refraction may be communicated by Heat, rapid Cooling, Pressure, and Induration 136 On Substances with Circular Double Refraction 136 chap. xvm. On the Polarization of Light 137 On the Polarization of Light by Double Refraction 138 CHAP. XIX. On the Polarization of Light by Reflexion 142 On the Law of the Polarization of Light by Reflexion 146 On the partial Polarization of Light by Reflexion 149 CHAP. XX. On the Polarization of Light by ordinary Refraction 152 CHAP. XXI. On the Colors of Crystallized Plates in Polarized Light 157 CHAP. XXII. On the System of Colored Rings in Crystals with one Axis 165 Polarizing Intensities of Crystals w-ith one Axis 172 CHAP. XXIII. On the Systems of Colored Rings in Crystals with two Axes ib. Polarizing Intensities of Crystals with two Axes 179 CHAP. XXIV. Interference of Polarized Light. — On the Cause of the Colors of Crystallized Bodies ib. CHAP. XXV On the Polarizing Structure of Analcimc , 183 CHAP. XXVI. On Circular Polarization 184 Circular Polarization in Fluids 188 Crystals which turn the Planes from Right to Left ib. Crystals which turn the Planes from Left to Right ib. CHAP. XXVII. On Elliptical Polarizaticon, and on the Action of Metals upon Light 190 On Elliptical Polarization 191 Order in which thee Metals polarize most Light in the Plane of Reflexion ib. Vlll CONTENTS* chap, xxvni. On the Polarizing Structure produced by Heat, Cold, Compression, Dilatation, and Induration 197 1. Transient Influence of Heat and Cold ib. (1.) Cylinders of Glass with one positive Axis of Double Refraction ib. (2.) Cylinders of Glacs with a negative Axis of Double Refraction 198 (3.) Oval Plates of Glass with two Axes of Double Re- fraction ib (4.) Cubes of Glass with Double Refraction 199 (5.) Rectangular Plates of Glass with Planes of no Double ' Refraction ib. (6.) Spheres of Glass, &c. with an infinite Number of Axes of Double Refraction 201 (7.) Spheroids of Glass with one Axis of Double Refrac- tion along the Axis of Revolution and two Axes along the Equatorial Diameters il>, (8.) Influence of Heat on regular Crystals 202 2. On the permanent Influence of sudden Cooling ib. 3. On the Influence of Compression and Dilatation 203 4. On the Influence of Induration 205 CHAP. XXIX. Phenomena of Composite or Tessclaled Crystals 206 CHAP. XXX. On the Dichroism.or Double Color, of Bodies; and the Absorption of Polarized Light 210 Colors of the two Images in Crystals with one Axis 211 Colors of the two Images in Crystals with two Axes 212 General Observations on Double Retraction 214 PART III. ON THE APPLICATION OF OPTICAL PRINCIPLES TO THE EXPLANATION OF NATURAL PHENOMENA. CHAP. XXXI. On unusual Refraction 215 CHAP. XXXII. OntheRainbow 223 CHAP. XXXIII. On Halos, Corona;, Parhelia, and Paraselenas 227 CHAP. XXXIV. On the Colors of Natural Bodies 235 CONTEXTS. 9 CHAP. XXXV. On Ihe Eye and Vision 240 . On the Phenomena and Laws of Vision 243 CHAP. XXXVI. On Accidental Colors and Colored Shadows 254 PART IV. ON OPTICAL INSTRUMENTS. chap, xxxvn. On Plane and Curved Mirrors 261 Kaleidoscope 262 Plane Burning Mirrors 264 Convex and Concave Mirrors 265 Cylindrical Mirrors 266 CHAP. XXXVIII. On Single and Compound Lenses 267 Burning and Illuminating Lenses 268 CHAP. XXXIX On Simple and Compound Prisms 270 Prismatic Lenses ib. Compound and Variable Prisms 271 Multiplying Glass 273 CHAP. XL. On the Camera Obscura and Camera Lucida 274 Magic Lantern 276 Camera Lucida 277 CHAP. XLI. On Microscopes .,..,, 279 Single Microscopes ib. Compound Microscopes 283 On Reflecting Microscopes 286 OnTestObjects 287 Rules for Microscopic Observations » ib. Solar Microscope 288 CHAP. XLII. On Refracting and Reflecting Telescopes 289 Astronomical Telescope ib. H 10 CONTENTS. Terrestrial Telescope 290 Galilean Telescope 291 Gregorian Reflecting Telescope ib. Cassegrainian Telescope 293 Newtonian Telescope 294 Sir William Herschel's Telescope 296 Mr. Ramage's Telescope , 297 CHAP. XLIII. On Achromatic Telescopes 297 On Achromatic Eye-pieces 300 Prism Telescope 302 Achromatic Opera Glasses with Single Lenses 304 Mr. Barlow's Achromatic Telescope ib. Achromatic Solar Telescopes with Single Lenses 305 On the Improvement of imperfectly Achromatic Telescopes . . . 306 NOTES. On the absolute Refractive powers of Bodies 315 Absorptive Power of Water ib. Sir David Brewster's analysis of the Spectrum ib. Melloni's experiments on the heating Powers of the Spectrum 316 Hypothesis of Undulations applied to explain Young's principle of Interference 318 On the translucency of gold leaf 320 Classification of Colored Bodies, by Sir David Brewster ib. Duration of Impressions on the Retina 321 Insensibility of the eyes of certain persons to particular Colors ib. A TREATISE ON OPTICS. INTRODUCTION. (1). Optics, from a Greek word which signifies to see, is that branch of knowledge which treats of the properties of light and of vision, as performed by the human eye. (2). Light is an emanation, or something which proceeds from bodies, and by means of which we are enabled to see them by the eye. All visible bodies may be divided into two classes — self-luminous and non-luminous. Self-luminous bodies, such as the stars, flames of all kinds, and bodies which shine by being heated or rubbed, are those which possess in themselves the property of discharging light. Non-luminous bodies are those which have not the power of discharging light of themselves, but which throw back the light which falls upon them from self-luminous bodies. One non-luminous body may receive light from another non-lumi- nous body, and discharge it upon a third ; but in every case the light must originally come from a self-luminous body. When a lighted candle is brought into a dark room, the form of the flame is seen by the light which proceeds from the flame itself; but the objects in the room are seen by the light which they receive from the candle, and again throw back ; while othon- objects, on which the light of the candle does not fall, receive light from the white ceiling and walls, and thus become visible to the eye. (3). All bodies, whether self-luminous or non-luminous, dis- charge light of the same color with themselves. A red flame or a red-hot body discharges red light ; and a piece of red cloth discharges red light, though it is illuminated by the white light of the sun. (4). Light is emitted from every visible point of a luminous or of an illuminated body, and in every direction in which the point is visible. If we look at the flame of a candle, or at a sheet of white paper, and magnify them ever so much, we shall not observe any points destitute of light. 12 INTRODUCTION. (5.) Light moves in straight lines, and consists of separate and independent parts, called rays of light. If we admit the light of the sun into a dark room through a small hole, it will illuminate a spot on the wall exactly opposite to the sun, — the middle of the spot, the middle of the hole, and the middle of the sun, being all in the same straight line. If there is dust or smoke in the room, the progress of the light in straight lines will be distinctly seen. If we stop a very small portion of the admitted light, and allow the rest to pass, or if we stop nearly the whole light, and allow only the smallest portion to pass, the part which passes is not in the slightest degree af- fected by its separation from the rest. The smallest portion of light which we can either stop or allow to pass is called a ray of light. (6). Light moves with a velocity of 192,500 miles in a second of time. It travels from the sun to the earth in seven minutes and a half. It moves through a space equal to the circumference of our globe in the 8th part of a second, a flight which the swiftest bird could not perform in less than three weeks. (7). When light falls upon any body whatever, part of it is reflected or driven back, and part of it enters the body, and is either lost within it or transmitted through it. When the body is bright and well polished like silver, a great part of the light is reflected, and the remainder lost within the silver, which can transmit light only when hammered out into the thinnest film. When the body is transparent, like glass or water, almost all the light is transmitted, and only a small part of it reflected. The light which is driven back from bodies is reflected according to particular laws, the considera- tion of which forms that branch of optics called catoptrics ; and the light which is transmitted through transparent bodies is transmitted according to particular laws, the consideration of which constitutes the subject of dioptrics. CHAP. I. INCIDENCE AND REFLEXION. 13 PART I. ON THE REFLEXION AND REFRACTION OF LIGHT. CATOPTRICS. (8.) Catoptrics is that branch of optics which treats of the progress of rays of light after they are reflected from sur- faces either plane or curved, and of the formation of images from objects placed before such surfaces. CHAP. I. REFLEXION BY SPECULA. AND MIRRORS. (9.) Any substance of a regular form employed for the pur- pose of reflecting light, or of forming images of objects, is called a speculum or mirror. It is generally made of metal or glass, having a highly polished surface. The name of mir- ror is commonly given to reflectors that are made of glass; and the glass is always quicksilvered on the back, to make it reflect more light. The word speculum is used to describe a reflector which is metallic, such as those made of silver, steel, or of grain tin mixed with copper. (10.) Specula or mirrors are either plane, concave, or convex. A plane speculum is one which is perfectly flat, like a look- ing-glass; a concave speculum is one which is hollow like the inside of a watch-glass ; and a convex speculum is one which is round like the outside of a watch-glass. As the light which falls upon glass mirrors is intercepted by the glass before it is reflected from the quick-silvered sur- face, we shall suppose all our mirrors to be formed of polished metal, as they are in almost all optical instruments. (11.) When a ray of light, A t),fig. 1., falls upon a plane speculum, M N, at the point D, it will be reflected or driven back in a direction D B, which is as much inclined to E D, a line perpendicular to M N, as the ray A D was ; that is, the angle B D E is equal to A D E, or the circular arc B E ^N is equal to E A. B 14 A TREATISE ON OPTICS. PART I. The ray A D is called the incident ray, and D B the re- flected ray, A D E the angle of incidence, and B D E the angle of reflexion ; and a plane passing through A]) and D B, or the plane in which these two lines lie, is called the plane of incidence, or the plane of reflexion. (12.) When the speculum is concave, as M N,fig. 2., then if C be the centre of the circle of which M N is a part, the incident ray A D and the reflected ray D B will form equal angles with the line C D, which is perpendicular to the small ^ portion of the speculum on which the ray falls at D. Hence in this case also the angle of incidence A D E is equal to the angle of reflexion B D E. (13.) When the speculum is convex, as M IS!, Jig. 3., let C be the centre of the circle of which M N forms a part, and C E a line drawn through D ; then the angle of incidence A D E will be equal to the angle of re- flexion B D E. These results arc found to be true by experiment; and they may be easily tal proved by admitting a ray of the sun's light through a hole in the window-shut- ter, and making it fall on the mirrors M N in the direction A D, when it will be seen reflected in the direction D B. If the incident ray A 1) is made to approach the perpendicular D E, the reflected ray D B will also approach the perpendicu- lar D E ; and when the ray A I) falls in the direction E D, it will be reflected in the direction I) E. In like manner when the ray A D approaches to D N, the ray D B will ap- proach to D M. (14.) As these results are true under all circumstances, we may consider it as a general law, that when light falls upon any surface, whether plane or curved, the angle of its reflex- ion is equal to the angle of its incidence. Hence we have a method of universal application for find- ing the direction of a reflected ray when we know the direc- tion of the incident ray. If A D, for example, figs. 1, 2, 3., is the direction in which the incident ray falls upon the mirror at D, draw the perpendicular D E in fig. 1., and in fig. 2 or fig. 3. draw a line from D to C, the centre of the curved sur- face M N ; and, having described a circle MBEAN round D as a centre, take the distance A E in the compasses and CHAP. I. PLANE MIRRORS. 15 carry it from E to B, and having 1 drawn a line from D to B, D B will be the direction of the reflected ray. Riflexion of Rays from Plane Mirrors. (15.) Reflexion of parallel rays. When parallel or equidis- tant rays, A D, A' lb', jig. 4., are incident upon a, plane mir- ing-. 4. rnr. M N, they will'continue to be parallel after reflexion. By the method already explained, describe arches of circles round D, D' as centres, and make the arch from E towards B equal to that between A D and D E, and also the arch from E' to- wards B' equal to that between A' D' and D' E' ; then drawing the lines D B, D' B', it will be found that those lines are par- allel. If the space between A D and A' D' is filled with other rays parallel to A D, so as to constitute a parallel beam or mass of light, A A' D' D, the reflected rays will be all parallel to B D, and will constitute a parallel reflected beam. The reflected beam, however, will be inverted ; for the side A D, which was uppermost before reflexion, will be undermost, as at D B, after reflexion. (16.) Reflexion of diverging rays. Diverging rays are those which proceed from a point, A, and separate as they ad- vance, like A D, A D', A D". When such rays fall upon a Fig. 5. \B' •E'NB IE' f st? A vO" \! \>r// Men N j>d\/ dY N D^^D^ fi 16 A TREATISE ON OPTICS. PART I. plane mirror M N,flg. 5., they will be reflected in directions) DB, D' B', D" B", making the angles B L) E, B' D' E', B" 1)" E" respectively, equal to A D E, A D' E', A D" E"; the lines I) E, D' E{ D" E" being drawn from the points 1), D', D", where the rays are incident, perpendicular to M N ; and by continuing the reflected rays backwards, they will be found to meet at a point A' as far behind the mirror l\l A" as A is before it; that is, if A N A' be drawn perpendicular to M N, A' N will be equal to A N. Hence the rays will have the same divergency after reflexion as they had before it. If we consider A D" D as a divergent beam of light included between A D and A D", then the reflected beam included between D B and D" B" will diverge from A', and will be in- verted after reflexion. (17.) Reflexion of converging- rays. Converging rays are those which proceed from several points A A' A", Jig. 0., towards one point B. When such rays fall upon a plane Ftg.9. mirror, M N, they will be reflected in directions D B', D' B', D" B', forming the same angles with the perpendiculars D E, D' E', D" E", as the incident rays did, and converging to a point B' as far before the mirror as the point B is behind it. If we consider A D D" A" as a converging beam of light, D" B' D will be its form after reflexion. In all these cases the reflexion does nothing more than invert the incident beam of light, and shift its point of diver- gence or convergence to the opposite side of the mirror. Reflexion of Rays from Concave Mirrors. (18.) Reflexion of parallel rays. Let M N, Jig. 7., be a concave mirror whose centre of concavity is C ; and let A D. AM, A N be parallel rays, or a parallel beam of light falling CHAP. r. CONCAVE MIRRORS. 17 upon it, at .and near to the vertex D. Then, since C M, C N are perpendicular to the surface ot" the mirror at the points M and N, C M A, C N A will be the angles of incidence of the rays A M, A N. Make the angles of reflexion C M F, C N F equal to C M A, C N A, and it will be found that the lines M F, N F meet at F in the line A D, and these lines M F, N F will be the reflected rays. The ray A C D being per- pendicular to the mirror at D, because it passes through the Fig. 7. . A - ...'■>€ Ar centre C, will be reflected in an opposite direction U F; so that all the three rays, A M, A D, and A N, will meet at one point, F. In like manner it will be found that all other rays between A M and A N, tailing upon other points of the mirror between M and N, will be reflected to the same point F. The point F, in which a concave mirror collects the rays which fall upon it, is called the focus, or fire-place, because the rays thus collected have the power of burning any inflammable body placed there. When the rays which the mirror collects are parallel, as in the present case, the point F is called its prin- cipal focus, or its focus for parallel rays. When we consider that the rays which form the beam AMNA occupy a large space before they fall upon the mirror M N, and by reflexion are condensed upon a small space at F, it is easy to understand how they have the power of burning bodies placed at F. Rule. — The distance of the focus F from the nearest point or vertex D of the mirror M N is in spherical mirrors, what- ever he their substance, equal to one half of C D, the radius of the mirror's concavity. The distance F D is called the '/m ncipal focal distance of the mirror. The truth of this rule may be found by projecting Jig. 7. upon a large scale, and by taking the points M N near to D. (19.) Reflexion of diverging rays. Let M N, fig. 8., be a concave mirror, whose centre of concavity is C ; and let rays A M, A D, A N, diverging or radiating from the point A, fall upon the mirror at the points, M, D, N, and be reflected from these points; M and N being near to D. The lines C M, C D, and C N being perpendicu- lar to the mirror at the points M, D, and N, we shall find B2 18 A TREATISE ON OPTICS. Fig. 8. TART I. the reflected rays M F, N F, by making the angle F M C equal to A M C, and F N C equal to AN C ; and the point F where these rays meet will be the focus where the diverging rays A M, A N are collected. By com paring T^. 7. with fig. 8. it is obvious that, as the incident ray A M in fig. 8. is nearer the perpendicular C M than the same ray is in fig. 7, the re- flected ray M F will also be nearer the perpendicular C M than the same ray in Jig. 7. ; and as the same is Uiie of the reflected ray N F, it follows that the point F must be nearer C in jig. 8. than in Jig. 7.; that is, in the reflexion of diverg- ing rays the focal distance I) F of the mirror is greater than its focal distance for parallel rays. If we suppose the point of divergence A, Jig. 8., or the radiant paint, as it is called, to approach to C, the incident rays A M, A N will approach to the perpendiculars C M, C N, and consequently the reflected rays M F, N F will also ap- proach to C M, C N ; that is, as the radiant point A approaches to the centre of concavity C, the focus F also approaches to it, so that when A reaches C, F will also reach C ; that is, when rays diverge from the centre, C, of a concave mirror, they will all be reflected to the same point. If the radiant point A passes C towards D, then the focus F will pass C towards A; so that if the light now diverges from F it will be collected in A, the points that were formerly the radiant points being now the foci. From this relation, or in- terchange, between the radiant points and the foci, the points /l and F have been called conjugate foci, because if either of them be the radiant point the other will be the focal point. If in Jig. 7. we suppose F to be the radiant point, then the focal point A will be at an infinite distance ; that is, the rays will never meet in a focus, but will be parallel, like M A, N A in fig. 7. In like manner it is obvious, that if the point F is at/, as in fig. 9., the reflected rays will be Ma, No; that is, they will diverge from some point, A', behind the mirror M N ; and as CHAP. I. CONCAVE MIRRORS. Fis- 9. 19 J"approaches to D, they will diverge more and more, as if the point A', from which they seem to diverge, approached to D. The point A' behind the mirror, from which the rays M a, N a seem to proceed, or at which they would meet if they moved backwards in the directions a M, a N is called their virtual focus, because they only tend to meet in that focus. In all these cases trie distance of the focus F may be deter- mined either by projection or by the following rule, the radius of the concavity of the mirror, C D, and the distance, A D, of the radiant point, being given. Rule. Multiply the distance, A D, of the radiant point from the mirror by the radius, C D, of the mirror, and divide this product by the difference between twice the distance of the radiant point and the radius of the mirror, and the quotient will be F D, the conjugate focal distance required. In applying Lliis rule we must observe, what will be readily seen from the figures, that if twice A D is less than C D (as at f, fig- 9.), the rays will not meet before the mirror, but will have a virtual focus behind it, the distance of which from D will be given by the rule. (20.) Reflexion of converging rays. Let M N,fig. 10., be a concave mirror whose centre of concavity is C, and let rays A M, A D, A N, converging to a point A' behind the mirror, fall upon iii" mirror at the points M, D, and N, and sutler reflexion at these points ; M and N being near to D. The lines C M, C D, and C N being perpendicular to the mirror at the points M, D, and N, we shall find the reflected rays M F and N F by making the angle F M C equal to A M C, and F N C equal to A N C ; and the point F, wh ere these rays meet, will be the focus where the converging rays A IM, A J\ are collected. By comparing Jig. 10. with, //<>-. 7. it will be manifest, that, as the incident ray A M in fig. 10. is farther from the perpendicular C M than the same ray A M in jig. 7., the reflected ray M F in fig. 10 20 A TREATISE OIV OPTICS. Fig. 10. 1'ART I. will also be farther from the perpendicular C M than the same ray in fig. 7. ; and as the same is true of the reflected ray N F, it follows that the point F must be farther from C in Jig. 10. than in Jig. 7. ; that is, in the reflexion of converging rays, the conjugate focal distance DF of the mirror is less than its distance for parallel rays. If we suppose the point of convergence A', fig. 10., to ap- proach to D, or the rays A M, A N to become more conver- gent, then the incident rays A M, AN will recede from the perpendiculars CM, C N ; and as the reflected rays M F, N F will also recede from C M, C N, the focus F will like- wise approach to D; and when A' reaches D, F will also reacl^ D. If the rays A M, A N become less convergent, that is, if their point of convergence A' recedes farther from D to the left, the focus F will recede from D to the right ; and when A' is infinitely distant, or when A M, A N are parallel, as in Jig. 7., F will be half-way between D and C. In these cases the place of the focus F will be found by the following rule. Rule. Multiply the distance of the point of convergence from the mirror by the radius of the mirror, and divide this product by the sum of twice the distance cf the radiant point and the radius C D, and the quotient will be the distance of the focus, or F D, the focus F being always in front of the mirror. Reflexion of Rays from Convex Mirrors. (21.) Reflexion of parallel rays. Let M N, Jig. 11., be a convex mirror whose centre is C, and let A M, AD, AN be parallel rays falling upon it. Continue the lines C M and C N to E, and M E, N E will be perpendicular to the surface of CONVEX MIRRORS. 21 'Jin mirror at the points M and N. The rays A M, AN will therefore be reflected in directions M 13, N B, the angles of reflexion E M B, E N B being equal to the angles of incidence E M A, E N A. By continuing the reflected rays B M, B N backwards, they will be found to meet at F, their virtual focus behind the mirror; and the focal distance D F for parallel rays Fig. 11. .--E A c '\5 \ Id A. \B A will he almost exactly one half of the radius of convexity C D, provided the points M and N are taken near D. (22.) Reflexion of diverging- rays. Let M N,.fig. 12., be a convex mirror, C its centre of convexity, and A M, A N rays diverging from A, which fall upon the mirror at the points M, N. The linens C M E and C N E will be, as before, per- pendicular to the' mirror at M and N ; and consequently, if we make the angles of reflexion E M B, p] N B equal to the angles of incidence E M A, E N A, M B, N B will be the re- flected rays which, when continued backwards, will meet at 22 A TRKATIsE OX OPTICS. F, their virtual focus behind the mirror. By comparing fig. 12. with fig. 11., it is obvious that the ray A M, fig. 12., is farther from M E than in fig. 11., and consequently the re- flected ray M B must also he farther from it. Hence, as the same is true of the ray N B, the point F, where these ravs meet, must be nearer to D in fig. 12. than in fig. 11. ; that is, in the reflexion of diverging rays, the virtual local distance D F is less than for parallel rays. For the same reason, if we suppose the point of divergence A to approach the mirror, the virtual locus F will also approach it; and when A arrives at D, F will also arrive at D. In like manner, if A recedes from the mirror, F will recede from it ; and when A is infinitely distant, or when the rays become parallel, as in fig. 11., F will be half-way between D and C. In all these cases, the focus is a virtual one behind the mirror.* CHAP. TI. IMAGES FORMED BY MIRRORS. (23.) The image of any object is a picture of it formed either in the air, or in the bottom of the eye, or upon a white ground, such as a sheet of paper. Images are generally form- ed by mirrors or lenses ; though they may be formed also by placing a screen, with a small aperture, between the object and the sheet of paper which is to receive the image. In order to understand this, let C D be a screen or window-shut- Fig. 13. ter with a small aperture, A, and E F a sheet of white paper placed in a dark room. Then, if an illuminated object, RGB, is placed on the outside of the shutter, we shall observe an in- verted image of this object painted on the paper at r g b. In order to understand how this takes place, let us suppose the object R B to have three distinct colors, red at R, green at G, and blue at B ; then it is plain that the red light from R will * For a discussion of the siibjccis in this chapter, see (in the College Edi tion) the Appendix of American Editor, Chapter I. CHAP. II. IMAGES FORMED BY MIRRORS. 23 pass in straight Hues through the aperture A, and fall upon the paper E F at r. In like manner the green light from G will fall upon the paper at g, and the blue light from B will iiill upon the paper at b ; thus painting upon the paper an in- verted image, rb, of the object, R B. As every colored point in the object R B has a colored point corresponding to it, and opposite to it on the paper E P, the image b r will be an ac- curate picture of the object R B, provided the aperture A is very small. But if we increase the aperture, the image will become less distinct ; and it will be nearly obliterated when the aperture is large. The reason of this is, that, with a large aperture, two adjacent points of the object will throw their light on the same point of the paper, and thus create confusion in the image. It is obvious from Jig. 13., that the size of the image br will increase with the distance of the paper E F behind the hole A. If A g is equal to A G, the image will be equal to the object; iffAif is less than AG, the image will be less than the object ; and if A g is greater than A G, the image will be greater than the object. As each point of an object throws out rays in all directions, it is manifest that those only which fall upon the small aper- ture at A concur in forming the image br; and as the num- ber of these rays is very small, the image b r must have very little light, and therefore cannot be used for any optical pur- poses. This evil is completely remedied in the formation of images by mirrors and lenses. (24.) Formation of images by concave mirrors. Let A B, fig. 14., be a concave mirror whose centre is C, and let M N be an object placed at some distance before it. Of all the rays emitted in every direction by the point M, the mirror re- ceives only those which lie between M A and M B, or a cone of rays M A B whose base is the spherical mirror, the section of which is A B. If we draw the reflected rays A m, Bm, for all the incident rays M A, M B, by the methods already described, we shall find that they will all meet at the point m, 24 A TREATISE ON OPTICS. PART I. and will there paint the extremity M of the object. In like manner, the cone of rays NAB flowing from the other ex- tremity N of the object will be reflected to a focus at n, and will there paint that point of the object. For the same reason, cones of rays flowing; from intermediate points between M and N will be reflected to intermediate points in the image between m and n, and m n will be an exact inverted picture of the object M N. It will also be very bright, because a great number of rays concur in forming each point of the image. The distance of the image from the mirror is found by the same rule which we have given for finding the locus of diverging rays, tin: points M, m in fig. 14. corresponding with A and F in. jig. 8. If we measure the relative sizes of the object M N and its image m n, we shall find that in every case the size of the image is to the size of the object as the distance of the image from the mirror is to the distance of the object from it. If the concave mirror A B is large, and if the object M N is very bright, such as a plaster of 1'aris statue strongly illu- minated, the image m n will appear suspended in the air; and a series of instructive experiments may be made by varying the distance of the object, and observing the variation in the size and place of the image. When the object is placed at ■m n, a magnified representation of it will be formed at JVI N. (25.) Formation of images by convex mirrors. In concave mirrors there is, in all cases, a real image of the object formed in front of the mirror, excepting when the object is placed be- tween the principal focus and the mirror, in which case it gives a virtual image formed behind it ; whereas in convex mirrors the image is always a virtual one formed behind the mirror. Let A B,fig. 15., be a convex mirror whose centre is C, and M N an object placed before it ; and let the eye of the observer be situated any- where in front of the mirror, as at E. Out of the greai number of rays which are emitted in every direction from the points M, N of the object, and are sub- sequently reflected from the mirror, a few only can enter the eye at E. Those which do enter the eye, such as D E, \ ! F E and G E, II E, will be reflected \; from the portions D F, G H of the mir- ror so situated with respect to the eye and the points M, N that the angles of incidence and reflexion will be equal. The ray M D will be reflected in a direction Fig. 15. CHAP. II. IMAGES IN PLANE MIKKORS. 25 D E, forming the same angle that M D does with the perpen- dicular C N, and the ray N G in the direction G E. In like manner, F E, H E will be the reflected rays corresponding to the incident ones M F, N H. Now, if we continue backwards the rays D E, F E, they will meet at m ; and they will there- fore appear to the eye to have come from the point m as their focus. For the same reason the rays G E, H E will appear to come from the point n as their focus, and m n will be the vir- tual image of the object M N. It is called virtual because it is not formed by the actual union of rays in a focus, and cannot be received upon paper. If the eye E is placed in any other position before the mirror, and if rays are drawn from M and N, which after reflexion enter the eye, it will be found that these rays continued backwards will have their virtual foci at m and n. Hence, in every position of the eye before the mir- ror, the image will be seen in the same spot m n. If we draw the lines C M, C N from the centre of the mirror, we shall find that the points m, n are always in these lines. Hence it is obvious that the image m n is always erect, and less than the object. It will approach to the mirror as the object M N approaches to it, and it will recede from it as M N recedes ; and when M N is infinitely distant, and the rays which it emits become parallel, the image m n will be half-way be- tween C and the mirror. In other positions of the object the distance of the image will be found by the rule already given for diverging rays falling upon convex mirrors. The size of the image is to the size of the object, as Cm, the distance of the image from the centre of the mirror, is to C M, the dis- tance of the object. In approaching the mirror, the image and object approach to equality ; and when they touch it, they are both of the same size. Hence it follows that objects are always seen diminished in convex mirrors, unless when they actually touch the mirror. (26.) Formation of images by plane mirrors. Let A B, Jig. 10., be a plane mirror or looking-glass, MN an object situated before it, and E the place of the eye ; then, upon the very same principles which we have explained for a convex mirror, it will be found that an image of M N will be formed at m n, the virtual foci m, n being determined by continuing back the reflected rays D E, F E till they meet at m, and G E, H E till they meet at n. If we join the points M, m and N, n, the lines M m, N n will C Fig. 16 26 A TREATISE ON OPTICS. PART I. be perpendicular to the mirror A B, and consequently parallel ; and the image will be at the same distance, and have the same position behind the mirror that the object has before it. Hence we see the reason why the images of all objects seen in a looking-glass have the same form and distance as the objects themselves.* DIOPTRICS. (27.) Dioptrics is that branch of optics which treats of the progress of those rays of light which enter transparent bodies and are transmitted through their substance. CHAP. III. REFRACTION. (28.) When light passes through a drop of water or a piece of glass, it obviously suffers some change in its direction, be- cause it does not illuminate a piece of paper placed behind these bodies in the same manner as it did before they were placed in its way. These bodies have therefore exercised some action, or produced some change upon the light, during its progress through them. In order to discover the nature of this change, let A B C D Fig- 17 - be an empty vessel, having a hole fj^ II in one of. its sides B D, and let a lighted candle S be placed within a few feet of it, so that a ray of its light S H may fall upon the bottom C 1) of the vessel, and form a round spot of light at a. The beam of C a, fi c m u - - lig-ht SHR a will be a straight line. Having marked the point a which the ray from S strikes, pour water into the vessel till it rises to the level E F. As soon as the surface of the water has become smooth, it will be seen that the round spot which was formerly at a is now at b, and that the ray SURA is bent at R; H R and R b being two straight lines meeting at R, a point in the surface of the water. Hence it follows, that all objects seen under water are not seen in their true direction by a person whose eye is not immersed in the water. If a fish, for example, is lying at b,fig. 17., it will be seen by an eye at S in the direc- tion S a, the direction of the refracted ray R S ; so that, in * For the formation of images by mirrors, see (in the College edition) the App«ndix of Am. # C D be now emptied, and let a bright object, such as a sixpence, be cemented on the bottom of it at a. If the observer places himself a few feet from the vessel, he will find a position where he will see the sixpence at a through the hole II. If water be now poured into the vessel up to E F, the observer will no longer see the sixpence; but if another sixpence is placed al a, and is moved towards b, it will become visible when it reaches b. Now, as the ray from the sixpence at /; reaches the eye, it must come out of the water at a point, R, m tin; surface, found by drawing a straight line, S II R, through the eye and the hole H; and conse- quently b R must be the direction of the ray, which makes the sixpence visible, before its refraction at R. But if this as^ 28 A TREATISE ON OPTICS. PART I. ray had moved onwards in a straight line, without being re- fracted at R, its path would have been b h ; whereas, in con- sequence of the refraction, its path is R II. Hence it follows, that when a ray of light, passing through any dense medium, such as water, &c, in a direction oblique or slanting to its surface, quits the medium at any point, and enters a rarer medium, such, as air, it is refracted from the line perpendicu- lar to the surface at the point where it quits it. When the ray SII R from the candle falls, or is incident upon the surface E F of the water, and is refracted in the di- rection R b, towards the perpendicular M N, the angle M RH which it makes with the perpendicular, is called the angle of incidence ; and the angle N R b, which the ray R b bent or refracted at R makes with the same perpendicular, is called the angle of refraction. The ray II R is called the incident ray, and R /; the refracted ray. But when the light comes out of the water from the sixpence at b, and is refracted at K in the direction R H, i R is the incident ray and R H the refracted ray. The angle N R b is the angle of incidence, and M R II the angle of refraction. Hence it follows, that when light passes out of a rarer into a denser medium, as from air hi water, the angle of incidence is greater than the angle of refraction; and when light passes out of a denser into a rarer medium, as out of water into air, the angle of incidence is less than the angle of refrac- tion: and these angles are so related to one another, that when the ray which was refracted in the one case becomes the incident ray, what was formerly the incident ray becomes the refracted ray. £29.) In order to discover the law, or rule, according to Pi „ m which the rays of light enter or quit water, or other refracting media, so that we may be able to determine the refracted ray when we know the di- rection of the incident ray, describe a circle M N upon a square board ABC D, fig. 18. standing upon a heavy pedestal P, and draw the two diameters M N, E F perpendicular to one another, and also to the sides, A B, A C of the piece of wood. Let a small tube, II R, be so made that it may be attached to the board along any radius IT R, II' R, or, what would be still better, that it may move freely round R as a centre. Let the board with its pedestal be CHAP. III. LAW OF REFRACTION. 29 placed in a pool or tub of water, or in a glass vessel of water, so that the surface of the water may coincide with the line E F without touching the end R of the tube II R. When the tube is in the position M R, perpendicular to the surface E F of the water, admit a ray of light down the tube, and it will be seen that it enters the water at R, and passes straight on to N, without suffering any change in its direction. Hence it fol- lows, that a ray of light incident ■perpendicularly on a re- fracting surface experiences no refraction or change in its direction. If we now place a sixpence at N, we shall see it through the tube M R; so that the rays from the sixpence quit the water at R, and proceed in the same straight line N R M. Hence a ray of light quitting a refracting surface perpendicularly undergoes no refraction or change, of direc- tion. If we now bring the tube into the position H R, and make a ray of light pass along it, the ray will be refracted at R in some direction R b, the angle of refraction N R b being less than the angle of incidence M R II. If we now with a pair of compasses, take the shortest distance b n of the point b from the perpendicular M N, and make a scale of equal parts, of which b n is one part, the scale being divided into tenths and hundredths, and if we set the distance H m upon this scale, we shall find it to be 1-330 of these parts, or li- nearly. If this experiment is repealed at any other position, H'R, of the tube where R b' is the refracted ray, we shall find that on a new scale, in which b' n' is one part, H' m' will also be 1-330 parts. But the lines H m, H'm' are called the sines of I he angles of incidence IIRM, H'RM, and bn, b' n' the sines of the angles of refraction b R N, b' R N. Hence it follows, that in water the sine of the angle of inci- dence is to the sine of the angle of refraction as 1*336 to 1, whatever be the position of the ray with respect to the surface E F of the water. This truth is called by optical writers the constant ratio of the sines. By placing a sixpence at b, we shall find that it will be seen through the tube when it has the position II R; and placing it at b', it will be seen through it in the position H'R. Hence, when light quits the surface of water, the sine of its angle or' incidence b R N will be to the sine of its angle of refraction H R M as 1 to 1-300, as these are the measures of the sines bn, H m; and since these are also the measures of b' n' H' m! upon another scale, in which b' n' is unity, we may conclude that, when light emerges from water into air, the sines of the angles of inci- dence and refraction are in the constant ratio of 1 to 1-330. If we make the same experiment with other bodies, we shall obtain different degrees of refraction at the same angles.; C2 30 A TREATrSE ON OPTICS. PART I but in every ease the sinos oi' the angles of incidence and refraction will b.-> found to have a constant ratio to each other. The number 1*336, which expresses this ratio for ukiter, is called the index of refract ion fur water, and sometimes its refractive power. (30.) As philosophers have determined the index of refrac- tion for a great variety of bodies, we are able, from those de- terminations, to ascertain the direction of any ray when re- fracted at any angle of incidence from the surface of a given body, either in entering or quitting it. Thus, in the case of water, let it he required to find the direction of a ray, H R,./?i,'. 18., after it is refracted at the surface E V of water : draw R M perpendicular to K F at, the point R, where the ray IT R enters the water, and from II draw II m perpendicular to M R. Take Hm in the compasses, and make a scale in which this distance occupies 1*336 parts, or l.' f nearly. Then, taking 1 on the same scale, place one foot of the compasses in the quadrant N P, and move that foot towards or from N till the other foot falls upon some one point n in the perpendicular RN, and in no other point of it. Let b be the point on which the first foot of the compasses is placed when the second falls upon n, then the line R b 'passing through this point will be the refracted ray corresponding to the incident ray II U, (31.) Table J. (Appendix) contains the index of refraction for some of the substances most interesting in optics. (32.) As the bodies contained in these tables have all dif- ferent densities, the indices of refraction annexed to their names cannot be considered as showing the relation of their absolute refractive powers, or the refractive powers of their ultimate particles. The small refractive index of hydrogen, for example, arises from its particles being at so great a distance from one another; and, if we take its specific gravity into account, we shall find that, instead of having a less refractive power than all other bodies, its ultimate particles exceed all other bodies in their absolute action upon light. Sir Isaac Newton has shown, upon the supposition that the ultimate particles of bodies are equally heavy, and that the law of the forces which different media exert is of the same form in all, that the^ absolute refractive power is equal to the excess of the square of the index of refraction above unity, divided by the specific gravity of the body. In this way Table II. (Appendix) has been calculated. Mr. Herschel has justly remarked, that if, according to the doctrines of modern chemistry, material bodies consist of a finite number of atoms, differing in their actual weight for every differently compounded substance, the intrinsic refrac- tive power of the atoms of any given medium will be the CHAP. IV. LENSES. 31 product arising from multiplying the number for the medium, in Table II. by the weight of its atom. (33.) In examining Table II., it appears that the substances which contain fluoric acid have the least absolute refractive power, while all inflammable bodies have the greatest. The high absolute refractive power of oil of cassia, which is placed above all other fluids, and even above diamond, indicates the great inflammability of its ingredients.f CHAP. IV.* REFRACTION THROUGH PRISMS AND LENSES. (34). Hy means of the law of refraction explained in the preceding pages, we are enabled to trace a ray of light in its r through any medium or body of any figure, or through any number of bodies, provided we can always find the incli- nation of the incident ray to that small portion of the surface where the ray either enters or quits the body. The bodies generally used in optical experiments, and in the construction of optical instruments, where the eflect is produced by refraction, are prisms, plane glasses, spheres, and lenses, a section of each of which is shown in the an- nexed figure. Fig. 19. 1. The most common optical prism, shown at A, is a solid having two plane surfaces A R, A S, which are called its re- fracting surf aces. The face R S, equally inclined to A It and A S, is called the base of the prism. ■ '2. A plane glass, shown at E, is a plate of glass with two plane surfaces, a l>. cd, parallel to each other. 3. A spherical lens, shown at C, is a sphere, all the points in its surface being equally distant from the centre < ). 4. A double convex lens, shown at D, is a solid tunned by two convex spherical surfaces, having their centres on oppo- site sides of tlie lens. When the radii of its two surfaces are equal, it is said to be equally convex; and when the radii are unequal, it is said to be an unequally convex lens. * For tin- subjects treated in this and in the preceding chapter, see (in the Coll -Hi' edition) tin: Appendix of Am. ed. chap. iii. t See Note No. I. at the close of author's Appendix. 32 A TREATISE ON OPTICS. PART 1 5. A plano-convex lens, shown at E, is a lens having one of its surfaces convex and the other plane. 6. A double concave lens, shown at F, is a solid bounded by two concave spherical surfaces, and may be either equally or unequally concave. 7. A plano-concave lens, represented at G, is a lens one of whose surfaces is concave and the other plane. 8. A meniscus, shown at II, is a lens one of whose surfaces is convex and the other concave, and in which the two surfaces meet if continued. As the convexity exceeds the concavity, it may be regarded as a convex lens. 9. A concavo-convex lens, shown at I, is a lens one of whose surfaces is concave and the other convex, and in which the two surfaces will not meet though continued. As the con- cavity exceeds the convexity, it may be regarded as a concave lens. In all these lenses a line, M N, passing through the centres of their curved surfaces, and perpendicular to their plane sur- faces, is called the axis. The figures represent only the sec- tions of the lenses, as if they were cut by a plane passing through their axis; but the reader will understand that the convex surface of a lens is like the outside of a watch-glass, and the concave surface like the inside of a watch-glass. In showing the progress of light through such lenses, and in explaining their properties, we shall still use the sections shown in the above figure ; for since every section of the same lens passing through its axis has exactly the same form, what is true of the rays passing through one section must be true of the rays passing through every section, and consequently through the whole surface. (35.) Refraction of light tlirough prisms. As prisms are introduced into several optical instruments, and are essential parts of the apparatus used for decomposing light and exam- ining the properties of its component parts, it is necessary that the reader should be able to trace the progress of light through their two refracii"-i;'siirf ices. Let A B C be a prism of plate glass whose index of refraction is 1-500, and let II R be a ray of light falling obliquely upon its first surface A B at the point R. Round R as a centre, and with any radius H R, de- scribe the circle H Mb. Through R %^ draw M R N perpendicular to A B, ,, and H m perpendicular to M R. The angle H R M will be the angle of in- cidence of the ray H R, and H m its sine, which in the present OITAV. IV. KKFKACTION BY PRISMS. 33 case is 1-500. Then having made a scale in which the dis- tance H m is 1-500, or 1^ parts, take 1 part or unity from the same scale, and having set one foot of the compasses on the circle somewhere about b, move it to different points of the circle till the other foot strikes only one point n of the line R N; the point b thus found will be that through which the refracted ray passes, R b will be'the refracted ray, and nRb the angle of refraction, because the sine b n of this angle has been made such that its ratio to H m, the sine of the angle of incidence, is as 1 to 1-500. The ray Rb thus refracted will go on in a straight line till it meets the second surface of the prism at R', where it will again suffer refraction in the direc- tion II' b'. In order to determine this direction, make R' II' equal to R II, and, with this distance as radius, describe the circle H' b'. Draw R' N perpendicular to AC, and II' m' perpendicular to R'N, and form a scale on which II' ml shall be 1 part, or 1-000, and divide it into tenths and hundredths. From this scale take in the compasses the index of refraction 1*500, or l! 2 of these parts; and having set one foot some- where in the line R' n', move it to different parts of it till the other foot falls upon some part of the circle about b', taking care that the point b' is such, that when one foot of the com- passes is placed there, the other foot will touch the line R' n', continued, only in one place. Join R' b'. Then, since H'R' m! is the angle of incidence on the second surface AC, and IV m' its sine, and since n' b', the sine of the angle b' R' n', has been made to have to II' in' the ratio of 1-500 to 1, b' R'n' will be the angle of refraction, and R' b' the refracted ray. If we suppose the original ray H R to proceed from a can- dle, and if we place our eye at b' behind the prism so as to receive the refracted ray b' R', it will appear as if it came in the direction D R' b\ and the candle will be seen in that di- rect ion; the angle II E D representing its angular change of direction, or the angle of deviation, as it is called. In the construction of fig. 20., the ray II R has been made to fall upon the prism at such an angle that the refracted ray R R' is equally inclined to the faces A B, AC, or is parallel to the base H G of the prism ; and it will be found that the angle of incidence on the face of the prism, H RB is equal to the a nolo of emergence b'R'C. Under these circumstances we shall find, by making the angle II RB either greater or less than it is in the figure, that th angle of deviation II E D is less than at any other angle of incidence. If we, therefore, place the eye behind the prism at b', and turn the prism round in the plane B A C, sometimes bringing A towards the eye and sometimes pushing it from it, we shall easily discover 34 A TREATISE ON OPTICS. FART I. the position where the image of the candle seen in the direc- tion b' I) has the least deviation. When this position is found, the angles II R B and b' R' C are equal, and R R' is parallel to B C, and perpendicular to A F, a line bisecting the refract- ing angle BAC of the prism. Hence it may be shown by the similarity of triangles, or proved by projection, that the angle of refraction b R a at the first surface is equal to E A F, half the refracting angle of the prism. But since B A F is known, the angle of refraction b Rn is also known; and the angle of incidence HRM being found by the preceding methods, we may determine the index of refraction for any prism by the following analogy. As the sine of the angle of refraction is to the sine of the angle of incidence, so is unity to the index of refraction ; or the index of refraction is equal to the sine of the angle of incidence divided by the sine of the angle of re- fraction. (36.) By this method, which is very simple in practice, we may readily measure the refractive powers of all bodies. It the body be solid, it must be shaped into a prism ; and if it is soft or fluid, it must be placed in the angle B A C of a hollow Fig. 2J. prism ABC, fig. 21., made by cement- ing together three pieces of plate glass, A B, A C, B C. A very simple hollow prism for this purpose may be made by fastening together at any angle two pieces of plate glass, A B, A C, with a bit of wax, F. A drop of the fluid may then be placed in the angle at A, where it will be retained by the force of capillary attraction. When light is incident upon the second surface of a prism, it may fall so obliquely that the surface is incapable of refract- ing it, and therefore the incident light is totally reflected from the second surface. As this is a curious property of light, we must explain it at some length. On the total Reflexion of Light. (37.) We have already stated, that when light falls upon the first or second surfaces of transparent bodies, a certain portion of it is reflected, and another and much greater portion transmitted. The light is in this case said to be partially re- flected. When the light, however, falls very obliquely upon the second surface of a transparent body, it is wholly reflected, and not a single ray suffers refraction, or is transmitted by the surface. Let A B C be a prism of glass, whose index of re- fraction is 1-500: let a ray of light G K,j%. 22., be refracted Fig. 22. TOTAL REFLEXION. 35 at K by the first surface A B, so as to fall on the point R of the second surface very obliquely, and in the direction K R. Upon R as a centre, and with any ra- dius, RH, describe the circle H *a MENF; then, in order to find the refracted ray corresponding to H R, make a scale on which H m is equal to 1, and take in the compasses 1-500 or 1| from that scale, and setting one foot in the quadrant E N, try to find some point in it, so that the other foot may fall only in one point of the radius R N. It will soon be seen that there is no such point, and that 1-500 is greater even than E R, the sine of an angle E R N of 90°. If the distance 1-500 in the compasses had been less than E R, the ray would have been refracted at R ; but as there is no angle of refraction whose sine is 1-500, the ray does not emerge from the prism, but suffers total reflexion at R in the direction It S, so that the angle of reflexion M R S is equal to the angle of incidence M R H. If we construct^. 22. so as to make the incident ray H R take different positions between M R and F It, we shall find that the refracted ray will take different positions between R N and R E. There will be some position of the incident ray about H R, where the re- fracted ray will just coincide with It E ; and that will happen when the quantity 1-500, taken from the scale on which II m is equal to 1, is exactly equal to R E, or radius. At all posi- tions of the incident ray between this line and F R, refraction will be impossible, and the ray incident at R will be totally reflected. It will also be found that the sine of the angle of incidence at R, at which the light begins to be totally reflect- ed, is equal to y.J^^, or -6G6, or |, which is the sine of 41° 48', the angle of total reflexion for plate glass. The passage from partial to total reflexion may be finely seen, by exposing one side, A C, of a prism A B C,fig. 20., to the light of the sky, or at night to the light reflected from a large sheet of white paper. When the eye is placed behind the other side, A B, of the prism, and looks at the image of the sky, or the paper, as reflected from the base, B C, of the prism, it will see when the angle of incidence upon B C is less than 41° 48', the faint light produced by partial reflexion ; but by turning the prism round, so as to render the incidence gradually more oblique, it will see the faint light pass sud- denly into a bright light, and separated from the faint light by 36 A TREATISE ON OPTICS. PART I. a colored fringe, which marks the boundary of the two reflex ions at an angle of 41° 48'. But, at all angles of incidence above this, the light will suffer total reflexion. Fig. 23. Aa' Refraction of Light through Plane Glasses. (38.) Let. M N,fig. 23., be the section of a plane glass with parallel faces; and let a ray of light, A B, fall upon the first surface at B, and be refracted into the direction B C : it will again be refracted at its emergence from the second surface at C, in a direction, C D, parallel to A B ; and to an eye at D it will ap- pear to have proceeded in a direction a C, which will be found by continu- ing D C backwards. It will thus appear to come from a point a below A, the point from which it was really emitted. This may be proved by projecting the figure by the method already described ; though it will be obvious also from the considera- tion, that if we suppose the refracted ray to become the inci- dent ray, and to move backwards, the incident ray will become the refracted ray. Thus the refracted ray B C, falling at equal angles upon the two surfaces of the plane glass, will suffer equal refractions at B and C, if we suppose it to move in op- posite directions ; and consequently the angles which the re- fracted rays BA, CD form with the two refracting surfaces will be equal, and the rays parallel. If we suppose another ray, A' B', parallel to A B, to fall upon the point B', it will suffer the same refraction at B' and C, and will emerge in the direction C I)', parallel to C D, as if it came from a point a'. Hence parallel rays falling upon a plane glass will retain their parallelism after passing through it. (39). If rays diverging from any point, A, fig. 24., such as &&• %*• A B, A B', are incident upon a ,-< n plane glass, M IN, they will be E=SLh ^^ refracted into the directions B C T B' C by the first surface, and C D, C D' by the second. By continuing C B, C B' backwards, they will be found to meet at a, a point farther from the glass than A. Hence, if we suppose ^JP the surface B B' to be that of standing water, placed horizon- CHAP. IV. CtlRVED SURFACES. 37 tally, an eye within it would see the point A removed to a, the divergency of the rays B C, B' C having been diminished by refraction at the surface B B'. But when the rays B C, B' C suffer a second refraction, as in the case of a plane glass, we shall find, by continuing D C, D' C backwards, that they will meet at b, and the object at A will seem to be brought nearer to the glass ; the rays C D, C D', by which it is seen, having been rendered more divergent by the two refractions. A plane glass, therefore, diminishes the distance of the diver- gent point of diverging rays. If we suppose D C, D' C to be rays converging to b, they will be made to converge to A by the refraction of the two surfaces; and consequently a plane glass causes to recede from it the convergent point of converging rays. If the two surfaces B B', C C are equally curved, the one being convex and the other concave, like a watch-glass, they will act upon light nearly like a plane glass ; and accurately like a plane glass, if the convex and concave sides are so re- lated that the rays B A, C D arc incident at equal angles on each surface : but this is not the case when the surfaces have the same centre, unless when the radiant point A is in their common centre. For these reasons, glasses with parallel sur- faces are used in windows and for watch-glasses, as they pro- duce very little change upon the form and position of objects seen through them. Refraction of Light through Curved Sttrfaces. (40). When we consider the inconceivable minuteness of the particles of light, and that a single ray consists of a suc- cession of those particles, it is obvious that the small part of any curved surface on which it falls, and which is concerned in refracting it, may be regarded as a plane. The surface of a lake, perfectly still, is known to be a curved surface of the same radius as that of the earth, or about 4000 miles ; but B square yard of it, in which it is impossible to discover any curvature, is larger in proportion to the radius of the earth than the small space on the surface of a lens occupied by a ray of light is in relation to the radius of that surface. Now. mathematicians have demonstrated that a line touching a curve at any point may be safely regarded as coinciding with an in- finitely small part of the curve ; so that when a ray of light, A B,^g-. 25., falls upon a curved refracting surface at B, its D 38 A TREATISE ON OFTICS. Fig. 25. angle of incidence must be considered as A B D, the angle which the ray A B forms with a line D C, perpendic- ular to a line M N, which touches, or is a tangent to, the curved surface at B. In all spherical surfaces, such as those of lenses, the tangent M N is perpendicular to the radius C B of the surface. Hence, in spherical surfaces the consideration of the tangent MN is unnecessary; because the radius C D, drawn through the point of incidence B, is the perpendicular from which the angle of incidence is to be reckoned. Refraction of Light through Spheres. (41.) Let M N be the section of a sphere of glass whose centre is C, and whose index of refraction is 1*500; and let parallel rays, fig. 20., H R, EPR', fall upon it at equal dis- Fiff. 20. tances on each side of the axis G C F. If the ray H R is in- cident at R, describe the circle H D b round R ; through C and R draw the line CRD, which will be perpendicular to the surface at R, and draw H in perpendicular to R D. Draw the ray R b r through a point b found by the method already ex- plained, and so that the sine b n of the angle of refraction b ft C may be 1 on the same scale on which II m is 1-500, or I 1 , ; then R b will bo the ray as refracted by the first surface of the sphere. In like manner draw R' r' for the refracted ray corresponding to H' R'. If we continue the rays R r, R'r', they will meet the axis at E, which will be the focus of parallel rays for a single con- vex surface RPR'; and the focal distance P E may be found by the following rule. Rule for finding the principal focus of a single convex surface. Divide the index of refraction by its excess above unity, and the quotient will be the principal focal distance, P E ; the radius of the surface, or C R, being 1. If C R is CHAP. IV. CURVED REFRACTING SURFACES. 39 given in inches, then we have to multiply the result by that number of inches. When the surface is that of glass, of which the index of refraction is 1*5, then the focal distance, P E, will always Be equal to thrice the radius, C R. Round r as a centre, with a radius equal to R H, describe the circle I)' 1/ h, and, by the method formerly explained, find a point b' in the circle, such that b' n', the sine of the angle of refraction b' rn', is 1-500 or 1^ on the same scale on which h m\ the sine of the angle of incidence, is 1 part, and r b' F will he the ray refracted at the second surface. In the same manner we shall find r' F to he the refracted ray correspond- ing to the incident ray RV, F being the point where r b' cuts the axis G E. Hence the point F will be the focus of paral- lel rays for the sphere of glass M N. If diverging rays fall upon the points R, R', it is quite clear, from the inspection of the figure, that their focus will be on some point of the axis G F more remote from the sphere than F, the distance of their focus increasing as the radiant point from which they diverge approaches to the sphere. When the radiant point is as far before the sphere as F is be- hind it, then the rays will be refracted into parallel directions, and the focus will be infinitely distant. Thus, if we suppose the rays F r, F r' to diverge from F, then they will emerge after refraction in the parallel directions R H, R' H'. If converging rays fall upon the points RR', it is equally manifest that their focus will be at some point of the axis, (i F, nearei the sphere than its principal focus F; and their convergency may be so great that their focus will fall within the sphere. All these truths may be rendered more obvious, and would be more deeply impressed upon the mind, by tracing rays of different degrees of divergency and convergency through the sphere, by the methods already so fully explained. (42.) In order to form an idea of the effect of a sphere made of substances of different refractive powers, in bringing parallel rays to a focus, let us suppose the sphere to have a radius of one inch, and let the focus F be determined as infg. 26., when the substances are, Index "f Dlatalloa, F Q. of the Refraction. Focui Ir'.m Ihe Sphere Tabaehi « i - 111145 4 inches. Water 1-3358 ' - 1 — Glass 1-50(1 - i — Zircon 2-000 . 6 — Hence we find that in tabasheer the distance F Q. is 4 inches; in water, 1 inch; in glass, half an inch; and in zircon, nothing; that -is, r and F coincide with Q, after a single refraction atR. 40 A TREATISE ON OPTICS. PART I. When the index of refraction is greater than 2-000, as in diamond and several other substances, the ray of light Rr will cross the axis at a point somewhere between C and Q. Under certain circumstances the ray R r will suffer, total re- flexion at r, towards another part of the sphere, where it will again suffer total reflexion, being carried round the circum- ference of the sphere, without the power of making its escape, till the ray is lost by absorption. Now, as this is true of every possible section of the sphere, every such ray, R r, incident upon it in a circle equidistant from the axis, G F, will suffer similar reflexions. Rule for finding the focus F of a sphere. The distance of the focus, F, from the centre, C, of any sphere may be thus found. Divide the index of refraction by twice its excess above 1, and the quotient is the distance, C F, in radii of the sphere. If the radius of the sphere is 1 inch, and its refrac- tive power 1-500, we shall have C F equal to 1J inches, and Q, F equal to half an inch. Refraction of Light through Convex and Concave Surfaces. (43.) The method of tracing the progress of a ray which enters a convex surface, is shown in Jig-. 26. for the ray H R, and of tracing one entering a concave surface of a rare me- dium, or quitting a convex surface of a dense one, is shown for the ray R r, in the same figure. When the ray enters the concave surface of a dense me- dium, or quits a similar surface, and enters the convex surface of arare medium, the method of tracing its progress is shown in fig. 'HI., where M N is a dense medium (suppose glass) with two concave surfaces, or a thick concave lens. Let C, C be the centres of the two surfaces lying in the axis C C, and H R, H' R' parallel rays incident on the first surface. As C R is perpendicular to the surface at R, H R C will be the angle of incidence; and if a circle is described with a radius CHAP. IV. REFRACTION THROUGH JJiEXSES. 41 It h, h m will be the sine of that angle. From a scale on which h m is 1*500, take in the compasses 1, and find some point, b, in the circle where, when one foot of the compasses is placed, the other will fall only on one point, n, of the per- pendicular C It: the line R 6 drawn through this point will be the refracted ray. By continuing this ray b R backwards, it will be found that it meets the axis at F. In like manner it will be seen that the ray H'R' will be refracted in the di- rection ft' r', as if it also diverged from F. Hence F will be the virtual focus of parallel rays refracted by a single concave surface, and may be found by the following rule. Rule for finding the principal focus of a single concave surface. Divide the index of refraction by its excess above unity, and the quotient will be the principal focal distance F E, the radius of the surface, or C E, being 1. If the radius C E is given in inches, we have only to multiply E F, thus obtained by that number of inches, to have the value of F E in inches. If, by a similar method, we find the refracted ray r d at the emergence of the ray r b from the second surface r r' of the lens, and continue it. backwards, it will be found to meet the axis at f; so that the divergent rays R r, RV are rendered still more divergent by the second surface, and /will be the focus of the lens M N. Refraction of Light through Convex Lenses. (44). Parallel rays. Rays of light falling upon a convex lens parallel to its axis are refracted in precisely the same manner as those which fall upon a sphere ; and the refracted ray may be found by the very same methods. But as a sphere has an axis in every possible direction, every incident ray must be parallel to an axis of it; whereas, in a lens which has only one axis, many of the incident rays must be oblique to that axis. In every case, whether of spheres or lenses, all the rays that pass along the axis sutler no refraction, because the axis is always perpendicular to the refracting surface. When parallel rays, It L, Jt C, R L, jig. 28., fall upon a double convex Ions, L L, parallel to its axis It F, the ray R C which coincides with tin? axis will pass through without suf- fering any refraction, but the other rays, R L, R L, will be refracted at each of the surfaces of the lens ; and the refract- ed rays corresponding to them, viz. L F, L F, will be found, by the method already given, to meet at some point, F, in the axis. When the rays are oblique to the axis, as S L, S L, T L, D2 42 A TREATISE ON OPTICS. Fig. 28. TART I. T L, the rays S C, T C, which pass through the centre, C, of the lens, will suffer refraction at each surface ; but as the two refractions are equal, and in opposite directions, the finally re- fracted rays Cf,Cf will be parallel to S C, T C. Hence, in considering oblique rays, such as S L, T L, we may regard lines Sf T/' passing through the centre, C,of the lens as the directions of the refracted rays corresponding to S C, T C. By the methods already explained, it will be found that S L, S L will be refracted to a common point,/, in the direction of the cen- tral ray S/, and T L, T L, to the point/'. The focal distance F C, or fC, may be found numerically by the following rule, when the thickness of the lens is so small that it may be neglected. Rule for finding the principal focus, or the focus of paral- lel rays, for a glass lens unequally convex. Multiply the radius of the one surface by the radius of the other, and divide twice this product by the sum of the same radii. When the lens is of glass and equally convex, the focal dis- tance will be equal to the radius. Rule for the principal focus of a plano-convex lens of glass. With either side of the lens turned to parallel rays, the distance of the focus, when the lens is thin, will be equal to twice the radius of the convex surface. (45.) Diverging rays. When diverging rays, R L, R L, fig. 29., radiating from the point R, fall upon the double con- vex lens L L, whose principal focus is at O and 0', their focus will be at some point F more remote than O. If R approaches to L L, the focus F will recede from L L. When R comen to P, so that P C is equal to twice the principal focal distance CHAP. IV. REFRACTION THROUGH LENSES. Fig. 29. L 43 R- ■" — .^^1- O K^ u-^Jc 1 ■ ■"P C 0, the focus F will be at P' as far behind the lens as the radiant point P is before it. When R comes to O', the focus F will be infinitely distant, or the rays L F, L F will be par- allel ; and when R is between O'-and C, the refracted rays will diverge and have a virtual focus before the lens. The focus F of a glass lens, when the thickness is small, will be found by the following rule. Rule for finding the focus of a convex lens for diverging rays. Multiply twice the product of the radii of the two sur- faces of the lens by the distance, R C, of the radiant point, for a dividend. Multiply the sum of the two* radii by the same distance R C, and from this product subtract twice the product of the radii, for a divisor. Divide the above dividend by the divisor, and the quotient will be the focal distance, C F, required. If the lens is equally convex, the rule will be this. Multi- ply the distance of the radiant point, or R C, by the radius of the surfaces, and divide thut product by the difference between the same distance and the radius, and the quotient will be the focal length, C F, required. When the lens is plano-convex, divide twice the product of the distance of the radiant point multiplied by the radius by the difference between that distance and twice the radius, and the quotient will be the distance of the focus from the centre of the lens. (46.) For converging rays. When rays, R L, R L, con- verging to a point/, fig. 30., fall upon a convex lens L L, they Fig. 30. f>-.+ - will be so refracted as to converge to a point or focus F nearer the lens than its principal focus O. As the point of con- vergence/recedes from the lens, the point F will also recede 44 A TREATISE ON OPTICS. from it towards O, which it just reaches when the point/ be- comes infinitely distant. When / approaches the lens, F also approaches it The focus F of a glass lens may be found when the thickness is small, by the following' rule : — Rule for finding the focus of converging rays. Multiply twice the product of the radii of the two surfaces of the lens by the distance /C of the point of convergence, for a divi- dend. Multiply the sum of the two radii by the same dis- tance/ C, and to this product add twice the product of the radii, tor a divisor. Divide the above dividend by the divisor, and the quotient will be the focal distance C F required. If the lens is equally convex, multiply the distance /C by the radius of the surface, and divide that product by the sum of the same distance and the radius, and the quotient will be the focal length F C required. When the lens is plano-convex, divide twice the product of the distance /C multiplied by the radius by the sum of that distance and twice the radius, and the quotient will be the focal distance F C required. Refraction of Light through Concave Lenses. (47.) Parallel rays. Let L L be a double concave lens, and * 31. R L, R L parallel rays incident upon it ; these rays will di- verge after refraction in the directions L r, L r, as if they radiated from a point F, which is the virtual focus of the lena The rule for finding F C is the same as for a convex lens. (48.) Diverging rays. When the lens L L receives the Fig. 32. rays R L, R L diverging from R, they will be refracted into REFRACTION THROUGH LENSES. 45 lines, Lr, Lr, diverging from a focus F, more remote from the lens than the principal focus O, and the focal distance, F (', will be found by the following rule : — Rulk for finding the focus of a concave lens of glass, for diverging rays. Multiply twice the product of the radii by the distance, R C, of the radiant point for a dividend. Mul- tiply the sum of the radii by the distance R C, and add to this twice the product of the radii, for a divisor. Divide the divi- (! 'ml by the divisor, and the quotient will be the focal distance. If the lens is equally concave, the rule will be this. Mul- tiply the distance of the radiant point by the radius, and divide the product by the sum of the same distance and the radius, and l lie quotient will be the focal distance. When the lens is plano-concave, multiply twice the radius by the distance of the radiant point, and divide this product by the sum of the same distance and twice the radius; the quotient will be the focal distance. (40). Converging rays. When rays, R L, R L, fig. 33., W converging to a point /, fall upon a concave lens, L L, they will be refracted so as to have their virtual focus at F, and the distance F C will be found by the rule given for convex lenses. The rule for finding the focus of converging rays is exactly the same as that for diverging rays in a double convex lens. When the lens is plano-concave, the rule for finding the focus of converging rays is the same as for diverging rays on a plano-convex lens. Refraction of Light through Meniscuscs and Concavo-con- vex Lenses of Class. (oO.) The general effect of a meniscus in refracting paral- lel, diverging, and converging rays, is the same as that of a convex lens of the same focal length ; and the general effect of a concavo-convex lens is the same as that of a concave lens of the same focal length. Rule /or a meniscus with parallel rays. Divide twice thr> product of the two radii by their difference, and the quotient will be the focal distance required. 46 A TREATISE ON OPTICS. PART I. Rule/o?- a meniscus with diverging rays. Multiply twice the distance of the radiant point by the product of the two radii for a dividend. Multiply the difference between the two radii by the same distance of the radiant point, and from this product take twice the product of the radii for a divisor. Divide the above dividend by this divisor, and the quotient will be the focal distance required. The truth of the preceding rules and observations is ca- pable of being demonstrated mathematically ; but the reader who has not studied mathematics may obtain an ocular den* i- stration of them, by projecting the rays and lenses in large diagrams, and determining the course of the rays after refrac- tion by the methods already described. We would recommend to him also to submit the rules and observations to the test of direct experiment with the lenses themselves. CHAP. V. ON THE FORMATION OF IMAGES BY LENSES, AND ON THEIR MAGNIFYING POWER.* (51.) We have already described, in Chapter II., the prin- ciple of the formation of images by small apertures, and by the convergency of rays to foci by reflexion from mirrors. Images are formed, by refraction, by lenses in the very same manner as they are formed, by reflexion, in mirrors ; and it is a universal rule, that when an image is formed by a convex lens, it is inverted in position relatively to the position of the object, and its magnitude is to that of the object as its distance from the lens is to the distance of the object from the lens. If M N is an object placed before a convex lens, L L, fig. 34., every point of it will send forth rays in every direction. Those rays which fall upon the lens L L will be refracted to foci behind the lens, and at such distances from it as may be determined by the Rules in the last chapter. Since the focus where any point cf the object is represented in ils image is in the straight line drawn from that point of the object through the middle point C of the lens, the upper end M of the object will be represented somewhere in the line M C m, and the lower end N somewhere in the line N C n, that is, at the See, in the College edition, Appendix of Am. ed. chap. iv. CHAP. V. IMAGES FORMED BY LENSES. 47 points m, n, where the rays L m, L m, L n, L n cross the li nes MCm,NCn. Hence m will represent the upper, and n the lower end of the object M N. It is also evident, that in the Fig. 34. n M ir ? > : ^5^J><^^C__ .^^Sy^^C^T^ m L -^M two triangles M C N, m C n, m n, the length of the image must be to M N the length of the object as C m, the distance of the image, is to C M, the distance of the object from the lens. We are enabled, therefore, by a lens, to form an image of an object at any distance behind the lens we please, greater than its principal focus, and to make this image as large as we please, and in any proportion to the object. In order to have the image large, we must bring the object near the lens, and in order to have it small, we must remove the object from the lens ; and these effects we can vary still farther, by using lenses of different focal lengths or distances. When the lenses have the same focus, we may increase the brightness of the image by increasing the size of the lens or the area of its surface. If a lens has an area of 12 square inches, it will obviously intercept twice as many rays proceed- ing from every point of the object as if its area were only 6 square inches; so that, when it is out of our power to in- crease the brightness of the object by illuminating it, we may always increase the brightness of the image by using a larger lens. (52.) Hitherto we have supposed the image m n to be re- ceived upon white paper, or stucco, or some smooth and white surface on which a picture of it is distinctly formed ; but if we receive it upon ground glass, or transparent paper, or upon a plate of glass one of whose sides is covered with a dried film of skimmed milk, and if we place our eye 6 or 8 inches or more behind this semi-transparent ground interposed at m n, we shall see the inverted image m n as distinctly as before. If we keep the eye in this position, and remove the semi- 48 A TREATISE ON OFPICS. PART I. transparent ground, we shall see an image in the air distinctly and more bright than before. The cause of this will be readily understood, if we consider that all the rays which form by their convergence the points m, n of the image mn, cross one another at m, n, and diverge from these points exactly in the same manner as they would do from a real object of the same size and brightness placed in m n. The image m n therefore of any object may be regarded as a new object; and by placing another lens behind it, another image of the image m a would be formed, exactly of the same size and in the same place as it would have been had m n been a real object. But since the new image of mn must be inverted, this new image will now be an erect image of the object M N, obtained by the aid of two lenses ; so that, by using one or more lenses, we can obtain direct or inverted images of any object at plea- sure. If the object M N is a movable one, and within our reach, it is unnecessary to use two lenses to obtain an erect image of it: we have only to turn it upside down, and we shall obtain, by means of one lens, an image erect in reality, though still inverted in relation to the object. (53.) In order to explain the power of lenses in magnifying objects and bringing them near us, or rather in giving mag- nified images of objects, and bringing the images near us, we must examine the different circumstances under which the same object appears when placed at different distances from the eye. If an eye placed at E looks at a man a b, Jig. 35., placed at a distance, his general outline only will be seen, Fig. 35. and neither his age, nor his features, nor his dress will be re- cognized. When he is brought gradually nearer to us, we dis- cover the separate parts of his dress, till at the distance of a few feet we perceive his features; and when brought still nearer, we can count his very eye-lashes, and observe the minutest lines upon his skin. At the distance E b the man is seen under the angle b E c, and at the distance E B he is seen under the greater angle B E A or b E A', and his appa- rent magnitudes, a b, A' b, are measured in those different positions by the angles b E a, B E A, or b E A'. The appa- rent magnitude of the smallest object may, therefore, be equal CHAP. V. MAGNIFYING POWER OF LENSES. 49 to the apparent magnitude of the greatest. The head of a pin, for example, may be brought so near the eye that it will appear to cover a whole mountain, or even the whole visible surface of the earth, and in this case the apparent magnitude of the pin's head is said to be equal to the apparent magnitude of the mountain, &c. Let us now suppose the man a b to be placed at the dis- tance of 100 feet from the eye at E, and that we place a con- vex glass of 25 feet focal distance, half-way between the ob- ject a b and the eye, that is 50 feet from each; then, as we have previously shown, an inverted image of the man will be formed 50 feet behind the lens, and of the very same size as the object, that is, six feet high. If this object is looked at by the eye, placed (J or 8 inches behinn it, it will be seen ex- ceedingly distinct, and nearly as well as if the man had been brought nearer from the distance of 100 feet to the distance of fi inches, at which we can examine minutely the details of bis persona] appearance, Now, in this case, the man, though not actually magnified, has been apparently magni- fied, because his apparent, magnitude is greatly increased, in the proportion nearly of inches to 100 feet, or of 200 to 1. But if, instead of a lens of 25 leet focal length, we make use of a lens of a shorter focus, and place it in such a position betwi en tiie eye and the man, that its conjugate foci may bo at the distance of 20 and 80 feet from the lens, that is, that the man is 20 feet before the lens, and his image 80 feet be- hind it, then the size of the image is four times that of the object, and the eye placed inches behind this magnified image will see it with the greatest distinctness. Now in this case the image is magnified 4 times directly by the lens, and 200 times by being brought 200 times nearer the eye; so that its apparent magnitude will be 800 times as large as before. If, on the other hand, we use a lens of a still smaller focal ii ii s tli, and place it in such a position between the eye and the man, that its conjugate foci may be at the distance of 75 a i id 'J5 fret, from the lens, that is, that the man is 75 feet be- fore the lens, and his image 25 feet behind it, then the size of tin' image will be only one third of the size of the object; but. though the image is thus diminished three times in size, yet its apparent magnitude is increased 200 times by being brought within 6 inches of the eye, so that it is still magni- fied, or its apparent magnitude is increased -1J", or fi7 times, nearly. At distances less than the preceding, where the focal length of the lens forms a considerable part of the whole dia- E >0 A TREATISE ON OPTICS. FART I. tance, the rule for finding the magnifying power of a lens, when the eye views, at the distance of G inches, the image formed by the lens, is as follows. From the distance between the image and the object in feet, subtract the focal distance of the lens in feet, and divide the remainder by the same focal distance. By this quotient divide twice the distance of the object in feet, and the new quotient will be the magnifying power, or the number of times that the apparent magnitude of the object is increased. When the focal length of the lens is quite inconsiderable, compared with the distance of the object, as it is in most cases, the rule becomes this. Divide the focal length_.pf the lens by the distance at which the eye looks at the image ; or, as the eye will generally look at it at the distance of 6 inches, in order to see it most distinctly, divide the focal length by (5 inches ; or, what is the same thing, double the focal length in feet, and the result will be the magnifying power. (54.) Here, then, we have the principle of the simplest telescope ; which consists of a lens, whose focal length ex- ceeds six inches, placed at one end of a tube whose length must always be six inches greater than the focal length of the lens. When the eye is placed at the other end of the tube, it will see an inverted image of distant objects, magnified in proportion to the focal length of the lens. If the lens has a focal length of 10 or 12 feet, the magnifying power will be from 20 to 24 times, and the satellites of Jupiter will be dis- tinctly seen through this single lens telescope. To a very short-sighted person, who sees objects distinctly at a distance of three inches, the magnifying power would be from 40 to 48. • A single concave mirror is, upon the same principle, a re- flecting telescope, for it is of no consequence whether the image of the object is formed by refraction or reflexion. In this case, however, the image m n, Jig. 14., cannot be looked at without standing in the way of the object ; but if the re- flection is made a little obliquely, or if the mirror is sufficiently large, so as not to intercept, all the light from the object, it may be employed as a telescope. By using his great mirror, 4 feet in diameter and 40 feet in focal length, in the way now men tioned, Dr. Herschel discovered one of the satellites of Saturn. But there is still another way of increasing the apparent magnitude of objects, particularly of those which are within our reach, which is of great importance in optics. It will be proved, when we come to treat of vision, that a good eye sees the visible outline of an object very distinctly when it is placed at a great distance, and that, by a particular power in the eye, we can accommodate it to perceive objects at differ- CTTAP. V. MAGNIFYING POWER OF LENSES. 51 ent distances. Hence, in order to obtain distinct vision of any object, wo have only to cause the rays which proceed from it to enter the eye in parallel lines, as if the object itself was very distant. Now, if we bring an object, or the. image of an object, very near to the eye, so as to give it great apparent magnitude, it becomes indistinct ; but if we can, by any contrivance, make the rays which proceed from it enter the eye nearly parallel, we shall necessarily see it distinctly. But we have already shown that when rays diverge from the focus of any lens, they will emerge from it parallel. If we, therefore, place an object, or an image of one, in the focus of a lens held close to the eye, and having a small focal distance, the rays will enter the eye parallel, and we shall see the ob- ject very distinctly, as it will be magnified in the proportion of its present short distance from the eye to the distance of six inches, at which we see small objects most distinctly. But this short distance is equal to the focal length of the lens, so that the magnifying power produced by the lens will be equal to six inches divided by the focal length of the lens. A lens thus used to look at or magnify any object is a single micro- scope ; and when such a lens is used to magnify the magnifi- ed image produced by another lens, the two lenses together constitute a compound microscope. When such a lens is used to magnify the image produced in the single lens telescope from a distant object, the two lenses together constitute what is called the astronomical re- fracting telescope ; and when it is used to magnify the image produced by a concave mirror from a distant object, the two constitute a reflecting telescope, such as that used by Le Maire and Herschel : and when it is used to magnify an en- larged image, M N, Jig. 14., produced from an object m n, placed before a concave mirror, the two constitute a reflect- ing microscope. All these instruments will be more fully described in a future chapter. CHAP. VI. SPHERICAL ABERRATION OF LENSES AND MIRRORS.* (.">.").) In the preceding chapters we have supposed that tho rays refracted at spherical surfaces meet exactly in a focus; but this is by no means strictly true : and if the reader has in any one case projected the rays by the methods described, he * For a discussion of these subjects, see (in the college edition) the Ap- pendix of Am. ed. chap. v. f$M 52 A TREATISE OI> OFTICS. must have scon that the rays nearest the axis of a spherical surface, or of a lens, are refracted to a focus more remote from the lens than those which are incident at a distance from the axis of the lens. The rules which we have given for the foci of lenses and surfaces are true for rays very near the axis In order to understand the cause of spherical aberration, let L L be a plano-convex lens one of whose surfaces is spherical, and let its plane surface LmLbe turned towards parallel rays R L, R L. Let R' L\ R' L' be rays very near the axis A F of the lens, and let P be their focus after refraction. Let R L, R L be parallel rays incident at the very margin of the lens, and it will be found by the method of projection that the corresponding refracted rays L/, hf will meet at a point/ much nearer the lens than F. In like manner intermediate rays between R L and R' L' will have their foci intermediate between ./"and F. Continue the rays hf hf, till they me< I at G and II a plane passing through F, and perpendicular to ¥ A. The distance fF is called the longitudinal spherical aberra- tion, and G II the lateral spherical aberration of the lens. In a plano-convex lens placed like that in the figure, the longitu- dinal spherical aberration fF is no less than 44 times m n the thickness of the lens. It is obvious that such a lens cannot form a distinct picture of any object in its focus F. If it is exposed to the sun, the central part of the lens L' m L' whose focus is at F, will form a pretty bright image of the sun at F ; but as the rays of the sun which pass through the outer part L L of the lens have their foci at points between /and F, the rays will, after arriving at those points, pass on to the plane Gil, and occupy a circle whose diameter is G II ; hence the image of the sun in the focus F will be a bright disc surrounded and rendered indistinct by a broad halo of light growing fainter and fainter from F to G and H. In like manner, every object seen through such a lens, and every image formed by it, will be rendered confused and indistinct by spherical aberration. These results may be illustrated experimentally by taking ru\V. VI. SPHERICAL ABERRATION. 6 •'* n ring of black paper, and covering up the outer parts of the face L L of the lens. This will diminish the halo G II, and the indistinctness of the image, and if we cover up all the lens excepting a small part in the centre, the image will be- come perfectly distinct, though less bright than before, and the focus will be at F. If, on the contrary, we cover up all the central part, and leave only a narrow ring at the circumfe- rence of the lens, we shall have a very distinct image of the sun formed about/ (56.) If the reader will draw a very large diagram of a plano-convex and of a double convex lens, and determine the refracted rays at different distances from the axis where par- allel rays fall on each of the surfaces of the lens, he will be able to verify the following results for glass lenses. 1. In a plano-convex lens, with its plane side turned to par- allel rays as in Jig. 35., that is, turned to distant objects if it is to form an image behind it, or turned to the eye if it is to be used in magnifying a near object, the spherical aberration will be 4o times the thickness, or 4J, times m n. 2. In a plano-convex lens, with its convex side turned to- wards parallel rays, the aberration is only l T y„ths of its thick- ness. In using a plano-convex lens, therefore, it should always be so placed that parallel rays either enter the convex surface or emerge from it. 3. In a double convex lens with equal convexities, the aber- ration is l r |;^ths of its thickness. 4. In a double convex lens having its radii as 2 to 5, the aberration will be the same as in a plano-convex lens in Rule I, if the side whose radius is 5 is turned towards parallel rays ; and the same as the plano-convex lens in Rule 2, if the side whose radius is 2 is turned to parallel rays. 5. The lens which has the least spherical aberration is a double convex one, whose radii are as 1 to 6. When the face Whose radius is I is turned towards parallel rays, the aberra- tion is only l ri { n ths of its thickness; but. when the side with the radius (i is turned towards parallel rays, the aberration is ^Vjhs of its thickness. These results are equally true of plano-concave and double concave lenses. If we suppose the lens of least spherical aberration to have its aberration equal to 1, the aberrations of the other lenses will be as follows : — Best form, as in Rule 5 1-000 Double convex or concave, with equal curvatures . 1-561 Plano-convex or concave in best position, as in Rule 2. 1-093 Plano-convex or concave in worst position, as in Rule 1. 4-206 E2 54 A TREATISE ON OPTICS. PART I. (57.) As the central parts of the lens L L, fig. 3G., refract the rays too little, and the marginal parts too much, it is evi- dent that if we could increase the convexity at n and diminish it gradually towards L, we should remove the spherical aber- ration. But the ellipse and the hyperbola are curves of this kind, in which the curvature diminishes from n to L; and mathematicians have shown how spherical aberration may be entirely removed, by lenses whose sections are ellipses or hy- perbolas. This curious discovery we owe to£)escartes. Fig. 38. If A LD L, for example, is an ellipse whose greater axis A D is to the distance between its foci F,f,as the index of re- fraction is to unity, then parallel rays R L, R L incident upon the elliptical surface L A L will be refracted by the single action of that surface into lines, which would meet exactly in the focus F, if there were no second surface intervening be- tween LAL and F. But as every useful lens must have two surfaces, we have only to describe a circle L a L round F as a centre, for the second surface of the lens L L. As all the rays refracted at the surface LAL converge accurately to F, and as the circular surface L a L is perpendicular to every one of the refracted rays, all these rays will go on to F without suf- fering any refraction at the circular surface. Hence it follows that a meniscus whose convex surface is part of an ellipsoid, and whose concave surface is part of any spherical surface whose centre is in the farther focus, will have no spherical aberration, and will refract parallel rays incident on its convex- surface to the farther focus. In like manner a concavo-convex lens, L L, whose concave Fig. 39. r~ ^~^tl±/^~~ "^\ /■ v- -^•"^"fiNv^^ ^/ CHAP. VI. LENSES WITHOUT ABERRATION. 55 surface L A L is part of the ellipsoid A L D L, and whose convex surface I a I is a circle described round the farther focus of the ellipse, will cause parallel rays R L, R L to di- verge in directions Lr, Lr, which when continued backwards will meet exactly in the focus F, which will be its virtual focus. If a plano-convex lens has its convex surface, L A L, part Fig. 40. of a hyperboloid formed by the revolution of a hyperbola whose greater axis is to the distance between the foci as unitv is to the index of refraction ; then parallel rays, R L, R L, falling perpendicularly on the plane surface will be refracted without aberration to the farther focus of the hyperboloid. The same property belongs to a plano-concave lens, having a similar hyperbolic surface, and receiving parallel rays on its plane surface. A meniscus with spherical surfaces has the property of re- fracting all converging rays to its focus, if its first surface is Fig. 41. convex, provided the distance of the point of convergence or divergence from the centre of the first surface is to the radius of the first surface as the index of refraction is to unity. 5G A TREATISE ON OPTICS. PART I. Thus, if M L L N is a meniscus, and R L, R L rays converg- ing to the point E, whose distance E C from the centre of the first surface L A L of the meniscus is to the radius C A or C Las the index of refraction is to unity, that is as 1-500 to 1, in glass ; then if F is the focus of the first surface, describe with any radius less than F A a circle M a N for the second surface of the lens. Now it will be found by projection that the rays R L, R L, whether near the axis A Eor remote from it, will be refracted accurately to the focus F, and as nil these rays fall perpendicularly on the second surface, they will still pass on without refraction to the focus F. In like manner it is obvious that rays F L, F L diverging from F will be re- fracted into RI^RL, which diverge accurately from the vir- tual focus. When these properties of the ellipse and hyperbola, and of the solids generated by their revolution, were first discovered, philosophers exerted all their ingenuity in grinding and polish- ing lenses with elliptical and hyperbolical surfaces, and various ingenious mechanical contrivances were proposed for this pur- pose. These, however, have not succeeded, and the practical difficulties which yet require to be overcome are so great, that lenses with spherical surfaces are the only ones now in use for optical instruments. But though we cannot remove or diminish the spherical aberration of single lenses beyond l r J () ths of their thickness, yet by combining two or more lenses, and making opposite aberrations correct each other, we can remedy this defect to a very considerable extent in some cases, and in other cases re- move it altogether. (58.) Mr. Herschel has shown, that if two plano-convex lenses A B, C D, whose focal lengths are as 2v$ to 1, are placed with their convex sides together, A B the least convex being next the eye when the combination is to be used as a micro- scope, the aberration will be only 0248, or one fourth of that ji Vo . 4 „ of a single lens in its best form. When this lens is used to form an image, A B must be turned to the object. If the focal lengths of the two lenses are equal, the spherical aberration will be 0*603, or a little more than one-half of a single lens in its best form. Mr. Herschel has also shown that the spherical aberration may be wholly removed by combining a meniscus C D with a double convex lens A B, as in Jiff s. 43. and 44., the lens A B being turned to the eye when it is used ^.lA?. VI. LENSES WITHOUT ABERRATION. 57 for a microscope, and to the object when it is to be used for forming images, or as a burning-glass. Fig. 44. The following are the radii of curvature for these lenses, as computed by Mr. Herschel ; the first supposes, as a condi- tion, that the focal length of the compound lens shall be as near 10000 as is consistent with correcting the aberration ; and the second, that the same focal length shall be the least possible : — Fig. 43. Focal length of the double convex ) , , n nnn , lens A B \ + WVW + Radius of its first or outer surface + 5-833 -f Radius of its second surface . . — 35-000 Focal length of the meniscus C D + 17-829 -f Radius of its first surface . . . -f- 3-688 -f Radius of its second surface . . + 6-291 + Focal length of the compound lens -f 6-407 + Fig. 44. 10000 5-833 35-000 5-497 2054 8-128 3-474 Spherical Aberration of Mirrors. (59.) We have already stated, that when parallel rays, A M, A N, are incident on a spherical mirror, M N, they are re- fracted to the same focus, F, only when they are incident very near the axis, AD. If F is the focus of those very near the Fig. 45. axis, such as A m, then the focus of those more remote, such as A M, will be at / between F and D, and/F will be the 58 A TREATISE ON OPTICS. PART I. longitudinal spherical aberration, which will obviously increase with the diameter of the mirror when its curvature remains the same, and with the curvature when its diameter is con- stant. The images, therefore, formed by mirrors will bo in- distinct, like those formed by spherical lenses, and the indis- tinctness will arise from the same cause. It is manifest that if M N were a curve of such a nature that a line, A M, parallel to its axis, A D, and another line, jTM drawn from M to a fixed point, /, should always form equal angles with a line, C M, perpendicular to the curve M N, we should in this case have a surface which would re- flect parallel rays exactly to a focus f, and form perfectly dis- tinct images of objects. Such a curve is the parabola ; and, therefore, if we could construct mirrors of such a form that their section M N is a parabola, they would have the invalua- ble property of reflecting parallel rays to a single focus. When the curvature of the mirror is very small, opticians have devised methods of communicating to it a parabolic figure ; but when the curvature is great, it has not yet been found practicable to give it this figure. In the same manner it may be shown, that when diverging rays fall upon a concave mirror of a spherical form, they will be reflected to different points of the axis ; and that if a sur- face could be formed so that the incident and reflected rays should form equal angles with a line perpendicular to the sur- face at the point of incidence, the reflected rays would all meet in a single point as their focus. A surface whose sec- tion is an ellipse has this property ; and it may be proved that rays diverging from one focus of an ellipse will be re- flected accurately to the other focus. Hence in reflecting microscopes the mirror should be a portion of an ellipsoid ; the axis of the mirror being the axis of the ellipsoid, and the object being placed in the focus nearest the mirror. On Caustic Curves formed by Reflexion and Refraction.* (60.) Caustics formed by reflexion. — As the rays incident on different points of a reflecting surface at different distances from its axis are reflected to different foci in that axis, it is evident that the rays thus reflected must cross one another at particular points, and wherever the rays cross they will illu- minate the white ground on which they are received with twice as much light as^ falls on other parts of the ground. These luminous intersections form curve lines, called caustic lines or caustic curves ; and their nature and form will, of * See (in the College edition) the Appendix of Am. ed. chap. v. CHAP. VI. CAUSTIC CURVES. 59 course, vary with the aperture of the mirror, and the distance of the radiant point. In order to explain their mode of formation and general properties, let M B N be a concave spherical mirror, Jig-. 46. , whose centre is C, and whose focus for parallel and central Fig. 46. 10 1110 9 rays is F. Let RMB be a diverging beam of light falling on the upper part, M B, of the mirror at the points 1, 2, 3, 4, &c. If we draw lines perpendicular to all these points from the centre C, and make the angles of reflexion equal to the angles of incidence, we shall obtain the directions and foci of all the reflected rays. The ray R 1, near the axis R B. will have its conjugate focus at f, between F and the centre C. The next ray, 112, will cut the axis nearer F, and so on with all the rest, the foci advancing from /to B. By drawing all the reflected rays to these foci, they will be found to intersect one another as in the figure, and to form by their intersections the caustic curve M f. If the light had also been incident on the lower part of the mirror, a similar caustic shown by a dotted lino would also have been formed between N and / If we suppose, therefore, the point of incidence to move from M to B, the conjugate focus of any two contiguous rays, or an infinitely slender pencil diverging from R, will move along the caustic from M to f. Let us now suppose the convex surface M B N of the mir- ror to be polished) and the radiant point R to be placed as fiir to the right hand of B as it is now to the left, it will be found, by drawing the incident and reflected rays, that they will di- verge after reflexion; and that when continued backwards they will intersect one another, and form an imaginary caustic dO A TREATISE ON OPTICS. situated behind the convex surface, and similar to the real caustic. If we suppose the convex mirror M B N to be completed round the same centre, C, as at M A N, and the pencil of rays still to radiate from R, they will form the imaginary caustic M/N smaller than MyiN, and uniting with it at the points M, N. Let the radiant point R be now supposed to recede from the mirror M B N, the line B/, which is called the tangent of the real caustic M/N, will obviously diminish, because the conjugate focus f will approach to F ; and, for the same reason, the tangent A /' of the imaginary caustic will in- crease. When R becomes infinitely distant, and the incident rays parallel, the points /,/', called the cusps of the caustic, will both coincide with F and F', the principal foci, and will have the very same size and form. But if the radiant point R approaches to the mirror, the cusp / of the real caustic will approach to the centre C, and the tangent B f will increase, the cusp f of the imaginary caustic will approach A, and its tangent Af, will diminish; and when the radiant point arrives at the circumference at A, the cusp f will also arrive at A, and the imaginary caustic will disappear. At the same time, the cusp / of the real caustic will be a little to the right of C, and its two opposite summits will meet in the radiant point at A. If we suppose the radiant point R now to enter within the circle A M B N, as shown in Jig. 47., so that R C is less than R A, a remarkable double caustic will be formed. This caustic will consist of two short ones of the common kind, CHAP. VI. CAUSTIC CURVES. 61 ar, b r, having their common cusp at r, and of two long branches, af, bf, which meet in a focus at f. When R C is greater than R A, the curved branches that meet at f behind the mirror will diverge, and have a virtual focus within the mirror. When R coincides with F, a point half-way between A and C, and the virtual principal focus of the convex mirror M A N, these curved branches become parallel lines ; and when R coincides with the centre C, the caustics disappear, and all the light is condensed into a single mathematical point at C, from which it again diverges, and is again reflected to the same point. In virtue of the principle on which these phenomena de- pend, a spherical mirror has, under certain circumstances, the paradoxical property of rendering rays diverging from a fixed point either parallel, diverging, or converging ; that is, if the radiant point is a little way within the principal focus of a mirror, so that rays very near the axis are reflected into par- allel lines, the rays which are incident still nearer the axis will be rendered diverging, and those incident farther from the axis will be rendered converging. This property may be distinctly exhibited by the projection of the reflected rays. Caustic curves are frequently seen in a very distinct and beautiful manner at the bottom of cylindrical vessels of china or earthenware that happen to be exposed to the light of the sun or of a candle. In these cases the rays generally fall too obliquely on their cylindrical surface, owing to their depth ; but this depth may be removed, and the caustic curves beau- tifully displayed, by inserting a circular piece of card or white paper about an inch or so beneath their upper edge, or by filling them to that height with milk or any white and opaque fluid. The following method, however, of ex- hibiting caustic curves I have found ex- ceedingly convenient and instructive. Take a piece of steel spring highly polished, such as a watch-spring, M N, fig. 48., and hav- ing bent it into a concave form as in the figure, place it vertically on its edge upon a piece of card or white paper A B. Let it then be exposed either to the rays of the sun, or those of any other luminous body, taking care that the plane of the card or the paper passes nearly through the sun ; and the two caustic curves shown in the figure will be finely displayed. By varying the size of the spring, and bending 62 A TREATISE ON OPTICS. it into curves of different shapes, all the variety of caustics, with their cusps and points of contrary flexure, will be finely exhibited. The steel may be bent accurately into different curves by applying a portion of its breadth to the reimircd curves drawn upon a piece of wood, and either cut or burned sufficiently deep in the wood to allow the edge of the thin strip of metal to be inserted in it. Gold or silver foil answers very well; and when the light is strong, a thin strip of mica will also answer the purpose. The best substance of all, however, is a thin strip of polished silver. (61.) Caustics formed by refraction. If we expose a globe of glass filled with water, or a solid spherical lens, or even the belly of a round decanter, filled with water, to the rays of the sun, or to the light of a lamp or candle, and re- ceive the refracted light on white paper held almost parallel to the axis of the sphere, or so that its plane passes nearly through the luminous body, we shall perceive on the paper a luminous figure bounded by two bright caustics, like aj and b f, Jig. 47., but placed behind the sphere, and forming a sharp cusp or angle at the point f, which is the focus of re- fracted rays. The production of these curves depends upon the intersection of rays, which, being incident on the sphere at different distances from the axis, are refracted to foci at dif- ferent points of the axis, and therefore cross one another. This result is so easily understood, and may be exhibited so clearlv, by projecting the refracted rays, that it is unneces- sary to say any more on the subject. Some of the phenomena of caustics produced by refraction may be illustrated experimentally in the following manner : — Take a shallow cylindrical vessel of lead, M N, two or three inches in diameter, and cut its upper margin, as shown in the figure, leaving two opposite projections, a c, b d, forming each about 10° or 15° of the whole circumference. Complete the circumference by cementing on the vessel two strips of mica, so as to substitute for the lead that lias been removed two transparent cylin- drical surfaces. If this vessel is filled with water, or any other transparent fluid, and a piece of card or white paper, A B C D, is held almost paral- lel to the surface of the water, and having its plane nearly passing through the sun or the candle, the caustics A F, D F will be finely displayed. By altering the curvature of the vessel, and that of the strips of mica, many interesting variations of the experiment may be made. Fiff. 49. CHAP. VII. ANALYSIS OF LIGHT BY THE PRISM. 63 PART II. PHYSICAL OPTICS. (62.) Physical Optics is that branch of the science which treats of the physical properties of light. These properties are exhibited in the decomposition and rccomposition of white light; in its decomposition by absorption ; in the inflexion or diffraction of light ; in the colors of thick and thin plates; and in the double refraction and polarization of light. CHAP. VII. ON THE COLORS OF LIGHT, AND ITS DECOMPOSITION. (63.) In the preceding chapters we have regarded light as n simple substance, all the parts of which had the same index of refraction, and therefore suffered the same changes when acted upon by transparent media. This, however, is not its constitution. White light, as emitted from the sun or from any luminous body, is composed of seven different kinds of light, viz., red, orange, yellow, green, blue, indigo, and violet; and this compound substance may be decomposed, or analyzed, or separated into its elementary parts, by two different pro- cesses, viz., by refraction and absorption. The first of these processes was that which was employed by Sir Isaac Newton, who discovered the composition of white light Having admitted a beam of the sun's light, SH, through a small hole, II, in the window-shutter, E F, of a darkened room, it, will go on in a straight line and form a round white spot at P. If we now interpose a prism, B A C, whose 1 refracting angle is B AC, so that this beam of light may fall on its first surface C A, and emerge at the same angle from its second surface 11 A in the direction g (J, and if we receive the refracted beam on the opposite wall, or rather on a white screen, M N, we should expect, from the principles already laid down, that the white beam which previously fell upon I' would suffer only a change in its direction, and fall somewhere upon M N, forming then- a round white spot ex- actly similar to that at P. But this is not the case. Instead of a white spot, there will he formed upon the screen M N an 64 A TREATTSK ON OHICS. TAUT II. oblong image K L of the sun, containing seven, colors, viz. red, orange, yellow, green, blue, indigo, and violet, the whole beam of light diverging from its emergence out of the prism at g, and being bounded by the lines g K, g L. This length- ened image of the sun is called the solar spectrum, or the prismatic spectrum. If the aperture H is small, and the dis tance g G considerable, the colors of the spectrum will be very bright. The lowest portion of it at L is a brilliant red. This red shades off by imperceptible gradations into orange, the orange into yellow, the yellow into green, the green into blue, the blue into a pure indigo, and the indigo into a violet. No lines are seen across the spectrum thus produced ; and it is extremely difficult for the sharpest eye to point out the bound- ary of the different colors. Sir Isaac Newton, however, by many trials, found the lengths of the colors to be as follows, in the kind of glass of which his prism was made. We have added the results obtained by Fraunhofer with flint glass. Newton. Fraunhofer Red 45 55 Orange 27 27 Yellow 48 27 Green 60 46 Blue 60 48 Indigo 40 47 Violet 80 109 Total length 360 360 CHAP. VII. RECoMrosmoN or lioiit. fi5 These colors are not equally brilliant. At the lower end, L, of the spectrum the red is comparatively faint, but grows brighter as it approaches the orange. The light increases gradually to the middle of the yellow space, where it is brightest; and from this it gradually declines to the upper or violet end, K, of the spectrum, where it is extremely faint. (64.) From the phenomena which we have now described, Sir Isaac Newton concluded that the beam of white light, S, is compounded of light of seven different colors, and that for each of these different kinds of light, the glass, of which his prism was made, had different indices of refraction ; the index of refraction for the red light being the least, and that of tiic violet the greatest. If the prism is made of crown glass, for example, the in- dices of refraction for the different colored rays will be as fol- lows : — Rod 1-5258 Blue 1-5360 Orange 1-52G8 Indigo 1.5417 yellow 1-5296 Violet 1-54G6 Green 1-5330 If we now draw the prism, B A C, on a great scale, and de- termine the progress of the refracted rays, supposed to be in- cident upon the same point of the first surface C A, by using tor each ray the index of refraction in the preceding table, we shall find them to diverge as in the preceding figure, and to form the different colors in the order of those in the spectrum. In order to examine each color separately, Sir Isaac made a hole in the screen M N, opposite the centre of each colored space ; and he allowed that particular color to fall upon a second prism, placed behind the hole. This light, when re- fracted by the second prism, was not drawn out into an oblong image as before, and was not refracted into any other colors. Hence he concluded that the light of each different color had the same index of refraction; and he called such light homo- geneous, or simple, white, light being regarded as heteroge- neous or compound. This important doctrine is called the different refrangibility of the rays of light. The different colors as existing in the spectrum are called primary colors ; and any mixtures or combinations of any of them are called secondary colors, because we can easily separate them into their primary colors by refraction through the prism. (65.) Having thus clearly established the composition of white light, Sir Isaac also proved, experimentally, that all the seven colors, when again combined and made to fall upon the F 2 66 A TREATISE ON OPTICS. PART II same spot, formed or recomposed white light. This important truth he established by various experiments; but the following method of proving it is so satisfactory, that no farther evidence seems to be wanted. Let the screen M N, Jig. 50., which re- ceives the spectrum, be gradually brought nearer the prism BAG, the spectrum K L will gradually diminish ; but though the colors begin to mix, and encroach upon one another, yet, even when it is brought close to the face B A of the prism, we shall recognize the separation of the light into its component colors. If we now take a prism, B a A, shown by dotted lines, made of the same kind of glass as B A C, and having its re- fracting angle A B a exactly equal to the refracting angle B A C of the other prism ; and if we place it in the opposite direction, we shall find that all the seven differently colored rays which fall upon the second prism, A B a, are again com- bined into a single beam of white light g P, forming a white circular spot at P, as if neither of the prisms had been inter- posed. The very same effect will be produced, even if the surfaces, A B, of the two prisms are joined by a transparent cement of the same refractive power as the glass, so as to re- move entirely the refractions at the common surface A B. In this state the two prisms combined are nothing more than a thick piece of glass, B C A a, whose two sides, A C, a B, are exactly parallel; and the decomposition of the light by the re- fraction of the first surface, A C, is counteracted by the oppo- site and equal refraction of the second surface, a B ; that is, the light decomposed by the first surface is recomposed by the second surface. The refraction and re-union of the rays in this experiment may be well exhibited by placing a thick plate of oil of cassia between two parallel plates of glass, and making a narrow beam of the sun's light fall upon it very ob- liquely. The spectrum formed by the action of the first sur- face will be distinctly visible, and the re-union of the colors by the second will be equally distinct. We may, therefore, consider the action of a plate of parallel glass on the sun's rays, that is, its property of transmitting them colorless, as a sufficient proof of tiie recom position of light. The same doctrine may be illustrated experimentally by mixing together seven different powders having the same colors as those of the spectrum, taking as much of each as seems to be proportional to the rays in each colored space. The union of these colors will be a sort of grayish-white, be- cause it is impossible to obtain powders of the proper colors. The same result will be obtained, if we take a circle of paper and divide it into sectors of the same size as the colored CHAP. VII. NEW ANALYSIS OP LIGHT. 67 spaces ; and when this circle is made to revolve rapidly, the effect of all the colors when combined will be a grayish-white. Decomposition of Light by Absorption. (60.) If we measure the quantity of light which is reflected from the surfaces and transmitted through the substance of transparent bodies, we shall rind that the sum of these quan- tities is always less than the quantity of light which falls upon the body. Hence we may conclude that a certain por- tion of light is lost in passing through the most transparent bodies. This loss arises from two causes. A part of the light is scattered in all directions by irregular reflexion from the imperfectly polished surface of particular media, or from the imperfect union of its parts; while another, and generally a greater portion, is absorbed, or stopped by the particles of the body. Colored fluids, such as black and red ink, though equally homogeneous, stop or absorb different kinds of rays, and when exposed to the sun they become heated in different degrees; while pure water seems to transmit all the rays equally,*' and scarcely receives any heat from the passing light of the sun. When we examine more minutely the action of colored glasses and colored fluids in absorbing light, many remarkable phenomena present themselves, which throw much light upon this curious subject. If we take a piece of blue glass, like that generally used for finger glasses, and transmit through it a beam of white light, the light will be a fine deep blue. This blue is not a simple homogeneous color, like the blue or indigo of the spec- trum, but is a. mixture of all the colors of white light which the glass has not absorbed ; and the colors which the glass has absorbed are those which the blue wants of white light, or which, when mixed with this blue, would form white light. In order to determine, what these colors are, let us transmit through the blue glass the prismatic spectrum K L, Jig. 50. ; or, what is the same thing, lot the observer place his eye be- hind the prism B A C, and look through it at the sun, or rather at a circular aperture made in the window-shutter of a dark room. He will then see through the prism the spec- trum K L as far below the aperture as it w r as above the spot P when shown in the screen. Let the blue glass be now inter- posed between the eye and the prism, and a remarkable spec- trum will be seen, deficient in a certain number of its difier- * See Note II., of Am. ed., which follows the author's Appendix. 68 A TREATISE o.N OPTICS. r ART II. ently colored rays. A particular thickness absorbs the middle of the red space, the whole of the orange, a great part of the green, a considerable part of the blue, a little of tlic indigo, and very little of the violet The yellow space, which has not been much absorbed, has increased in breadth. It occu- pies part of the space formerly covered by the orange on one side, and part of the space formerly covered by the green on the other. Hence it follows, that the blue glass has absorbed the red light, which, when mixed with the yellow light, con- stituted orange, and has absorbed also the blue light, which, when mixed with the yellow, constituted the part of the green space next to the yellow. We have therefore, by ab- sorption, decomposed green light into yellow and blue, and orange light into yellow and red; and it consequently follows, that the orange and green rays of the spectrum, though they cannot be decomposed by prismatic refraction, can be decom- posed by absorption, and cictually consist of two different colors possessing the same degree of refrangibility. Differ- ence oj color is therefore not a test of difference, of refrangi- bility, and the conclusion deduced by Newton is no longer admissible as a general truth: "That to the same degree of refrangibility ever belongs the same color, and to the same color ever belongs the same degree of refrangibility." With the view of obtaining a complete analysis of the spec- trum, I have examined the spectra produced by various bodies, and the changes which they undergo by absorption when viewed through various colored media, and I find that, the- color of every part of the spectrum may be changed not only in intensity, but in color, by the action of particular media ; and from these observations, which it would be out of place here to detail, I conclude that the solar spectrum consists of three spectra of equal lengths, viz. a red spectrum, a yellow spectrum, and a blue spectrum. The primary red spectrum has its maximum of intensity about the middle of the red space in the solar spectrum, the primary yellow spectrum has its maximum in the middle of the yellow space, and the primary blue spectrum has its maximum between the blue and the indigo space. The two minima of each of the three primary spectra coincide at the two extremities of the solar spectrum. From this view of the constitution of the solar spectrum we may draw the following conclusions: — 1. Red, yellow, and blue light exist at every point of the solar spectrum. 2. As a certain portion of red, yellow, and blue constitute white light, the color of every point of the spectrum may be CHAP. VII. NEW ANALYSIS OF LIGHT. 69 considered as consisting of Ihe predominating color at any point mixed with white light In the red space there is more red than is necessary to make white light with the small por- tions of yellow and blue which exist there; in the yellow space there is more yellow than is necessary to make white light with the red and blue ; and in the part of the blue space which appears violet there is more red than yellow, and hence the excess of red forms a violet with the blue. 'S. By absorbing the excess of any color at any point of the spectrum above what is necessary to form white light, we may actually cause white light to appear at that point, and this white light will possess the remarkable property of re- maining white after any number of refractions, and of being decomposable only by absorption. Such a white light I have succeeded in developing in different parts of the spectrum. These views harmonize in a remarkable manner with the hypothesis of three colors, which has been adopted by many philosophers, and which odiers had rejected from its incom- patibility with the phenomena of the spectrum. The existence of three primary colors in the spectrum, and the mode in which they produce by their combination the seven secondary or compound colors which are developed by the prism, will be understood from Jig. 51. where M N is the prismatic spectrum, consisting of three primary spectra of the .same lengths, M N, viz. a red, a yellow, and a blue spectrum. The red spectrum has its maximum intensity at R; and this intensity may be represented by the distance of the point R from J\l N. The intensity declines rapidly to M and slowly to N, at both of which points it vanishes. The yellow spec- trum has its maximum intensity at Y, the intensity declining to zero at M and N ; and the blue lias its maximum intensity at B, declining to nothing at M and N. The general curve which represents the total illumination at any point will be outside of these three curves, and its ordinate at any point will be equal to the sum of the three ordinates at the same point. Thus the ordinate of the general curve at the point Y will be equal to the ordinate of the yellow curve, which we 70 A TREATISE ON OPTICS. TART II. may suppose to be 10, added to that of the red curve, which may be 2, and that of the blue, which may be 1. Hence the general ordinate will be 13. Now, if we suppose that 3 parts of yellow, 2 of red, and 1 of blue make white, we shall have the color at Y equal to 3 -f 2 -f- 1> equal to parts of white mixed with 7 parts of yellow ; that is, the compound tint at Y will be a bright yellow without any trace of red or blue. As these colors all occupy the same place in the spectrum, they cannot be separated by the prism ; and if we could hud a colored glass which would absorb 7 parts of the yellow, we should obtain at the point Y a white light which the prism could not decompose.* CHAP. VIII. ON THE DISPERSION OF LIGHT. In the preceding observations, we have considered the pris- matic spectrum, K I \ fig. 50., as produced by a prism of glass having a given refracting angle, B AC. The green ray, or g G, which, being midway between g K and g L, is called the mean ray of the spectrum, has been refracted from P to G, or through an angle of deviation, P g G, which is called the mean refraction or deviation, produced by the prism. If we now increase the angle B A C of the prism, we shall in- crease the refraction. The mean ray g G will be refracted to a greater distance from P, and the extreme rays g L, g K, to a greater distance in the same proportion ; that is, if g G is refracted twice as much, g L and g K will also be refracted twice as much, and consequently the length of the spectrum K L will be twice as great. For the same reason, if we diminish the angle B A C of the prism, the mean refraction and the spectrum will diminish in the same proportion ; but, whatever be the angle of the prism, thej^ength K L will al- ways bear the same proportion to G P, the mean refraction. Sir Isaac Newton supposed that prisms made of all sub- stances whatever, produced spectra bearing the same propor- tion to the mean refraction as prisms of glass; and it is a re- markable circumstance, that a philosopher of such sagacity should have overlooked a fact so palpable, as that different bodies produced spectra whose lengths were different, when the mean refraction was the same. The prism BA(! being supposed to be made of crown * See Nole III., by Am. ml., following the author's Appendix. CHAP. VIII. DISPERSION OF LIGHT. 71 glass, let us take another of flint glass or white crystal, with such a refracting angle that, when placed in the position B A C, the light enters and quits it at equal angles, and re- fracts the mean ray to the same point G. The two prisms ought, therefore, to have the same mean refraction. But when we examine the spectrum produced by the flint glass prism, we shall find that it extends beyond K and L, and is evidently longer than the spectrum produced by the crown glass prism. Hence flint glass is said .to have a greater dispersive power than crown glass, because at the same angle of mean refrac- tion it separates the extreme rays of the spectrum, g L, g K, farther from the mean ray g G. In order to explain more clearly what is the real measure of the dispersive power of a body, let us suppose that in the crown glass prism, B A C, the index of refraction for the ex- treme violet ray, gK, is 1-5460, and that for the extreme red ray, g L, 1-5258; then the difference of these indices, or •0208, would be a measure of the dispersive power of crown glass, if it and all other bodies had the same mean refraction : but as this is far from being the case, the dispersive power must be measured by the relation between -0208 and the mean refraction, or 1*5330, or to the excess of this above unity, viz., -5:330, to which the mean refraction is always pro- portional. For the purpose of making this clearer, let it be required to compare the dispersive powers of diamond and crown glass. The index of refraction of diamond for the ex- treme violet ray is 2- 107, and for the extreme red, 2411, and the difference of these is -0500, nearly three times as great as ■0208, the same difference for crown glass; but then the dif- ference between the sines of incidence and refraction, or the excess of the index of refraction above unity, or 1-439, is also about three times- as great as the same difference in crown glass, viz., 5330; and, consequently, the dispersive power of diamond is very Little greater than that of crown glass. The two dispersive powers are as follows: — Crown Glass Diamond DispersiYe Power = 0-0396 = 00388 This similarity of dispersive power might be proved experi- mentally, by taking a prism of diamond, which, when placed at B A C in Jig. 50., produced the same mean refraction as the green ray g G. It would then be seen that the spectrum which it produced was of the same length as that produced by the prism of crown glass. Hence the splendid colors which distinguish diamond from every other precious stone 72 A TKfiATlSE ON OPTICS. PART II. are not owing to its high dispersive power, but to its great mean refraction. As the indices of refraction given in our table of refractive powers are nearly suited to the mean ray of the spectrum, we may, by the second column of the Table of the Dispersive Powers of Bodies, given in the Appendix, No. I., obtain the approximate indices of refraction tor the extreme red and the extreme violet rays, by adding half of the number in the col- umn to the mean index of refraction for the index of refrac- tion oi' the v it .lot, and subtracting half of the same number for the index of the red ray. The measures in the table are given for the ordinary light of day. When the sun's light is used, and when the eye is screened from the middle rays of the spectrum, the red and violet may be traced to a much greater distance from the mean ray of the spectrum. When the index of refraction for the extreme ray is thus known, we may determine the position and length of the spectra produced by prisms of different substances, whatever be their refracting angle, whatever be the positions of the prism, and whatever be the distance of the screen on which the spectrum is received. If we take a prism of crown glass, and another of flint glass, with such refracting angles that they produce a spec- trum of precisely the same length, it will be found, that when the two prisms are placed together with their refracting angles in opposite directions, they will not restore the refracted pencil to the state of white light, as happens in the combination of two equal prisms of crown or two equal prisms of flint glass. The white light V,fig. 50., will be tinged on one side with purple, and on the other with green light. This is called the secondary spectrum, and the colors secondary colors ; and it is manifest that they must arise from the colored spaces in the spectrum of crown glass not being equal to those in the spec- trum of flint glass. In order to render this curious property of the spectrum very obvious to the eye, let two spectra of equal length be formed by two hollow prisms, one containing oil of cassia, and the other sulphuric acid. The oil of cassia spectrum will resemble A R,fig- 52., and the sulphuric acid spectrum C D. In the former, the red, orange, and yellow spaces are less than in the latter, while the blue, indigo, and violet spaces are greater ; the least refrangible rays being, as it were, contract- ed in the former and expanded in the latter, while the most refrangible rays are expanded in the one and contracted in the other. In consequence of this difference in the colored spaces, the middle or mean ray m n does not pass through the same CHAP. VIII. DISPERSION OF LIGHT. 73 color in both spectra. In the oil of cassia spectrum it is in the blue space, and in the sulphuric acid spectrum it is in the Violet Indigo Blue. Fig. 52. green space. As the colored spaces have not the same ratio to one another as the lengths of the spectra which they com- pose, this property lias been called the irrationality of disper- sion, or of the colored spaces in the spectrum. In order to ascertain whether any prism contracts or ex- pands the least refrangible rays more than another, or which of them acts most on green light, take a prism of each with such angles that they correct each other's dispersion as much as possible, or that they produce spectra of the same length. If, through the prisms placed with their refracting angles in opposite directions, we look at the bar of the window parallel to the base of the prism, we shall see its edges perfectly free from color, provided the two prisms act equally upon green light. But if they act differently on green light, the bar will have a fringe of purple on one side, and a fringe of green on the other; and the green fringe will always be on the same side of the bar as the vertex of the prism which contracts the yel- low space and expands the blue and violet ones. That is, if the prisms are flint and crown glass, the uncorrected green fringe will be on the lower side of the bar when the vertex of the flint glass^rism points downwards. Flint glass, therefore, has a less action upon green light than crown glass, and con- tracts in a greater degree the red and yellow spaces. See Appendix, No. II. 74 A TREATISE ON OPTICS. TART II. CHAP. IX. ON THE PRINCIPLE OF ACHROMATIC TELESCOPES. In treating of the progress of rays through lenses, it was taken for granted that the light was homogeneous, and that every ray that had the same angle of incidence had also the same angle of refraction ; or, what is the same thing, that every ray which fell upon the lens had the same index of re- fraction. The observations in the two preceding chapters have, however, proved that this is not true, and that, in the case of light falling upon crown glass, there are rays with every possible index of refraction from 1*5258, the index of refraction for the red, to 1-5466, the index of refraction for the violet rays. As the light of the sun, by which all the bodies of nature are rendered visible, is white, this property of light, viz. the different refrangibility of its parts, affects greatly the formation of images by lenses of all kinds. In order to explain this, let L L be a convex lens of crown glass, and R L, It L rays of white light incident upon it par- JL-,0 allel to its axis R r. As each ray R L of white light consists of seven differently colored rays having different degrees of refrangibility or different indices of refraction, it is evident that all the rays which compose R L cannot possibly be re- fracted in the same direction, so as to fall upon one point. The extreme red rays, for example, in R L, R L, whose index of refraction is 1*5258, if traced through the lens by the method formerly given, will be found to have their focus in r, and C r will be the focal length of the lens for red rays. In like manner the extreme violet rays, which have a greater index of refraction, or 1*5466, will be refracted to a focus v much nearer the lens, and C v will be the focal length of the lens for violet rays. The distance v r is called the chromatic aber- ration, and the circle whose diameter is a b passing through the focus of the mean refrangible rays at o, is called the circle of least aberration. These effects may be shown experimentally by exposing the CHROMATIC ABERKATIOX. 70 lens LLtothe parallel rays of the sun. If we receive the image of the sun on a piece of paper placed between o and C, the luminous circle on the paper will have a red border, be- cause it is a section of the cone Lab L, the exterior rays of which La, L/> are red; but if the paper is placed at any greater distance than o, the luminous circle on the paper will have a violet border, because it is a section of the cone I abl', the exterior rays of which ai, bl' are violet, being a contin- uation of the violet rays L«, Lv. As the spherical aberration <>f the lens is here combined with its chromatic aberration, the undisguised effect of the latter will be better seen by tailing a large convex Jeas L L, and covering up all the cen-* tral part, leaving only a small rim round its circumference at L L, through which the rays of light may pass. The refrac- tion of the differently colored rays will be then finely dis- played by viewing the image of the sun on the different sides of a b. It is clear from these observations that the lens will form a violet image of the sun at v, a red image at r, and images of the other colors in the spectrum at intermediate points be- tween r and v; so that if we place the eye behind these images, we shall see a confused image, possessing none of that sharpness and distinctness which it would have had if formed only by one kind of rays. The same observations are true of the refraction of white light by a concave lens; only in this case the parallel rays which such a lens refracts diverge, as if they proceeded from separate foci, v and r, in front of the lens. If we now place behind L L a concave lens G G of the .same glass, and having its surfaces ground to the same cur- vature, at is obvious that since v is its virtual focus for violet, and r its virtual focus for red rays, if the paper is held at a b, the focus of the mean refrangible rays, where the violet and red rays cross at a and b, the image will be more distinct than in any other position ; and when rays converge to the focus of any concave lens, they will be refracted into parallel direc- tions; that is, the concave lens will refract these converging rays into the parallel lines Gl, G I, and they will again form white light. That the red and violet rays will be thus re- united in one, viz. G I, may be proved by projecting them ; but it is obvious also from the consideration that the two lenses L L, G G actually form a piece of parallel glass, the outer concave surface of G G being parallel to the outer convex sur- face of L L. (07.) But though we have thus corrected the color produced by L L, by means of the lens G G, we have done this by a 76 A TREATISE ON OPTICS. PART II. useless combination ; since the two together act only like a piece of plane glass, and are incapable of forming an image. If we make the concave lens G G, however, of a longer focus than L L, the two together will act as a convex lens, and will form images behind it, as the rays G I, G I will now converge to a focus behind L L. But as the chromatic aberration of the lens G G will now be less than that of L L, the one will not correct or compensate the other ; so that the difference be- tween the two aberrations will still remain. Hence it is im- possible, by means of two lenses of the same glass, to form an image which shall be free from color. As Sir Isaac Newton believed that all substances whatever produced the same quantity of color, or had the same chro- matic aberration when formed into lenses, he concluded that it was impossible, by the combination of a concave with a con- vex glass, to produce refraction without color. But we have already seen that the premises from which this conclusion was drawn are incorrect, and that bodies have different dispersive powers, or produce different degrees of color at the same mean refraction. Hence it follows that different lenses may produce the same degree of color when they have different focal lengths; so that if the lens L L is made of croivn glass, whose ind#x of refraction is 1*519, and dispersive power 0*036, and the lens G G of flint glass, whose index of refraction is 1*589, and dispersive power 0*0398, and if the focal length of the convex crown-glass lens is made 4J- inches, and that of the concave flint-glass lens 7§ inches, they will form a lens with a focal length of 10 inches, and will refract white light to a single focus free of color. Such a lens is called an achro- matic lens ; and when used as a telescope, with another glass to magnify the colorless image which it forms of distant ob- jects, it constitutes the achromatic telescope, one of the greatest inventions of the last century. Although Newton, reasoning from his imperfect knowledge of the dispersive power of bodies, pronounced such an invention to be hopeless ; yet, in a short time after the death of that great philosopher, it was accomplished by a Mr. Hall, and afterwards by Mr. Dollond, who brought it to a high degree of perfection. The image formed by an achromatic lens thus constructed would have been perfect if the equal spectra formed by the crown and flint glass were in every respect similar : but as we have seen that the colored spaces in the one are not equal to the 'colored spaces in the other, a secondary spectrum is left ; and therefore the images of all luminous objects, when seen through such a lens, will be bordered on one side with a purple fringe, and on the other with a. green fringe. If two CHAP. IX. ACHROMATIC TELESCOPE. 77 substances could be found of different refractive and dispersive powers, and capable of producing equal spectra, in whicb the colored spaces were equal, a perfect achromatic lens would be produced : but, as no such substances have yet been found, philosophers have endeavored to remove the imperfection by other means ; and Doctor Blair had the merit of surmounting the difficulty. He found that muriatic acid had the property of producing' a primary spectrum, in which the green rays were among the most refrangible, something like C H,Jig: 52., as in crown glass. But as muriatic acid has too low a refrac- tive and dispersive power to fit it for being used as a concave lens along with a convex one of crown glass, he therefore conceived the idea of increasing the refractive and dispersive powers of the muriatic acid, by mixing it with metallic solu- tions, such as muriate of antimony ; and he found he could do this to the requisite extent without altering its law of disper- sion, or the proportion of the colored spaces in its spectrum. By inclosing, therefore, muriate of antimony, L L, between two convex lenses of crown glass, as AB, CD in Jig. 54., Doctor Blair succeeded in refracting parallel rays R A, R B Fig. 54. ELD to a single focus F, without the least trace of secondary color. Before he discovered this property of the muriatic acid, he had contrived another, though a more complicated combination, for producing the same effect; but as he prefer- red the combination which we have described, and employed it in the best aplanatic object-glasses which he constructed, it is unnecessary to dwell any longer upon the subject In these observations, we have supposed that the lenses which are combined have no spherical aberration ; but though this is not the case, the combination of concave with convex- surfaces, when properly adjusted, enables us completely to correct the spherical along with the chromatic aberration of lenses. In the course of an examination of the secondary spectra produced by different combinations, I was led to the conclu- sion that there may be refraction without color, by means of two prisms, and that two lenses may converge white light tu G2 78 A TREATISE ON OPTICS. PART II. one focus, oven though the prisms and the lenses are made of the same kind of glass. When one prism of a different angle is thus made to correct the dispersion of another prism, a ter- tiary spectrum is produced, which depends wholly on the angles at which the light is refracted at the two surfaces of the prisms. See Treatise on New Philosophical Instru- ments , p. 400. CHAP. X. ON THE PHYSICAL PROPERTIES OF THE SPECTRUM. (68.) In the preceding chapter we have considered only those general properties of the solar spectrum on which the construction of achromatic lenses depends. We shall now proceed to take a general view of all its physical properties. On the Existence of Fixed Lines in the Spectrum. In the year 1802, Dr. Wollaston announced that in the spectrum formed by a fine prism of Hint glass, free from veins, when the luminous object was a slit, the twentietli of an inch wide, and viewed at the distance of 10 or 12 feet, there were two fixed dark lines, one in the green and the other in the blue space. This discovery did not excite any attention, and was not followed out by its ingenious author. Without a knowledge of Dr. Wollaston's observation, the late celebrated M. Fraunhofer, of Munich, by viewing through a telescope the spectrum formed from a narrow line of solar light by the finest prisms of Hint glass, discovered that the surface of the spectrum was crossed throughout, its whole length by dark lines of different breadths. None oftlie.se lines coincide with the boundaries of the colored spaces. They are nearly 600 in number: the largest of them subtends an angle of from 5" to 10". From their distinctness, and the taciiity with which they may be found, seven of these lines, viz. B, C, D, E, F, G, II, have been particularly distinguished by M. Fraunhofer. Of these B lies in the red space, near its outer end; C, which is broad and black, is beyond the middle of the red ; D is in the orange, and is a strong double line, easily seen, the two lines being nearly of the same size, and separated by a bright one; E is in the green, and consists of several, the middle one being the strongest ; F is in the blue, and is a very strong line; G is in the indigo, and H in the CHAP. X. ON LINES IN THE SPECTRUM. 79 violet. Besides these lines there are others which deserve to be noticed. At A is a well defined dark line within the red Fig. 55. space, and half-way between A and B is a group of seven or eight, tbrming together a dark band. Between B and C there are 9 lines; between C and D there are 30; between D and E there are 84 of different sizes. Between E and b there are 24, at b there are three very strong lines, with a fine clear space between the two widest ; between b and F there are 52; between F and G 185; and between G and H 190, many being accumulated at G. These lines are seen with equal distinctness in spectra pro- duced by all solid and fluid bodies, and, whatever be the lengths of the spectra and the proportion of their colored spaces, the lines preserve the same relative position to the boundnries of the colored spaces; and therefore their propor- tional distances vary with the nature of the prism by which they are produced. Their number, however, their order, and their intensity are absolutely invariable, provided light coming either directly* or indirectly from the sun be employed. Similar bands are perceived in the light of the planets and fixed stars, of colored flames, and of the electric spark. The spectra from the light of Mars and from that of Venus con- tain the lines 1), E, b, and F in the same positions as in sun- light. In the spectrum from the light of Sirius, no fixed lines could be perceived in the orange and yellow spaces; but in the green there was a very strong streak, and two other very strong ones in the blue. They had no resemblance, however, to any of the lines in planetary light. The star Castor gives a spectrum exactly like that of Sirius, the streak in the green bein9., where they were first pub- lished, as communicated to me by Sir Humphry. t For the recent observations of Siguor Melloni, see Note IV. of Am. eil. which follows author's Appendix. CHAP. X. MAGNETIC RAYS. 83 in the solar spectrum, one on the red side which favors oxy- genation, and the other on the violet side which favors dis- oxygenation. M. Ritter also found that phosphorus emitted white fumes in the invisible red ; while in the invisible violet, pbosphorus in a state of oxygenation was instantly extin- guished. In repeating the experiments with muriate of silver, M. Seebeck found that its color varied with the colored space in which it was held. In and beyond the violet, it was reddish brown ; in the blue, it was blue or bluish grey ; in the yellow, it was white, either unchanged or faintly tinged with yellow ; and in and beyond the red it was red. In prisms of flint glass, the muriate was decidedly colored beyond the limits of the spectrum. Without knowing what had been done by Ritter, Dr. Wol- laston obtained the very same results respecting the action of violet light on muriate of silver. In continuing his experi- ments, he discovered some new chemical effects of light upon gum guaiacum. Having dissolved some of this gum in alco- hol, and washed a card with the tincture, he exposed it in the different colored spaces of the spectrum without observing any change of color. He then took a lens 7 inches in diame- ter, and having covered the central part of it. so as to leave only a ring of one tenth of an inch at its circumference, he could collect the rays of any color in a focus, the focal dis- tance being about 24|- inches for yellow light. The card washed with guaiacum was then cut in small pieces, which were placed in the different rays concentrated by the lens. In the violet and blue rays it acquired a green color. In the yellow no effect was produced. In the red rays, pieces of the card already made green lost their green color, and were re- stored to their original hue. The guaiacum card, when placed in carbonic acid gas, could not be rendered green at any dis- tance from the lens, but was speedily restored from green to yellow by the red rays. Dr. Wollaston also found that the back of a heated silver spoon removed the green color as ef- lectually as the red rays. On the Magnetizing Power of the Solar Rays. (73.) Dr. Morichini, more than twenty years ago, announced that the violet rays of the solar spectrum had the power of magnetizing small steel needles that were entirely free from magnetism. This effect was produced by collecting the violet rays in the focus of a convex lens, and carrying the focus of these rays from the middle of one half of the needle to the I 84 A TREATISE ON OPTICS. FART II. extremities of that half, without touching the other half. When this operation had been performed for an hour, the needle had acquired perfect polarity. MM. Carpa and Ridol.i repeated this experiment with perfect success ; and Dr. Mori- chini magnetized several needles in the presence of Sir H. Davy, Professor Playfair, and other English philosophers. M. Berard at Montpclier, M. Dhombre Firmas at Alais, and pro- fessor Configliachi at Pavia, having failed in producing the same effects, a doubt was thus cast over the accuracy of pre- ceding researches. A few years ago, Dr. Morichini's experiment was restor< 1 to credit by some ingenious experiments by Mrs. Somerville. Having covered with paper half of a sewing needle, about an inch long, and devoid of magnetism, and exposed the other half uncovered to the violet rays, the needle acquired mag- netism in about two hours, the exposed end exhibiting north polarity. The indigo rays produced nearly the same efiett, and the blue and green produced it in a less degree. \\ he n the needle was exposed to the yellow, orange, red, or calorific rays beyond the red, it did not receive the slightest magetism, although the exposures lasted for three days. Pieces of clock and watch springs gave similar results ; and when the violet ray was concentrated with a lens, the needles, &c, were magnetized in a shorter time. The same effects were pro duced by exposing the needles half covered with paper to the sun's rays transmitted through glass colored blue with cobalt. Green glass produced the same effect. The light of the sun transmitted through blue and green riband produced the same effect as through colored glass. When the needles thus cov- ered had hung a day in the sun's rays behind a pane of glass, their exposed ends were north poles, as formerly. In repeating Mrs. Somerville's experiments, M. Baumgart- ner of Vienna discovered that a steel wire, some parts of which were polished, while the rest were without lustre, be- came magnetic by exposure to the white light of the sun ; a north pole appearing at each polished part, and a south pole at each unpolished part. The cflect was hastened by concen- trating the solar rays upon the steel wire. In this way he ob- tained 8 poles on a wire eight inches long. He was not able to magnetize needles perfectly oxidated, or perfectly polished, or having polished lines in the direction of their lengths. About the same time, Mr. Christie of Woolwich found that when a magnetized needle, or a needle of copper or glass, vi- brated by the force of torsion in the white light of the sun, the arch of vibration was more rapidly diminished in the sun's light than in the shade. The effect was greatest on the mag- CHAP. X. MAGNETIC RAYS. 85 netizcd needle. Hence he concludes that the compound solar rays possess a very sensible magnetic influence. These results have received a very remarkable confirmation from the experiments of M. fiarlocci and M. Zantedeschi. Professor Barlocci found that an armed natural loadstone, which could carry \\ Roman pounds, had Its power nearly doubled by twenty-four hours' exposure to the strong light of the sun. M. Zantedeschi found that an artificial horse-shoe Loadstone, which carried 13£ oz., carried 3^ more by three days' exposure, and at last supported 31 oz., by continuing it in the sun's light He found, that while the strength in- creased in oxidated magnets, it diminished in those which were not oxidated, the diminution becoming insensible when the loadstone was highly polished. He now concentrated the solar rays upon the loadstone by means of a lens ; and he found that, both in oxidated and polished magnets, they ac- quire strength when their north pole is exposed to the sun's rays, and lose strength when the south pole is exposed. He found likewise that the augmentation in the first case ex- ceeded the diminution in the second. M. Zantedeschi re- peated the experiments of Mr. Christie on needles vibrating in the sun's light ; and he found that, by exposing the north pole of a needle a foot long, the semi-amplitude of the last oscillation was G° less than the first ; while, by exposing the south pole, the last oscillation became greater than the first. JV1. Zantedeschi admits that he often encountered inexplicable anomalies in these experiments.* Decisive as these results seem to be in favor of the mag- netizing power both of violet and white light, yet a series of apparently very well conducted experiments have been lately published by MM. Riess and Moser,f which cast a doubt over the researches of preceding philosophers. In these experi- ments, they examined the number of oscillations performed in a given time before and after the needle was submitted to the influence of the violet rays. A focus of violet light concen- trated by a lens 1-2 inches in diameter, and 2*3 inches in focal length, was made to traverse one' half of the needle 200 times; and though this experiment was repeated with differ- ent needles, at different seasons of the year, and different hours of the day, yet the duration of a given number of oscil- lations was almost exactly the same after as before the experi- ment Their attempts to verify the results of Eaumgartner were equally fruitless; and they therefore consider themselves * Edinburgh Journal of Science, New Series, No. V., p. 76. t Id. No. IV., p. 225. H I 88 TREATISE ON OPTICS TART II. as entitled to reject totally a discovery, which for seventeen years has at different times disturbed science. " The small variations," they observe, " which are found in some of our experiments, cannot constitute a real action of the nature of that which was observed by MM. Morichini, Baumgartner, &c, in so clear and decided a manner." CHAP. XI. ON THE INFLEXION OR DIFFRACTION OF LIGHT. (74.) Having thus described the changes which light expe- riences when refracted by the surfaces of transparent bodies, and the properties which it exhibits when thus decomposed into its elements, we shall now proceed to consider the phe- nomena which it presents when passing near the edges of bodies. This branch of optics is called the inflexion or the diffraction of light. This curious property of light was first described by Gri- maldi in 1665, and afterwards by Newton ; but it is to the late M. Fresnel that we are indebted for a most successful and able investigation of the phenomena. In order to observe the action of bodies upon the light which passes near them, let a lens L L, of very short focus, fig. 56., be fixed in the window-shutter, M N, of a dark room ; Fig. 5C. and let R L L be a beam of the sun's light, transmitted through the lens. This light will be collected into a focus at F, from which it will diverge in lines FC, FD, forming a circular image of light on the opposite wall. If a small hole, about the fortieth of an inch in diameter, had been fixed in the win- dow-shutter in place of the lens, nearly the same divergent CHAP. XI. INFLEXION OF LIGHT. 87 beam of light would have been obtained. The shadows of all bodies whatever held in this light will be found to be sur- rounded with three fringes of the following colors, reckoning from the shadow: — First fringe. — Violet, indigo, pale blue, green, yellow, red. Second fringe. — Blue, yellow, red. Third fringe. — Pale blue, pale yellow, pale red. In order to examine these fringes, we may either receive them on a smooth white surface as Newton did, or adopt the method of Fresnel, who looked at them with a magnifying glass, in the same manner as if they had been an image formed by a lens. This last method is decidedly the best, as it enables the observer to measure the fringes, and ascertain the changes which they undergo under different circum- stances. I .et a body B be now placed at the distance B P from the focus, and let its shadow be received on the screen C D, at a fixed distance from the body B, and the following phenomena will be observed : — 1. Whatever be the nature of the body B with regard to its density or refractive power, whether it is platina or the pith of a rush, whether it is tabasheer or chromate of lead, the fringes surrounding its shadow will be the very same in mag- nitude ami in color, and the colors will be those given above. •J. It' the light R L is homogeneous light of the different colors in the spectrum, the fringes will be of the same color as the light RL; and they will be broadest in red light, smallest in violet, and of intermediate sizes in the interme- diate colors. 3. The body B continuing fixed, let us either bring the screen C D nearer to B, or bring the lens with which we view the fringes nearer to B, so as to see them at different distances behind B. It will be found that they grow less and less as they approach the edge of B, from which they take their rise. But if we measure the distances of any one fringe from the shadow at different distances behind B, we shall find that the line joining the same point of t lie fringe is nut ;i straight line, but a hyperbola whose vertex is at the edge of the body; so that the same fringe is not formed by the same light at all distances from the body, but resembles a caustic curve formed by the intersection of different rays. This cu- rious fact we have endeavored to represent in the figure by the hyperbolic curves joining the edge of the body B and the fringes which are shown by dotted lines. 4. Hitherto we have supposed that B has been held at the same distance from F; but lot it now be brought to b, much *> 88 A TREA.TTSK ON OPTJCR. V.W.T U, nearer F, and let the screen C D be brought to c d, so that bg is equal B G. In this new position, where nothing lias been changed but the distance from F, the fringes will be found greatly increased in breadth, their relative distances from each other and from the margin of the shadow remain- ing the same. The influence of distance from the radiant point F on the size of the fringes, or on the quantity of inflexion, is shown in the following results obtained by M. Fresnel: — F6 FB a '•( the inflect l! behind the 4 inches. 20 feet. Angular iiiflrxion of the red rayi of the fir.-t Cringe. 3!J inches. 39 12' 6' 3 55 When we consider that the fringes are largest in red, and smallest in violet light, it is easy to understand the cause of their colors in white light; for the colors seen in this case arise from the superposition of fringes of all the seven colors; tiiat is, if the eye could receive all the seven differently color- ed fringes at once, these colors would form by their mixture the actual colors in the fringes seen by white light. Hence we see why the color of the first fringe is violet near the shadow, and red at a greater distance ; and why the blending of the colors beyond the third fringe forms white light, in- stead of exhibiting themselves in separate tints. Upon measuring the proportional breadths of the fringes with great care, Newton found that they were as the num- bers 1, >/ 3, s/ .'» y/ji an d tuc i r intervals in the same pro- portion. Besides the external fringes which surround all bodies, Grimaldi discovered within the shadows of long and narrow bodies a number of parallel streaks or fringes alternately light and dark. Their number grew smaller as the body tapered; and Dr. Young remarked that the central line Va»s always white, so that there must always be an odd number of white stripes, and an even number of dark ones. At the angular termination of bodies these fringes widen and become convex to the central white line; and when the termination is rect- angular, what arc called the crested fringes of Grimaldi are produced. The phenomena exhibited by substituting apertures of various forms in place of the body B are very interesting. When the aperture is circular, such as that formed in a piece of lead with a small pin, and when a lens is placed behind it so as to view the shadow at different distances, the aperture will be seen surrounded with distinct rings, which contract CHAP. XI. INFLEXION OF LIGHT. 89 and dilate, and change their tints in the most beautiful man- ner. When the aperture is one thirtieth of an inch, its dis- tance F B from the luminous point 6 feet 6 inches, and its distance from the focus of the eye-lens, or B G, 24 inches, the following series of rings was observed : — 1st order. White, pale yellow, yellow, orange, dull red. 2d order. Violet, blue, whitish, greenish yellow, yellow, bright orange. 3d order. Purple, indigo blue, greenish blue, bright green, yellow green, red. 4th order. Bluish green, bluish white, red. 5th order. Dull green, faint bluish white, faint red. 6th order. Very faint green, very faint red. 7th order. A trace of green and red. When the aperture B is brought nearer to the eye-lens whose focus is supposed to be at G, the central white spot grows less and less till it vanishes, the rings gradually closing in upon it, and the centre assuming in succession the most brilliant tints. The following were the tints observed by Mr. Herschel; the distance betwedjh the radiant point F and the focus G of the eye-lens remaining constant, and the aperture, supposed to be at B, being gradually brought nearer to G :— Color of the Central spot. Character of (he rings which surround thf central spot. 24 in 18 13 5 10 925 910 8-75 8-36 800 7-75 TOO 063 (V00 .vs:, 5-50 500 ■1-75 4-50 400 385 3 50 1 White. White. j Yellow. J Intense orange. Deep orange red. Brilliant blood red Deep crimson red. Deep purple. Very sombre violet Intense indigo blue Pure deep blue. Sky blue. Bluish white. Very pale blue. Greenish white. Yellow. Orange yellow. Scarlet. Rod. Blue. Dark blue. Kings as described above. First two rings confused. Red of 3d, and green of 4th order, splendid. Inner rings diluted. Red and green of the outer rings good. All the rings much diluted. Rings all very dilute. Rings all very dilute. Rings all very dilute. Rings all very dilute. A broad yellow ring. A pale yellow ring. A rich yellow. A ring of orange, with a sombre space. Orange red, with a pale yellow space. \ crimson red ring. Purple, with orange yellow. HI no, orange. Bright blue, orange red, pale yellow, white Pale yelloW, violet, pale yellow, white. White, mdigo, dull orange, white. White, yellow, blue, dull red. |Orange. light blue, violet, dull orange. Il2 H 90 A TREATISE ON OPTICS. PART IT. When two small apertures are used instead of one, and the rings examined by the eye-lens as before, two systems of rings will be seen, one round each centre ; but, besides the rings, there is another set of fringes which, when the aper- tures are equal, are parallel rectilineal fringes equidistant from the two centres, and perpendicular to the line joining these centres. Two other sets of parallel rectilineal fringes diverge in the form of a St. Andrew's cross from the middle point between the two centres, and forming equal angles be- tween the first set of parallel fringes. If the apertures are unequal, the two systems of rings are unequal, and the first set of parallel fringes become hyperbolas, concave towards the smaller system of rings, and having the aperture in their common focus.* The finest experiments on this subject are those of Fraun- hofer ; but a proper view of them would require more space than we can spare, f CHAP. XI 1. ON THE COLORS OF THIN PLATES. (75.) When light is either reflected from the surfaces of transparent bodies, or transmitted through portions of them with parallel surfaces, it is invariably white, for all the dif- ferent thicknesses of such bodies as we are in the habit of seeing. The thinnest films of blown glass, and the thinnest films of mica generally met with, will both reflect and trans- mit white light If we diminish, however, the thickness of these two bodies to a certain degree, we shall find that, in- stead of giving white light by reflexion and transmission, the light is in both cases colored. Mr. Boyle seems first to have observed that thin bubbles of the essential oils, spirit of wine, turpentine, and soap and water, exhibited beautiful colors ; and he succeeded in blow- ing glass SO thin as to show the same tints. Lord Brereton had observed the colors of the thin oxidated films which the action of the weather produces upon glass; and Dr. Hooke obtained films so equally thin that they exhibited over their whole surface the same brilliant color. Such pieces of mica may be produced at the edges of plates quickly detached from a mass; but they may be more readily obtained by * Hersr.hel's Treatise on Light, § TH/i. } See Edinburgh Encyclopedia, art. Optics, Vol. XV., p. 550. UHAr. XII. COLORS OF THIN TLATF.S. 91 sticking one side of a plate of mica to a piece of sealing-wax, and tearing it away with a sudden jerk. Some extremely thin films will then be left on the wax, which will exhibit the liveliest colors by reflected light. If we could produce a film of mica with only one tenth part of the thickness of that which produces a bright blue color, this film would reflect no light at all, and would appear black if viewed by reflexion against a black body. But though no such film has ever been obtained, or is likely to be obtained by any means with which we are acquainted, yet accident on one occasion produced solid fibres as thin, and actually incapable of reflecting light. This very remarkable fact occurred in a crystal of quartz of a smoky color, which was broken in two. The two surfaces of fracture were absolutely black ; and the blackness appeared, at first sight, to be owing to a thin film of opaque matter which had insinuated itself into the crevice. This opinion, however, was untenable, as every part of the surface was black, and the two halves of the crystals could not have stuck together had the crevice extended across the whole section. Upon examin- ing this specimen with care, I ibund that the surface was per- fectly transparent by transmitted light, and that the blackness of the surfaces arose from their being entirely composed of a fine down of quartz, or of short and slender filaments, whose diameter was so exceedingly small that they were incapable of reflecting a single ray of the strongest light. The diameter of these fibres was so small, that, from principles which we shall presently explain, they could not exceed the one third of the millionth part of an inch. This curious specimen is in the cabinet of her grace the duchess of Gordon.* I have another small specimen in my own possession ; and I have no doubt that fractures of quartz and other minerals will yet be found which shall exhibit a fine down of different colors de- pending on their size. The colors thus produced by thinness, and hence called the colors of thin plates, are best observed in fluid bodies of a viscous nature. If we blow a soap-bubble, and cover it with a clear glass to protect it from currents of air, we shall ob- serve, after it has grown thin by standing a little, a great many concentric colored rings round the top of it. The color in the centre of the rings will vary with the thickness; but as the bubble grows thinner the rings will dilate, the central spot will become white, then bluish, and then black, after which the bubble will burst, from its extreme thinness at the place of the black spot. The same change of color with the See Edinburgh Journal of Science, No. I., p. lUtf. 92 A TREATISE ON OPTICS. TAET II. thickness may be seen by placing a thick film of an evapora- Me fluid upon a clean plate of glass, and watching the effects of the diminution of thickness which take place in the course of evaporation. The method used by Sir Isaac Newton for producing a thin plate of air, the colors of which he intended to investigate, is shown in fig. 57., where L L is a plano-convex lens, the Fig. 57. radius of whose convex surface is 14 feet, and 11 a double convex lens, whose convex surfaces have a radius of 50 feet each. The plane side of the lens L L was placed downwards, so as to rest upon one of the surfaces of the lens 1 I. These lenses obviously touch at their middle points; and if the upper one is slowly pressed against the under one, there will be seen round the point of contact a system of circular color- ed rings, extending wider and wider as the pressure is in- creased. In order to examine these rings under different degrees of pressure, and when the lenses L L, I I are at different distances, three clamp-screws, p,p,p, should be em- ployed, as shown in fig. 58., by turning which we may pro- duce a regular and equal pressure at the point of contact. When we look at these rings through the upper lens, so ;is to see those formed by the light reflected from the plate of ail- between the lenses, we may observe seven rings, or rather seven circular spectra or orders of colors, as described by Newton in the first two columns of the following Table ; the colors being very distinct in the first three spectra, but growing more and more diluted in the others, till they almost entirely dis- appear in the seventh spectrum. When we view the plate of air by looking through the un- der lens 1 1 from below, we observe another set of rings or spectra formed in the transmitted light. Only five of these transmitted rings are distinctly seen, and their colors, as ob- served by Newton, are given in the third column of the fol- lowing table; but they are much more taint than those seen by reflexion. By comparing the colors seen by reflexion with those seen by transmission, it will be observed that the color transmitted is always complementary to the one reflected, or which, when mixed with it, would make white light. CHAP. XII. COLORS OF THIN PLATES. 93 Table of the Colors of Thin l'latcs of Air, Water, and Glass. First Spectrum j order] of Colora Second Spectrum order of Colors. Third Spectrum or order of Colors. Fourth I Spectrum or order of Colors. Fifth f Spectrum ) or order I of Colors. [ Sixth f Spectrum J or order j of Colors. (^ Seventh ( Spectrum J or order I of Colors. { \ ery black Black Beginning of black Blue White Yellow Orange Red Violet Indigo Blue Cireen Yellow < )range Bright red Scarlet Purple Indigo Blue Green Yellow Red Bluish red White Yellowish red Black Violet Blue 2: 74 9 White Yellow Red Violet Blue Green Yellow Red Bluish green Bluish green Green Yellowish ) green ) Bed Red Bluish green Greenish blue Red Greenish blue Bc<: Gre< uish ) blue ■$ Ruddy white Red 1H 12) 14 154 16} 17| 184 I9:j 21 •>•) i -■-I ii 23 -. 25| 274 29 32 l\ 14 3| 5* 6 6? 9f 10l 111 12i 13 13? 141 If 3 4 a 54 7? 8,-t 9 9? 104 Hi 111 12= 152. 164 W« 18A 20| 2l| 24 1344 14] 15rV 10} 174 18? 20? 34 85f 30 404, 25^ 26 1 27 30j !<; ■)"j 34f 393 •22 23§ 26 29| 34 44 48^ 71 77 534 -u\ 45| 49| 94 A TREATISE ON OPTICS. PART II. The preceding colors are those which are seen when light is reflected and transmitted nearly perpendicularly ; but Sir Isaac Newton found that when the light was reflected and transmitted obliquely, the rings increased in size, the same color requiring a greater thickness to produce it. The color of any film, therefore, will descend to a color lower in, or nearer the beginning of, the scale, when it is seen obliquely. Such are the general phenomena of the colored rings when seen by white light. When we place the lenses in homoge- neous light, or make the different colors of the solar spectrum pass in succession over the lenses, the rings, which are always of the same color as the light, will be found to be largest in red light, and to contract gradually as they are seen in all the succeeding colors, till they reach their smallest size in the violet rays. Upon measuring their diameters, Newton found them to have the following ratio in the different colors at their boundaries : — Yellow. Orean. Blue. Indigo. Violet 0-865 0-K>5 0-763 0-711 0-681 Since white light is composed of all the preceding colors, the rings seen by it will consist of all the seven differently color- ed systems of rings superposed as it were, and forming, by their union, the different colors in the Table. In order to explain this, we have constructed the annexed diagram, fig. 59., on the supposition that eacli ring or spectrum has the Fig. 59. I same breadth in homogeneous light which it actually has when it is formed between surfaces nearly flat, or when the thickness of the plate varies with the distance from the point of contact.* Let us then suppose that we form such a * This supposition is made in order to simplify the diagram. CHAP. XII. COLORS OF THIN PLATES. 95 system of rings with the seven colors of the spectrum, and that a sector is cut out of each system, and placed, as in the figure, round the same centre C. Let the angle of the red sector be 50°, of the orange 30°, the yellow 40°, the green 60°, the blue 60°, the indigo 40°, and the violet 80°, being 360° in all, so as to complete the circle. From the centre C set off the first, second, and third rings in all the sectors, with radii corresponding to the values in the preceding small Table. Thus, since the proportional diameters of the ex- treme red and the extreme orange are 1 and 0-924, the mid- dle of the red will be in the middle between these numbers, or 0962; and consequently the proportional diameter, or the radius of the first red ring for the middle of the red space R, will be - 962. In like manner, the radius for the orange will be 0-904, for the yellow 0-855, for the green 0-794, for the blue 0-737, for the indigo 0-696, and for the violet 0-655. Let the red rings be colored red as they appear in the experiment, the orange rings orange, and so on, each color resembling that of the spectrum as nearly as possible. If we now suppose all these colored sectors to revolve rapidly round C as a centre, the effect of them all, thus mixed, should be the production of the colored rings as seen by white light. As the diameter of each ring varies from the beginning of the red space to the end of it, and so on with all the colors, the portion of the ring in each sector should be part of a spiral, and all these separate parts should unite in forming a single spiral, the red forming the commencement, and the violet the termination of the spiral for each ring. This diagram enables us to ascertain the composition of any of the rings seen in white light. Let it be required, for ex- ample, to determine the color of the ring at the distance C m from the centre, rn being in the middle of the second red ring. Round C as a centre, and with the radius C m, describe a cir- cle, m n o p, and it will be seen from the different colors through which it passes what is its composition. It passes nearly through the very brightest* part of the second red ring, at m, and through a pretty bright part of the orange. It passes nearly through the bright part of the yellow, a1 I ■ ; through the brightest part of the green ; through a less bright part of the blue; through a dark part of the indigo, at p; and through the darkest part of the third violet ring. If we knew the exact law according to which the brightness of any fringe varied from its darkest to its brightest point, it would thus be easy to ascertain with accuracy the number of rays * In the figure, the brightest part is the most shaded. i 98 A TREATISE ON orTTCS- VAUT II. of each color which entered into the composition of any of the rings seen by white light In order to determine the thickness of the plate of air by which eacli color was produced, Newton found the squares of the diameters of the brightest parts of each to be in the arithmetical progression of the odd numbers, 1, 3, 5, 7, 9, &c., and the squares of the diameters of the obscurest parte in the arithmetical progression of the even numbers, 2, 4, 6, 8, LO; and as one of the glasses was plane, and the other spherica , their intervals at these rings must be in the same progression. He then measured the diameter of the fifth dark ring, and found that the thickness of the air at the darkest part of the first dark ring, made by perpendicular rays, was the f,^!,,,,, part of an inch. He then multiplied this number by the pro- gression 1, 3, 5, 7, 9, &.c, and 2, 4, 6, 8, 10, and obtained the following results : — Thlc) mrss of thn nir at the Thirknwj of th «*l luminous part. First Ring - - 1 1 1 "MOTS' "IlTtTnoTf 01 Second Ring - - - 3 1TT!.0 0(S' 1 1 -'B.TlOfT Third Ring - - S i - ff.H ti a 1 1 TSVn n 1 T^.ooo 8 i iTj.rroiT When Newton admitted water between the lenses, he found the colors to become fainter, and the rings smaller; and upon measuring the thicknesses of water at which the same rings were produced, he found them to be nearly as the index of refraction for air is to the index of refraction for water, that is, nearly as 1-000 to 1-336. From these data he was enabled to compute the three last columns of the Table given in page 93, which show the thicknesses in millionth parts of an inch at which the colors are produced in plates of air, water, and glass. These columns are of extensive use, and may be regarded as presenting us with a micrometer for measuring minute thicknesses of transparent bodies by their colors, when all other methods would be inapplicable. We have already seen that when the thickness of the film of air is about , ■ ';,,,, , d th of an inch, which corresponds to the seventh ring, the colors cease to become visible, owing to the union of all the separate colors forming white light; but when the rings are seen in homogeneous light, they appear in much greater numbers, a dark and a colored ring succeeding each other to a considerable distance from the point of contact. In this case, however, when the rings are formed between object glasses, the thickness of the plate of air increases so rapidly that the outer rings crowd upon one another, and cease to become visible from this cause. This effect would CHAP. XIII. COLORS OF THICK PLATES. 97 obviously not be produced if they were formed by a solid film whose thickness varied by slow gradations. Upon this prin- ciple, Mr. Talbot has pointed out a very beautiful method of exhibiting these rings with plates of glass and other sub- stances even of a tangible thickness. If we blow a glass ball so thin that it bursts,* and hold any of the fragments in the light of a spirit lamp with a salted wick, or in the light of any of the monochromatic lamps which I have elsewhere de- scribed, all of which discharge a pure homogeneous yellow light, the surface of these films will be seen covered with fringes alternately yellow and black, each fringe marking out by its windings the lines of equal thickness in the glass film. Where the thickness varies slowly, the fringes will be broad and easily seen ; but where the variation takes place rapidly, the fringes are crowded together, so as to require a micro- scope to render them visible. If we suppose any of the films of glass to be only the thousandth part of an inch thick, the rings which it exhibits will belong to the 89th order ; and if a large rough plate of this glass could be got with its thick- ness descending to the millionth part of an inch by slow gra- dations, the whole of those 89 rings, and probably many more, would be distinctly visible to the eye. In order to produce such effects, the light would require to be perfectly homoge- neous. The rings seen between the two lenses are equally visible whether air qr any other gas is used, and even when there is no gas at all ; for the rings are visible in the exhausted re- ceiver of an air-pump. CHAP. XIII. ON THE COLORS OF THICK PLATES. (76.) The colors of thick plates were first observed and described by Sir Isaac Newton, as produced by concave glass mirrors. Admitting a beam of solar light, R, into a dark room, through an aperture a quarter of an inch in diameter formed in the window-shutter M N, he allowed it to fall ujjon a glass mirror, A B, a quarter of an inch thick, quicksilvered behind, having its axis in the direction R r, and the radius of the curvature of both its surfaces being equal to its distance behind th" aperture. When a sheet of paper was placed on the window-shutter M N, with a hole in it to allow the sun- * Films of mica answer the purpose still better. 98 A TREATISE ON OPTICS. PART II. beam to pass, he observed the hole to be surrounded with four or Jive colored rings, with sometimes traces of a sixth Fig. 60. M/i A ]|= i'Yn\\ f\ ; '• : .i^ V J A B dfe and seventh. When the paper was held at a greater or a less distance than the centre of its concavity, the rings became more dilu.te, and gradually vanished. The colors of the rir.gs succeeded one another like those in the transmitted system in thin plates, as given in column 3d of the Table in page 'J3. When the light R was red the rings were red, and so on with the other colors, the rings being largest in red and smallest in violet light. Their diameters preserved the same propor- tion as those seen between the object glasses ; the squares of the diameters of the most luminous parts (in homogeneous light) being as the numbers 0, 2, 4, 6, &c, and the squares of the diameters of the darkest parts as the intermediate num- bers 1, 3, 5, 7, &c. With mirrors of greater thickness the rings grew less, and their diameters varied inversely as the square roots of the thickness of the mirror. When the quick- silver was removed, the rings became fainter ; and when the back surface of the mirror was covered with a mass of oil of turpentine, they disappeared altogether. These facts clearly prove that the posterior surface of the mirror concurs with the anterior surface in the production of the rings. When the mirror A B is inclined to the incident beam R r, the rings grow larger and larger as the inclination increases, and so also does the white round spot ; and new rings of color emerge successively out of their common centre, and the white spot becomes a white ring accompanying them, and the incident and reflected beams always fall upon the opposite parts of this white ring, illuminating its perimeter like two mock suns in the opposite parts of an iris. The colors of these new rings were in a contrary order to those of the former. The Duke de Chaulnes observed similar rings upon the sur- face of the mirror when it was covered with gauze or muslin, or with a skin of dried skimmed milk ; and Sir W. Herschel noticed analogous phenomena when he scattered hair-powder CHAP. XIII. COLORS OF THICK PLATES. 99 in the air before a concave mirror on which a beam of light was incident, and received the reflected light on a screen. (77.) The method which I have found to be the most sim- ple for exhibiting these colors, is to place the eye immediately behind a small llame from a minute wick fed with oil or wax, so that we can examine them even at a perpendicular inci- dence. The colors of thick plates may be seen even with a common candle held before the eye at the distance of 10 or 1:2 teet from a common pane of crown glass in a window that lias accumulated a little line dust upon its surface, or that has on its surface a fine deposition of moisture. Under these circumstances they are very bright, though they may be seen even when the pane of glass is clean. The colors of thick plates may, however, be best displayed, and their theory best studied, by using two plates of glass of equal thickness. The phenomena thus produced, and which presented themselves to me in 1817, are highly beautiful, and, as Mr. Herschel has shown, are admirably fitted for illus- trating the laws of this class of phenomena. In order to ob- tain plates of exactly the same thickness, I formed out of the same piece of parallel glass two plates, A B, C D, and having placed between them two pieces of soft wax, I pressed them to the distance of about one tenth of an inch from each other ; and by pressing above one piece of wax more than another, I was able to give the two plates any small incli- nation I chose. Let A B, C D then be a section of the two plates, thus inclined, at right angles to the com- mon section of their surfaces, and let R S be a ray of light incident nearly in a vertical direction and proceeding from a candle, or, what is better, from a circular disc of condensed light subtending an an- gle of 2° or 3°. If we place the eye behind the plntes, when they are parallel we shall see only an image of the circular disc; but when they are inclined, as in the figure, we shall observe in the direction V R several reflected images in a row besides the direct image. The first or the brightest of these will be seen crossed with fifteen or sixteen beautiful fringes or bands of color. The three central ones consist of blackish or whitish stripes ; and the exterior ones of brilliant bands of- red and green light. The direction of these bands is always parallel to the common section of the inclined Fig. 61. 100 A TREATISE ON OPTICS. plates. These colored bands increase in breadth by diminish- ing the inclination of the plates, and diminish by increasing their inclination. When the light of the luminous circular object falls obliquely on the first plate, so that the plane of in- cidence is at right angles to the section of the plates, the fringes are not distinctly visible across any of the images; but their distinctness is a maximum when the plane of inci- dence is parallel to that section. The reflected images of course become more bright, and the tints more vivid, as the angle of incidence becomes greater; when the angle of inci- dence increases from 0° to 90°, the images that have suffered the greatest number of reflexions are crossed by other fringes inclined to thein at a small angle. If we conceal the bright light of the first image so as to perceive the image formed by a second reflexion within the first plate, and if we view the image through a small aperture, we shall observe colored bands across the first image far surpassing in precision of outline and richness of coloring any analogous phenomenon. When these fringes are again concealed, others are seen on the image immediately behind them, and formed by a third reflexion from the interior of the first plate. If we bring the plate C I) a little farther to the right hand, and make the ray R S fall first upon the plate C 1), and be afterwards reflected back upon the first plate A B, from both the surfaces of C D, the same colored bands will be seen. The progress of the rays through the two plates is shown in the figure. When the two plates have the form of concave and convex lenses, and are combined, as in the double and triple achro- matic object glass, a series of the most splendid systems of rings are developed ; and these are sometimes crossed by others of a different kind. I have not yet had leisure to pub- lish an account of the numerous observations I have made on this curious class of phenomena. In viewing films of blown glass in homogeneous yellow light, and even in common day-light, Air. Talbot has observed that when two films are placed together, bright and obscure fringes, or colored fringes of an irregular form, are produced between them, though exhibited by neither of them separately. I €ttAr. XIV. COLORS OP FIBRES. 101 CHAP. XIV. ON THE COLORS OF FIBRES AND GROOVED SURFACES. (78.) When we look at a candle or any other luminous body through a plate of glass covered with vapor or with dust in a finely divided state, it is surrounded with a corona or ring of colors, like a halo round the sun or moon. These rings increase as the size of the particles which produce them is diminished; and their brilliancy and number depend on the uniform size of these particles. Minute fibres, such as those of silk and wool, produce the same series of rings, which increase as the diameter of the fibres is less ; and hence Dr. Young proposed an in- strument called an eriometer, for measuring the diameters of minute particles and fibres, by ascertaining the diameter of any one of the series of rings which they produce. For this purpose, he selected the limit of the first red and green ring as the one to be measured. The eriometer is formed of a piece of card or a plate of brass, having an aperture about the fiftieth of an inch in diameter in the centre of a circle about half an inch in diameter, and perforated with about eight small holes. The fibres or particles to be measured are fixed in a slider, and the eriometer being placed before a strong light, and the eye assisted by a lens applied behind the small bole, the rings of colors will be seen. The slider must then be drawn out or pushed in till the limit of the red and green ring coincides with the circle of perforations, and the index will then show on the scale the size of the particles or fibres. The seed of the lycoperdon bovista was found by Dr. Wol- laston to be the 8500dth part of an inch in diameter ; and as this substance gave rings which indicated 3^ on the scale, it follows that 1 on the same scale was the 29750th part of an inch, or the 30,000dth part. The following Table contains some of Dr. Young's measurements, in thirty-thousandths of an inch : — Milk diluted indistinct . Dust of lycoperdon bovista Bullock's blood . . . Smut of barley . . . Blood of a marc . . . Human blood diluted with water Pus Silk Beaver's wool .... Mole's fur ... . 3 3A Ah Ci fi 12 13 16 12 Shawl wool ]9 Saxon wool ...... 22 Lloneza wool 25 Alpacca wool 2(> Farina of lanrestinus . . 20 Ryeland VI ' ino wool . . 27 Merino So ith Down . . 28 Seed of lycopodium ... 32 South Down ewe .... 39 Coarse wool 40 Ditto from some worsted . GO 102 A TREATISE ON OPTICS. PART II. (79.) By observing the colors produced by reflexion from the fibres which compose the crystalline lenses of the eyes of fishes and other animals, I have been able to trace these fibres to their origin, and to determine the number of poles or septa to which they are related. The same mode of observation, and the measurement of the distance of the first colored image from the white image, has enabled rue to determine the diameters of the fibres, and to prove that they all taper like needles, diminishing gradually from the equator to the poles of the lens, so as to allow them to pack into a spherical su- perficies as they converge to their poles or points of origin. These colored images, produced by the fibres of the lens, lie in a line perpendicular to the direction of the fibres, and by taking an impression on wax from an indurated lens the colors are communicated to the wax. In several lenses I observed colored images at a great distance from the common image, but lying in a direction coincident with that of the fibres; and from this I inferred, that the fibres were crossed by joints or lines, whose distance was so small as the ll,(K)0dtii part of an inch ; and I have lately found, by the use of very powerful microscopes, that each fibre has in this case teeth like those of a rack, of extreme minuteness, the colors being produced by the lines which form the sides of each tooth. (80.) In the same class of phenomena we must rank the principal colors of mother-of-pearl. This substance, obtained from the shell of the pearl oyster, has been long employed in the arts, and the fine play of its colors is therefore well known. In order to observe its colors, take a plate of regularly formed mother-of-pearl, with its surfaces nearly parallel, and grind these surfaces upon a hone or upon a plate of glass with the powder of schistus, till the image of a candle reflected from the surfaces is of a dull reddish-white color. If we now place the eye near the plate, and look at this reflected image, C, we Fig. C2. M shall see on one side of it a prismatic image, A, glowing with all the colors of the rainbow, and forming indeed a spectrum of the candle as distinct as if it had been formed by an equi- lateral prism of flint glass. The blue side of this image is CHAP. XIV. COLORS OF MOTHER-OF-PEARL. 103 next the image C, and the distance of the red part of the image is in one specimen 7° 22' ; but this angle varies even in the same specimen. Upon first looking into the mother-of- pearl, the image A may be above or below C, or on any side of it; but, by turning the specimen round, it may be brought either to the right or left hand of C. The distance A C is smallest when the light of the candle tails nearly perpen- dicular on the surface, and increases as the inclination of the incident ray is increased. In one specimen it was 2° 7' at nearly a perpendicular incidence, and 9° 14' at a very great obliquity. On the outside of the image A there is invariably seen a mass, M, of colored light, whose distance M C is nearly double A C. These three images are always nearly in a straight line, but the angular distance of M varies with the angle of incidence according to a law different from that of A. At great angles of incidence the nebulous mass is of a beautiful crimson color; at an angle of about ?,1° it becomes green ; and nearer the perpendicular it becomes yellowish- white, and very luminous. If we now polish the surface of the mother-of-pearl, the ordinary image C will become brighter and quite white, but a second prismatic image, B, will start up on the other side of C, and at the. same distance from it. This second image has in all other respects the same pro- perties as the first. Its brightness increases with the polish of the surface, till it is nearly equal to that of A, the lustre of which is slightly impaired by polishing. This second image is never accompanied, like the first, with a nebulous mass M. If we remove the polish, the image B vanishes, and A resumes its brilliancy. The lustre of the nebulous mass M is improved by polishing. If we repeat these experiments on the opposite side of the specimen, the very same phenomena will be observed, with this difference only, that the images A and M are on the op- posite side of C. In looking through the mother-of-pearl, when ground ex- tremely thin, nearly the same phenomena will he observed. The colors and the distances of the images are the same ; but the nebulous mass M is never seen by transmission. When the second image, B, is invisible by reflexion, it is exceedingly bright when seen by transmission, and vice versa. In making these experiments, I had occasion to fix the mother-of-pearl to a goniometer with a cement of resin and boes'-wax ; and upon removing it, 1 was surprised to see the whole surface of the wax shining with the prismatic colors of 104 a tkkattst: on orrics. taut ir. the mother-of-pearl. I at first thought that a small film of the substance had been left upon the wax ; but this was soon found to be a mistake, and it became manifest that the mother- of-pearl really impressed upon the cement its own power of producing the colored spectra. When the unpolished mother- of- pearl was impressed on the wax, the wax gave only one image, A ; and when the polished surface was used, it gave both A and B: but the nebulous image M was never exhibited by the wax. The images seen in the wax are always on the opposite side of C, from what they are in the surface that is impressed upon it. The colors of mother-of-pearl, as communicated to a soil surface, may be best seen by using black wax ; but I have transferred them also to balsam of Tolu, realgar, fusible metal, and to clean surfaces of lead and tin by hard pressure, or the blow of a hammer. A solution of gum arabic or of isinglass, when allowed to indurate upon a surface of mother- of-pearl, takes a most perfect impression from it, and exhibits all the communicable colors in the finest manner, when seen either by reflexion or transmission. By placing the isinglass between two finely polished surfaces of good specimens of mother-of-pearl, we shall obtain a film of artificial mother-of- pearl, which when seen by single lights, such as that of a candle, or by an aperture in the window, will shine with the brightest hues. If, in this experiment, we could make the grooves of the one surface of mother-of-pearl exactly parallel to the grooves in the other, as in the shell itself, the images, A and B, formed by each surface, would coincide, and only two would be ob- served by transmission and reflexion : but, as tins cannot be done, four images are seen through the isinglass film, and also four by reflexion ; the two new ones being formed by re- flexion from the second surface of the film. From these experiments it is obvious that the colors under our consideration are produced by a particular configuration of surface, which, like a seal, can convey a reverse impres- sion of itself to any substaiiee capable of receiving it. By examining this surface with microscopes, i discovered in almost every specimen a grooved structure, like the delicate texture of the skin at the top of an infant's finger, or like the section of the annual growths of wood, as seen upon a dressed plank of fir. These may sometimes be seen by the naked eye, but they are often so minute that 13000 of them arc contained in an inch. The direction of the grooves is always at right angles to the line M A C B, fig. 62. ; and hence in irregularly formed mother-of-pearl, where the grooves are often circular, CHAP. XIV. COLOHS OF MOTHEK-OF-PEARL- 105 and having every possible direction, the colored images A, B arc irregularly scattered round the common image C. If the re, accordingly, circular, the series of prismatic 5, A B, would form a prismatic ring round C, provided ioves retained the same distance. The general distance rrorn the 200th to the 5000th of an inch, and the distance of the prismatic images from C increases as the grooves become closer. In a specimen with 2500 in an inch, the distance AC was 3° 41* ; and in a specimen of about 5000 it was about 7° 22*. These grooves are obviously the sections of all the con- c< ntric strata of the shell. When we use the actual surface of any stratum, none of the colors A, B are seen, and we ob- serve only the mass of nebulous light M occupying the place of tin; principal image C. Hence we see the reason why the pearl gives none of the images A, B, why it communicates nunc ef it.s colors to wax, and why it shines with that delicate white light which gives it all its value. The pearl is formed of concentric spherical strata, round a central nucleus, which .Sir Everard Home conceives to be one of the ova of the fish. None of the edges of its strata are visible, and as the strata .have parallel surfaces, the mass of light M is reflected exactly like the image C, and occupies its place; whereas in the mother-of-pearl it is reflected from surfaces of the strata, in- clined to the general surface of the specimen which reflects the image C. The mixture of all these diffuse masses of nebulous light, of a pink and green hue, constitutes the beau- tiful white of the pearls, in bad pearls, where the colors are too blue or too pink, one or other of these colors has pre- dominated. If we make an oblique section of a pearl, so as to exhibit a sufficient number of concentric strata, with their edges tolerably close, we should observe all the communicable colors of mother-of-pearl.* These phenomena may be observed in many other shells besides that of the pearl-oyster; and in every case we may distinguish communicable from incommunicable colors, by placing a film of fluid or cement between the surface and a plate of class. The communicable colors will all disappear from the tilling up of the grooves, and the incommunicable colors will be rendered more brilliant (81.) Mr. Herschel has discovered in very thin plates of mother-of-pearl another pair of nebulous prismatic images, more distant Iron C than A ami B, and also a pair of fainter nebulous images, the line joining which is always at right ' Svi! Kilinhuri'h Journal of Science, No. XII., p. 577. I I 10G A TREATISE ON OPTICS. TART II. angles to the line joining the first pair.* These images ire seen by looking through a thin piece of mother-of-pearl, cut parallel to the natural surface of the shell, and between the 70th and the 300dth of an inch thick. They arc much larger than A and B; and Mr. Herschel found that the line joining them was always perpendicular to a veined structure which goes through its substance. The distance of the red part of the image from C was found to be 16° 29',/ and the veins which produced these; colors were so small that 3700 of them were contained in an inch. We have represented them in fig. 63. as crossing the ordinary grooves which give the com- municable colors. Mr. Herschel describes them as crossing these grooves at all angles, " giving the whole surface much the appearance of a piece of twilled silk, or the larger waves of the sea intersected with minute ripplings." The second pair of nebulous images seen by transmission must arise from a veined structure exactly perpendicular to the first, though the structure has not yet been recognized by the microscope. The structure which produces the lightest pair Mr. Herschel lias found to be in all cases coincident with the plane passing through the centres of the two systems of polarized rings. The principle of the production of color by grooved sur- faces, and of the communicability of these colors by pressure to various substances, has been happily applied to the arts by John Barton, Esq. By means of a delicate engine, operating by a screw of the most accurate workmanship, he has suc- ceeded in cutting grooves upon steel at the distance of from * In a specimen now before us, the line joining the two faintest nebulous images is at right angles to the line joining A and II. CHAP. XIV. COLORS OF GROOVED SURFACES. 107 the 2000th to the 10,000th of* an inch. These lines are cut with the point of a diamond ; and such is their perfect paral- lelism and the uniformity of their distance, that while in mother-of-pearl we see only one prismatic image, A, on each side of the common image, C, of the candle, in the grooved steel surfaces 6, 7, or 8 prismatic images are seen, consisting of spectra, as perfect as those produced by the finest prisms. Nothing in nature or in art can surpass this brilliant display of colors ; and Mr. Barton conceived the idea of forming but- tons for gentlemen's dress, and articles of female ornament covered with grooves, beautifully arranged in patterns, and shining in the light of candles or lamps with all the hues of the spectrum. To these he gave the appropriate name of Iris ornaments. In forming the buttons, the patterns were drawn on steel dies, and these, when duly hardened, were used to stamp their impressions upon polished buttons of brass. In day-light the colors on these buttons are not easily distinguish- ed, unless when the surface reflects the margin of a dark ob- ject seen against a light one ; but in the light of the sun, and that of gas-flame or candles, these colors are scarcely if at all surpassed by the brilliant flashes of the diamond. The grooves thus made upon steel are, of course, all trans- ferable to wax, isinglass, tin, lead, and other substances; and by indurating thin transparent films of isinglass between two of these grooved surfaces, covered with lines lying in all di- rections, we obtain a plate which produces by transmission the most extraordinary display of prismatic spectra that has ever been exhibited. (82.) In examining the phenomena produced by some of the finest specimens of Mr. Barton's skill, which he had the kindness to execute for this purpose, I have been led to the observation of several curious properties of light. In mother- of-pearl, well polished, the central image, C, of the candle or luminous object is always white, as we should expect it to be, in consequence of being reflected from the flat and polished surfaces between the grooves. In like manner, in many specimens of grooved steel the image C is also perfectly white, and the spectra on each side of it, to the amount of six or eight, are perfect prismatic images of the candle; the image A, which is nearest C, being the least dispersed, and all the rest in succession more and more dispersed, as if they were formed by prisms of greater and greater dispersive powers, or greater and greater refracting angles. These spec- tra contain the fixed lines and all the prismatic colors; but the red or least refrangible spaces are greatly expanded, and the 108 A TREATISE ON OPTICS. TART II. violet or most refrangible spaces greatly contracted, even more than in the spectra produced by sulphuric acid. In examining some of these prismatic images which seemed to be defective in particular rays, I was surprised to find that, in the specimens which produced them, the image C reflected from the polished original surface of the steel was itself slightly colored; that its tint varied with the angle of inci- dence, and had some relation to tiie defalcation of color in the prismatic images. In order to observe these phenomena through a great range of incidence, I substituted for the can- dle a long narrow rectangular aperture, formed by nearly closing the window-shutters, and I then saw at one view the 6tate of the ordinary image and all the prismatic images. In order to understand this, let A B, Jig. 64, be the ordinary Fig. 04. .., „ , A. , » m ... a —a. »,cli a o a ,a ,.,ci«-a •* •i — iwriwi—iiTn *■ 7-LJntJ i'-rU itU i l_l i-_l it, l_l m-L. W F V b B b li /.' F image of the aperture reflected from the flat surface of the steel which lies between the grooves, and a b, a' b\ a" b", &c, the prismatic images on each side of it, every one of these images forming a complete spectrum with all its difierent colors. The image A B was crossed in a direction perpen- dicular to its length with broad colored fringes, varying in CHAP. XIV. COLORS OF GROOVED SURFACES. 109 their tints from 0°. to 90° of incidence. In a specimen with 1000 grooves in an inch, the following were the colors dis- tinctly seen at different angles of incidence : — White - - Yellow - - Reddish orange Pink - - - Junction of pink blue - - Brilliant blue Whitish - - Yellow - - Pink - - - Junction of pink blue - - iogla <>r Lncidei - 90° 0' - 80 30 - 77 30 - 76 20 5 40 and " and 74 30 71 64 45 59 45 58 10 Blue - - - - Bluish green Yellowish green Whitish green - Whitish yellow - Yellow - - - Pinkish yellow - Pink red - - - Whitish pink Green - - - Yellow - - - Reddish - - - Angle of inciilciKO. 56° C ■ 54 30 - 53 15 - 51 - 49 - 47 15 - 41 - 36 - 31 - 24 - 10 - These colors are those of the reflected rings in thin plates. If we turn the steel plate round in azimuth, the very same colors appear at the same angle of incidence, and they suffer no change cither by varying- the distance of the steel plate from the biminous aperture, or the distance of the eye of the observer from the grooves. In the preceding table there are four orders of colors; but in some specimens there are only three, in others two, in others one, and in some only one or two tints of the first order are developed. A specimen of 500 grooves in an inch gave only the yellow of the first order through the whole quadrant of incidence. A specimen of 1000 grooves gave only one complete order, with a portion of the next. A specimen of 3333 grooves gave only the yellow of the first order. A spe- cimen of 5000 gave a little more than one order ; and a spe- cimen of 10,000 grooves in an inch gave also a little more than one order. In fig, (;!. we have represented the portion of the quadrant of incidence from about 22° to 76°. In the first spectrum, a b a t>, v v is the violet side of it, and r r the red side of it, and between these are arranged all the other colors. At m, at an incidence of 74°, the violet light is obliterated from the spectrum ab\ and at n, at an incidence of 66°, the red rays are obliterated ; the intermediate colors, blue, green, &c, being obliterated at intermediate points between m and n. In the second spectrum, a' b' a' b\ the violet rays are obliterated at in' at an incidence of 66° 20', and the red, at n' at an inci- dence of 50°. In the third spectrum, a" b" a" b", the violet rays uro obliterated at in" at 57°, and the red at n" at 41° K 110 A TREATISE ON OPTICS. PART II. 35'; and in the fourth spectrum, the violet rays are oblit- erated at m'" at 48°, and the red at n'" at 23° 30". A simi- lar succession of obliterated tints takes place on all the pris- matic images at a lesser incidence, as shown at nv, f»'r'; the violet being obliterated at p and p!, and the red at v and v', and the intermediate colors at intermediate points. In this second succession the line ft v begins and ends at the same angle of incidence as the line m" n" in the third prismatic image, a" b", and the line p' v' in the second prismatic image corresponds with in"' n'" on the fourth prismatic image. In all these cases, tlie tints obliterated in the direction mn pv, &c, would, if restored, form a complete prismatic spectrum whose length is m n // v, &c. Considering the ordinary image as white, a similar oblitera- tion of tints takes place upon it. The violet is obliterated at o about 76°, leaving pink, or what the violet wants of white light; and the red is obliterated at p at 74°, leaving a bright blue. The violet is obliterated at q and s, and the red at r and t, as may be inferred from the preceding Table of colors. The analysis of these curious and apparently complicated phenomena becomes very simple when they are examined by homogeneous light. The effect produced on red light is re- presented in Jig. 65., where A B is the image of the narrow Fig C 5, aperture reflected from the original ^ _ surface of the steel, and the four images on each side of it. correspond with the prismatic images. All these nine images, however, consist of homogene- ous red light, which is obliterated, or nearly so, at the fifteen shaded rectan- gles, which are the minima of the new series of periodical colors which cross both the ordinary and the lateral images. The centres p, r, t, n, >', &c, of these rectangles correspond with the points marked with the same letters in Jig. 64, ; and if we had drawn the same figure for violet light, the centres of the rectangles would have been all higher up in the figure, and would have corresponded with o, q, s, m, f, &c. in fig. 64. The rectangles should have been shaded off" to represent the phenomena accurately, but the only object of the figure is to show to the eye the position and relations of the minima. P v t Il\T\ XV. THEORY OF FITS. Ill If we cover the surface of the grooved steel with a fluid, so as to dimmish the refractive power of the surface, we de- vekope more orders <>f colors on the ordinary image, and a greater number of minima on the lateral images, higher tints being produced at a given incidence. But, what is very re- markable, in grooved surfaces when the ordinary image is perfect! v white, and when the spectra are complete without any obliteration of tints, the application of fluids to the grooved surface developes colors on the ordinary image, and a corresponding obliteration of tints on the lateral images. The following Table contains a few of the results relative to the ordinary image : — N umbel < ( an inch. Maximum tj.t without a 11, .. I. Maximum tint with fluid*. 312. 3333 I Perfectly while. < Gamboge yellow { of the iirst order. t 1. Water, tinge of yellow. ) 2. Alcohol, tinge of yellow. ( 3. Oil of cassia, faint reddish yellow. C 1. Wa'er, pinkish red (Iirst order). J 2. Alcohol, reddish pink, i 3. Oil of cassia, bright blue (second order). Phenomena analogous to those above described take place upon the grooved surfaces of gold, silver, and calcareous spar ; and upon the surfaces of tin, isinglass, realgar, &c, to which the grooves have been transferred from steel. For an account of the phenomena exhibited by several of these substances, I must refer the reader to the original memoir in the Philosophical Transactions for 1829. CHAP. XV. ON FITS OF REFLEXION AND TRANSMISSION, AND ON THE INTERFERENCE OF LIGHT. (8.3.) In the preceding chapters we have described a very extensive class of phenomena, all of which 6eem to have the same origin. From his experiments on the colors of thin and of thick plates, Newton inferred that they were produced by a singular property of the particles of light, in virtue of which they possess, at different p>oiiits of their path, fits or dispositions to be reflected from or transmitted by transparent bodies. Sir Isaac does not pretend to explain the origin of these fits, or the cause which produces them ; but we may turm a tolerable idea of them by supposing that each particle 112 A TREATISE ON OITICS. TART II. of light, after its discharge from a luminous body, revolves round an axis perpendicular to the direction of its motion, and presenting alternately to the line of its motion an attractive and a repulsive pole, in virtue of which it will lie refracted if the attractive pole is nearest any refracting- surface oil which it falls, and reflected if the repulsive pole is nearest that sur- face. The disposition to be refracted and reflected will or' course increase and diminish as tbe distance of either polo from the surface of the body is increased or diminished. A less scientific idea may be formed of this hypothesis, by sup- posing a body with a sharp and a blunt end passing through space, and successively presenting its sharp and blunt ends to the line of its motion. When the sharp end encounters any soft body put in its way, it will penetrate it ; but when the blunt end encounters the same body, it will be reflected or driven back. To explain this more clearly, let R,jfa\ Of'., be a ray of light falling upon a refracting surface MJV, and transmitted by that surface. It is clear that it must have met the surface M N when it was nearer its fit of transmission than its fit of reflexion; but whether it was exactly at its fit of transmission, or a little from it, it is put, by the action of the surface, into the same state as if it had begun its fit of trans- mission at t. Let us suppose that, after it has moved through a space equal to t r, its fit of reflexion takes place, the fit of trans- mission always recommencing at t V, &c. and that of reflexion at r r', &c. ; then it is obvious, that if the ray meets a second transparent surface at 1 1', &c, it will be transmitted, and if it meets it at rr', &c, it will be reflected. The spaces t V, t! t" are called the intervals of the fits of transmission, and r r', r' r" the intervals of the fits of re- flexion. Now, as the spaces 1 1', rr', &c. are supposed equal for light of the same colors, it is manifest that, if MN be the first surface of a body, the ray will be transmitted if the thickness of the body is t f, 1 1", &c. ; that is, 1 1','2 t V, :< / /'. 4 1 I', or any multiple whatever of the interval of a fit of easy transmission. In like manner the ray will be reflected if the thickness of the body is t r, t r> ; or, since t V is equal to r r', if the thickness of the body is 4 / 1', 11 1 1\ 2', t /', ;!', / V. If the body M N, therefore, had parallel surfaces, and if the eye were placed above it so as to receive the rays reflected perpendicularly, it would, in every case, see the surface M A' by the portion of light uniformly reflected from that surface ; CHAP. XV. THEORY OF FITS. 113 but when the thickness of the body was 1 1',2 t V, 3 tt', 4 1 1\ or KKK) / t', the eye would receive no rays from the second Burface, because they are all transmitted; and in like manner, if the thickness was \ l /', 1.] / V, 2h 1 t\ or 1000£ 1 t', the eye would receive all the light reflected from the second surface, in 'cause it is all reflected. When this reflected light meets the first surface M N, on its way to the eye, it is all trans- mitted, because it is then in its fit of transmission. Hence, in the first case, the eye receives no light from the second surface, and in the second case, it receives all the light from the second surface. If the body had intermediate thicknesses between t V and 2 t V, &c, as £ 1 1', then a portion of the light would be reflected from the second surface, increasing as the thickness increased from t V to H t V, and diminishing again as the thickness increased from \\Tk' to 2 ft'. But let us now suppose that the plate whose surface is M N is unequally thick, like the plate of air between the two lenses, or a film of blown glass. Let it have its thickness varying like a wedge M N P,fig- 67. Let 1 1', rr' be the in- tervals of the fits, and let the eye be placed above the wedge as before. It is quite clear that near the point N the light that falls upon the second surface N P will be all transmitted, as it is in a fit of transmission ; but at the thickness t r the light R will be reflected by the second surface, because it is then in its fit of reflexion. In like manner the light will be trans- mitted at t\ again reflected at ?•', and again transmitted at t " ; so that the eye above M N will see a series of dark and luminous bands, the middle of the dark ones being at N, t', t" in the line N P, and of the luminous ones at r, ?•', &c. in the same line. Let us suppose that the figure is suited to red homogeneous light, t V being the interval of a fit for that species of rays ; then in violet light, V, the interval of the fits will be less, as rp. If we therefore use violet light, the in- K2 114 A TRF^ATfSF. ON OPTICS. PART TT, terval of whose fits is rp, a smaller serins of violet and ob- scure bands or fringes will be seen, whose obscurest points are at N, i-', r", &c, and whose brightest points are at p, 'p, &c. In like manner, with the intermediate colors of the spectrum, bands of intermediate magnitudes will be formed, having their obscurest points between r' and V, r" and /", and their brightest points between p and r, p and r', &c. ; and when white light is used, all these differently colored bands will be seen forming fringes of the different orders of colors given in the Table in page (K<. IF M N P, in place of being the section of a prism, were the section of one half of a plano-concave lens, whose centre is N, and whose concave sur- face has an oblique direction somewhat like N P, the direction of the colored bards will always be perpendicular to the radius N M, or willue regular circles. For the same reason, the colored bands are circular in the concave lens of air be- tween the object glasses; the same colors always appearing at the same thickness of the medium, or at the same distance from the centre. By the same means Sir Isaac Newton explained the colors of thick plates, with this difference, that the fringes are not in that case produced by the light regularly refracted and re- flected at the two surfaces of the concave mirror, but by the light irregularly scattered by the first surface of the mirror in consequence of its imperfect polish; for, as he observes, "there is no glass or speculum, how well soever polished, but, besides the light which it retracts and reflects regularly, scatters every way irregularly a faint light, by means of which the polished surface, when illuminated in a dark mem by a beam of the sun's light, may be easily seen in all posi- tions of the eye." The same theory of fits affords a ready explanation of the phenomena of double and equally thick plates, which we have described in another chapter. There are other phenomena of colors, however, to which it is not equally applicable ; and it has accordingly been, in a great measure, superseded by the doctrine of interference, which we shall now proceed to explain. (83.) In examining the black and white stripes within the shadows of bodies as formed by inflexion, Dr. Young found that when he placed an opaque screen either a few inches be- fore or a few inches behind one side of the inflecting body, B, fig. 56., so as to intercept all the light on that side by receiv- ing the edge of the shadow on the screen, then all the fringes in the shadow constantly disappeared, although the light still passed by the other edge of the body as before. Hence he CHAP. XV. ON THE INTERFERENCE OF LIGHT. 115 Concluded that, the light which passed on both sides was ne- cessary to the production of the fringes ; a conclusion which he might have deduced also from the known fiict, that when i te body was above a certain size, fringes never appeared in its shadow. In reasoning upon this conclusion, Dr. Young was led to the opinion, that the fringes within the shadow were produced by the interference of the rays bent into the shadow by one side of the body 1$ with the rays bent into the shadow by the other side. In order to explain the law of interference indicated in this experiment, let us suppose two pencils of light to radiate from two points very close to each other, and that this light falls upon the same spot of a piece of paper held parallel to the line joining the points, so that the spot is directly opposite the point which bisects the distance between the two radiant points. In this case they may be said to interfere with one another ; because the pencils would cross one another at that spot if tlic paper were removed, and would diverge from one another. The spot will, therefore, be illuminated with the sum of their lights ; and in this case the length of the paths of the two pencils of light is exactly the same, the spot on the paper being equally distant from both the radiant points. Now, it has been found that when there is a certain minute difference between the lengths of the paths of the two pencils of li 1. Rhomb with obtuse summit, jig. 72. — Carbonate of lime (Iceland spar. — Carbonate of lime and iron. — Carbonate of lime and mag- nesia. — Phosphato-arseniate of lead. — Carbonate of zinc. — Nitrate of soda. — Phosphate of lead. — Ruby silver. — Levyne. — Tourmaline. — Rubellite. — Alum stone. — Dioptase. — Quartz. 2. Rhomb with acute summit, Jig. 73. — Corundum. — Sapphire. —Ruby. — Cinnabar. — Arseniate of copper. 3. Regular Hexahedral Prism, jig. 74 — Emerald. — Beryl. — Phosphate of lime (apatite) — Nepheline. — Arseniate of lead. -f- Hydrate of magnesia. 4. Octohedron with a square base, jig. 75, -f Zircon. -j- Oxide of tin. -j- Tungstate of lime. — Mellite. — Molybdate of lead. — Octohedrite. — Prussiate of potash. — Cyanide of mercury. *& I 130 A TREATISE ON OPTICS. Fig. 75. Fig. 76. PART II. 5. Right Prism with a square base, Jig. 76. Sulphate of nickel and cop- per. — Hydrate of strontia. + Apophyllite of utoe. -f Oxahverite. -j- Superacetate of copper and lime. + Titanite. -flee (certain crystals). — Idocrase. — Wernerite. — Paranthine. — Meionite. — Somervillite. — Edingtonite. — Arseniate of potash. — Sub-phosphate of potash. — Phosphate of ammonia and magnesia. In all the preceding crystals, and in the primitive forms to which they belong, the line A X is the axis of figure and of double refraction, or the only direction aiong which there is no double refraction. On the Law of Double Refraction in Crystals with'one Negative Axis. f (93.) In order to give a familiar explanation of the law of Fig. 77. double refraction, let us suppose that a rhomb of Iceland spar is turned in a lathe to the form of a sphere, as shown in Jig. 77., AX being the axis of both the rhomb and the sphere. If we now make a l ay pass along the axis A X, after grinding or polishing a small flat surface at A and X, perpendicular to A X, we shall find that there is no double refraction ; the ordinary and extraordinary ray forming a single ray. Hence, The index of refraction along ) 1-654 for ordinary ray. the axis A X will be - \ 1-654 for extraordinary ray 0-000 difference. CHAP. XVII. NEGATIVE DOUBLE REFRACTION. 131 If we do the same at any point, a, about 45° from the axis, we shall have The index of refraction along the line R a b O, which is nearly perpen- dicular to the face of the rhomb, 1654 for ordinary ray. 1-572 for extraordinary i ray. 0.082 difference. If we do the same at any point of the equator C D, in- clined 90° to the axis, we shall have The index of refraction per- ) 1-654 for ordinary ray. pendicular to the axis, ) 1.483 for extraordinary ray. 0-171 difference. Hence it follows that the index of extraordinary refraction decreases from the axis A X to the equator C D, or to a line perpendicular to the axis, where it is the least. The index of extraordinary refraction is the same at all equal angles with the axis A X ; and hence, in every part of a cir- cle described on the surface of the sphere round the pole A or X, the index of extraordinary refraction has the same value, and consequently the double refraction or separation of the rays will be the same. In crystals, therefore, with one axis of double refraction, the lines of equal double refraction are circles parallel to the equator or circle of greatest double refraction. The celebrated Huygens, to whom we owe the discovery of the law of double refraction in crystals with one axis, has given the lbllowing method of determining the index of ex- traordinary refraction at any point of the sphere, when the ray of light is incident in a plane passing through the axis of the crystal A X : — Let it be required, for example, to determine the index of refraction for the extraordinary ray Rab, Jig. 77., A X being the axis, and C I) the equator of the crystal ; the ordinary index of refraction being known, and also the least extra- ordinary index of refraction, or that which takes place in the equator. In calcareous spar these numbers are 1-654 and 1.483. From O set off in the lines O C, O D continued, O c, O (I, so that O C or O D is to O c or O d as ttshx is to T .^ 3 , or as -604 is to -674 ; and through the points A, c, X, d, draw an ellipse, whose greater axis is c d, and whose lesser axis is A X. The radius O a of the ellipse will be what is called the reciprocal of the index of refraction at a ; and as we can find O a, either by projecting the ellipse on a large scale, or by calculation, we have only to divide 1 by O a to have that mm 132 A TREATISE»ON OFTICS. TART II. index. In the present case O a is 636, and .-^ is equal to 1-572, the index required. As the index of extraordinary refraction thus found always diminishes from the pole A to the equator C D, and is always equal to the index of ordinary refraction minus another quantity depending on the difference between the radii of the circle and those of the ellipse, the crystals in which this takes place may be properly said to have negative double refraction. . In order to determine the direction of the extraordinary refracted ray, when the plane of incidence is oblique to a plane passing through the axis, the process, either by projec- tion or calculation, is too troublesome to be given in an ele- mentary work. In every case the force which produces the double refrac- tion exerts itself as if it proceeded from the axis. Every plane passing through the axis is called a principal section of the crystal. On the Law of Double Refraction in Crystals with one Positive Axis. (94.) Among the crystals best fitted for exhibiting the phenomena of positive double refraction is rock crystal or quartz, a mineral which is generally found in six-sided Fig. 78. prisms, like Jig. 78., terminated with six-sided pyra- A mids, E, F. If we now grind down the summits A and X, and replace them by faces well polished, and per- pendicular to the axis AX; and if we transmit a ray through these faces, so that it may pass along \W^j. the axis A X, we shall find that there is no double ^ refraction, and that the index of refraction is as x follows : — Index of refraction along > 1-5494 for ordinary ray. the axis AX - - £ 1-5484 for extraordinary ray 0-0000 difference. If we now transmit the ray perpendicularly through the parallel faces E F, which are inclined 38° 20' to the axis A X, the plane of its incidence passing through A X, we shall obtain the following results : — Index of refraction perpen- dicular to the faces of 3& D the pyramid 1-5484 for ordinary ray. 1-5544 for extraordinary ray. 00060 difference. CHAP. XVII. DOUBLE REFRACTION. 133 In like manner, it will be found that when the ray passes perpendicularly through the faces C D, perpendicular to the axis A X, the index of extraordinary refraction is the greatest, viz. Index of refraction perpen- 1 1 . 5m f()r 0rdj Ihe prism CD - \ r5582 for extraor ^ ar y ™Y- 0-0098 difference. Hence it appears that in quartz the index of extraordinary refraction increases from the pole A to the equator C D, whereas it diminished in calcareous spar, and the extraordi- nary ray appears to be drawn to the axis. In this case the variation of the index of extraordinary refraction will be represented by an ellipse, Ac Xd, whose greater axis coincides with the axis A X of double refraction, as in fig. 79., and O C will be to Oc as T .T!^ ?T is to t . 5 t^2' or as "6458 is to -6418. By determining, therefore, the radius O a of the ellipse for any ray R Z» a, and dividing 1 by it, we shall have the index of extraordinary refraction for that ray. As the index of extraordinary refraction is always equal to the index of ordinary refraction, plus another quantity de- pending on the difference between the radii of the circle and the ellipse, the crystals in which this takes place may properly be said to have positive double refraction. On Crystals with two Axes of Double Refraction. (95.) The great variety of crystals, whether they are mineral bodies or chemical substances, have two axes of double refraction, or two directions inclined to each other along which the double refraction is nothing. This property of possessing two axes of double refraction I discovered in 1815, and I found that it helonged to all the crystals which arc included in the prismatic system of Mohs, or whose primitive forms are, A right prism, base a rectangle. base a rhomb. base an oblique parallelogram. Oblique priKm, base a rectangle. base a rhomb. base an oblique parallelogram. Octohedron, base a rectangle batM) a rhomb. M 134 A TREATISE ON OPTICS. PAKT II. In all these primitive forms there is not a single pre-emi- nent line or axis about which the figure is symmetrical. The following is a list of some of the most important crys- tals, with their primitive forms according to Hairy, and the inclination of the two lines or axes along which there is no double refraction : — Glauberite - - ■ Nitrate of potash Arragonite - - Sulphate of baryta Mica - - - - 2° or 3° Oblique prism, base a rhomb. 5° 20' Octohedron, base a rectangle, 18 18 Octohedron, base a rectangle. 37 42 Right prism, base a rectangle. 45 Right prism, base a rectangle. Right prism, base an oblique parallelogram. 65 Octohedron, base a rectangle. 80 30 Prismatic system of Mobs. 90 Oblique prism, base a rhomb. 60 Sulphate of lime - Topaz - - - - Carbonate of potash Sulphate of iron - In crystals with one axis of double refraction, the axis has the same position whatever be the color of the pencil of light jtfhich is used; but in crystals with two axes, the axes change tfcir position according to the color of the light employed, so that the inclination of the two axes varies with differently colored rays. This discovery we owe to Mr. Herschel, who found that in tartrate of potash and soda (Rochelle salts) the inclination of the axis for violet light was about 56°, while in red light it was about 76°. In other crystals, such as nitre, the inclination of the axes for the violet rays is greater than for the red rays ; but in every case the line joining the extremity of the axes for all the different rays is a straight line. In examining the properties of Glauberite, I found that it had two axes for red light inclined about 5°, and only one axis for violet light. It was at first supposed that in crystals with two axes, one of the rays was refracted according to the ordinary law of the sines, and the other by an extraordinary law ; but Mr. Fresnel has shown that both the rays are refracted according to laws of extraordinarv refraction. On Crystals with innumerable Axes of Double Refraction. (96.) In the various doubly refracting bodies hitherto men- tioned, the double refraction is related to one or more axes ; but I have found that in analcime there are several planes, along which if the refracted ray passes, it will not suffer double refraction, however various be the directions in which CHAP. XVII. DOUBLE REFIMCTIOX. 135 it is incident. Hence we may consider each of these planes as containing an infinite number of axes of double refraction, or rather lines in which there is no double refraction. When the ray is incident in any other direction, so that the refracted ray is not in one of these planes, it is divided into two rays by double refraction. No other substance has yet be»£ 138 A TREATISE ON OPTICS. PART II. These two beams, O o, E e, Jig. 81., are therefore said to be polarized, or to be beams of polarized light, because they have sides or poles of different properties ; and planes passing" through the lines A B, C D, or A' B', C D', are said to be the planes of polarization of each beam, because they have the same property, and one which no other plane passing through the beam possesses. Now, it is a curious fact, that if we cause the two polarized beams O o, E e to be united into one, or if we produce iliem by a thin plate of Iceland spar, which is not capable of sepa- rating them, we obtain a beam which has exactly the same properties as the beam A B C D of common light Hence we infer, that a beam of common light, A B C D, consists of tico beams of polarized light, whose planes of po- larization, or whose diameters of similar properties, arc at right angles to one another. If O o is laid upon E e, it will produce a figure like A B C D, and we, therefore, re- present common light by such a figure. If we place O o above E e, so that the planes of polarization A' B' and C D' coincide, then we shall have a beam of polarized light twice as luminous as either O o or E e, and possessing exactly the same properties ; for the lines of similar property in the one beam coincide with the lines of similar property in the other. Hence it follows that there are three ways of converting a beam of common light, A B C D, into a beam or beams of po- larized light. 1. We may separate the beam of common light, A B C D, into its two component parts, O o and E e. 2. We may turn round the planes of polarization, A B, C D, till they coincide or are parallel to each other. Or, 3. We may absorb or stop one of the beams, and leave the other, which will consequently be in a state of polarization. The first of these methods of producing polarized light is that in which we employ a doubly refracting crystal, which we shall now consider. On the Polarization of Light by Double Refraction. (99.) When a beam of light suffers double refraction by a negative crystal, as Iceland spar, Jig. 71., where the ray R r is incident in the plane of the principal section, or, what is the same thing, in a plane passing through the axis, the two pencils r O, r E are each polarized ; the plane of polarization of the ordinary ray, r O, coinciding with the principal section, and the plane of polarization of the extraordinary ray, r E, being CHAP. XV XII. POLARIZATION OF LIGHT. 130 at right angles to the principal section. In fig. 82., if O be made to denote a section of the ordinary beam r O, fig. 71., E, the diameter of which is drawn at right angles to that of O, will represent a section of the extraordinary beam r E. Fig. 82. Fig. 83. If the beam of light R r is incident upon a positive crystal, like quartz, O of Jig. 83., will be the symbol of the ordinary ray, and E that of the extraordinary ray. The phenomena which arise from this opposite polarization of the two pencils may be well seen in Iceland spar. For this purpose let A r X be the principal section of a rhomb of Ice- land spar, fig. 84., through the axis A X, and perpendicular to one of the liices, and let A' F X' be a similar section of an- other rhomb, all the lines of the one being parallel to all the lines of the other. A ray of light, R r, incident perpendicu- larly at r, will be divided into two pencils ; an ordinary one, Fig. 84. Fig. 85. A V "1 i / c A: I" n J Oo E- A f N ■1 c U X i 1 G o fc H 1C Co Oe - D, and an extraordinary one, r C. Tbe ordinary ray falling on the second crystal at G, again suffers ordinary refraction, and emerges at K an ordinary ray, O o, represented by the symbol O, fig. 8'2. In like manner the extraordinary ray, r C, falling on the second crystal at F, again suffers extraordinary 140 A TREATISE ON OPTICS. FART If. refraction, and emerges at II an extraordinary ray, E r, repre- sented by E, Jig. 82. These results are exactly the same as if the two crystals had formed a single crystal by being united at their surfaces C X, A' G, either by natural cohesion or by a cement Let the upper crystal A X now remain fixed, with the same ray R r falling upon it., and let the second crystal A' A' be turned round 90°, so that its principal section is perpendicular to that of the upper oar-, as shown in Jig. 85. ; then the ray r D ordinarily refracted by the first rhomb will he extraordi- narily refracted by the second, and the ray r C extraordinarily refracted by the first rhomb will be ordinarily refracted by the second. The pencils or images formed from the ray R r, in the two positions shown in Jigs. 84. and 85., may be thus described as marked in the figures : — O is the pencil refracted ordinarily by the first rhomb. E is the pencil refracted extraordinarily by the first rhomb. o is the pencil refracted ordinarily by the second rhomb. € is the pencil refracted extraordinarily by the second rhomb. O o is the pencil refracted ordinarily by both rhombs in ^.84. E e is the pencil refracted extraordinarily by both rhombs in Jig. 84. O e is the pencil refracted ordinarily by the first, and ex- traordinarily by the second rhomb in Jig, 85. E o is the pencil refracted extraordinarily by the first, and ordinarily by the second rhomb in fig. 85. In both the cases shown in figs. 84. and 85., when the planes of the principal sections of the two rhombs are either parallel, as in fig. 84., or perpendicular to each other, as in fig. 85., the lower rhomb is not capable of dividing into two any of the pencils which fall upon it; but in every other po- sition between the parallelism and the perpendicularity of the principal sections, each of the pencils formed by the first rhomb will be divided into two by the second. In order to explain the appearances in all intermediate po- sitions, let us suppose that the ray R r proceed:- from a round aperture, like one of the circles at A, Jig. 8(1, and that the eye is placed behind the two rhombs at II is, fig. 84., so as to see the images of this aperture. Let the two images shown at A, fig. 86., be the appearance of the aperture at R, seen through one of the rhombs by an eye placed behind C D,Jig. 84., then ii, Jin ■ 8< i., will represent the images seen through the two rhombs in the position in fig. 84., their distance being doubled, from suffering the same quantity of double refraction twice. If we now turn CHAP. XVIII. POLARIZATION OF LIGHT. 141 the second rhomb, or that nearest the eye, from left to right, two faint images will appear, sis at C, between the two bright ones, which will now be a little fainter. By continuing to A B Fig. 80. E F turn, the four images will bo all equally luminous, as at D ; they will next appear as at E ; and when the second rhomb has moved round 90°, as in fig. 85., there will be only two images ot' equal brightness, as at P. Continuing to turn the second rhomb, two faint images will appear, as atG; by a farther rotation, they will be all equally bright, as at H ; farther on they will become unequal, as at I ; and at 180° of revolution, when the planes of the principal section are again parallel, and the axes A X, A' X' at right angles nearly to each other, all the images will coalesce into one bright image, as at K, having double the brightness of either of those at A, B, or P, and four times the brightness of any one of the four at I) and IL If wo bow follow any one of the images A, B from the po- sition in fig. 84., where the principal sections are inclined 0° to one another, to the position in jig. 85., where it disappears at F, we shall find that its brightness diminishes as the square of the cosine of the angle formed by the principal sections, while the brightness of any image, from its appearance be- tween B and C, fig. 8fi., to its greatest brightness at F, in- creases as the square of the sine of the same angle. By considering the preceding phenomena it will appear, that whenever the plane of polarization of a polarized ray, whether ordinary or extraordinary, coincides with or is parallel to the principal section, the ray will be refracted ordinarily ; and whenever the plane of polarization is perpendicular to the principal section, it will be refracted extraordinarily. In all intermediate positions it will sutler both kinds of refraction, and will be doubly refracted; the ordinary pencil being the brightest if the plane of polarization is nearer the position of parallelism than that of perpendicularity, and the extraordi- nary peneil the brightest if the plane of polarization is nearer the position of perpendicularity than that of parallelism. At equal distances from both these positions, the ordinary and ex- traordinary images are equally bright. 142 A TREATISE ON OPTICS. PABT II. (100.) It does not appear from the preceding experiments that the polarization of the two pencils is the eflect of any po- larizing force resident in the Iceland spar, or of any change produced upon the light. The Iceland spar has merely sepa- rated the common light into its two elements, according to a different law, in the same manner as a prism separates all the seven colors of the spectrum from the compound white beam by its power of refracting these elementary colors in different degrees. The re-union of the two oppositely po- larized pencils produces common light, in the same manner as the re-union of all the seven colors produces white light. The method of producing polarized light by double refrac- tion is of all others the best, as we can procure by this means from a given pencil of light a stronger polarized beam than in any other way. Through a thickness of three inches of Ice- land spar we can obtain two separate beams of polarized light one third of an inch in diameter ; and each of these beams contains half the light of the original beam, excepting the small quantity of light lost by reflexion and absorption. By sticking a black wafer on the spar opposite either of these beams, we can procure a polarized beam with its plane of po- larization either in the principal section or at right angles to it. In all experiments on this subject, the reader should re- collect that every beam of polarized light, whether it is pro- duced by the ordinary or the extraordinary refraction, or by positive or negative crystals, has always the same properties, provided the plane of its polarization has the same direction. CHAP. XIX* ON THE POLARIZATION OF LIMIT BY REFLEXION. (101.) In the year 1810, the celebrated French philosopher M. Mains, while looking through a prism of calcareous spar at the light of the setting sun reflected from the windows of the Luxembourg palace in Paris, was led to the curious dis- covery, that a beam of light reflected from glass at, an angle of 56°, or from water at an angle of 53°, possessed tin' very same properties as one of the lays formed by a. rhomb of cal careous spar; that is, that it was wholly polarized, having its plane of polarization coincident with or parallel to the plane of reflexion. * For the formula relating to this chapter, see (in the College edition,) Appendix' of Am. ed., Chap. VI. CHAP. XIX. POLARIZATION BY REFLEXION. 143 This most curious and important fact, which he found to be true when the light was reflected from all oilier transparent or opaque bodies, excepting metals, gave birth to all those dis- coveries which have, in our own day, rendered this branch of knowledge one of the most interesting, as well as one of the most perfect, of the physical sciences. lu order to explain this and the other discoveries of Malus, let C D, Jig. 87., be a tube of brass or wood, having at one end of it a plate of glass, A, not quicksilvered, and capable of Fig. 87. turning round an axis, so that it may form different angles with the axis of the tube. Let D G be a similar tube a little smaller than the other, and carrying a similar plate of glass B. If the tube D G is pushed into C 1), we may, by turning the one or the other round, place the two glass plates in any po- sition in relation to one another. Let a beam of light, 11 r, from a candle or a hole in the window-shutter, fall upon the glass plate A, at an angle of 56° 4-7 ; and let the glass be so placed that the reflected ray r s may pass along the axis of the two tubes, and fall upon the second plate of glass B at the point s. If the ray r s falls upon the second plate B at an angle of 56° 45' also, and if the plane of reflexion from this plate, or the plane passing through a- E and s r, is at right angles to the plane of reflexion from the first plate, or the plane passing through r 11, r s, the ray r s will not suffer reflexion from B, or will be so faint as to be scarcely visible. The very same thing will happen if r s is a ray polarized by double refraction, and having its plane of po- larization in the plane passing through r R, r s. Here then we have a new property or test of polarized light, — that it will not suffer reflexion from a plate of glass B, when incident at an angle of 56° 45', and when the plane of incidence or re- flexion is at right Jingles to the plane of polarization of the ray. If we now turn round the tube D G with the plate B, ^ • 0^»^ fM ) im mmfm 144 A TREATISE ON OPTICS. TART II. without moving the tube C D, the last reflected ray «E will become brighter and brighter till the tube has been turned round 90°, when the plane of reflexion from B is coincident with or parallel to that from A. In this position the reflected ray s E is brightest. JJy continuing to turn the tube D G, the ray s E becomes fainter and fainter, till, after being turned 90° farther, the ray s E is faintest, or nearly vanishes, which hap- pens when the plane of reflexion from 15 is perpendicular to that from A. After a farther rotation of 90°, the ray s E \\ HI recover its greatest brightness; and when, by a still farther rotation of 9U J , the tube I) G and plate 15 arc brought back into their first position, the ray s 1! will again disappear. These effects may be arranged in a table, as follows : — ray •!: reflected 91)0 At angles between 90° and IMP 180° At angles between 1?0° ami <>70° •270° At angles between 270° and 3C0° 3(i0° or OO At angles between 0° and i)0° . 9(10 Scarcely visible Tbe image grows brighter and brighter Brightest The image grows fainter and fainter Scarcely visible The imagegrows brighter and brighter Brightest The image grows fainter and fainter Scarcely visible If we now substitute in place of the ray r s one of the po- larized rays or beams formed by Iceland spar, so that its plane of polarization is in the plane Rr«, it will experience Ihe very same changes as the ray R r does when polarized by re- flexion from A at an angle of 56° 45'. Hence it is manifest, that a ray reflected at 50° 45' from glass has all the properties of polarized light as produced by double refraction. (102.) In the preceding observations, the ray R r is sup- posed to be reflected only from the first surface of the glass ; but Malus found that the light reflected from the second sur- face of the glass was polarized at the same time with that re- flected from the first, although it obviously suffers reflexion at a different angle, viz. at an angle equal to the angle of refrac- tion at the first surface. The angle of 56° 45', at which light is polarized by re- flexion from glass, is called its maximum polarizing angle, be- cause the greatest quantity of light is polarized at that angle. When the light was reflected at angles greater or less than 56° 45', Malus found that a portion of it only was polarized, the remaining portion possessing all the properties of common light. The polarized portion diminished as the angle of inci- dence receded on either side from 56° 45', and was nothing at 0°, or a perpendicular incidence, and also nothing at 90°, or the most oblique incidence. CHAP. XIX. POLARIZATION BY REFLEXION. 145 In continuing his experiments on this subject, Malus found that the angle of maximum polarization varied with different bodies ; an^ after measuring it in various substances, he con- cluded that it follows neither the order of the refractive powers nor that of the dispersive, powers, but that it is a prop- erty of bodies independent of the other modes of action which they exercise upon light. After he had determined the angles under which complete polarization takes place in different bodies, such as glass and water, he endeavored to ascertain the angle at which it took place at their separating surfaces when they were put in contact. In this inquiry, however, he did not succeed ; and he remarks, " that the law according to which this last angle depends on the first two remains to be determined." If a pencil or beam of light reflected at the maximum po- larizing angle from glass and other bodies were as completely polarized as a pencil polarized by double refraction, then the two pencils would have been equally invisible when reflected from the second plate, B, at the azimuths 90° and 270° ; but this is not the case : the pencil polarized by double refraction vanishes entirely when it passes through a second rhomb, even if it is a beam of the sun's direct light ; whereas the pencil polarized by reflexion vanishes only if its light is faint, and if the plates A and B have a low dispersive power. When the sun's light is used, there is a large quantity of unpoWized light, and this unpolarized light is greatly increased when the plates A and B have a high dispersive power. This curious and most important fact was not observed by Malus. A very pleasing 1 and instructive variation of the general ex- periment shown in jig. 87. occurred to me in examining this subject. If, when the plates of glass A and B have the position Bhown in the figure where the luminous body from which the ray 8 E proceeds is invisible, we breathe gently upon the plate B, the ray s E will be recovered, and the luminous body from which it proceeds will be instantly visible. The cause of this is obvious : a thin film of water is deposited upon the glass by breathing, and as water polarizes light at an angle of about b'S° 11', the glass 15 Bhould have been inclined at an angle of 53° 11' to the my r s, in order to be incapable of retlecting the polarized ray ;* but as it is inclined 50° 45' to the incident ray rs, it has the power of reflecting a portion of the ray r s. If the glass B is* now placed at an angle of 5-'3° 11' to the ray r s, it will then reflect a portion of the polarized ray r s to * We neglect the consideration of the separating surface of the water and glass, and suppose the glass B to he opaque. N I 146 A TREATISE ON OITICS. PART II. the eye at E ; but if we breathe upon the glass B, the re- flected light will disappear, because the reflecting' surface is now water, and is placed at an angle of 53° 11', the polarizing angle for water. If therefore we place two glass plates at B, the one inclined 56° 45', and the other 53° 11', to the beam r s, sufficiently large to fall upon both, the luminous object will be visible in the one but not in the other ; but if we breathe upon the two plates, we shall exhibit the paradox of reviving an invisible image, and extinguishing a visible one by the same breath. This experiment will be more striking if the ray r s is polarized by double refraction. On the Law of the Polarization of Light by Reflexion. (103.) From a very extensive series of experiments made to determine the maximum polarizing angles of various bodies, both solid and fluid, I was led, in 1814, to the following simple law of the phenomena : — The index of refraction is the tangent of the angle of po- larization. In order to explain this law, and to show how to find the polarizing angle for any body whose index of refraction is known, let M N be the surface of any transparent body, such as water. From any point, r, draw r A perpendicular to M N, Fig. 88. fig. 88., and round r as a centre de- scribe a circle, M AND. From A draw A F, touching the circle at A, and from any scale on which A r is 1 or 10 set off A F equal to 1-336 or 13-36, the index of refraction for water. From F draw F r, which will be the incident ray that will be polarized by reflexion from the water in the direc- tion r S. The angle A r R will be 53° 11', or the angle of maximum polarization for water. This angle may be obtained more readily by looking for 1-336 in the column of natural tangents in a book of logarithms, and there will be found opposite to it the corresponding angle of 53° 11'. If we calculate the angle of refraction T r D, corresponding to the angle of incidence A r R, or determine it by projection, we shall find it to be 36° 49'. From the preceding law we may draw the following con- clusions : — , 1. The maximum polarizing angle, for all substances what- ever, is the complement of the angle of refraction. Thus, in water, the complement of 36° 49' is 53° 11', the polarizing angle. CIIAI*. XIX. rOiARIZATION BY REFLEXION. 147 2. At the polarizing angle, the sum of the angles of inci- dence and n fraction is a right angle, or OCR Thus, in water, the angle of incidence is 53° 11', and that of refraction 30° 49', and their sum is 90°. 3. When a ray of light, R r, is polarized by reflexion, the re- flected ray, r S, tortus a right angle with the refracted ray, r T. When light is reflected at the second surface of bodies, the law of polarization is as follows: — The index of refraction is the cotangent of the angle of polarization. In order to determine the angle in this case, let M N be the second surface of any body such as water. From r draw r A perpendicular to M N, fig. 89., and round r describe the circle M AND. From A draw A F, touching the circle at A, and upon a scale in which r N is 1 take A F equal to -7485, that is to t.ttb B jv the reciprocal of the index of refraction,* and from F draw Fr ; the ray Rr will be polarized when reflected in the direction r S. The maximum polarizing angle A r R will be 36° 49', exactly equal to the angle of refraction of the first surface. Hence it follows, 1. That the polarizing angle at the second surface of bodies is equal to the complement of the polarizing angle at the first, or to the angle of refraction at the first surface. The reason is, therefore, obvious why the portions of a beam of light re- elected at the first and second surfaces of a traiisparent parallel plate are simultaneously polarized. 2. That the angle which the reflected ray r S forms with the refracted ray r T is a right angle. The laws of polarization now explained are applicable to the separating surfaces of two media of different refractive powers. If the uppermost fluid is water, and the undermost -glass, then the index of refraction of their separating surface is equal to J:' ', to the greater index divided by the lesser, which is 11 il~). l>y using this index it will be found that the polarizing angle is l* : '17'. When the ray moves from the less refractive substance into the greater, aa from water to glass, as in the preceding case, we must make use of the law and the method above ex- plained tor the first surface of Ixxlics; but. when the ray moves from the greater refractive body into the less, as from oil of cassia to glass, we must use the law and method for tlie second surface of bodies. » The tangent of an angle to radius 1, is the reciprocal of the cotangent. 148 A TREATISE ON OPTICS. TART II If we lay a parallel stratum of water upon glass whose index of refraction is 1 -508, the ray reflected from the refract-' ing surfaces will be polarized when the angle of incidence upon the first surface of tiie water is 90° (104.) The preceding observations are all applicable to white light, or to the mo.-i luminous rays of the spectrum; but, as every different color has a different index of refraction, the law enables us to determine the angle of polarization for every diflerent color, as in the following table, where it is supposed that the most luminous ray of the spectrum is the mean one :- - Water Plate Glass Oil of Cassia lied rays Mean rays Violet rays Red rays Mean rays Violet rays Red rays Mean rays Violet rays 1-330 1-336 1-342 1-51 1-52 1-535 1-59 1-642 1-687 53° 4' 53 11 19 31 -If) 55 57 40 - 21 4 L° 15' 21' 24' The circumstance of the different rays of the spectrum being polarized at diflerent angles, enables us to explain the existence of unpolarized light at the maximum polarizing angle, or why the ray s E, in Jig. 87., never wholly vanishes. If we were to use red light, and set the two plates at angles of 50° 34', the polarizing angle of glass for red light, then the pen- cil s E would vanish entirely. But when the light is white, and the angle at which the plates are set is 56° 45', or that which belongs to mean or yellow rays, then it is only the yellow rays that will vanish in the pencil s E. A small portion of red and a small portion of violet will be reflected, because the glasses are not set at their polarizing angles ; and the mixture of these two colors will produce a purple color, which will be that of the unpolarized light which remains in the pencil s E. If we place the plates at the angle belonging to the ral ray, then the red only will vanish, and the color of the unpo- larized light will be bluish green. If we place the plates at the angle corresponding with the blue light, then the blue only will vanish, and the unpolarized light will be of a reddish cast. In oil of cassia, diamond, chromate of lead, realgar, specidar iron, and other highly dispersive substances, the coloi of the unpolarized light is extremely brilliant and beautiful Certain doubly refracting crystals, such as Iceland spar CHAP. xix. PARTIAL POLARIZATION. 149 ckromate of lend, Sic., have different polarizing angles on dif- ferent surfaces, and in different directions on the same sur- face ; but there is always one direction where the polariza- tion is not affected by the doubly refracting force, or where the tangent of the polarizing angle is equal to the index of ordinary refraction. On the partial Polarization of Light by Reflexion. (105.) If, in the apparatus in fig. 87., we make the ray R r fall upon the plate A at an angle greater or less than 56° 45', then the ray s E will not vanish entirely ; but, as a consider- able part of it will vanish like polarized light, Malus called it ■partially polarized light, and considered it as composed of a portion of light perfectly polarized, and of another portion in the state of common light. He found the quantity of polar- ized light to diminish as tiie angle of incidence receded from that of maximum polarization. M. Biot and M. Arago also maintained that partially polar- ized light consisted partly of polarized and partly of common light ; and the latter announced that, at regular angular dis- tances above and below the maximum polarizing angle, the reflected pencil contained the same proportion of polarized light. In St. GobiiVs glass he found that the same proportion of light was polarized at an angle of incidence of 82° 48' as at 24° 18' ; in water he found that the same proportion was polarized at 16° 12' as at 86° 31' ; but he remarks, " that the mathematical law which connects the value of the quantity of polarized light with the angle of incidence and the refractive power of the body has not yet been discovered." In the investigation of this subject, I found that though there was only one angle at which light could be completely polarized by one reflexion, yet it might be polarized at any angle of incidence by a sufficient number of reflexions, as shown in the following Table. BELOW THE F0LARI7.IN8 ANGLE. A HOVE THE POLARIZING ANGLE. No. of il which tin Light No. of Angle at which thi' Litfht Reflc-xlunn. la polari Reflaxlona. ik ponvliid, 1 56° 45' 1 56° 45' 2 50 26 2 6:2 30 3 46 SO 3 65 33 4 43 51 4 67 33 5 41 43 5 6.9 1 6 40 6 70 9 7 38 33 7 71 5 8 37 20 8 71 51 N2 150 A TREATISE ON OPTICS. l'ART II. In polarizing light by successive reflexions, it is not neces- sary that the reflexions be performed at the some angle. Some of them may be above and some below the polarizing angle, or all the reflexions may be performed at different angles. From the preceding facts it follows as a necessary conse- quence, that partially polarized light, or light reflected at an angle different from the polarizing angle, has suffered a physi- cal change, which enables it to be more easily polarized by a subsequent reflexion. The light, tor example, which remains unpolarized after five reflexions at 70°, in place of being com- mon light, has suffered such a physical change that it is capa- ble of being completely polarized by one reflexion more at 70°. This view of the subject has been rejected by M. Arago, as incompatible with experiments and speculations of his own ; and, in estimating the value of the two opinions, Mr. Herschel has rejected mine as the least probable. It will be seen, how- ever, from the following facts, that it is capable of the most rigorous demonstration. It- does not appear, from the preceding inquiries, how a beam of common light is converted into polarized light by reflexion. By a series of experiments made in 1829, I have been able to remove this difficulty. It has been long known that a polar- ized beam of light has its plane of polarization changed by re- flexion from bodies. If its plane is inclined 45° to the plane of reflexion, its inclination will be diminished by a reflexion at 80°, still more by one at 70°, still more by one at 60° ; and at the polarizing angle the plane of the polarized ray will be in the plane of reflexion, the inclination commencing again at reflexions above the polarizing angle, and increasing till at 0°, or a perpendicular incidence, the inclination is again 45°.* I now conceived a beam of common light, constituted as in Jig. 81., to be incident on a reflecting surface, so that the plane of reflexion bisected the angle of 90° which the two planes of polarization, A B, C D, formed with each other, as shown in Jig. IK)., No. 1., where M N is the plane of reflexion, and A B, C D the planes of polarization of the beam of white light, each inclined 45° to M N. By a reflexion from glass, where the index of refraction is 1-525, at 80°, the inclination of A B to M N will be 33° 13', as in No. 2., instead of 45° ; and in like manner the inclination of C D to M N will bo 33° 13', in place of 45° ; so that the inclination of A B to C D in * The rule for finding the inclination is this : — Find the sum of the angles of incidence and refraction, and also their difference ; divide the cosine of the former by the cosine of the latter, and the quotient will be the tangent of the inclination required. CHAP. XIX. PARTIAL POLARIZATION. 151 place of 90° is 00° 26', as in No. 2. At an incidence of 65° tin: inclination of A B to C i) will be 25° 30', as in No. 3. ; and at the polarizing angle of 50° 45' the planes A B, C D of the two beams will be parallel or coincident, as in No. 4. At incidences below 56° 45' the planes will again open, and their Fig. 90. No. 4. AC inclination will increase till at 0° of incidence it is 90°, as in No. 1., having been 25° 36' at an incidence of about 48° 15', as in No. 3., and 66° 26' at an incidence of about 30°, as in No. 2. In the process now described, we see the manner in which common light, as in No. 1., is converted into polarized light, as in No. 4., by the action of a reflecting surface. Each of the two planes of its component polarized beams is turned round into a state of parallelism, so as to be a beam with only- one plane of polarization, as in No. 4. ; a mode of polariza- tion essentially different in its nature from that of double re- fraction. The numbers in fig. 90. present us with beams of light in different stages of polarization from common light in No. 1. to polarized light in No. 4. In No. 2. the beam has made a certain approach to polarization, having suffered a physical change in the inclination ot its planes; and in No. 3. it has made a nearer approach to it. Hence we discover the whole mystery of partial polarization, and we see that par- tially polariztd light is light whose planes of polarization are inclined at angles less than 90° and greater than 0°. The influence of successive reflexions is therefore obvious. A reflexion at 8(Q° will turn the planes, as in fig. 90., No. 2. ; another reflex ion at 80° will bring them closer ; a third still closer ; and so 1 &n : and though they never can by this process be brought ir.'to a state of exact parallelism, as in No. 4. (which can on ly be done at the polarizing i ngle), yet they can be brought infinitely near it, so that tin beam will appear as completely polarized as if it had been reflected at the polar- izing angle. The correctness of my former experiments and views is, therefore, demonstrated by the preceding analysis of common light. It is manifest from these views that partially polarized light ^m 152 A TREATISE ON OPTICS. PAKT II. does not contain a single ray of completely polarized light ; and yet if we reflect it from the second plate B, in fig. 87., at the polarizing angle, a certain portion of it will disappear as if it were polarized light, a result which led to the mistake of Mains and others. The light which thus disappears may be called apparently polarized light ; and I have explained in another place* how we may determine its quantity at any angle of incidence, and for any refractive medium. The fol- lowing Table contains some of the results for glass, whose in- dex of refraction is 1-525. The quantity of reflected light is calculated by a rule given by M. Fresnel. Angles of Inclination of the Planes Quantity of reflected Quantity of polarized Incidence. C D, fis ■ BO. Ray* out of looo. Rays out «jf 1000. 0° 90° 0' 43-23 0- 20 80 26 43-41 7-22 40 47 22 49-10 33-25 56 45' 79-5 79-5 70 37 41 162-67 129-8 80 66 26 391-7 156-6 85 78 24 616-28 123-75 90 90 1000- 0- CHAP. XX. ON THE POLARIZATION OF LIGHT BY ORDINARY REFRACTION. (106.) Although it might have been presumed that the lio-ht refracted by bodies suffered some change, corresponding to that which it receives from reflexion, yet it was not until 1811 that it was discovered that the refracted portion of the beam contained a portion of polarized light.f To explain this property of light, let R r, fig. 91., be a beam of light incident at a great angle, between 80° and 90°, on a horizontal plate of glass, No. 1. ; a portion of it will be reflected at its two surfaces, r and a, and the refracted beam a is found to contain a small portion of polarized light. If this beam a falls upon a second plate, No. 2., parallel to the first, it will suffer two reflexions; and the refracted pencil b will contain more polarized light than a. In like manner, by transmitting it through the plates Nos. 3, 4, 5, and ♦See Phil. Transactions, 1830, p. 76., or Edinburgh Journal of Science, New Series, No. V., p. 160. t This discovery was made by independent observation by Malus, Biot, and the author of this work. CFIAP. XX. POLARIZATION BY REFRACTION. 153 0., the last refracted pencil, fg, will be found to consist entirely, so fa: as the eye can judge, of polarized light. But, what is very interesting, the beam fg is not polarized in the plane of refraction or reflexion, but in a plane at right angles to it; that is, its plane of polarization is not represented by A' B' Fig. 91. Jig: 81., as is the ordinary ray in Iceland spar, or as light polarized by reflexion, but by C D' like the extraordinary ray in Iceland spar. From a great number of experiments, I tound that the light of a wax candle at the distance of 10 or 12 feet was polarized at the following angles, by the fol- lowing number of plates of crown glass. No. of Plates of Crown Ulass. Observed \ugWa ai win. h Hi.- Pencil is polarised. No. of Plates of Crown (iIukm. ObserVfii ktkg)M at which the Pencil is polarized. 8 12 16 21 24 79° 11' 74 69 4 63 21 60 8 27 31 35 41 47 57° 10' 53 28 50 5 45 35 41 41 ^ It follows from the above experiments, that if we divide the number 41 84 by any number of crown glass plates, we shall have the tangent of the angle at which the beam is polarized,, by that number. Hence it is obvious that the power of polarizing the re- fracted light increases with the angle of incidence, being no- thing or a minimum at a perpendicular incidence, or 0°, and the greatest possible or a maximum at 90° of incidence. I found, likewise, by various experiments, that the power of po- larizing the light at any given angle increased with the re- fractive power of the body, and consequently that a smaller number of plates of a highly refracting body was necessary than of a refracting body of low power, the angle of incidence being the same. As Malus, Biot, and Arago considered the beams a, b, &c, before they were completely polarized, as partially polarized, /mW ' x m j tm v mirmn***** ■ -■'-■■••■• ir»4 A TREATISE ON Ol'TICS. I'ART II. 0.11(1 as consisting of a portion of polarized and a portion of unpolarized light; so, on the other hand, I concluded from the following reasoning that the unpolarized light had Buffered a physical change, which made it approach to the state of com- plete polarization. For since sixteen plates are required to polarize completely a beam of light incident at an angle of 69°, it is clear that eight plates will not polarize the whole beam at the same angle, but will leave a portion unpolarized. Now, if this portion were absolutely unpolarized like common light, it would require to pass through other sixteen plates, at an angle of 69°, in order to be completely polarized ; but the truth is, that it requires to pass through only eight plates to be completely polarized. Hence I conclude that the beam has been nearly half polarized by the first eight plates, and the polarization completed by the other eight. This conclusion, though rejected by both the French and English philosophers, is capable of rigid demonstration, as will appear from the fol- lowing observations. In order to determine the change which refraction produced in the plane of polarization of a polarized ray, 1 used prisms and plates of glass, plates of water, and a plate of a highly re- fractive metalline glass ; and I found that a refracting surface produced the greatest change at the most oblique incidence, or that of 90° ; and that the change gradually diminished to a perpendicular incidence, or 0°, where it was nothing. I found also that the greatest effect produced by a single plate of glass was about 16° 39', at an angle of 80° ; that it was 3° 54' at an angle of 55°, 1° 12' at an angle of 35°, and 0° at an angle of 0°.* A beam of common light, therefore, constituted as in Jig. 92., No. 1., with each of its planes A B, C D inclined 45° to No. 2 Fig. 92. No. 4. N the plane of refraction, will have these planes opened 16° 39' ♦The rule for finding the inclination after a single refraction is as fol- lows: — Find the difference between the angles of incidence and refraction, and take the cosine of this difference. This number will be the cotangent of the inclination required ; and twice this inclination will be the inclina- tion of A 15 to C D. CHAP. XX. TOLARIZATION BY REFRACTION. 155 each, by one plate of glass at an incidence of 8G° ; that is, their inclination, in place of 90°, will be 123° 18', as in No. 2. By the action of two or three plates more they will be opened wider, as in No. 3. ; and by 7 or 8 plates they will be opened to near 180°, or so that A B, C D nearly coincide, as in No. 4., so as to form a single polarized beam, whose plane of polarization is perpendicular to the plane of refraction. I have shown, in another place,* that these planes can never be brought into mathematical coincidence by any number of re- fractions ; but they approach so near to it that the pencil is, to all appearance, completely polarized with lights of ordinary strength. All the light polarized by refraction is only par- tially polarized, and it has the same properties as that which is partially polarized by reflexion. A certain portion of the light of a beam thus partially polarized, will disappear when reflected at the polarizing angle from the plate B, fig. 87. ; and this quantity, which 1 have elsewhere shown how to cal- culate, is given in the following table for a single surface of glass, whose index of refraction is 1-525. >S„c'. Incllantioll of the PUnea of I oiarlsationA D,UD, Jig W. Uuauiiiy of transmitted Kays nut ut 1000. Quantity of pnlnriied Kitya out r,f 1000. u° 90° 0' 95677 20 90 26 956-59 7-22 40 92 950-90 32-2 56° 45' 94 58 920-5 79-5 70 98 56 837-33 129-8 80 40' 104 55 608-3 156-7 85 108 44 383-72 123-7 90 112 58 Although the quantity of light polarized by refraction, as given in the last column of this Table, is calculated by a formula essentially different from that by which the quantity of light polarized by reflexion was calculated ; yet it is cu- rious to see that the two quantities are precisely equal. Hence we obtain the following law : — When a ray of common light is reflected and refracted by any surface, the quantity of light polarized by refraction is exactly equal to that polarized by reflexion. This law is not at all applicable to plates, as it appeared to be from the experiments of M. Arago. When the preceding metbod of analysis is applied to the light reflected by the second surfaces of plates, we obtain the following curious law : — * See Phil. Transactions, 1830, p. 137., or Edinburgh, Journal of Science, New Scries, No. VI., p. 218. — b MHMipijHlMBHpMMjMpM 156 A TREATISE ON OPTICS. PART II, A pencil of light reflected from the second surfaces of transparent plates, and reaching the eye after two refrac~ lions and an intermediate reflexion, contains at all angles of incidence, from 0° to the maximum polarizing angle, a por- tion of light polarized in the plane of reflexion. Above the polarizing angle, the part of the pencil polarized in the plane of reflexion diminishes, till the incidence becomes 78° 7' in glass, when it disappears, and the whole pencil has the character of common light. Above this last angle the pencil contains a quantity of light polarized perpendicularly to the plane of reflexion, which increases to a maximum, and then diminishes to nothing at 90°.* (107.) As a bundle of glass plates acts upon light, and po- larizes it as effectually as reflexion from the surface of glass at the polarizing angle, we may substitute a bundle of glass plates in the apparatus, fig. 87., in place of the plates of glass A, B. Thus, if A {Jig. 93.) is a bundle of glass plates which Fig. 93. polarizes the transmitted ray s t, then, if the second bundle B is placed as in the figure, with the planes of refraction of its plates parallel to the planes of refraction of the plates of A, the ray s t will penetrate the second bundle ; and if s t is in- cident on B at the polarizing angle, not a ray of it will be re- flected by the plates of B. If B is now turned round its axis, the transmitted light v w will gradually diminish, and more and more light will be reflected by the plates of the bundle, till, after a rotation of 90°, the ray v w will disappear, and all the light will be reflected. By continuing to turn round B, the ray v w will re-appear, and reach its maximum brightness at 180°, its minimum at 270°, and its maximum at 0°, after having made one complete revolution. By this apparatus we may perform the very same experi- ments with refracted polarized light that we did with reflected polarized light in the apparatus of fig. 87. We have now described two methods .of converting com- mon light into polarized light : 1st, By separating by double refraction the two oppositely polarized beams which constitute common light ; and, 2dly, By turning round, by the action of * See Phil. Trans. 1830, p. 145. ; or Edinburgh Journal of Science, No. VI., p. 234. New Series. CHAP. XXI. COLORS OF CRYSTALLIZED PLATES. 157 the reflecting and refracting forces, the planes of both these beams till they coincide, and thus form light polarized in one plane. Another method still remains to be noticed ; namely, to disperse or absorb one of the oppositely polarized beams which constitute common light, and leave the other beam po- larized in one plane. These effects may be produced by agate and tourmaline, &c. (108.) If we transmit a beam of common light through a plate of agate, one of the oppositely polarized beams will be converted into a nebulous light in one position, and the other polarized beam in another position, so that one of the polar- ized beams with a single plane of polarization is left. The same effect may be produced by Iceland spar, arragonite, and artificial salts prepared in a particular manner, to produce a dispersion of one of the oppositely polarized beams.* When we transmit common light through a thin plate of tourmaline, one of the oppositely polarized beams which con- stitute common light is entirely absorbed in one position, and the other in another position, one of them always remaining with a single plane of polarization. Hence, plates of agate and tourmaline are of great use, either in affording a beam of light polarized in one plane, or in dispersing and absorbing one of the pencils of a compound beam, when we wish to analyze it, or to examine the color or properties of one of the pencils seen separately. CHAP. XXI. ON THE COLORS OF CRYSTALLIZED PLATES IN POLARIZED LIGHT. (109.) The splendid colors, and systems of colored rings, produced by transmitting polarized light through transparent bodies that possess double refraction, are undoubtedly the most brilliant phenomena that can be exhibited. The colors pro- duced by these bodies were first discovered by independent observation, by M. Arago and the author of this volume ; and they have been studied with great success by M. 13iot and other authors. In order to exhibit these phenomena, let a polarizing ap- paratus be prepared, similar in its nature to that in fig. 87. ; but without the tubes, as shown in Jig. 94., where A is a plate * See Edinburgh Encyclopwdia, vol. xv. pp. 000, 001. p. 140. o Phil. Trans. 1819, p 158 A TREATISE ON OPTICS. PART II. of glass which polarizes the ray R r, incident upon it at an angle of 56° 45', and reflects it polarized in the direction r s, where it is received by a second plate of glass, J?, whose plane of reflexion is at right angles to that of the plate A, and which reflects it to the eye at O, at an angle of 56° 45'. In Fig. 94. order that the polarized pencil r s may be sufficiently brilliant, ten or twelve plates of window glass, or, what is better still, of thin and well-annealed flint glass, should be substi- tuted in place of the single plate A. The plate or plates at A are called the polarizing plates, because their only use is to furnish us with a broad and bright beam of polarized light. The plate B is called the analyzing plate, because its use is to analyze, or separate into its parts, the light transmitted through any body that may be placed between the eye and the polarizing plate. If the beam of light R r proceeds from the sky, which will answer well enough for common purposes, then an eye placed at O will see, in the direction O s, the part of the sky from which the beam R r proceeds. But as r s will be polarized light if it is reflected at 56° 45' from A, almost none of it will be reflected to the eye at O from the plate B ; that is, the eye at O will see, upon the part of the sky from which R r pro- ceeds, a black spot ; and when itvdocs not see this black spot, it is a proof that the plates A and B are not placed at the proper inclinations to each other. When a position is found, either by moving A or B, or both, at which the black spot is darkest, the apparatus is properly adjusted. (110.) Having procured a thin film of sulphate of lime or mica, between the 20th and the 60th of an inch thick, and which may be split by a fine knife or lancet from a mass of any of these minerals in a transparent state, expose it, as shown at C E D F, so that the polarized beam r s may pass through it perpendicularly. If we now apply the eye at O, and look towards the black spot in the direction O s, we shall see the surface of the plate of sulphate of lime entirely cov- ered with the most brilliant colors. If its thickness is per- CHAP. X\r. COLONS OF CRYSTALLIZED PLATES. 159 feotly uniform throughout, its tint will be perfectly uniform; but if it has different thicknesses, every different thickness will display a different color — some red, some green, some blue, and some yellow, and all of the most brilliant descrip- tion. If we turn the film C E D F round, keeping it perpen- dicular to the polarized beam, the colors will become less or more bright without changing their nature, and two lines, CD, EF at right angles will be found, so that when either of them is in the plane of reflexion rsO,no colors whatever are perceived, and the black spot will be seen as if the sulphate of lime had not been interposed, or as if a piece of common glass had been substituted for it. It will also be observed, by continuing the rotation of the sulphate of lime, that the colors again begin to appear ; and reach their greatest brightness when either of the lines G H, L K, which are inclined 45° to C D, E F, are in the plane of reflexion r s O. The plane Rr s, or the plane in which the light is polarized, is called the plane of primitive polarization ; the lines CD, E F, the neutral axes ; and G 11, K L, the depolarizing axes, because they de- polarize, or change the polarization of the polarized beam r s. The brilliancy or intensity of the colors increases gradually, from the position of no color, to that in which it is the most brilliant. Let us now suppose the plate C E D F to be fixed in the po- sition where it gives the brightest color ; namely, when G H is perpendicular to the plane of primitive polarization Rr s, or parallel to the plane r s O, and let the color be red. Let the analyzing plate L be made to revolve round the ray r s, beginning its motion at 0°, and preserving always the same inclination to the ray r s, viz. 56° 45'. The brightest red being now visible at 0°, when the plate B begins to move from its position shown in the figure, its brightness will gradually diminish till B has turned round 45°, when the red color will wholly disappear, and the black spot in the sky be seen. Be- yond 45° a faint green will make its appearance, and will be- come brighter and brighter till it attains its greatest bright- ness at. 90°. Beyond 90° the green becomes paler and paler till it disappears at 135°. Here the red again appears, and reaches its maximum brightness at 180°. The very same changes are repeated while the plate B passes from 180° round to its first position at 360° or 0°. From this experi- ment it appears, that when the film C E D F alone revolves, only one color is seen ; and when the plate B only revolves, two colors are seen during each half of its revolution. If we repeat the preceding experiment with films of differ- ent thicknesses, that give different colors, we shall find that the m$3ESS3gg£XSm | ■ .-:■-:•■--- A TREATISE ON OPTICS. TART II. 1G0 two colors arc always complementary to each other, or to- gether make white light. (111.) In order to understand the cause of these beautiful phenomena, let the eye be placed between the film and the plate B, and it will be seen that the light transmitted through the film is white, whatever be the position of the film. The separation of the colors is therefore produced, or the white light is analyzed, by reflexion from the plate B. Now, sul- phate of lime is a doubly refracting crystal ; and one of its neutral axes, C D, is the section of a plane passing through its axis, while E F is the section of a plane perpendicular to the principal section. Let us now suppose either of these planes, for example E F, to be placed, as in the figure, in the plane of po- larization R r s of the polarized light; then this ray will not be doubled, but will pass into the ordinary ray of the crystallized film ; and falling upon B, it will not suffer reflexion. In like manner, if C D is brought into the plane Rr s, it will pass entirely into the ordinary ray, which, falling upon B, will not suffer reflexion. In these two positions of the film, therefore, it forms only a single image or beam ; and as the plane of po- larization of this image or beam is at right angles to the plane of reflexion from B, none of it is reflected to the eye at O. But in every other position of the doubly refracting film C E D F, it forms two images of different intensities, as may be inferred from fig. 86. ; and when cither of the depolarizing axes G H or K L is in the plane of primitive polarization, the two images are of equal brightness, and are polarized in op- posite planes ; one in the plane of primitive polarization, and the other at right angles to it. Now, one of these images is red, and the other green, for reasons which will be afterwards explained ; and as the green is polarized in the plane of primi- tive polarization R r s, it does not suffer reflexion from the plate B ; while the red, being polarized at right angles to that plane, is reflected to the eye at O, and is therefore alone seen. For a similar reason, when B is turned round 90°, the red will not suffer reflexion from it ; while the green will suffer re- flexion, and be transmitted to the eye at O. In this case the plate B analyzes the compound beam of white light trans- mitted through the film of sulphate of lime, by reflecting the half of it which is polarized in the plane of its reflexion, and refusing to reflect the other half, which is polarized in an op- posite plane. If the two beams had been each white light, as they are in thick plates of sulphate of lime, in place of seeing two different colors during the revolution of the plate B, the reflected pencil s O would have undergone different variations of brightness, according as the two oppositely polarized beams CHAP. XXI. COLORS OF CRYSTALLIZED PLATES. 161 of white light were more or less reflected by it ; the positions of greatest brightness being those wher,e the red and green colors were the brightest, and the darkest points being those where no color was visible. (112.) The analysis of the white beam composed of two beams of red and green light, has obviously been effected by the power of the plate to rijlecl the one and to transmit or refract the other ; but the same beam may be analyzed by va- rious other methods. If we make it pass through a rhomb of calcareous spar sufficiently thick to separate by double refrac- tion the red from the green beam, we shall at the same time see both the colored beams, which we could not do in the for- mer case ; the one forming the ordinary, and the other the ex- traordinary image. Let us now remove the plate B, and sub- stitute for it a rhomb of calcareous spar, with its principal sec- tion in the plane of reflexion r s O, or perpendicular to the plane of primitive polarization R r s, and let the rhomb have a round aperture in the side farthest from the eye, and of such a size that the two images of the aperture, formed by double refraction, may just touch one another. Remove the film CEDF, and the eye placed behind the rhomb will see only the extraordinary image of the aperture, the ordinary one having vanished. Replace the film, with its neutral axes as in the figure, parallel and perpendicular to the plane Rrs, and no effect will be produced ; but if either of the depolarizing axes are brought into the plane R r s, the ordinary image of the aperture will be a brilliant red, and the extraordinary image a brilliant green ; the double refraction of the rhomb having separated these two differently colored and oppositely polarized beams. By turning round the film, the colors will vary in brightness ; but the same image will always have the same color. If we now keep the film fixed in the position that gives the finest colors, and move the rhomb of calcareous spar round, so that its principal section shall make a complete revolution, we shall find that, after revolving 45° from its first position, both images become white. After revolving 90°, the ordinary image that was formerly red is now green, and the extraordinary image that was formerly green is now red. The two images become again white at 135°, 225°, and 315° ; and at 150°, the ordinary image is again red, and the extraordi- nary one green ; and at 270°, the ordinary image is green, and the other red. If we use a large circular aperture on the face of the rhomb, the ordinary and extraordinary images O, E will over- lap each other, as in fig. 95. ; the overlapping parts at FG being pure white light, and the parts at C and D having the 2 162 A TREATISE ON OITICS. colors above described. This experiment affords ocular de- monstration that the two colors at C and D are comple- mentary, and form white light. The analysis of the compound beam transmitted by the sul- phate of lime may also be effected by a plate of agate, or by any of the crystals, artificially prepared for the purpose of dispersing one of the component beams. The agate being placed between the eye and the film CEDF, it will disperse into nebulous light the red beam, and enable the green one to reach the eye ; while in another position it will scatter the green beam, and allow the red light to reach the eye. With a proper piece of agate this experiment is both beautiful and instructive ; as the nebulous light, scattered round the bright imao-e, will be green when the distinct image is red, and red when the distinct image is green. The analysis may also be effected by the absorption of tour- maline and other similar substances. In one position the tour- maline absorbs the green beam, and allows the red to pass ; while in another position it absorbs the red, and suffers the green to pass. The yellow color of the tourmaline, however, is a disadvantage. The analysis may also be performed by a bundle of glass plates, such as A or B, fig. 93. In one position such a bundle will transmit all the red, and reflect all the green; while in another position it will transmit all the green, and reflect all the red, in the opposite manner, but according to the same rules as the analyzing plate B, fig. 94. (113.) Tn all those experiments the thickness of the sul- phate of lime has been supposed such as to give a red and a green tint ; but if we take a film 0-00040 of an English inch thick, and place it at C F D F in fig. 94., it will produce no colors at all, and the black spot in the sky will be seen, what- ever be the position of the film. A film 0*00124 thick will give the white of the first order in Newton's scale of colors, given in p. 93 ; and a plate 0-01818 of an inch thick, and all plates of greater thickness, will give a white composed of all the colors. Films or plates of intermediate thicknesses CHAP. XXI. COLORS OF CRYSTALLIZED PLATES. 163 between 000124 and 0-01818 will give all the intermediate colors in Newton's Table between the white of the first order and the white arising from tbe mixture of all the colors. That is, the colors reflected to the eye at O will be those in column 2d, while the colors observed by turning round the plate E will be those in column 3d ; the one set of colors correspond- ing to the reflected tints, and the other to the transmitted tints of thin plates. In order to determine the thickness of a film of sulphate of lime which gives any particular color in the Table, we must have recourse to the numbers in the last col- umn for glass, which has nearly the same refractive power as sulphate of lime. Suppose it is required to have the thickness which corresponds to the red of the first spectrum or order of colors. The number in the column for glass, opposite red, is 5| ; then, since the white of the first order is produced by a fiim 0-00124 of an inch thick, the number corresponding to which is 3f in the column for glass, we say, as 3f is to 5^, so is 0-00124 to 0-00211, the thickness which will give the red of the first order. In the same mann#r, by having the thick- ness of any film of this substance, we can determine the color which it will produce. Since the colors vary with the thickness of the plate, it is manifest, that if we could form a wedge of sulphate of lime, with its thickness varying from 0-00124 to 0-01818 of an inch, we should observe at once all the colors in Newton's Table in parallel stripes. An experiment of the same kind may be made in the following manner : — Take a plate of sulphate of lime, M N, Jig. 96., whose thickness exceeds 0-01818 of an Fig. 96. inch. Cement it with isinglass on a plate of glass ; and placing it upon a fine lathe, turn out of it with a very sharp tool a concave of hollow surface between A and B, turning it so thin at the centre that it either begins to break or is on the eve of breaking. If the plate M N is now placed in water, the water will after some time dissolve a small portion of its .substance, and polish the turned surface to a certain degree. If the plate is now held at C E D F, Jig. 94., we shall see all the colors in Newton's Table in the form of concentric rings, iwmzmm: 164 A TREATISE CX OPTICS. PAKT II as shown in the figure. If the thickness diminishes rapidly the rings will he closely packed together, but if the turned surface is large, and the thickness diminishes slowly, the col- ored bands will be broad. In place of turning out the con- cavity, it might be done better by grinding it out, by applying a convex surface of great radius, and using the finest emery. When the plate M N is thus prepared, we may give the most perfect polish to the turned surface by cementing upon it a plate of glass with Canada balsam. The balsam will dry, and the plate may be preserved for any length of time. By the method now described, the most beautiful patterns, such as are produced in bank-notes, &c, may be turned upon a plate of sulphate of lime 0-01818 of an inch thick, cemented to glass. All the grooves or lines that compose the pattern may be turned to different depths, so as to leave different thicknesses of the mineral, and the grooves of different depths will all appear as different colors, when the pattern is held in the apparatus in fig. 94. Colored drawings of figures and landscapes may in like manner be executed, by scraping away the mineral to the thickness that will give the required colors; or the effect may be produced by an etching ground, and using water and other fluid solvents of sulphate of lime to reduce the mineral to the required thicknesses. A cipher may thus be executed upon the mineral ; and if we cover the surface upon which it is scratched, or cut, or dissolved, with a balsam or fluid of exactly the same refractive power as the sulphate, it will be absolutely illegible by common light, and mav be distinctly read in polarized light, when placed at C E I) F in Jig. 94. As the colors produced in the preceding experiments vary with the different thicknesses of the body which produces them, it is obvious that two films put together, as they lie in the crystal with similar lines coincident or parallel, will pro- duce a color corresponding to the sum of their thicknesses, and not the color which arises from the mixture of the two colors which they produce separately. Thus, if we take two films of sulphate of lime, one of which gives tiie orauui of the first order, whose number in the last column in Newton's Table, p. 93., is 5^, while the other gives the red of the 2d order, whose number is 11;.' ; then by adding those numbers, we get 17, which corresponds in the Table to greenish yellow of the 3d order. But if the two plates are crossed, so that similar lines in the one are at right angles to similar lines in the other, then the tint or color whieh they produce will he that which belongs to the difference of their thicknesses. Thus, in the present case, the difference of the above numbers is 0^, which CHAP. XXT. COLORS OF CRYSTALLIZED PLATES. 165 corresponds in the Table to a reddish violet of the second oilier, [f the plates which are thus crossed are equally thick, and produce the same colors, they will destroy each other's effects, and blackness will be produced ; the difference of the numbers in the Table being 0. Upon this principle, we may produce colors by crossing plates of such a thickness as to give no colors separately, provided the difference of their thickness does not exceed - 01818 ; for if the difference of their thickness is greater than this, the tint will be white, and beyond the limits of the Table. If the polarized light employed in the preceding experi- ments is homogeneous, then the colors reflected from the plate B will always be those of the homogeneous light employed. In red light, for example, the colors or rather shades which succeed each other, with different thicknesses of the mineral, will be red at one thickness, black at another, red at another, and black at another, and so on with all the different colors. If we place the specimen shown in fit?. 95. in violet light, the rings A B will be less than in red light; and in intermediate colors they will be of intermediate magnitudes, exactly as in the rings of thin plates formerly described. When white light is used, all the different sets of rings are combined in the very same manner as we have already explained, in thin plates of air, and will form by their combinations the various colored ring's in Newton's Table. CHAP. XXII. ON THE SYSTEM OF COLORED RINGS IN CRYSTALS WITH ONE AXIS. (114.) Tn all the preceding experiments the film CEDF must be held at such a distance from the eye, or from the plate B, that its surface may be distinctly seen, and in the apparatus used by different philosophers this distance was considerable. In the year 1813 I adopted another method, namely, that of bringing the film or crystal to be examined as close to the eye as possible, a very small plate, B, not above one fourth of an inch, being interposed, as in Jig. 94., between the crystal and the eye, to reflect the light transmitted through the crystal. By this means I discovered the systems of rings formed along the axes of crystals with one and two axes, which form the most splendid phenomena in optical science, and which by their analysis have led philosophers to the most important dis- coveries, f *!9***W Fig. 97. 1G6 A TREATISE ON OrTICS. PART II. I discovered them in ruby, emerald, topaz, ice, nitre, and a great variety of other bodies, and Dr. Wollaston afterwards observed them in Iceland spar. In order to observe the system of rings round a single axis of double refraction, grind down the summits or obtuse angles A X of a rhomb of Iceland spar, fig. 72., and replace them by plane and polished surfaces perpendicular to the axis of double refraction A X. But as this is not an easy operation without the aid of a lapidary, I have adopted the following method, which enables us to transmit light along the axis A X without injuring the rhomb. Let C D E F, Jig. 97., be the principal section of the rhomb; cement upon its surfaces C D, F E, with Canada balsam, two prisms, DLK, FG II, having the angles LDK, G F H each equal to about 45° ; and by let- ting fall a ray of light perpendicu- larly upon the face D L, it will pass ^E along the axis A X, and emerge per- pendicularly through the face F G. Let the rhomb thus prepared be held in the polarized beam r s,Jig. 94., so that r s may pass along the axis A X, and let it be held as near the plate B as possible. When the eye is held very near to B, and looks along O s as it were through the reflected image of the rhomb C E, it will perceive along its axis A X a splendid system of colored rings resembling that shown in Jig. 98., intersected by a rectangular black cross, A B C D, the arms of which meet at the centre of the rings. The colors in these rings are exactly the same as those in Newton's Table of colors, and consequently the *• cnAP. xxrr. kings in crystals with one axis. 167 same as the system of rings seen by reflexion from the plate of air between the 'object glasses. If we turn the rhomb round its axis, the rings will suffer no change ; but if we fix the rhomb, or hold it steadily, and turn round the plate B, then, in the azimuths 0°, 90°, 180°, and 270° of its revolution, we shall see the same system of rings; but at the intermediate azimuths of 45°, 135°, 225°, and 315°, we shall see another system, like that in fig. 99., in which all the colors are com- plementary to those in Jig. 93., being- the same as those seen Fig. 99. 5SN in the rings formed by transmission through the plate of air. The superposition of these two systems of rings would repro- duce white light If, in place of the glass plate B, we substitute a prism of calcareous spar, that separates its two images greatly, or a rhomb of great thickness, we shall see in the ordinary image the first system of rings, and in the extraordinary image the second system of complementary rings, when the principal section of the prism or rhomb is in the plane r s O as formerly described. As the light which forms the first system of rings is polarized in an opposite plane to that which forms the second system, we may disperse the one system by agate, or absorb it by tovrmaline, and thus render the other visible, the first or the second system being dispersed or absorbed according to the position of the agate or the tourmaline. If we split the rhomb of calcareous spar,^£\ 97., into two plates by the fissure M N, and examine the rings produced by each plate separately, we shall find that the rings produced by each plate are larger in diameter than those produced by the whole rhomb, and that the rings increase in size as the thick- ness of the pin to diminishes. It will also be found that the circular area contained within any one ring is to the circular $*,v>fcj**wMv»t' '-"■■"■'-■••■- 168 A TREATISE ON OPTICS. PART IX. area of any other ring - , as the number in Newton's Table cor- responding to the tint of the one ring is to the number corre- sponding to the tint of the other. If we use homogeneous light, we shall find that the rings are smallest in violet light and largest in red light, and of in- termediate sizes in the intermediate colors, consisting always of rings of the color of the light employed, separated by black rings. In white light all the rings formed by the seven dif- ferent colors are combined, and constitute the colored system above described, according to the principles which were fully explained in Chapter XII. (115.) All the other crystals which have one axis of double refraction, give a similar system of rings along their axis of double refraction ; but those produced by the positive crystals, such as zircon, ice, &c., though to the eye they differ in no respect from those of the negative crystals, yet possess dif- ferent properties. If we take a system of rings formed by ice or zircon, and combine it with a system of rings of the very same diameter formed by Iceland spar, we shall find that the two systems destroy one another, the one being nega- tive and the other positive; an effect which might have been expected from the opposite kinds of double refraction possessed by these two crystals. If we combine two plates of negative crystals, such as Ice- land spar and beryl, the system of rings which they produce will be such as would be formed by two plates of Iceland spar, one of which is the plate employed, and the other a plate which gives rings of the same size as the plate of beryl. But if we combine a plate of a negative crystal with a plate of a positive crystal, such as one of Iceland spar with one of zir- con or ice, the resulting system of rings, in place of arising from the sum of their separate actions, will arise from their difference ; that is, it will be equal to the system produced by a plate of Iceland spar whose thickness is equal to the differ- ence of the thicknesses of the plate of Iceland spar employed, and another plate of Iceland spar that would give rings of the same size as those produced by the zircon or ice. These experiments of combining rings are not easily made, unless we employ crystals which have external faces perpen- dicular to the axis of double refraction, such as the variety of Iceland spar called spath calcaire basee, some of the micas with one axis, and well crystallized plates of ice, &c. When two such plates cannot be obtained, I have adjusted the axes of the two plates so as to coincide, by placing between them, at their edges, two or three small pieces of soft wax, by press- CHAP. XXII. RINGS IN CRYSTALS WITH ONE AXIS. 169 ing which in different directions, we may produce a sufficiently accurate coincidence of the systems of rings to establish the preceding conclusions. If, when two systems of rings are thus combined, either botli negative or both positive, or the one negative and the other positive, we interpose between the plates which produce them crystallized films of sulphate of lime or mica, we shall produce the most beautiful changes in the form and character of the rings. This experiment I found to be particularly splendid when the film was placed between two plates of the spatk calcaire basee of the same thickness, and taken from the same crystal. By fixing them permanently with their faces parallel, and leaving a sufficient interval between them for the introduction of films of crystals, I had an apparatus by which the most splendid phenomena were produced. The rings were no longer symmetrical round their axis, but exhib- ited the most beautiful variety of forms during the rotation of the combined plates, all of which are easily deducible from the general laws of double refraction and polarization. The table of crystals that have negative double refraction shows the bodies that have a negative system of ring's ; and the table of positive crystals indicates those that Have a posi- tive system of rings. (116.) The following is the method which I have used for distinguishing whether any system of rings is positive or negative. Take a film of sulphate of lime, such as that shown at C E D F, fig. 94, and mark upon its surface the lines or neutral axes CD, EF as nearly as may be. Fix this film by a little wax on the surface, L 1) or F G, fig. 97., of the rhomb which produces the negative system of rings. If the film produces alone the red of the second order, it will now, when combined with the rhomb, obliterate part of the red ring of the second order, either in the two quadrants A C, B D, fig. 98., or in the other two, A D, C B. Let it obliterate the red in A C, B D; then if the line C D,fig. 94, of the film crosses these two quadrants at right angles to the rings, it will be the principal axis of the sulphate of lime ; but if it crosses the other two quadrants, then the line E F, which crosses the quadrants A C, B D, will be the principal axis of sulphate of lime, and it should be marked as such. We shall suppose, however, that C D h;is been proved to be the principal axis. Then, if we wish to examine whether any other system of rings is positive or negative, we have only to cross the rings with the axis C D, by interposing the film: and if it obliter- ates the red ring of the second order in the quadrant which it P 170 A TREATISE ON OITICS. PART H. crosses, the system will be negative ; but if it obliterates the same ring in the other two quadrants which it does not cross, then the system will be positive. It is of no consequence what color the film polarizes, as it will always obliterate the tint of the same nature in the system of rings under exan> ination. (117.) In order to explain the formation of the systems of rings seen along the axes of crystals, we must consider the two causes on which they depend ; namely, the thickness of the crystal through which the polarized light passes, and the inclination of the polarized light to the axis of double retrac- tion or the axis of the rings. We have already shown how the tint or color varies with the thickness of the crystallized body, and how, when we know the color for one thickness, we may determine it for all other thicknesses, the inclination of the ray to the axis remaining always the same. We have now, therefore, only to consider the effect of inclination to the axis. It is obvious that along the axis of the crystal, where the two black lines A B, C T>,Jig. 98., cross each other, there is neither double refraction nor color. When the polarized ray is slightly inclined to the axis, a faint tint appears, like the blue in the first order of Newton's scale ; and as the incli- nation gradually increases, all the colors in Newton's table are produced in succession, from the very black of the first order up to the reddish white of the seventh order. Here, then, it appears that an increase in the inclination of the polarized light to the axis corresponds to an increase of thickness ; so that if the light always passed through the same thickness of the mineral, the different colors of the scale would be pro- duced by difference of inclination alone. Now, it is found by experiment, that in the same thickness of the mineral, the numerical value of the tints, or the numbers opposite to the tints in the last column of Newton's table, vary as the square of the sine of the inclination of the polarized ray to the axis. Hence it follows, that at equal inclinations the same tint will be produced ; and consequently, the similar tints will be at equal distances from the axis of the rings, or the lines of equal tint or rings will be circles whose centre is in the axis. Let us suppose that at an inclination of 30° to the axis we observe the blue of the second order, the numerical value of whose tint is 9 in Newton's table, and that we wish to know the tint which would be produced at an inclination of 45°. The sine of 30° is -5, and its square -25. The sine of 45° is -7071, and its square .5. Then we say, as -25 is to 9, so is -5 to 18, which in the table is the numerical value of the red of the third CHAP. XXII. RINGS IX CRYSTALS WITH ONE AXIS. 171 order. If we suppose the thickness of the mineral to be in- creased at the inclinations 30° and 45°, then the numerical value of the tint would increase in the same proportion. It is obvious from what has been said, that the polarizing force, or that which produces the rings, vanishes when the double refraction vanishes, and increases and diminishes with the double refraction, and according to the same law. The polarizing force, therefore, depends on the force of double re- fraction; and we accordingly rind that crystals with high double refraction have the power of producing the same tint, either at much less thicknesses, or at much less inclinations to the axis. In order to compare the polarizing intensities of different crystals, the best way is to compare the tints which they produce at right angles to the axis where the force of double refraction and polarization is a maximum, and with a given thickness of the mineral. Thus, in the case given above, we may find the tint at right angles to the axis, by- taking the square of the sine of 90°, which is 1 ; so that we have the following proportion : as "25 is to 9, so is 1 to 36, the value of the maximum tint of calcareous spar at right angles to the axis, upon the supposition that a tint of the value of 9 was produced at an inclination of 30°. If we have measured the thickness of Iceland spar at which the tint 9 was produced, we are prepared to compare the polarizing intensity of Iceland spar with that of any other mineral. Thus, let us take a plate of quartz, and let us suppose that at an inclination of 30°, and with a thickness fifty-one times as great as that of the plate of Iceland spar, it produces a yellow of the first order, whose value is about 4. Then to find the tint at 90°, or at right angles to the axis, we say, as the square of the sine of 30°, or -25, is to 4, so is the square of the sine of 90°, or 1, to 16, the tint at 90°, or the green of the third order. Now the polarizing power or intensity of the Iceland spar w r ould have been to that of the quartz as 36 to 16, or 2\ times as great, if the thickness of the two minerals had been the same; but as the thickness of the quartz was 51 times as great as that of the Iceland spar, the polarizing intensity of the Iceland spar will Ik; ~)1 multiplied by 2\ times, or 115 times as great as that of quartz. The intensities for various crystals have been determined by several observers, but the following have been given by Mr. Herschel : — .■•••■-.■■---■■- 172 A TREATISE ON OPTICS. TART II. Polarizing Intensities of Crystals with One Axis. Iceland spur Hydrate of strontia Tourmaline ... Hyposulphatc of lime - Quartz Apophyllite, 1st variety Camphor .... Vesuvian - - - Apophyllite, 2d variety 3d variety 35801 1246 851 470 312 111!) 101 41 33 3 0-000028 0-000802 0-001175 0-00212!) 0003024 0-009150 0-009856 o-o24 no 0-H30374 0-366620 The above measures are suited to yellow light, and the numbers. in the second column show the proportions of the thicknesses of* the different substances that produce the same tint The polarizing force of Iceland spar is so enormous at right angles to the axis, that it is almost impracticable to pre- pare a film of it sufficiently thin to exhibit the colors in New- ton's table. CHAP. XXIII. ON THE SYSTEMS OF COLORED RINGS IN CRYSTALS WITH TWO AXES. (118.) It was long believed that all crystals had only one axis of double refraction ; but, after I discovered the double system of rings in topaz and other minerals, I found that these minerals had two axes of double refraction as well as of polar- ization, and that the possession of two axes characterized the great body of crystals which are either formed by art, or which occur in the mineral kingdom. The double system of rings, or rather one of the sets of the double system of rings in topaz, first presented itself to me when I was looking along the axis of topaz, which reflected a part of the light of the sky that happened to be polarized, so that they were seen without the aid either of a polarizing or an analyzing plate. In this and some other minerals, however, the axes of double refraction are so much inclined to one an- other, that we cannot see the two systems of rings at once. I shall therefore proceed to explain them as exhibited by nilre, in which I also discovered them and examined many of their properties. CRAP. XXIII. RINGS IN' CRYSTALS WITH TWO AXES. 173 Nitre, or saltpetre, is an artificial substance which crystal- lizes in six-sided prisms with angles of about 120°. It belongs to the prismatic system of Mobs, and has therefore two axes of double refraction along which a ray of light is not divided into two. These axes are each inclined about 2£° to the axis of the prism, and about 5° to each other. If, therefore, we cut off a piece of a prism of nitre with a knife driven by a smart blow from a hammer, and polish two flat surfaces perpendicu- lar to the axis of the prism, so as to leave a thickness of the sixth or eighth of an inch, and then transmit the polarized light r s, Jig. 94., along the axis of the prism, keeping the crystal as near to the plate B as possible on one side, and the eye as near it as possible on the other, we shall see the double system of rings, A B, shown in fig. 100., when the plane pass- ing through the two axes of nitre is in the plane of primitwe Fig. loo. Fig. 101. polarization, or in the plane of reflexion r s O, fig. 94., and the system shown in fig. 101. when the same plane is inclined 45° to either of these planes. In p;is^in ; from the state of fig. 100. to that of fig. 101., the black I in assume the forms shown mfigs. 102. and 103. These systems of rings have, generally speaking, the same colors as those of thin plates, or as those of the systems of nngs round one axis. The orders of colors commence at the P2 i^mmri *X(1%Jfc?iMb\* -■■■■■■-•■ ~ 174 A TREATISE ON OPTICS. PART II. centres A and B of each system ; but at a certain distance, which in Jig. 100. corresponds to the sixth ring, the rings, in Fig. 102. Fig. 103. place of returning and encircling each pole A and B, encir- cle the two poles as an ellipse does its two foci. When we diminish the thickness of the plate of nitre, the rings enlarge ; the fifth ring will then surround both poles. At a less thickness, the fourth ring will surround them, till at last all the rings will surround both poles, and the system will have a great resemblance to the system surrounding one axis. The place of the poles A, B never changes, but the black lines A B, C D become broad and indefinite ; and the whole system is distinguished from the single system principally by the oval appearance of the rings. If we increase the thickness of the nitre, the rings will di- minish in size ; the colors will lose their resemblance to those of Newton's scale; and the tints do not commence at the poles A, B, but at virtual poles in their proximity. The color of the rings within the two polos is red, and without them blue ; and the great body of the rings is pink and green. As the same color exists in every part of the same curve, the curves have been called isochromatic lines, or lines of equal tint. The lines or axes along which there is no double refraction or polarization, and whose poles are A, B, fig. 100., have been called optical axes, or axes of no polarization, or axes of compensation, or resultant axes ; because they have been found not to be real axes, but lines along which the op- CHAP. XXIII. RINGS IN CRYSTALS WITH TWO AXES. 175 jwsite actions of other two real axes have been compensated, or destroy one another. (119.) In various crystallized bodies, such as nitre and ar- ragonite, whore the inclination of the resultant axes, A, B, fig. 100., is small, the two systems of rings may he easily seen at the same time; but when the inclination of the result- ant axes is great, as in topaz, sulphate of iron, &c, we can only see one of the systems of rings, which may be done most advantageously by grinding and polishing two parallel faces perpendicular to the axis of the rings. In mica and topaz, and various other crystals, the plane of most eminent cleavage is equally inclined to the two resultant axes ; so that, in such bodies the systems of rings may be readily found and easily exhibited. t Let M N, for example, fig. 104., be a plate of topaz, cut or split so as to have its face perpendicular to the axis of the Fig. 104. prism in which this body crystallizes. If we place this plate, fig. 104., in the apparatus/"-. 94. so that the polarized ray rs, Fig. 105 fin- 94-1 passes along the line A BeE, fig. 104., and if the eye receives this ray when reflected from the analyzing plate B, it will see in the direction of that ray a system of oval rings, like i that in fig. 105. In like manner, if | the polarized I ight is transmitted along I the line did 1), the eye will sec an- 1 other sv.-tem perfectly similar to the B first. The lines A BeE and C Ed 1) ar.-. therefore, the resultant axes of topaz. The angle A 13 C will be found equal to about 121° 10' ; but if we compute the inclination of the refract- ed rays B tl, B e, we shall find it, or the angle d B e, to be only 05° ; which is, therefore, the inclination of the op* tical or resultant axes of topaz. ■jHpMMn ■gMgHMMMMMPp 176 A TREATISE ON OPTICS. PART II. If wo suppose the plate of nitre fixed in any of the positions which give any of the rings shown in Jigs. 100, 101, 102, or 103., then, if we turn round the plate B, we shall observe in the azimuths of 90° and 270° a system of rings complement- ary to each, in which the black cross in Jig. 100. and the black hyperbolic curves in jigs. 101. 103. are white, all the other dark parts light, and die red green, the green red, &.c. as in the single system of rings with one axis. In the preceding observations we have supposed the polari- zation of the incident light, and the analysis of the transmitted light, to be necessary to the production of the rings ; but in certain cases they may be shown by common light with the analyzing plate, or by polarized light without the analyzing plate B, and in some cases without either the light being po- larized or analyzed. If in topaz, for example. Jig. 104., we allow common light to fall in the direction A B, so as to be refracted along Be, one of the resultant axes, and subsequently reflected at c from the second surface, and reaching the eye at c, we shall see, ailer reflexion from the analyzing plate, the system of rings in fig. 105. ; or if A B is polarized light, the rings will be seen by the eye at c without an analyzing plate. There are several other curious phenomena seen und« r these circumstances, which I have described in the l'hil. Transactions for 1814, p. 203. 211. I have found some crystals of nitre which exhibit their rings without the use either of polarized light or an analyzing plate; and Mr. Herschel has found the same property in some crystals of carbonate of potash. (120.) When the preceding phenomena are seen by polar- ized homogeneous light, in place of white light, the rings are bright curves, separated by dark intervals; the curves having always the color of the light employed. In many crystals the difference in the size of the rings seen in different colors is not very great, and the poles A, 6 of the two systems do not greatly change their place; but Mr. Herschel found that there were crystals, such as tartrate of potash and soda, in which the variation in the size of the rings was enormous, being greatest in red, and least in violet light, and in which the distance A B,figs. 100. 101., or the inclination of the resultant axes, varied from 56° in violet light to 76° in red, the inclina- tion having intermediate values for intermediate colors, and the centres of all the different systems lying in the line A B. When all these systems of rings are combined, as they are in using white light, the system of rings which they form is ex- ceedingly irregular, the two oval centres, or the halves of the first order of colors, being drawn out with long spectra or CHAP. XXIII. RINGS IN CRYSTALS WITH TWO AXES. 177 tails of re. I, green, and violet light, and the ends of all the other rings being red without the resultant axes, and hlue within. ' Mr. Hcrschel found other crystals in which the rings are smallest in red, and largest in blue light, and in which the inclination of the axes or A B is least in red, and greatest in vbdci light In all crystals of this kind, the deviation of the tints, or the colors of the rings seen in white light, from Newton's Table is very considerable, and may be calculated from the preceding principles. This deviation i found to be very great, even in crystals with one axis of double refraction and one system of rings, such as apophyllite where the rings have scarcely any otiier tints than a succession of greenish yellow, and reddish purple ones. By viewing these rings in homogeneous light, Mr. IJersehel has found that the system is a negative one for the rays at the one end of the spectrum, a positive one for the rays at the other end of the spectrum, and that there are no rings at all in yellow light. A similar and equally curious anomaly I have found in glauberite, which is a crystal which has two axes of double refraction, or two systems of rings for red light, and one nega- tive system for violet light. (121). All the singularities of these phenomena disappear, and may be rigorously calculated by supposing the resultant axes of crystals where there are two, or the single axis where there is one, with a system of riugs deviating from Newton's scale, as merely apparent axes, or axes of compensation, pro- duced by the opposite action of two or more rectangular axes, the principal one of which is the line bisecting the angle formed by the two resultant axes. Upon this principle, I have shown that all the phenomena presented by such crystals may be computed with as much accuracy as we can compute the motions of the heavenly bodies. The method of doing this may be understood from the fol- _/,.;„ ]( l( ;_ lowing observations. Let A C B D, fig. 106., be a crystal with two axes 'turned into a sphere. Let P, 1* be the \ poles of the axes, the point bisecting \ them, and A B a line passing through —j'D O, and perpendicular to CD, a line J passing through I', P. Let us suppose / an axis to pass through O, perpendicu- lar to the plane AC B J), then we may Hi account for all the phenomena of such crystals, by supposing the axis at O to be the principal one, () I J 178 A TREATISE ON OPTICS. PART II. and the other axis to be along either of the diameters A B or C D. If we take C D, then the axe? O and C D must be both of the same name, either both positive or both negative ; but it' we take A B, the axes must be one positive and the other negative; or, what is perhaps the simplest supposition for il- Lion, we shall suppose the two rectangular axes which produce all the phenomena to be AB, CD, either both positive or l>oth negative, leaving out the one at O. Supposing A O B, C P P D to be projections of great circles of the sphere, then P, P are the points where the axis A B destroys the effect of the axis C D; that is, where the tints produced by each axis must be equal and opposite. Now, if we suppose the arch C P to be 00°, then, since A P is 90°, it follows that the axis C I) produces at GO the same tint that -A B does at 90°, and consequently the polarizing intensity of C D will be to that of A B as the square of the sine of 90° is to the square of the sine of 60°, or as 1 to 0-75, or as 100 to 75. The polarizing force of each axis being thus determined, it is easy to find the tint which will be produced by each axis separately at any given inclination to the axis, by the method formerly explain- ed. Let E be any point on the surface of the sphere, and let the tints produced at that point be 9, or the blue of the second order, by C D, and 10, or the green of the third order, by A B. Let the inclination of the planes passing through A E, C E, or the spherical angle C E A be determined, then the tint at the point E will correspond to the diagonal of a parallelogram whose sides are 9 and 10, and whose angle is double the angle C E A. This law, which is general, and applies also to double refraction, has been confirmed by Biot and Fresnel, the last of whom has proved that it coincides rigorously with the law deduced from the theory of waves. If the axes A B, C 1) are equal, it follows that they will produce the same tint at equal inclinations; that is, they will compensate each other only at one point, viz. O, and will pro- duce round O a system of colored rings, the very same as if O were a single axis of double refraction of an opposite name to A U,CD. If the axis A B has exactly the same propor- tional action that C D has upon each of the differently colored rays, a compensation will take place tor each color exactly at (), the centre of the resultant systems of rings, and the colors will be exactly those of Newton's scale. But if each axis ex- ercises a different proportional action upon the colored rays, a compensation will take place at O for some of the rays (for violet, for example), while the compensation for red will take place on each side of O ; consequently, in such a case the CHAP. XXIV. INTERFERENCE OF POLARIZED LIGHT. 179 crystal will have one axis for violet light, and two axes for red light, like glauberite. The phenomena of apnphyllite may, in a similar manner, be explained by two equal negative axes, A B, C D, and a positive axis at O. According to this method of combining the action of dif- ferent rectangular axes, it follows that three equal and rectan- gular axes, either all positive or all negative, will destroy one another at every point of the sphere, and thus produce the very same effect as if the crystal had no double refraction and polarization at all. Upon this principle I have explained the absence of double refraction in all the crystals which form the tessular system of Mohs, each of the primitive forms of which has actually three similarly situated and rectangular axes. If one of these axes is not precisely equal to the other, and the crystallization not perfectly uniform, traces of double refrac- tion will appear, which is found to be the case in muriate of soda, diamond, and other bodies of this class. (122.) The following table contains the polarizing inten- sities of some crystals with two axes, as given by Mr. Her- schel : — Polarizing Intensities of Crystals with Two Axes. Value of hiphest Tint. Thicknesses that prodocs the same Tint. 0-000135 0-000526 0-000765 0-001920 0-004021 7400 1900 1307 521 249 Anhydrite, inclination of axes 43° 48' Mica, inclination of axes 45° - - - Sulphate of baryta Heulandite (white), inclination of) axes 54° 17' $ 1 CHAP. XXIV. INTERFERENCE OF rOLARIZED LIGHT. — ON THE CAUSE OF THE COLORS OF CRYSTALLIZED liODIL.S. (123.) Having thus described the principal phenomena of the colors produced by regularly crystallized bodies that pos- sess one or two axes of double refraction, we shall proceed to explain the cause of these remarkable phenomena. Dr. Young had the great merit of applying the doctrine of interference to explain the colors produced by double refrac- tion. When a pencil of light tails upon a thin plate of* 180 A TREATISE ON OPTICS. PART II. doubly refracting crystal, it is separated into two, which move through the plate with different velocities, corresponding to the different indices of refraction for the ordinary and extra- ordinary ray. In calcareous spar, the ordinary ray moves with greater velocity than the extraordinary one; and there- fore they ought to interfere with one another, and in homo- geneous light produce rings consisting of bright and dark cir- cles round the axis of double refraction. According to this doctrine, however, the rings ought to be produced in common as well as in polarized light; but as this was not the case, Dr. Young's ingenious hypothesis was long neglected. The sub- ject was at last taken up by Messrs. Fresnel and Arago, who displayed great address in their investigation of the subject, and succeeded in showing how the production of the rings de- pended on the polarization of the incident pencil and its sub- sequent analysis by a reflecting plate or a doubly retracting prism. The following are the laws of the interference of polarized light as discovered by MM. Fresnel and Arago: — 1. When two rays polarized in the name plane interfere with each other, they will produce by their interference fringes of the very same kind as if they xoere common light. This law may be proved by repeating the experiments on the inflexion of light, mentioned in Chap. XL, in polarized in place of common light; and it will be found that the very same fringes are produced in the one case as in the other. 2. When two rays of light are polarized at right angles to each other, they produce no colored fringes in the same cir- cumstances under which two rays if common light woidd produce them. When the rays are polarized at angles inter- mediate between 0° and 99°, they produce fringes of inter- mediate brightness, the fringes bring totally obliterated at 90°, and recovering their greatest brightness at 0°, as in Law 1. In order to prove this law, MM. Fresnel and Arago adopted several methods, the simplest of which is the following, em- ployed by the latter. Having made'^wo fine slits in a thin plate of copper, lie placed the copper behind the focus F of a lens, as in Jig. 56., and received the shadow of the copper upon the screen C D, where the fringes produced by the inter- ference of the rays passing through the two slits were visible. In order, however, to observe the fringes more accurately, he viewed them with an eye-glass, as formerly described. He next prepared a bundle of transparent plates, like either of those shown at A and B,fig. 93., made of fifteen thin films of mica or plane glass, and he divided this bundle into two, by CHAP. XXIV. INTERFERENCE OF POLARIZED LIGHT. 181 a sharp cutting instrument. At the line of division these bundles had as nearly as possible the same thickness, and they were capable of polarizing completely light incident upon them at an angle of 30°. These bundles were then placed before the slits so as to receive and transmit the rays from the focus F at an incidence of 30°, and through portions of the mica in each bundle that were very near to each other pre- vious to their separation. The bundles were also fixed to re- volving frames, so that, by turning either bundle round, their planes of polarization could be made either parallel or at right angles to each other, or could be inclined at any intermediate angle. When the bundles were placed so as to polarize the rays in parallel planes, the fringes were formed by the slits exactly as when the bundles were removed; but when the rays were polarized at 90°, or at right angles to each other, the fringes wholly disappeared. In all intermediate positions the fringes appeared with intermediate degrees of brightness. 3. Two rays originally polarized at right angles to each other may be subsequently brought into the same plane of po- larization, icithout acquiring the power of forming fringes by their interference If, in the preceding experiment, a doubly refracting crystal be placed between the eye and the copper slits, having its principal section inclined 45° to either of the planes of polari- zation of the interfering rays, each pencil will be separated into two equal ones polarized in two rectangular planes, one of which planes is the principal section itself. Two systems of fringes ought, therefore, to be produced ; one system from the interference of the ordinary ray from the right hand slit with that of the ordinary ray from the left hand slit, and an- other system from the interference of the extraordinary ray from the right hand slit with the extraordinary ray from the left hand slit ; but no such fringes are produced. 4. Two rays polarized at right angles to each other, and afterwards brought into similar planes of polarization, pro- duce, fringes by their interference like rays of common light, provided they belong to a pencil, the whole of which was originally polarized in the same plane. 5. In the phenomena of interference produced by rays that have suffered double refraction, a difference of half an undu- lation must be allowed, as one of the pencils is retarded by that quantity from some unknown cause. The second of these laws affords a direct explanation of the fact which perplexed Dr. Young, that no fringes are ob- served when light is transmitted through a thin plate possess- ing double refraction The two pencils thus produced do not - • ' - - --•-'■-■ 182 A TREATISE ON OPTICS. H form fringes by their interference, because they are polarized in opposite planes* The production of the fringes by the action of doubly re- fracting crystals on polarized light may be thus explained, Let M N,fig. 107., be a section of the plate of sulphate of Fig 107 1 irne, C E D F, Jig. 94., and B the ana- lyzing plate. Let Rrbea polarized // ray incident upon the plate M N, and // let O and E he the ordinary and ex- SmL traordinary rays produced by the double refraction of the plate M N. When the plate M N is in such a po- sition that either of its neutral axes C D, E F, fig. 94., are in the plane of primitive polarization of the ray Rr,fig. 107., then one of the pencils will not suf- fer reflexion by the plate B, and consequently only one of the rays will be reflected. Hence it is obvious that no colors can be produced by interference, because there is only one ray. But in every other position of the plate M N, the two rays, O s, E s, will be reflected by the plate B; and being polarized by the plate in the same plane, they will^by Law L, interfere, and produce a color or a fringe corresponding to the retardation of one of the rays within the plate, arising from the difference of their velocities. If we call d the interval of retardation within the plate M N, we must add to it half an undulation to get the real interval, as one of the rays passes from the or- dinary to the extraordinary state. If we now suppose the plate B to make a revolution of 90°, M N remaining fixed, then the ray E will be reduced to the ordinary state ; and con- sequently we must subtract half an undulation from d, the in- terval of retardation within the plate, to have the real differ- ence of the intervals of retardation. Hence the two intervals of retardation will differ by a whole undulation ; and conse- quently the color produced when the plate B has been turned round 90°, will be complementary to that which is produced when the plate B has the position shown in fig. 107. If we suppose the rays E and O to be received upon and analyzed by a prism of Iceland spar, we shall have two or- dinary rays interfering to form the colors in one image, and two extraordinary rays interfering to produce the complement- ary colors in the other image. CHAP. XXV. POLARIZING STRUCTURE OF ANALCIME. 183 CHAP. XXV. ON THE POLARIZING STRUCTURE OF ANALCIME. (124.) In a preceding chapter I have mentioned the very remarkable double refraction which is possessed by analcime. This mineral, which is also called cubizite, has been regarded by mineralogists as having the cube for its primitive form ; but if this were correct, it should have exhibited no double refrac- tion. Analcime has certainly no cleavage planes, and it must be regarded at present as forming in this respect as great an anomaly in crystallography as it does in optics by its extra- ordinary optical phenomena. The most common form of the analcime is the solid called the icositetrahedron, which is bounded by twenty-four equal and similar trapezia ; and we may regard it as derived from the cube, by cutting oft" eacli of its angles by three planes equally inclined to the three faces which contain the solid angle. If we now conceive the cube to be dissected by planes passing through all the twelve diagonals of its six faces, each of these planes will be found to be a plane of no double re- fraction, or polarization ; that is, a ray of polarized light trans- mitted in any direction whatever, provided it is in one of these planes, will exhibit none of the polarized tints when tlie crystal is placed in the apparatus, Jiff. 94. These planes of no double refraction are shown by dark lines in Jigs. 1-08. and Fig. 108. 109. If the polarized ray is in- cident in any direction which is out of these planes, it will be divided into two pencils, and exhibit the finest tints, all of which are related to the planes of no double refraction. The double refraction is sufficiently great to admit a distinct separation of the images when the incident ray passes through any pair of the four planes which are adjacent to the three axes of the solid, or of \ the cube from which it is derived. * The least refracted image is the extraordinary one ; and conse- quently the double refraction is negative in relation to the axes to which the doubly refracted ray is perpendicular. 184 A TREATISE O I* OPTICS. PART II. In all other doubly refracting crystals, eacli particle lias the same force of double refraction ; but in the analcime, the double refraction of each particle varies with the square of its distance from the planes already described. The beautiful distribution of the tints shown in Jigs. 108. and 109. cannot, of course, be exhibited to the eye at once, but are deduced by transmitting polarized light, in every direction through the mineral. In several of the crystals, the tints rise to the third and fourth order ; but when the crystals are very small, the tints do not exceed the white of the first order. The tints are ex- actly those of Newton's scale, which indicates that they are not the result of opposite and dissimilar actions. In Jigs. 108. and 109. the tints are represented by the taint shaded lines having their origin from the planes where the double refrac- tion disappears. The preceding property of analcime is a simple and easily applied mineralogical character, which would identity the most shapeless fragment of the mineral. The abbe Hauy first observed in this mineral its property of yielding no electricity by friction, and derived the name of analcime from its want of this property. When we consider that the crystal is intersected by numerous planes, in whicli the ether does not exist at all, or has its properties neutralized by opposite actions, we may ascribe to this cause the difficulty with which friction decomposes the natural quantity of elec- tricity residing in the mineral. chap. xxvi. ON CIRCULAR POLARIZATION. (125). In all crystals with one axis there is neither double refraction nor polarization along the axis; and this is indicated in the system of rings, by the disappearance of all light in the centre of the rings at the intersection of the black cross. When we examine, however, the system of rings produced by a plate of rock crystal whose faces are perpendicular to the axis, we find that the black cross is obliterated within the inner ring, which is occupied with a uniform tint of red, green, or blue, according to the thickness of the plate. This effect will be seen in fig. 110. M. Arago first observed these colors in 1811. He found that when they were analyzed by a prism of Iceland spar, the two images had complementary CHAP. XXVI. ON CIRCULAR POLARIZATION. 165 colors, and that the colors changed, descending in Newton's Fig. no. scale as the prism revolved ; so that if the color of the extraordinary image was red, it became in succession orange, yellow, green, and violet. From this result he concluded, that the differently colored rays had been polarized in dif- ferent planes, by passing along the axis of the rock crystal. In this state of the subject, it was taken up by M. Biot, who investigated it with much sagacity and success. Let C E D F be the plate of quartz, fig. 94., along whose axis a polarized ray, r s, is transmitted. When the eye is placed at O, above the analyzing plate fixed as in the figure, it will see, for example, a circular red space in the centre of the rings. If we turn the quartz round its axis, no change whatever takes place ; but if we turn the plate B from right to left, through an angle of 100° for example, we shall observe the red change to orange, yellow, green, and violet, the latter having a dark purple tinge. If we now cut from the same prism of rock crystal another plate of twice the thickness, and place it in the apparatus, the plate B remaining where it was left, we shall find that its tint is different from that of the former plate ; but by turning the plate B 100° farther, we shall again bring the tint to its least brightness, viz., a sombre violet. By a plate thrice as thick, the least brightness will be obtained by tinning the plate B 100° farther, and so on, till, when the thickness is very great, the plate B may have made several complete revolutions. Now, it might happen that a thickness had been taken, so that the rotation of B which pro- duced the sombre violet was 360°, or terminated in the point 0°, from which it set out, which would have perplexed the observer, if he had not made the succession of experiments which we have mentioned. This phenomenon will be better understood, by supposing that we take a plate of qvartz Jjlh of an inch thick, and use the different homogeneous rays of the spectrum in succession. Beginning with red, we shall find that the red light in the centre of the rings has its maximum brightness when the plate B is at 0° of azimuth, as in Jig. 94. If we turn B from right to left, the red tint will gradually decrease, and after a rotation of 17i° the red tint will wholly vanish, having reach- ed its minimum. With a plate - 2 3 ths thick, the red will vanish at 35°, every additional thickness of the 25th of an inch renuirintr an additional rotation of 1T. J >°. If the light is 1 ° Q:> .•••'■•"■"''"■*■ 18fi A TREATISE ON OPTIC3. PAR* II. violet, the same thickness, viz., J,th of an inch, will require a rotation of 41° to make it vanish, every additional 25th of an inch of thickness requiring a rotation of 41° more. (126.) The rotations for different colors corresponding to 1 millimetre, or ^th of an inch of quartz, are as follows: — Homogeneous Ray. Extreme red Mean red Limit of red and orange - Mean orange Limit of orange and yellow Mean yellow Limit of yellow and green- Mean green ... 170 30' lit 00 JO 2D '21 24 22 w ■u 00 25 40 27 51 Homogc lUy. Limit of green and blue - Mean blue Limit of blue ar.d indigo .Mean indigo .... Limit of indigo and violet Mean violet Extreme violet • • • 31)003' 19 :i4 :;i 3o 07 37 41 40 53 41 05 Upon trying various specimens of quartz, M. Biot found that there were several in which the very same phenomena were produced by turning the plate B from left to right. Hence, in reference to this property, quartz may be divided into right-handed and left-handed quartz. From these interesting facts it follows, that, in passing along the axis of quartz, polarized Kght comports itself, at its egress from the crystal, as if its pianos of polarization revolved in the direction of a spiral within the crystal, in some speci- mens from right to left, and in others from left to right. " To conceive this distinction," says Mr. Herschel, "let the reader take a common cork-screw, and holding it with the head to- wards him, let him turn it in the usual manner as if to pene- trate a cork. The head will then turn the same way as the plane of polarization of a ray, in its progress from the spec- tator through a right-handed crystal, may be conceived to do. If the thread of the cork-screw were reversed, or were what is termed a left-handed thread, then the motion of the head as the instrument advances would represent that of the plane of polarization in a left-handed specimen of rock crystal." From the opposite characters of these two varieties of quartz, it follows, that if we combine a plate of right-handed with a Elate of left-handed quartz, the result of the combination will e that of a plate of the thickest of the two, whose thickness is equal to the difference of the two thicknesses. Thus, if a plate Ty'.th of an inch thick of right-handed quartz is combined with a plate Aths thick of left-handed quartz, the same colors will be produced as if we used a plate T^ths of an inch thick of left-handed quartz. When the thicknesses are equal, the plates of course destroy each other's effects, and the sys- tem of rings with the black cross will be distinctly seen. (127.) In examining the phenomena of circular polarization, t irvv. XXVI. ON CIRCULAR POLARIZATION. 187 in the amethyst, I found that it possessed the power in the same specimen of turning the planes of polarization both from right to left and from left to right, and that it actually consisted of alternate strata of right and left-handed quartz, whose planes rvere parallel to the axis of double refraction of the prism. When we cut a plate perpendicular to the axis of the prism, we therefore cut across these strata, as shown in Jig. 111., which exhibits sections of the strata which occur Fig. 111. opposite the three alternate faces of the six-sided prism. The shaded lines are those which turn the planes of polariza- tion from right to left, while the inter- mediate unshaded ones and the three un- shaded sectors turn them from left to right. These strata are not united to- gether like the parts of certain composite crystals, whose dissimilar faces are brought into mechanical contact ; for the right and left-handed strata destroy each other at the middle line between each stratum, and each stra- tum has its maximum polarizing force in its middle line, the force diminishing gradually to the lines of junction. In some specimens of amethyst the thickness of these strata is so minute, that the action of the right-handed stratum ex- tends nearly to the central line of the left-handed stratum, and vice versa, so as nearly to destroy each other ; and hence in such specimens we see the system of colored rings with the black cross almost entirely uninfluenced by the tints of circular polariztntion. A vein of amethyst, therefore, ^'-.th of an inch thick in the direction of the axis, may be so thin in a direction perpendicular to the axis that the arc of rotation for the red ray may be 0° ; and we shall have the curious phe- nomenon of a plate which polarizes circularly only the most refrangible rays of the spectrum. By a greater degree of thinness in the strata, the plate would be incapable of polar- izing circularly the yellow ray ; and by a greater thinness still, there would be no action on the violet light. These feeble actions, however, might be rendered visible at great thicknesses of the mineral. We may therefore conclude that the axes of rotation in amethyst vary from 0° to each of the numbers in the preceding table, according to the thickness of the strata. The coloring matter of the amethyst I have found to be curiously distributed in reference to these views; but I must refer to the original memoir for farther information.* * Edinburgh Transactions, vol. ix. p. 139. 188 A TREATISE ON OPTICS. PART II. M, Biot maintained that this remarkable property of quartz resided in its ultimate particles, and accompanied them in all their combinations. I have found, however, that it is not pos- sessed by opal, tabasheer, and other silicious bodies, and that it disappears in melted quartz. Mr. Herschel also found that it does not exist in a solution of silica in potash. Hitherto no connexion could be traced between the right and left-handed structure in quartz, and the crystalline form of the specimens which possess- ed these properties. Mr. Herschel, however, dis- covered that the plagiedral quartz which contains unsymmetrical faces, x x x,jig. 112., turns the planes of polarization in the same direction in which these faces lean round the summits A x x, a x x. Circular Polarization in Fluids. (128.) The remarkable property of polarizing light circu- larly occurs in a feeble degree in certain fluids, in which it was discovered by M. Biot and Dr. Seebeck. Mr. Herschel has found it in camphor in a solid state, and I have discovered it in certain specimens of unannealed glass. If we take a tube six or seven inches long, and fill it with oil of turpentine, and place it in the apparatus, Jig. 94, so that polarized light transmitted through the oil may be reflected to the eye from the plate B, we shall observe the complementary colors and a distinct rotation of the plane of polarization from right to left. Other fluids have the property of turning the planes of polari- zation from left to right, as shown in the following table, which contains the results of M. Biot's experiments. Crystals which turn the Planes from Right to Left Rock crystal Oil of turpentine Solution of 1753 parts of artificial camphor in 17359 of alcohol ' Essentia! oil cf laurel. turpentine. 18° 25' 16 01 Relative Th-rk- 1 68$ Crystals which turn the Planes from Left to Right. Rock crystal Essential oil of lemons - - - Concentrated syrup (from sugar) Arc of Rotation for every 251 li of an inch in Thk-kncM. Relative Thick- DOM that produce, the same KUVct. 18°25' 26 33 1 38 4i CHAP. XXVI. ON CIRCULAR POLARIZATION. 189 In examining these phenomena, M. Fresnel discovered that in quartz they were produced by the interference of two pencils formed by double refraction along the axis of the quartz. He succeeded in separating - these two pencils, which differ both from common and polarized light. They differ from polarized light, because when either of them is doubled by a doubly refracting crystal, the pencil or image never vanishes during the revolution of the crystal. They differ from com- mon light, because when they suffer two total reflexions from glass, at an angle of about 54°, the one will emerge polarized in a plane inclined 45° to the right, and the other in a plane 45° to the left, of the plane of total reflexion. M. Fresnel has also discovered the following properties of a circularly polarized ray : — When it is transmitted through a thin doubly refracting plate parallel to its axis, it is divided into two pencils with complementary colors ; and these colors will be an exact quarter of a tint, or an order of colors, either higher or lower in Newton's scale, than the color which the same crystallized plate would have given by polarized light. M. Fresnel also proved that a circularly polarized ray, when transmitted along the axis of rock crystal, will not exhibit the complementary colors when analyzed. (129.) In the prosecution of this curious subject, M. Fresnel discovered the following method of producing a ray possessing all the above properties, and therefore exactly similar to one of the pencils produced by circular double refraction. Let ABCD, fig. 113., be a parallelopiped of crown glass, whose index of refraction is 1*510, and whose angles A B C, A D C are each 54^°. If a common polarized ray, R r, is incident perpendicularly upon A B, and emerges perpendicularly from C D, after having suffered two total reflexions at E and F, at angles of 54j ; and if these reflexions are performed in a plane inclined 45° to the plane of polarization of the ray, the emer- gent ray F G will have all the properties of a circularly polarized ray, resembling in every respect one of those produced by double refraction along the axis of rock crystal. But as this circularly polarized ray may be restored to a single plane of polarization, inclined 45° to the plane of reflexion, by two total reflexions at 54£°, it follows, and 1 have verified the re- sult by observation, that if the parallelopiped ABCD is sufficiently long, the pencil will emerge circularly polarized, Fig. 11.1. 190 A TREATISE ON OPTICS. TART II. at 2, 6, 10, 14, 18 reflexions, and polarized in a single plane after 4, 8, 12, 10, 20 reflexions. M. Fresnel proved that the ray R r would emerge at G, circularly polarized by three total reflexions at 09° 12', and four total reflexions at 74° 42'. Hence, according to the pre- ceding reasoning, the ray will be circularly polarized by 9, ]•">, 21, 27, &-C. reflexions at u'!J° 12', and restored to common polarized light at 0, 12, 18, and 24 reflexions at the same angle; and it will be circularly polarized by 12,20,28, 36, &.c. reflexions at 74° 42', and be restored to common polar- ized light by 8, 10,. 24, 32, &c. reflexions. I have found that circular polarization can be produced by 2^, 1\, 12^, &c. reflexions, or any other number which is a multiple of 2\; for though we cannot see the ray in the mid- dle of a reflexion, yet we can show it when it is restored to a single plane of polarization, at 5, 10, 15 reflexions.* When we use homogeneous light, we find that the angle at which circular polarization is produced is different for the differently colored rays; and hence these different rays cannot be restored to a single plane of polarization at the same angle of reflexion. Complementary colors will therefore be produced, such as I described long ago, and which, I believe, have not been ob- served by any other person.f These colors are essentially different from those of common polarized light, and will be understood when we come to explain those of elliptical polar- ization. CHAP. XXVII. ON ELLIPTICAL POLARIZATION, AND ON THE ACTION OF METALS UPON LIGHT. On Elliptical Polarization. (130.) The action of metals upon light has always present- ed a troublesome anomaly to the philosopher. Malus at first announced that they produced no effect whatever; but he afterwards found that the difference between transparent and metallic bodies consisted in this, — that the former reflect all the light which they polarize in one plane, and refract all the light which they polarize in an opposite plane ; while metallic bodies reflect what they polarize in both planes. Before 1 was * See Phil. Transactions, 1830, p. 301. t See Phil. Transactions, 1830, p. 309. 325. CHAP. XXVII. ELLIPTICAL POLARIZATION. 191 acquainted with any of the experiments of Malus, I had found* that light was modified by the action of metallic bodies; and that, in all the metals which I tried, a great portion of light was polarized in the plane of incidence. In February, 1815, I discovered the curious property possessed by silver and gold and other metals, of dividing polarized rays into their comple- mentary colors by successive reflexions : but I was misled by some of the results into the belief, that a reflexion from a metallic surface had the same effect as a certain thickness of a crystallized body; and that the polarized tints varied with the angle of incidence, and rose to higher orders, by increasing the number of reflexions. M. Biot, in repeating my experi- ments, and in an elaborate investigation of the phenomena,! was misled by the same causes, and has given a lengthened detail of experiments, formulae, and speculations, in which all the real phenomena are obscured and confounded. Although I had my full share in this rash generalization, yet I never viewed it as a correct expression of the phenomena, and I have repeatedly returned to the subject with the most anxious desire of surmounting its difficulties. In this attempt I have succeeded ; and 1 have been enabled to refer all the phenomena of the action of metals to a new species of polarization, which I have called elliptical polarization, and which unites the two classes of phenomena which constitute circular and rectilineal polarization. (131.) In the action of metals upon common light, it is easy to recognize the fact announced by Malus, that the light which they reflect is polarized in different planes. I have found that the pencil polarized in the plane of reflexion is always more intense than that polarized in the perpendicular plane. The difference between these pencils is least in silver, and greatest in galena, and consequently the latter polarizes more light in the plane of reflexion than silver. The following table shows the effect which takes place with other metals : — Order in which the Metals polarize most Light in the Plane of Reflexion. Galena. Lead. Grey cobalt. Arsenical cobalt. Iron pyrites. Antimony. Steel. Copper. Zinc. Tin plate. Speculum metal. Brass. Platinum. Grain tin. Bismuth. Jewellers' gold. Mercury. Fine gold. Common silver. Pure silver. Total reflexion from glass. * Treatise on New Philos. Instruments, p. 317. and Preface. t Traite de Physique, torn. iv. p. 579. 600. 192 A TREATISE ON OPTICS. PART II. By increasing the number of reflexions, the whole of the incident light may be polarized in the plane of reflexion. Eight reflexions from plates of steel, between 60° and 80°, polarize the whole light of a wax candle ten feet distant. Au increased number of reflexions [above 36] is necessary to do this with pure silver; and in total reflexions from glass, where the circular polarization begins, and where the two pencils are equal, the effect cannot be produced by any number of reflexions. In order to examine the action of metals upon polarized light, we must provide a pair of plates of each metal, flatly ground and highly polished, and each at least 14 inch long and half an inch broad. These parallel plates should be fixed upon a goniometer, or other divided instrument, so that one of the plates can be made to approach to or recede from the other, and so that their surfaces can receive the polarized ray at different angles of incidence. In place of giving the plates a motion of rotation round the polarized ray, I have found it better to give the plane of polarization of the ray a motion round the plates, so that the planes of reflexion and of polari- zation may be set at any required angle. The ray reflected from the plates one or more times is then analyzed, either by a plate of glass or a rhomb of Iceland spar. When the plane of reflexion from the plates is either par- allel or perpendicular to the plane of primitive polarization, the reflected light will receive no peculiar modification, ex- cepting what arises from their property of polarizing a portion of light in the plane of reflexion. But in every other position of the plane of reflexion, and at every angle of incidence, and after any number of reflexions, the pencil will have re- ceived particular modifications, which we shall proceed to explain. One of these, however, is so beautiful and striking, as to arrest our immediate attention. When the plates are silver or gold, the most brilliant complementary colors are seen in the ordinary and extraordinary images, changing with the angle of incidence and the number of reflexions. These colors are most brilliant when the plane of reflexion is in- clined 45° to the plane of incidence-, and they vanish when the inclination is 0° or 90°. All the other metals in the table, p. 191, give analogous colors ; but they are most brilliant in silver, and diminish in brilliancy from silver to galena. In order to investigate the cause of these phenomena, let us suppose steel plates to be used, and the plane of the polar- ized ray to be inclined 45° to the plane of reflexion. At an incidence of 75° the light has suffered some physical change, CHAP. XXVII. ELLIPTICAL POLARIZATION. 193 which is a maximum at that, angle. It is not polarized light, because it does not vanish during the revolution of the ana- lyzing plate. It is neither partially polarized light nor com- mon light; because, when we reflect it a second time at 75°, it is restored to light polarized in one plane. If we transmit the light reflected from steel at 75° along the axis of Iceland spar, the system of rings shown in fig. 98. is changed into the system shown in fig. 114., as if a thin film of a crystallized Fig. 1 14. body which polarizes the blue of the first order had crossed the system. If we substitute fur the calcareous spar films of sulphate of lime which give different tints, we shall find that these tints are increased in value by a quan- tity nearly equal to a quarter of a tint, according as the metallic action coin- cides with or opposes that of the crys- tal. It was on the authority of this experiment that I was led to believe that metals acted like crystallized plates. And when I found thai tlic colors were better developed and more pure after successive reflexions, I rashly concluded, as M. Biot also did after me, that each successive reflexion corresponded to an additional thickness of the film. In order to prove the error of this opinion, let us transmit the light reflected 2, 4, 6, 8 times from steel at 75° along the axis of Iceland spar, and we shall find that the system of rings is perfect, and that the whole of the light is polarized in one plane ; a result absolutely incompatible with the supposition of the tints rising with the number of reflexions. At 1, 3, 5, 7, 9, 11 reflexions, the ligh. when transmitted along the axis of Iceland spar will produce an effect equal to nearly a quarter of a tint, beyond which it never rises. I now conceived that light reflected 1, 3, 5, 7, 9 times from steel at 75° resembled circularly polarized light. In circularly polarized light produced by two total reflexions from glass, the ray originally polarized -f 45° to the plane of reflexion is, by the two reflexions at the same angle, restored to ligfct polarized — 45° to the plane of reflexion; whereas in steel, a ray polar- ized -f- 45°, and reflected once from steel at 75°, is restored by another reflexion at 75° to light polarized — 17°. With different metals the same effect is produced, but the inclination of the plane of polarization of the restored ray is different, as the following table shows : — R »::M4WV«4^1**3rf?*«^*»«^^ r«*Ty*l*«T*i*;>vr -r*w ■* 194 A TREATISE ON OFTICS. From glass - Pure silver Common silver Fine gold - - Jewellers' gold Grain lin - - Brass - - - Tin Plate - - Copper - - - Mercury - - Platinum - - arm red Ray. 45° 0' : 39 48 ! 3(5 1 35 33 33 32 31 29 o 26 1 22 | FART II. Kny. Bismuth - - - Speculum metal Zinc - - - - Steel - - - - Iron pyrites - ■ Antimony - - Arsenical cobalt Cobalt - - - ■ Lead - - - ■ Galena - - • Specular iron, 21° (V 21 19 10 17 14 It; 15 13 12 30 11 2 In total reflexions, or in circular polarization, the circularly polarized ray is restored to a single plane by the same number of reflexions and at the same angle at which it received cir- cular polarization, whatever be the inclination of the plane of the second pair of reflexions to the plane of the first pair; but in metallic polarization, the angle at which the second re- flexion restores the ray to a single plane of polarization varies with the inclination of the plane of the second reflexion to the plane of the first reflexion. In the case of total reflexions, this angle varies as the radii of a circle ; that is, it is always the same. In the case of metallic polarization, it varies as the radii of an ellipse. Thus, when the plane of the polarized ray is inclined 45° to the plane of primitive polarization, the ray reflected once at 75° will be restored to polarized light at an incidence of 75° ; but when the two planes are parallel to one another, the restoration takes place at 80° ; and when they are perpendicular, at 70° ; and at intermediate angles, at in- termediate inclinations. For these reasons, I have called this kind of polarization elliptic polarization. We have already seen that light polarized + 45° is ellipti- cally polarized by 1, 3, 5, 7 reflexions from steel at 75°, and restored to a single plane of polarization by 2, 4, 6, 8 reflexions at the same angle ; and we have stated that the ray restored by two reflexions has its plane of polarization brought into the state of — 17°. The following are the inclinations of this plane to the plane of reflexion, by different numbers of re- flexions from steel and silver : — No. of Re- flexions. Inclination of the Plane of the polarized Ray. No. of Re- flexions. Inclination of the Plane of the polarized Ray. Steel. Silver. Sir el. Silver. 2 4 6 8 — 17° 0' + 5 22 — 138 + 30 — 38° 15' + 31 52 — 26 6 + 21 7 10 12 18 36 — 0° 9' + 3 — + —16° 56' + 13 30 — (i 42 + 47 CHAP. XXVII. ELLIPTICAL TOLARIZATIOX. 195 Those results explain in the clearest manner why common light is polarized by steel after eight reflexions, and by silver not till after thirty-six reflexions. Common light consists of two pencils, one polarized -f- 45°, and the other — 45°; and steel brings these planes of polarization into the plane of re- flexion after eight reflexions, while silver requires more than thirty-six reflexions to do this. (132.) The angles at which elliptical polarization is pro- duced by one reflexion may be considered as the maximum polarizing angles of the metal, and their tangents may be considered as the indices of refraction of the different metals, as shown in the following table : — Grain tin - - - - Mercury - - - - Galena .... Iron pyrites - - - Grey cobalt - - - Speculum metal Antimony melted - Steel Bismuth - - - - Pure silver - - - Zinc Tin plate hammered Jewellers' gold - - „■!(• ••< Maximum Index of tabu zation. Refraction 78<-> 30 4-915 78 27 4-893 7;s 10 4-773 77 30 4-511 76 56 4-309 76 4011 7.") 35 3-814 i.) 3-732 74 50 3-689 73 3-271 7-2 30 3172 70 50 2-879 70 45 2-864 Elliptical polarization may be produced by a sufficient num- ber of reflexions at any given angle, either above or below the maximum polarizing angle, as shown in the following table for Steel : — which Elliptical l'olariza- which the Tenril ia re- stored to a single Plane. of Incidence 3 9 15 &c. 6 12 18 &c. 86o o' 2J 74 12J &c. 5 10 15 &c. 84 2 6 10 &c. 4 8 12 &c. 82 20 14. 44 7J&c. 3 6 9 &c. 79 1 3 5 &c. 2 4 6 &c. 75 14 44 7J &c. 3 6 9 &c. 67 40 2 6 10 &c. 4 8 12 &c. 60 20 24 7J 124 &c. 5 10 15 &c. 56 25 3 9 15 &c. 6 12 18 &c. 52 20 When the number of reflexions is an integer, it is easily understood how an elliptically polarized ray begins to retrace its course, and to recover its state of polarization in a single plane, by the same number of reflexions by which it lost it ; but it is interesting to observe, when the number of reflexions 196 A TREATISE ON OPTICS. PART II. is 1',, 2L or any other mixed number, that the ray must have acquired its elliptical polarization in the middle of the second and third reflexion ; that is, when it had reached its greatest depth within the metallic surface it then begins to resume its state of polarization in a single plane, and recovers it at the end of 8, 5, and 7, reflexions. A very remarkable effect takes place when one reflexion is made on one side of the maxi- mum polarizing angle, and one on the other side. A ray that lias received partial elliptical polarization by one reflexion at 35° does not acquire more elliptic polarization by a reflexion at 54°, but it retraces its course and recovers its state of single polarization. By a method which it would be out of place to explain here, I have determined the number of points of restoration which can occur at different angles of incidence from 0° to 90°, for any number of reflexions ; and I have represented them in Jig. 115., where the arches I, I., II, II., &c. represent the quadrant of incidence, for one, two, &c. reflexions; C being the point of 0°, and B that of 90° of incidence. In the quadrant, I, I. there is no point of restoration. In II, II. there is only one point or node of restoration, viz. at 73° in silver. In III, III. there are two points of restoration, because a ray elliptically polarized by one and a half reflexion will be re- stored by three reflexions at 63° 43' beneath the maximum polarizing angle, and at 79° 40' above that angle. It may also be shown that for IV. reflexions there arc 3 points of restora- tion, for V. reflexions 4 points ; and for VI. reflexions 5 points, as shown in the figure. The loops or double curves are drawn to represent the intensity of the elliptic polarization which has its minimum at 1, 2, 3, &c, and its maximum in the middle of the unshaded parts. If we now use homogeneous light, we shall find that the loops have different sizes in the different colored rays, and that their minima and maxima are different, CRAP. XXVII. ELLIPTICAL POLARIZATION. 197 Hence, in the Vlth quadrant, C B for example, there will be loops of all the different colors, viz. CI; 1, 2; 2, 3; 3, 4, &c. ; overlapping one another, and producing by their mixture those beautiful complementary colors which have already been mentioned. For a more full account of this curious branch of the subject of polarization, I must refer the reader to the Philosophical Transactions, 1830; or to the Edinburgh Journal of Science, Nos. VII. and VIII. new scries, April, 1831. CHAP. XXVIII. ON THE POLARIZING STRUCTURE PRODUCED BY HEAT, COLD, COMPRESSION, DILATATION, AND INDURATION. The various phenomena of double refraction, and the sys- tems of polarized rings with one and two axes of double re- fraction, and with planes of no double refraction, may be pro- duced either transiently or permanently, in glass and other substances, by heat and cold, rapid cooling, compression and dilatation, and induration. 1. Transient Influence of Heat and Cold. (1.) Cylinders of glass with one positive axis of double refraction. (133.) If we take a cylinder of glass, from half an inch to an inch in diameter, or upwards, and about half an inch or more in thickness;, and transmit heat from its circumference to its centre, it will exhibit when exposed to polarized light, in the apparatus, fig. 94., a system of rings with a black cross, exactly similar to those in fig 98. ; and the complementary system shown in Jig. 99. will appear by turning round the plate B 90°. In this case we must hold the cylinder at the distance of 8 or 10 inches from the eye, when the rings will appear as it were in the inside of the glass. If we cover up any portion of the surface of the glass cylinder, we shall hide a corresponding portion of the rings, so that the cylinder has its single axis of double refraction fixed in the axis of its figure, and not in every possible direction parallel to that axis as in crystals. By crossing the rings with a plate of sulphate of lime, as formerly explained, we shall find that it depresses the tints in the two quadrants which the axis of the plate crosses ; and R2 MMMMtMMMMMlllMi ■■■■■■■■■■IMHH 198 A TREATISE ON OPTICS. PART II. consequently that the system of rings is negative, like that of calcareous spar. As soon as the heat reaches the axis of the cylinder, the rings begin to lose their brightness, and when the heat is uniformly diffused through the glass, they disappear entirely. (2.) Cylinders of glass with a negative axis of double refraction. (134.) If a similar cylinder of glass is heated uniformly in boiling oil, or otherwise brought to a considerably high tem- perature, and is made to cool rapidly by surrounding its cir- cumference with a good conductor, it will exhibit a similar system of rings, which will all vanish when the glass is uni- formly cold. By crossing these rings with sulphate of lime, they will be found to be positive, like those of ice and zircon ; or the same thing may be proved by combining this system of rings with the preceding system, when they will be found to destroy one another. In both these systems of rings, the numerical value of the tint or color at any one point varies as the square of the dis- tance of that point from the axis. By placing thin films of sulphate of lime between two of these systems of rings, very beautiful systems may be produced. (3.) Oval plates of glass tvith two axes of double refraction. (135.) If we take an oval plate A B D C, fig. 116., and perform with it the two preceding experi- ments, we shall find that it has in both cases two axes of double refraction, the principal axis passing through O, being negative when it is heated at its circumference, and positive when cooled at its circumference. The curves A B, C 1), correspond to the black ones in C fig- 101., and the distance m n to the inclina- tion of the resultant axes. The effect shown in Jig. 110. is that which is produced by inclining m n 45° to the plane of primitive polarization; but when m n is in the plane of prim- itive polarization, or perpendicular to ii, the curves A B, C D, will form a black cross, as in fig. 100. In all the preceding experiments, the heat and cold might have been introduced and conveyed through the glass from each extremity of the axis of the cylinder or plate. In this case the phenomena would have been exactly the same, but the axes that were formerly negative will now be positive, and vice versa. EU) CHAP. XXVIII. POLARIZATION BY HEAT. 199 (4.) Cubes of glass with double refraction. (I'Mj.) When the shape of the glass is that of a cube, the rings have the form shown in fig. 117. and when it is a paral- lelopiped with its length about three times its breadth, the Fig. us. rings have the form shown in Jig. 118. the curves of equal tint near the angles being circles, as shown in both the figures. (5.) Rectangular plates of glass with planes of no double refraction. (137.) If a well annealed rectangular plate of glass, E FD C, is placed with its lower edge C D on a piece of iron ABDC fig. 119., nearly red hot, and the two together are placed in the J^F. 119. apparatus, Jig. 94., so that C 1) may be inclined 45° to the plane of primitive polarization, and that polarized light may reach the eye at O from every part of the glass, we shall ob- serve the following phenomena. The instant that the heat enters the surface C 1), fringes of brilliant colors will be seen parallel to C 1), and almost at the same time before the heat has reached the upper surface E F, or even the central line a b, similar fringes will appear at E F. Colors at first faint blue, and then white, yellow, orange, &c, all spring up at a b ; and these central colors will be divided from those at the edges by two dark lines, M N, O P, in which there is neither double refraction nor polarization. These lines correspond with the black curves in fig. 101. and fig. 116., and Hie struc- ture between M N and OP is negative, like that of cal- careous spar ; while the structures without M N and O P are positive, like those of zircon. The tints thus developed are those of Newton's scale, and are compounded of the different 200 A TREATISE ON orTICS. PART II. sets of tints that would be given in each of the homogeneous rays of the spectrum. In these plates there is obviously an infinite number of axes in the planes passing through M N, O P, and all the tints, as well as the double refraction, can be calculated by the very same laws as in regular crystals, mutatis mutandis. If the plate E F D C is heated equally all round, the fringes are produced with more regularity and quickness; and if the plate, first heated in oil or otherwise, is cooled equally all round, it will develope the same fringes, but the central ones at a b will in this last case be positive, and the outer ones at E F and C D negative. Similar effects to those above described may be produced in similar plates of rock salt, obsidian, fluor spar, copal, and other solids that have not the doubly refracting structure. A series of splendid phenomena are produced by crossing simi- lar or dissimilar plates of glass when their fringes are developed. When similar plates of glass, or those in which the fringes are produced by heat, as in fig. 119., are crossed, the curves or lines of equal tint at the square of intersection, A B C D, jig. 120., *-,>. ion. w iU DG hyperbolas. The tint at the centre will be the difference of the central tints of each of the two plates, and the tints of the succeeding lry- B perbolas will rise gradually in the scale above that central tint. If the tints produced by each plate are precisely the same, and the plates of the same shape, the central tints will destroy each other, the hyperbolas will be equilateral ones, and the tints will gradually rise from the zero of Newton's scale. When dissimilar plates are crossed, as in Jig. 121., viz. one in which the fringes are produced by heat with one in which they are produced by cold, the lines of equal tint in the square of intersection ABCD {fig- 121.), will be ellipses. The tii^ts in the centre will be equal to the sum of the separate tints, and the tints formed by the combination of the externa] friugi s will be equal to their difference. If the plates and their tints are perfectly equal, the lines of equal tint will be circles. The beauty of these combinations can be understood only from col- ored drawings. When the plates are combined lengthwise, they add to or subtract from each other's effect, according as similar or dissimilar fringes arc opposed to one another. CHAP. XXVIII. TOLARIZATION BY HEAT. Fig. 121. 201 (6.) Spheres of glass, <$c. with an infinite number of axes of double refraction. (139.) If we place a sphere of glass in a glass trough of hot oil, and observe the system of rings, while the heat is passing to the centre of the sphere, we shall find it to be a regular system, exactly like that in fig. 98. ; and it will suffer no change by turning the sphere in any direction. Hence the sphere has an infinite number of positive axes of double re- fraction, or one along each of its diameters. If a very hot sphere of glass is placed in a glass trough of cold oil, a similar system will be produced, but the axes will all be negative. (7.) Spheroids of glass with one axis of double refraction along the axis of revolution and two axes along the equa- torial diameters. (139.) If we place an oblate spheroid in a glass trough of hot oil, we shall find that it has one axis of positive double refraction along its shorter axis, or that of revolution ; but if we transmit the polarized light along any of its equatorial diameters, we shall find that it has two axes of double refrac- tion, the black curves appearing as in fig. 116. when the axis of revolution is inclined 45° to the plane of primitive polari- zation, and changing into a cross when the axis is parallel or perpendicular to the plane of primitive polarization. The very same phenomena will be exhibited with a prolate spheroid, only the black cross opens in a different plane when the two axes are developed. Opposite systems of rings will be developed in both these cases, if hot spheroids are plunged in cold oil. 202 A TREATISE ON OPTICS. The reason of using oil is to enable the polarized light to pass through the spheres or spheroids without refraction. The oil should have a refractive power as near as possible to that of the glass. A number of very curious phenomena arise from heating and cooling glass tubes, or cylinders, along their axes ; the most singular variations taking place according as the heat and cold are applied to the circumference, or to the axis, or to both. (8.) Influence of heat on regular crystals. (140.) The influence of uniform heat and cold on regular crystals is very remarkable. M. Fresnel found that heat dilates sulphate of lime less in the direction of its principal axis than in a direction perpendicular to it ; and professor Mitscher- lich has found that Iceland spar is dilated by heat in the di- rection of its axis of double refraction, while in all directions at right angles to this axis it contracts ; so that there must be some intermediate direction in which there is neither contrac- tion nor dilatation. Heat brings the rhomb of Iceland spar nearer to the cube, and diminishes its double refraction. In applying heat to sulphite of lime, professor Mitscherlich found that the two resultant axes (P, P, fig. 106.) gradually approach as the heat increases, till they unite at O, and form a single axis. By a still farther increase of heat they open ■out on each side towards A and B. A very curious fact of an analogous kind I have found in glauberite, which has one axis of double refraction for violet, and two axes for red light. With a heat below that of boiling water, the two resultant axes (P, P, Jig. 106.) unite at O, and, by a slight increase of heat, the resultant axes again open out, one in the direction O A, and the other in the direction O B. By applying cold, the single axis for violet light at O opened out into two at P and P. At a certain temperature the violet axis also opened out into two, in the plane A B. 2. On the permanent Influence, of sudden Cooling. (141.) In March, 1814, I found that glass melted and^sud- denly cooled, such as prince Rupert's drops, possessed a per- manent doubly refracting structure;* and in December, l8l4, Dr. Seebeck published an account of analogous experiments with cubes of glass. Cylinders, plates, cubes, spheres, and spheroids of glass, with a permanent doubly refracting struc- * Letter to Sir Joseph Ranks, April 8. 1814. Phil. Trans. 1814. CHAP. XXVIII. POLARIZATION BY SUDDEN COOLING. 203 ture, may be funned by bringing the glass to a red heat, and cooling it rapidly at its circumference, or at its edges. As these solid bodies often lose their shape in the process, the symmetry of their structure is affected, and the system of rings or fringes injured ; so that the phenomena are not pro- duced so perfectly as during the transient influence of heat and cold. It is often necessary, too, to grind and polish the surfaces afresh : an operation during which the solids are often broken, in consequence of the state of constraint in which the particles are held. An endless variety of the most beautiful optical figures may be produced by cooling the glass upon metallic patterns (metals being the best conductors) applied symmetrically to each surface of the glass, or symmetrically round its circum- ference. The heat may be thus drawn oft" from the glass in lines of any form or direction, so as to give any variety what- ever to its structure, and, consequently, to the optical figure which it produces when exposed to polarized light. (142.) In all doubly refracting crystals the form of the rings is independent of the external shape of the crystal ; but in glass solids that have received the doubly refracting structure, either transiently or permanently, from heat, the rings depend entirely on the external shape of the solid. If, in fig. 119., we divide the rectangular plate E F D C into two equal parts through the line a b, each half of the plate will have the same structure as the whole, viz. a negative and two positive structures, separated by two dark neutral lines. In like manner, if we cut a piece of a tube of glass, by a notch, through its circumference to its centre, or if we alter the shape of cylindrical plates and spheres, &c., by grinding them into different external figures, we produce a complete change upon the optical figures which they had previously exhibited. 3. On the Influence of Compression and Dilatation. (143.) If we could compress and dilate the various solids above mentioned with the same uniformity with which we can heat and cool thorn, we should produce the same doubly re- fracting structures which have been described, compression and dilatation always producing opposite structures. The influence of compression and dilatation may be well exhibited by taking a strip of glass, A B D C, fig. 1'22.. and bending it by the force of the hands. When it is held in the apparatus, fig. 94., with its edge A B inclined 45° to the plane of primitive polarization, the whole thickness of the glass will be covered with colored fringes, consisting of a negative set A TREATISE ON OI*TICS. TAUT II. 204 separated from a positive set by the dark neutral line M N. The fringes on the convex side A B are negative, and those on the concave side positive. As the bending force increases, the tints increase in number; and as it diminishes, they di- minish in number, disappearing entirely when the plate of glass recovers its shape. The tints, which are those of New- ton's scale, vary with their distances from M N ; and when two such plates as that shown in Jig. 122. cross each other, they produce in the square of intersection rectilineal fringes parallel to the diagonal of the square which joins the angles where the two concave and the two convex sides of the plates meet. When a plate of bent glass is made to cross a plate crys- tallized by heat, and suddenly cooled, the fringes in the square of intersection are parabolas, whose vertex will be towards the convex side of the bent plate, if the principal axis of the other plate is positive, but towards the concave side, if that axis is negative. The effects of compression and dilatation may be most dis- tinctly seen by pressing or dilating plates or cylinders of calves'-feet jelly or soft isinglass. By the application of compressing and dilating forces, I have been able to alter the doubly refracting structure of regularly crystallized bodies in every direction, increasing or diminishing their tints according to the direction in which the forces were applied.* The most remarkable influence of pressure, however, is that which it produces on a mixture of resin and white wax In all the cases hitherto mentioned of the artificial production of double refraction, the phenomena are related to the shape of the mass in which the change is induced : but I have been able to communicate to the compound above mentioned a double refraction, similar to that which exists in the particles of crystals. The compressed mass has a single axis of double refraction in every parallel direction, and the colored rings are produced by the inclination of the refracted ray to the axis according to the same law as in regular crystals. If we * See Edinburgh Transactions, vol. viii. p 281. CHAP. XXVIII. rOLARIZATION BY INDURATION. 205 remove the compressed film, any portion of it will be found to have one axis of double refraction like portions of a film of any crystal with one axis. The important deductions which this experiment authorizes will be noticed at the conclusion of this part of the work. 4. On the Influence of Induration. (144.) In 1814 I had occasion to make some experiments on the influence of induration in communicating double refraction to soft solids. When isinglass is dried in a glass trough of a circular form, it exhibits a system of tints with the black cross exactly like negative crystals with one axis. When a thin cylindrical plate of isinglass is indurated at its circumference, it. produces a system of rings with one positive axis. If the trough in the first of these experiments and the plate in the second are oval, two axes of double refraction will be ex- hibited. When jelly placed in rectangular troughs of glass is grad- ually indurated, we have a positive and a negative structure developed, and these are separated by a black neutral line. If the bottom of the trough is taken out, so as to allow the induration to go on at two parallel surfaces, the same fringes are produced as in a rectangular plate of glass heated in oil, and subsequently cooled. Spheres and spheroids of jelly may be made by proper in- duration to produce the same effects as spheres and spheroids of glass when heated or cooled. The lenses of almost all animals possess the doubly refracting structure. In some there is only one structure, which is generally positive. In others there are tw o structures, a positive and a negative one ; and in many there are three structures, a negative between two positive, and a positive between two negative structures. In some instances we have two structures of the same name together. By the process of induration we may remove en- tirely the natural structure of the lens, especially when it is spherical or spheroidal, and superinduce the structure arising from induration. 1 have now before me a spheroidal lens of the boneto fish, with one beautiful system of rings along the axis of the spheroid, and two systems along the equatorial diameters. I have also several indurated lenses of the cod, that display in the finest manner their doubly refracting structure. S 206 A TREATISE ON OPTICS. PART II. CHAP. XXIX. PHENOMENA OF COMPOSITE OR TESSEEATED CRYSTALS. (145.) In all regularly formed doubly refracting crystals, the separation of the two images, the size of the rings, and the valse of the tints, are exactly the same in all parallel direc- tions. If two crystals, however, have grown together with their axes inclined to one another, and if we cut a plate out of these united crystals so that the eye cannot distinguish it from a plate cut out of a single crystal, the exposure of such a crystal to polarized light will instantly detect its composite nature, and will exhibit to the eye the very line of junction. This will be obvious upon considering that the polarized ray has different inclinations to the axis of each crystal, and will therefore produce different tints at these different inclinations. Hence the examination of a body in polarized light furnishes us with a new method of discovering structures which can- not be detected by the microscope, or any other method of observation. A very fine example of this is exhibited in the bipyramidal sulphate of potash, which Count Bournon and other crystal- lographers regarded as one simple crystal, whose primitive form was the bipyramidal dodecahedron, like the crystal shown in fig- 112- But by cutting a plate perpendicular to the axis of the pyramid, and exposing it to polarized light, I found it to be composed of several crystals, all united so as to form the regular figure above represented. The crystal has two axes of double refraction, and the plane passing through the two axes of one, is inclined 60° to the plane passing through the two axes of each of the other two. So that when we incline the plate, each of the three combined crystals displays different colors. I have found many remarkable structures of this kind in the mineral kingdom, and among artificial salts ; but two of these are so interesting as to merit particular notice. (146.) The apophyllite from Faroe generally crystallizes in right-angled square prisms, and splits with great facility into plates by planes perpendicular to the axis of the prism If we remove with a sharp knife the uppermost slice, or the un- dermost, it will be found to have one axis of double refraction, and to give the single system of rings shown in fig. 98. If we remove other slices in the same manner, we shall find that when exposed to polarized light they exhibit the curious tesselated structure shown in Jig. 123. The outer case, MONP, consists of a number of parallel veins or plates. In CHAP. XXIX. COMPOSITE OR TESSELATED CRYSTALS. 207 Fig. 123. Fig. 124. the centr.* is a small lozenge, ab cd, with one axis of double refraction, and round it are four crystals, A, B, C, D, with two axes of double refraction, the plane pass- ing through the axes of A and D being perpendicular to the plane passing through the axes of B and C ; and the former plane being in the direction M N, and the latter in the direction O P. When the polarized light is trans- mitted through the faces of certain prisms, the beautiful tesselated figure shown in fig. 124 is ex- hibited, all the differently shaded parts shining with the most splendid colors. As the prism has every- where the same thickness, it is obvious that the doubly refracting force varies in different parts of the crystal ; but this variation takes place in such a symmetrical manner in rela- tion to the sides and ends of the prism, as to set at defiance all the Fig. 125. recognized laws of crystallography. With the view of observing the form of the lines of equal co- lor, I immersed the crystal in oil, and transmitted the polar- ized light in a direc- tion parallel to a di- agonal of the prism; the effect then exhib- ited is shown in fisc. 125., where ABCD is the crystal ; A C, and B D, its edges, where the thickness is nothing, and m a the edge through which the di- agonal of the prism passes. Now, it is 5 obvious, that if this had been a regu- lar crystal, the lines of equal tint or of equal double refraction would have been all straight lines parallel to AC P or B D; but in the apophyllite they present the most singular irregularities, all of which are, however, symmetrically re- 208 A TREATIBE ON OPTICS. PART II. lated to certain fixed points within the crystal. In the middle of the crystal, half way between m and n, there are only //re frinj ea or orders of colors; at points equi-distant from this there are six fringes, the sixth returning into itself in the form of an oval. At other two equidistant points near m and ?i, the 3d, 4th, and 5th fringes are singularly serrated, and the 6th and 7th fringes return into themselves in the form of a square ; beyond this, near m and n, there are only four fringes, in consequence of the fifth returning into itself! (147.) A composite structure of a very different kind, but extremely interesting from the effects which it produces, is exhibited in many crystals of Iceland spar, which are inter- sected by parallel films or veins of various thicknesses, as shown in fig. 128. These thin veins or strata are perpendic- ular to the short diagonals E F, G H of the faces of the rhomb, and parallel to the edges EG, F H. When we look perpendicularly through the faces A E B F, DGCH, the light will not pass through any of the planes eb eg, A BCD, afh d, and consequently we shall only see two images of any object ^JT just as if the planes were not there. But if we look through any of the other two pair of parallel faces, we shall observe the two common images at their usual distance; and at a much greater distance, two secondary images, one on each side of the com- mon images. In some cases there &rejbnr, and in other cases six, secondary images, arranged in two lines; one line being on each side of the common images, and perpendicular to the line joining their centres. When the interrupting planes are numerous, and especially when they are also found perpen- dicular to the short diagonals of the other two far.es of the rhomb that meet at B, the obtuse summit, the secondary images are extremely numerous, and sometimes arranged in pyramidal heaps of singular beauty, vanishing, and reappear- ing, and changing their color and the intensity of their light, by every inclination of the plate. If the light of the luminous object is polarized, the phenomena admit of still greater va- riations. When the strata or veins are thick, the images are not colored, but have merely at their edges the colors of re- fracted light. Mai us considered these phenomena as produced by fissures or cracks within the crystal, and he regarded the colors as those of thin plates of air or space ; but I have found that they are produced by veins or twin crystals firmly united together so as to resist separation more powerfully than the natural CHAP. X\1X. COMPOSITE OR TESSELATED CRYSTALS. 209 cleavage planes, and I have found this both crystallographically, by measuring the angles of tlie veins, and optically, by ob- serving the system of rings seen through the veins alone. This composite structure will be understood from fig. 127., where A B D C is the principal section of a rhomb of Iceland spar whose axis is A D. The form and position of one of the intersecting veins or rhomboidal plates, is shown at M m N n, but greatly thicker than it actually is; the angles AmM, and D n N, being 141° 44'. A ray of common light R b, incident on the face A C at b, will be refracted in the lines b c, b d. These rays entering the vein M m N n, at c and d, will be again refracted doubly ; but as the vein is so thin as to produce the complementary colors of polarized light by the interference of the two pencils which compose each of the pencils c e, df, these colors will depend on the thickness of the vein M N, and on the inclination of the ray to the axis of the plate M N. These double pencils will emerge from the vein at e,f, and will be refracted again as in the figure into the pencils e m, i n,fo,J'p; the colors ofen,fo, being complementary to those of em,fp* That the multiplication and color of the images are owing to the causes now explained may be proved ocularly, as I have done, by dividing rhombs of calcareous spar, and inserting between them, or in grooves cut in a single plate of calcareous spar, a thin film of sulphate of lime or mica. In this way all the phenomena of the natural compound crystal may be reproduced in the artificial one, and we may give great variety to the phenomena by inserting thin films in different azimuths round the polarized pencils b c, b d, and at different inclinations to the axis of double refraction. The compound crystal shown in fig. 127. is in reality a natural polarizing apparatus. The part of the rhomb AmNC, polarizes the incident light R b. The vein M N is the thin crystallized vein whose colors are to be examined; and the part BM n D, is the analyzing rhomb. Various other minerals and artificial crvstals are intersected S2 210 A TREATISE ON OPTICS. PART II. with analogous veins, and produce analogous phenomena. There are several composite crystals which exhibit remarkable peculiarities of structure, and display curious optij*tl" phe- nomena by polarized light. The Brazilian topaz is one of those which is worthy of notice, and whose properties I have explained by colored drawings, in the second volume of the Cambridge Transactions. For a full account of the properties of composite crystals, and of the multiplication of images by the crystals of cal- careous spar that are intersected by veins, we refer the reader to the Edinburgh Transactions, vol. ix. p. 317., and the Phil. Trans., 1815, p. 270. ; or to the Edinburgh Encyclopedia, art. Optics. CHAP. XXX ON THE DICHROISM, OR DOUBLE COLOR, OF BODIES J AND THE ABSORPTION OF POLARIZED LIGHT. (148.) If a crystallized body has a different color in different directions when common light is transmitted through its substance, it is said to possess dichroism, which signifies two colors. Dr. Wollaston observed this property long ago in the muriate of •palladium and potash, which appeared of a deep red color along the axis, and of a vivid green in a transverse direction ; and M. Cordier observed the same change of color in a mineral called iolite, to which Iiaiiy gave the name of dichroite. Mr. Herschel has observed a similar fact in a variety of sub-oxysulphate of iron, which is of a deep blood red color along the axis, and of a light green color perpen- dicular to the axis. In examining this class of phenomena, I have found that they depend on the absorption of light, being regulated by the inclination of the incident ray to the axis of double refraction, and on a difference of color in the two pencils formed by double refraction. In a rhomb of yellow Iceland spar, the extraordinary image was of an orange yellow color, while the ordinary image was yellowish white along the axis. The color and intensity of the two pencils were the same, and the difference of color and intensity increased with the inclination to the axis. When the two images overlapped each other, their combined color was the same at all angles with the axis, and this color was that of the mineral. If we expose the rhomb to polarized light, its color will be orange yellow in the position where the ordinary image vanishes, and yellowish white in the position where the extraordinary image vanishes. The crystals in the CHAP. XXX. ABSORPTION OF POLARIZED LIGHT. 211 following Table possess the same properties, the ordinary and extraordinary images having the colors opposite to their names : — Colors of the two linages in Crystals with ONE Axis. Principal Section in Plane Principal 8-ection, perpendicular Names of Cryntaln. 01 Polarization. to Plan ■_• of Polarization. Zircon. Brownish white. Deeper brown. Sapjjhire. Yellowish green. Blue. Ruby. Pale yellow. Bright pink. Emerald. Yellowish green. Bluish green. Emerald. Bluish green. Yellowish green. Beryl, blue. Bluish white. Blue. Beryl, green. Whitish. Bluish green. Beryl, yellowish ) green. ) Pale yellow. Pale green. Rock crystal, near- ) ly transparent. ^ Whitish. Faint brown. Rock crystal, yellow. Yellowish white. Yellow. Amethyst. Blue. Pink. Amethyst. Greyish white. Ruby red, Amethyst. Reddish yellow. Bluish green. Tourmaline. Greenish white. Bluish green. Rubellitn. Reddish white. Faint red. Fdocrase. Yellow. Green. Mcllite. Yellow. Bluish white. Apatite lilac. Bluish. Reddish. Apatite olive. Bluish green. Yellowish green. Orange yellow. Phosphate of lead Bright green. Iceland spar. Orange yellow. Yellowish white. Octohednte. Whitish biown. Yellowish brown. (149.) When the crystals have two axes of double refrac- tion, the absorption of the incident rays produces a variety of phenomena, at and near the two resultant axes. These phe- nomena are finely displayed in iolite. This mineral, which crystallizes in six and twelve-sided prisms, is of a deep blue color when seen along the axis, and of a brownish yellow when seen in a direction perpendicular to the axis of the prism. When we look along the resultant axes which are inclined 62° 50' to one another, we see a system of rings which are pretty distinct when the plate is thin; but when it is thick, and when the plane passing through the axes is in the plane of primitive polarization, branches of blue and white light are seen to diverge in the form of a cross from the centre of the system of rings. This curious effect is shown in fig. 128., where P, P', are the centres of the two systems of rings, O the principal negative axis of the crystal, and C D the plane passing through the axes. The blue branches, which are shaded in the figure, are tipped with purple at their summits P, P', and are separated by whitish light in some specimens, ?&* MM 212 A TRKATISF. ON OPTICS. PART II and by bluish light in others. From P and P' to O, the white or yellowish light becomes more and more blue, and at O it is quite blue; while from P and P' to C and D it becomes more and more yellow, and at C and D it is quite yel- low, the yellow being almost equally bright in the plane A C B D, perpendic- ular to the principal axis O. When the plane C D is perpendicular to the plane of primitive polarization, the poles P, P' are marked with patches of white or yellowish light, but everywhere else the light is a deep blue. When examined by common light, we find that the ordinary image is brownish yellow at C and 1), and the extraordinary one faint blue ; the former acquiring some blue rays, and the latter some yellow ones from to D, and from A to B where there is still a great difference in the color of the imagea The yellow image becomes fainter from A to P and P', and from B to P and P', where it changes into blue, the feeble blue image being gradually reinforced by other blue rays till the intensity of the two blue images is nearly equal. The faint blue image increases in intensity from C to P, and from D to P', and the yellow one acquires an accession of blue light, and becomes bluish white from P and P' to O; the ordinary image is whitish, and the other a deep blue ; but the white- ness gradually diminishes towards O, where the two images are almost equally blue. The following table will show that this property exists in many other crystals: — Colors of the two Images in Crystals with two Axes. Names of Crystals. Plane of Axi« in piano of Polarizati in. Plana of axia perpendicular lo Plan.: of Polarization. Topaz blue. While. Blue. green. White. Green. greenish blue Reddish grey. Blue. pink. Pink. White. pink yellow. Pink. Yellow. yellow. Yellowish white. Orange. Sulphate of baryta. yellowish } purple. $ Lemon yellow. Purple. yellow. Lemon yellow. Yellowish white. orange yellow Gamboge yellow. Yellowish white. Cvanite. White." Blue. Dichroite. Blue. Yellowish white. Cymophane. Yellowish white. Yellowish. Epidote olive green. Brown. Sap green. whitish green Pink white. Yellowish white. Mica. Reddish brown. Reddish white. CHAP. XXX. ABSORPTION OF POLARIZED LIGHT. 213 The following table shows the color of the images in crys- tals with two axes which have not been examined. Names of Crystal*. Axis of Prism in Plane of Axis of Prism perpendicular to Plane of Polarization. Mica. Blood red. Pale greenish yellow. Acciaic of copper. Blue. Greenish yellow. Muriate of copper.* Greenish white. Blue. Olivine. Bluish green. Greenish yellow. Sphene. Yellow. Bluish. Nitrate of copper. Bluish white. Blue. Chromate of lead. Orange. Blood red. Staurotitle. Brownish red. Yellowish white. Augile. Blood red. Bright green. Anhydrite. Bright pink. Pale yellow. Axinite. Reddish white. Yellowish white. Diallage. Brownish white. White. Sulphur. Yellow. Deeper yellow. Sulphate of strontia. Blue. Bluish white. cobalt. Pink. Brick red. Olivine. Brown. Brownish white. Tn the last nine crystals in the preceding table, the tints are not given in relation to any fixed line. The following list contains the colors of the two pencils, in crystals, whose number of axes is not yet known. Phosphate of iron. Actynolite. Precious opal. Serpentine. Asbestos. Blue carbonate of ) copper. \ Octohedrite (one axis.) Chloride of gold and ) sodium. ) and I ammonium. and Fine blue.t Green. Yellow. Dark green. Greenish. Violet blue. Whitish brown. Lemon yellow. ' Lemon yellow. Lemon yellow. Bluish white. Greenish white. Lighter yellow. Lighter green. Yellowish. Greenish blue. Yellowish brown. Deep orange."] |s Deep orange. > |° Deep orange. J s. potassium. (150.) By the application of heat to certain crystals, I have been able to produce a permanent difference in the color of the two pencils formed by double refraction. This experiment may be made most easily on Brazilian topaz. In one of these topazes, in which one of the pencils was yellow and the other pink, I found that a red heat acted more powerfully upon the extraordinary than upon the ordinary pencil, discharging the yellow color entirely from the one, and producing only a slight * The colors are given iu relation to tiie short diagonal of its rltoinboidal base, t When the axis of the prism is in the piano of polarization. 214 A TREATISE o.N OPTICS. change upon the pink lint of the other. When the topaz was hot, it was perfectly colorless, and, during the process of cool- ing, it gradually acquired a pink tint, which could not be modified or renewed by the most intense heat. In various topazes, the color of whose two pencils was exactly the same, heat discharges more of the color from one pencil than the other, and thus gives them the power of absorbing light in reference to the axes of double refraction. General Observations on Double Refraction. (151.) The various facts which have been explained in the pieceding chapters, enable us to form very plausible opinions respecting the origin and nature of the doubly refracting structure. The particles of bodies reduced to a state of fluidity by heat, and prevented by the same cause from com- bining into a solid body, exhibit no double refraction ; and, in like manner, the particles of crystallized bodies, including metals when existing in a state of solution, exhibit no double refraction. As soon, however, as cooling in the one case, and evaporation in the other, permits the particles to combine in virtue of their mutual affinities, these particles have, subse- quent to the action of the forces by which they combine, ac- quired the doubly refracting structure. This effect may be accounted for in two ways; either by supposing that the par- ticles have originally a doubly refracting structure, or that they have no trace of such a structure. On the first of these suppositions, we must ascribe the disappearance of the double refraction in the fluid mass, and, in the solution, to the opposite action of the particles, which must have had an axis in every possible direction; but as no double refraction is visible, it is more philosophical to suppose that none exists in the particles. On the second supposition, then, that the particles have no doubly refracting structure, it is easily understood how it may be produced by the compression of any two particles brought together by attraction ; for each particle will have an axis of double refraction in the direction of the line joining their centres, as if they had been compressed by an external force. By following out this idea, which I have done elsewhere,* I have shown how the various phenomena may be explained by the different attractive forces of three rectangular axes, which may produce a single negative axis, a single positive axis, or * Phil. Transactions, 1829, or Edinburgh Journal of Science, new scries vol. vi. p. 328—337. I CHAP. XXXI. UNUSUAL REFRACTION. 215 two axes, either both positive or both negative, or the one positive and the other negative. The influence of heat, in changing the intensity of the two axes of sulphate of lime, and in removing one of the axes, or in creating a new one, admits of an easy explanation on these principles. PART III. ON THE APPLICATION OF OPTICAL PRINCIPLES TO THE EXPLANATION OF NATURAL PHENOMENA. CHAP. XXXI. ON UNUSUAL REFRACTION. (152.) The atmosphere in which we live is a transparent mass of air possessing the property of refracting light. We learn from the barometer that its density gradually diminishes as we rise in the atmosphere, and, as we know from direct experiment that the refractive power of air increases with its density, it follows, that the refractive power of the atmosphere is greatest at the earth's surface, and gradually diminishes till the air becomes so rare as scarcely to be able to pro- duce any effect upon light. When a ray of light falls ob- liquely upon a medium thus varying in density, in place of being bent at once out of its direction, it will be gradually more and more bent during its passage through it, so as to move in a curve line, in the same manner as if the medium had consisted of an infinite number of strata of different re- fractive powers. In order to explain this, let E, Jig. 129., be Fig 129. 216 A TREATISE 01V Ol'TICS. PART III. the earlh, surrounded with an atmosphere ABCD, consisting of four concentric strata of different densities and different refractive powers. The index of refraction for air at the earth's surface being 1-000,294, let us suppose that the index of the other three strata is 1 -000,200, 1 -000,120, 1-000,050. Let BED bo the horizon, and let a ray S ??, proceeding from the sun under the horizon, fall on the outer stratum at n, whose index of refraction is 1-000,050. Drawing the per- pendicular E n in, find by the rule formerly given the angle of refraction, E n a, corresponding to the angle of incidence S n in. When the ray n a falls on the second stratum at a, whose index of refraction is 1-000,120, we may in like manner, by drawing a perpendicular E«p, find the refracted ray a b. In the same way, the refracted rays b c and c d may be found. The same ray S n will therefore liave been re- fracted in a polygonal line n a b c d, and as it reaches the eye in the direction cd, the sun will be seen in the direction dc S', elevated above the horizon, by the refraction of the atmosphere, when it is still-betow it. In like manner it might be shown that the sun appears above the horizon by refraction, when he is actually below it at sunset. Although the rays of light move in straight lines in vacuo and in all media of uniform density, yet, on the surface of the globe, the rays proceeding from ;i distant object, must neces- sarily move in a curve line, because they must pass through portions of air of different densities and refractive powers. Hence it follows that, excepting in a vertical line, no object, whether it is a star or planet beyond our atmosphere, or is actually within it, is seen in its real place. Excepting in astronomical and trigonometrical observations, where the greatest accuracy is necessary, this refraction of the atmosphere does not occasion any inconvenience. But since the density of the air and its refractive power vary greatly when heated or cooled, great local heats or local colds will produce great changes of refractive power, and give rise to optical phenomena of a very interesting kind. Such phenom- ena have received the name of unusual refraction, and they are sometimes of such an extraordinary nature as to resemble more the effects of magic than the results of natural causes. (153.) The elevation of coasts, mountains, and ships, when seen over the surface of the sea, has long been observed and known by the name of looming. Mr. Huddart described several cases of this kind, but particularly the very interesting one of an inverted image of a ship seen beneath the real ship. J)r. Vince observed at Ramsgate a ship, whose topmasts only were seen above the horizon ; but he at the same time ob- CHAI\ XXXI. UNUSUAL REFRACTION. 217 served, in the field of the telescope through which he was looking, two images of the complete ship in the air, both di- rectly above the ship, the uppermost of the two being erect, and the other inverted. lie then directed his telescope to another ship whose hull was just in the horizon, and he ob- served a complete inverted image of it; the mainmast of which just touched the mainmast of the ship itself. The first of these two phe- nomena is shown in fig. 130. in which A is the real ship, and B, C the images seen by unusual refraction. Upon looking at another ship, Dr. Vince saw inverted images of some of its parts which sud- denly appeared and vanished, " first ap- pearing," says he, " below, and running up very rapidly, showing more or less of the masts at different times as they broke out, resembling in the swiftness of their breaking out the shooting of a beam of the aurora borealis." As the ship continued to descend, more of the image gradually ap- peared, till the image of the whole ship =rr was at last completed, with the mainmasts ~ in contact. When the ship descended still lower, the image receded from the ship, but no second image was seen. Dr. Vince observed another case, shown in fig. Fig. 131. 131., in which the sea was distinctly seen between the ships B, C. As the ship A came above the horizon, the image C gradually disappeared, and during this time the image B descended, but the ship did not seem so near the horizon as to bring the mainmasts together. The two images were visible when the whole ship was beneath the horizon. Captain Scoresby, when navigating the Greenland seas, observed several very interesting cases of unusual refraction. On the 28th of June, 1820, he saw from the mast-head eighteen sail of ships at the distance of about twelve miles. One of them was drawn out, or lengthened, in a vertical direction ; another was con- tracted in the same direction ; one had an inverted image immediately above it ; and other two had two distinct inverted T 218 A TREATISE ON OPTICS. TART III. images above them, accompanied with two images of the strata of ice. In 1822, Captain Scoresby recognized Iris father's ship, the Fame, by its inverted image in the air, although the ship itself was below the horizon. He afterwards found that the ship was seventeen miles beyond the horizon, and its dis- tance thirty miles. In all these cases, the image was directly above the object; but on the 17th of September, 1818, MM. J urine and Soret observed a case of unusual refraction, where the image was on one side of the object. A bark about 4(XX) toises distant was seen approaching Geneva by the left bank of the lake, and at the same moment there was seen above the water an image of the sails, which, in place of following the direction of the bark, receded from it, and seemed to ap- proach Geneva by the right bank of the lake ; the image sail- ing from east to west, while the bark was sailing from north to south. The image was of the same size as the object when it first receded from the bark, but it grew less and less as it receded, and was only one-half that of the bark when the phenomenon ceased. While the French army was marching through, the sandy deserts of Lower Egypt, they saw various phenomena of un- usual refraction, to which they gave the name of mirage. When the surface of the sand was heated by the sun, the land seemed to be terminated at a certain distance by a general inundation. The villages situated upon eminences appeared to be so many islands in the middle of a great, lake, and under each village there was an inverted image of it. As the army approached the boundary of the apparent inundation, the imaginary lake withdrew, and the same illusion appeared round the next village. M. Monge, who has described these appearances in the Memoires sur FEgypte, ascribes them to reflexion from a reflecting surface, which he supposes to take place between two strata of air of different densities. One of the most remarkable cases of mirage was observed by Dr. Vince. A spectator at Ramsgale sees the tops of the four turrets of Dover Castle over a hill between Ramsgate and Dover. Dr. Vince, however, on the 6th of August, 1806, at seven p. m., saw the whole of Dover Castle, as if it had been brought over and placed on the Ramsgate side of the hill. The image of it was so strong that the hill itself was not seen through the image. The celebrated fata morgana, which is seen in the straita of Messina, and which for many centuries astonished the vul- gar and perplexed philosophers, is obviously a phenomenon of this kind. A spectator on an eminence in the city of Reggio. with his back to the sun and his face to the sea, and when tha cii.vp. xxx r. UNUSUAL REFRACTION. 219 Fig. 132. rising sun shines from that point whence its incident ray forms an angle of about 45° on the sea of Iteggio, sees upon the water numberless series of pilasters, arches, castles well delineated, regular columns, lofty towers, superb palaces with balconies and windows, villages and trees, plains with herds and Hocks, armies of men on foot and on horseback, all passing rapidly in succession on the surface of the sea. These same objects are, in particular states of the atmosphere, seen in the air, though less vividly ; and when the air is hazy, they are seen on the surface of the sea, vividly colored, or fringed with all the prismatic colors. (154.) That the phenomena above described are generally produced by refraction through strata of air of different den- sities may be proved by various experiments. In order to illustrate "this, Dr. Wollaston poured into a square phial {fig. 132.) a small quantity of clear syrup, and above this he poured an equal quantity of water, which grad- ually combined with the syrup, as seen at A. The word Syrup upon a card held behind the bottle appeared erect when seen through the pure syrup, but inverted, as represented in the figure, when seen through the mixture of water and syrup. Dr. Wollaston then put nearly the same quantity of rectified spirit of wine above the water, as in the same figure at B, and he saw the appearance there represented, the true place of the word Spirit, and the inverted and erect images below. Analogous phenomena may be seen by looking at objects over the surface of a hot poker, or along the surface of a wall or painted board heated by the sun. The late Mr. II. Blackadder has described some phenomena both of vertical and lateral mirage as seen at King George's Bastion, Leith, which are very instructive. The extensive bulwark, of which this bastion forms the central part, is formed oi huge blocks of cut sandstone, and from this to the eastern end the phenomena are best seen. To the east of the tower the bulwark is extended in a straight line to the distance of 500 feet. It is eight feet high towards the land, with a foot- way about two feet broad, and three feet from the ground. The parapet is three feet wide at top, and is slightly inclined towards the sea. When the weather is favorable, the top of the parapet re- sembles a mirror, or rather a sheet of ice ; and if m this state another person stands or walks upon it, an observer at a little tmUm 220 A TREATISE ON OPTICS. PART III. distance will pro an inverted imago of the person under him. If, while standing on the footway another person stands on it also, but at some distance, with his face turned towards the sea, his image will appear opposite to him, giving the appear- ance of two persons talking or saluting each other. If, again, when standing on the footway, and looking in a direction from the tower, another person crosses the eastern extremity of the bulwark, passing through the water-gate, either to or from the sea, there is produced the appearance of two persons moving in opposite directions, constituting what has been termed a lateral mirage : first one is seen moving past, and then the other in an opposite direction, with some interval be- tween them. In looking over the parapet, distant objects are seen variously modified ; the mountains (in Fife) being con- verted into immense bridges ; and on going to the eastward extremity of the bulwark, and directing the eye towards the tower, the latter appears curiously modified, part of it being as it were cut off and brought down, so as to form another small and elegant tower in the form of certain sepulchral monuments. At other times it bears an exact resemblance to an ancient altar, the fire of which seems to burn with great intensity.* (15o.) In order to explain as clearly as possible how the erect and inverted image of a ship is produced as in fig. 131., let S P {Jig. 133.) be a ship in the horizon, seen at E by Fig. 133. means of rays S E, P E passing in straight lines through a track of air of uniform density lying between the ship and the eye. If the air is more rare at c than at a, which it may be from the coldness of the sea below a, its refractive power will * Edinburgh Journal of Science, No. V. p. 13 CHAP. XXXI. ITXrsVAL REFRACTION*. 221 be less at c than at a. In this case, rays S d, P c, which, nailer ordinary circumstances, never could have reached the eye at E, will be bent into curve lines P c, S d ; and if the variation of density is such that the uppermost of these rays S d crosses the other at any point x, then S d will be under- most, and will enter the eye E as if it came from the lower end of the object. If E p, E s, are tangents to these curves or rays, at the point where they enter the eye, the part S of the ship will be seen in the direction E s, and the part P in the direction E p; that is, the image s p will be inverted. In like manner, other rays, Sra, Pffl, may be bent into curves S n E, P m E, which do not cross one another, so that the tangent E s' to the curve or ray S n will still be uppermost, and the tangent E // undermost. Hence the observer at E will see an erect image of the ship at s' p' above the inverted image s p, as ini Jig: K31. It is quite clear that the state of the air may be such as to exhibit only one of these images, and that these a ppearanccs may be all seen when the real ship is beneath the horizon. In one of captain Scoresby's observations we have seen that the ship was drawn out, or magnified, in a vertical direc- tion, while another ship was contracted or diminished in the same direction. If a cause should exist, which is quite pos- sible, which elongated the ship horizontally at the same time that it elongatetd it vertically, the effect would be similar to that of a convex lens, and the ship would appear magnified, and might be recognized at a distance far beyond the limits of unassisted vision. This very case seems to have occurred. On the 26th July, 1798, at Hastings, at five p. m. Mr. Latham saw the French coast, which is about 40 or 50 miles distant, as distinctly as through the best glasses. The sailors and fish- ermen could not at first be persuaded of the reality of the ap- pearance ; but as the cliffs gradually appeared more elevated, they were so convinced that they pointed out and named to Mr. Latham the different places which they had been accustomed to visit : such as the bay, the windmill at Boulogne, St. Vallery, and other places on the coast of Picardy. All these places appeared to them as if they were sailing at a small distance into the harbor. From the eastern cliff or hill, Mr. Latham saw at once Dungeness, Dover cliffs, and the French coast, all the way from Calais, Boulogne, on to St. Vallery, and, as some of the fishermen affirmed, as far as Dieppe. The day was ex- tremely hot, without a breath of wind, and objects at some distance appeared greatly magnified. This class of phenomena may be well illustrated, as I have T2 222 A TREATISE ON OITIOS. PART IF I. elsewhere* suggested, by holding a mass of heated iron above a considerable thickness of water, placed in a glass trough, with plates of parallel glass. By withdrawing the heated iron, the gradation of density increasing downwards, will be accompanied by a decrease of density from the surface, and through such a medium the phenomena of the mirage may be seen. (156.) That some of the phenomena ascribed to unusual refraction are owing to unusual reflexion, arising from differ- ence of density, cannot, we think, admit of a doubt. If an observer beyond the earth's atmosphere at S, fig. 129., were to look at one composed of strata of different refractive powers as shown in the figure, it is obvious that the light of the sun would be reflected at its passage through the boundary of each stratum, and the same would happen if the variation of re- fractive power were perfectly gradual. Well described cases of this kind are wanting to enable us to state the laws of the phenomena; but the following fact, as described by Dr. Buchan, is so distinct, as to leave no doubt respecting its ori- gin. "Walking on the cliff," says he, "about a mile to the east of Brighton, on the morning of the 18th of November, 1804, while watching the rising of the sun, I turned my eyes directly towards the sea just as the solar disc emerged from the surface of the water, and saw the face of the cliff on which I was standing represented precisely opposite to me at some distance on the ocean. Calling the attention of my companion to this appearance, we soon also discovered our own figures standing on the summit of the opposite apparent cliff, as well as the representation of a windmill near at hand. The reflected images were most distinct precisely opposite to where we stood, and the false cliff seemed to fade away, and to draw near to the real one, in proportion as it receded towards the west. This phenomenon lasted about ten minutes, till the sun had risen nearly his own diameter above the sea. The whole then seemed to be elevated into the air, and suc- cessively disappeared, like the drawing up of a drop scene in a theatre. The surface of the sea was covered with a dense fog of many yards in height, and which gradually receded before the rays of the sun. The sun's light fell upon the cliif at an incidence of about 73° from the perpendicular." + Edinburgh Encyclopedia, art. Meat. CHAP. XXXII ON THE RAINBOW. 223 CHAP. XXXII. ON THE RAINBOW.* (157.) The rainbow is, as every person knows, a luminous arch extending across the region of the sky opposite to the sun. Under very favorable circumstances, two bows are seen, the inner and the outer, or the primary and the secondary, and within the primary rainbow, and in contact with it, and without the secondary one, there have been seen supernu- merary bows. The primary or inner rainbow, which is commonly seen alone, is part of a circle whose radius is 42°. It consists of seven differently colored bows, viz. violet, which is the inner- most, indigo, blue, green, yellow, orange, and red, which is the outermost. These colors have the same proportional breadth as the spaces in the prismatic spectrum. This bow is, therefore, only an infinite number of prismatic spectra, ar- ranged in the circumference of a circle; and it would be easy, by a circular arrangement of prisms, or by covering up all the central part of a large lens, to produce a small arch of exactly the same colors. All that 'we require, therefore, to forma rainbow, is a great number of transparent bodies capable of forming a great number of prismatic spectra from the light of the sun. As the rainbow is never seen, unless when rain is actually falling between the spectator and the sky opposite to the sun, we are led to believe that the transparent bodies required are drops of rain which we know to be small spheres. If we look into a globe of glass or water held above the head, and oppo- site to the sun, we shall actually see a prismatic spectrum re- flected from the farther side of the globe. In this spectrum the violet rays will be innermost, and the spectrum vertical. If we hold the globe horizontal on a level with the eye, so as to see the sun's light reflected in a horizontal plane, we shall see a horizontal spectrum with the violet rays innermost. In like manner, if we hold a globe in a position intermediate be- tween these two, so as to see the sun's light reflected in a plane inclined 45° to the horizon, we shall perceive a spec- trum inclined 45° to the horizon with the violet innermost. Now, since in a shower of rain there are drops in all positions relative to the eye, the eye will receive spectra inclined at all angles to the horizon, so that when combined they will form the large circular spectrum which constitutes the rainbow. * In the College edition, see Appendix of Am.ed. chap. vii. ¥%%%&& 224 A TREATISE ON OPTICS. PART III. Tc explain this more clearly, let E, F,fig. 134., be drops of rain exposed to the sun's rays, incident upon them in the Fig. 134. directions R E, R F ; out of the whole beam of light which falls upon the drop, those rays which pass through or near the axis of the drop will be refracted to a focus behind it, but those which fall on the upper side of the drop will be refracted, the red rays least, and the violet most, and will fall upon the back of the drop with an obliquity such that many of them will be reflected, as shown in the figure. These rays will be again refracted, and will meet the eye at O, which will per- ceive a spectrum or prismatic image of the sun, with the red space uppermost, and the violet undermost. If the sun, the eye, and the drops E, F, are all in the same vertical plane, the spectrum produced by E, F will form the colors at the very summit of the bow as in the figure. Let us now suppose a drop to be near the horizon, so that the eye, the drop, and the sun, are in a plane inclined to the horizon ; a ray of the sun's light will be reflected in the same manner as at E, F, with this difference only, that the plane of reflexion will be in- clined to the horizon, and will form part of the bow distant from the summit. Hence, it is manifest, that the drops of rain above the line joining the eye and the upper part of the rain- bow, and in the plane passing through the eye and the sun, will form the upper part of the bow : and the drops to the right and left hand of the observer, and without the line join- ing the eye and the lowest part of the bow, will form the lowest part of the bow on each hand. Not a single drop, therefore, between the eye and the space within the bow is concerned in its production : so that, if a shower were to fall regularly from a cloud, the rainbow would appear before a single drop of rain had reached the ground. Tf we -compute the inclination of the red ray and the violet chap, xxxrr. ON THE KAINBOW. 225 ray to the incident rays R E, R F, we shall find it to be 42° 2' for the red, and 40° 17' for the violet, so that the breadth of the rainbow will be the difference of those numbers, or 1° 45', or nearly three times and a half the sun's diameter. These results coincide so accurately with observation, as to leave no doubt that the primary rainbow is produced by two refractions and one intermediate reflexion of the rays that fall on the upper sides of the drops of rain. It is obvious that the red and violet rays will suffer a second reflexion at the points where they are represented as quitting the drop, but these reflected rays will go up into the sky, and cannot possibly reach the eye at O. But though this is the case with rays that enter the upper side of the drop as at E F, or the side farthest from the eye, yet those which enter it on the under side, or the side nearest the eye, may after two re- flexions reach the eye, as shown in the drops II, G, where the rays R, R enter the drops below. The red and violet rays will be refracted in different directions, and after being twice reflected will be finally refracted to the eye at O ; the violet forming- the upper part, the red the under part of the spectrum. If we now compute the inclination of these rays to the inci- dent rays R, R, we shall find them to be 50° 57' for the red ray, and 54° 7' for the violet ray ; the difference of which or 3° 10' will be the breadth of the bow, and the distance be- tween the bows will be 8° 55'.* Hence it is clear that a secondary bow will be formed exterior to the primary bow, and with its colors reversed, in consequence of their being produced by two reflexions and two refractions. The breadth of the secondary bow is nearly twice as great as that of the primary one, and its colors must be much fainter, because it consists of light that has suffered two reflexions in place of one. (15S.) Sir Isaac Newton found the semi-diameter of the in- terior bow to be 42°, its breadth 2° 10', and its distance from the outer bow 8° 30'; numbers which agree so well with the calculated results as to leave no doubt of the truth of the ex- planation which has been given. But if any farther evidence were wanted, it may be found in the fact, whicli I observed in 181% that the light of both the rainbows is wholly polarized in planes passing- through the eye and the radji of the arch. This result demonstrates that the bows are formed by reflexion at or near the polarizing angle, from the surface of a trans- parent body. The production of artificial rainbows by the spray of a waterfall, or by a shower of drops scattered by a mop, or forced out of a syringe, is another proof of the pre- * No correction for the sun's apparent diameter, it) here made. TRKATISE ON OPTICS. PART nr. ceding explanation. Lunar rainbows are sometimes seen, but the colors are faint, and scarcely perceptible. In 1814, 1 saw, at Berne, a fog-bow, which resembled a nebulous arch, in which the colors were invisible. (159.) On the 5th of July, 1828, I observed three supernu- merary bows within the primary bow, each consisting of green and red arches, and in contact with the violet arch of the primary bow. On the outside of the outer or secondary bow I saw distinctly a red arch, and beyond it a very feint green one, constituting a supernumerary bow, analogous to those within the primary rainbow. Dr. Halley has shown that the rainbow formed by three re- flexions within the drops will encircle the sun itself, at the distance of 40° 20', and that the rainbow formed by four re- flexions will likewise encircle him at the distance of 45° 33'. The rainbows formed by five reflexions will be partly covered by the secondary bow. The light which forms these three bows is obviously too faint to make any impression on our organs, and these rainbows have therefore never been observed. JVlany peculiar rainbows have been seen and described. On the 10th August, 1605, a faint rainbow was seen at Chartres, crossing the primary rainbow at its vertex. It was formed by reflexion from the river. On the 6th August, 1699, Dr. Halley, when walking on the walls of Chester, observed a remarkable rainbow, shown in fig. 135., where A B C is the primary bow, D H E the second- ary one, and AFHGC the new bow intersecting the second- ary bow D II E, and dividing it nearly into three parts. Dr, Halley observed the points F, G to rise, and the arch F G gradually to contract, till at length the two arches F H G and F G coincided, so that the secondary iris for a great space lost its colors, and appeared like a white arch at the top. The new bow, A H C, had its colors in the 6ame order as the primary CHAP. XXXIII. ON HALOS AND PARHELIA. 227 one ABC, and consequently the reverse of the secondary bow ; and on this account the two opposite spectra at G and F counteracted each other, and produced whiteness. The sun at this time shone on the river Dee, which was unruffled, and Dr. Halley found that the bow A H C was only that part of the circle of the primary bow that would have been under the castle bent upwards by reflexion from the river. A third rainbow seen between the two common ones, and not con- centric with them, is described in Rozier's Journal, and is doubtless the same phenomenon as that observed by Dr. Halley. Red rainbows, distorted rainbows, and inverted rainbows on the grass, have been seen. The latter are formed by the drops of rain suspended on the spiders' webs in the fields. CHAP. XXXIII. ON HALOS, CORON.E, IWRHELIA, AND PARASELENiE. (160.) When the sun and moon are seen in a clear sky, they exhibit their luminous discs without any change of color, and without any attendant phenomena. In other conditions of the atmosphere, the two luminaries not only experience a change of color, but are surrounded with a variety of luminous circles of various sizes and forms. When the air is charged witli dry exhalations, the sun is sometimes as red as blood. When seen through watery vapors, he is shorn of his beams, but preserves his disc white and colorless; while, in another state of the sky, I have seen the sun of the most brilliant salmon color. When light fleecy clouds pass over the sun and moon, they are often encircled with one, two, three, or even more, colored rings, like those of thin plates ; and in cold weather, when particles of ice are floating in the higher re- gions, the two luminaries are frequently surrounded with the most complicated phenomena, consisting of concentric circles, circles passing through their discs, segments of circles, and mock suns or moons, formed at the points where these circles intersect each other. The name halo is given indiscriminately to these phenom- ena, whether they are seen round the sun or the moon. They arc called parhelia when seen round the sun, and paraselene; when seen round the moon. The small halos seen round the sun and moon in fine wea- ther, when they are partially covered with light fleecy clouds, have been also called corona. They are very common round ■m& 228 A TREATISE ON OPTICS. PART III. the sun. though, from the overpowering brightness of his rays, they are best seen when he is observed by reflexion from the surface of still water. In June, 1692, Sir Isaac Newton ob- served, by reflexion in a vessel of standing water, tbree rings of color round the sun, like three little rainbows. The colors- of the first or innermost were blue next the sun, red without, and white in the middle between the blue and red. The colors of the second ring were purple and blue within, and pale red without, and green in the middle. The colors of the third ring were pale blue within and pale red without. The colors and diameters of the rings are more particularly given as fol- lows : — 1st Ring - Blue, white, red - Diameter, 5° to 6°. 3d Ring - Pale blue, pale red - Diameter, 12°. On the 19th February, 1664, Sir Isaac Newton saw a halo round the moon, of two rings, as follows : — 1st Ring - White, bluish green, yellow, red - Diameter, 3° 2d Ring - Blue, green, red Diameter, 5^° Sir Isaac considers these rings as formed by the light pass- ing through very small drops of water, in the same manner as the colors of thick plates. On the supposition that the glob- ules of water are the 500th of an inch in diameter, he finds that the diameters of the rings should be as follows : — 1st Red ring Diameter, 1\° 2d Red ring Diameter, 1 \° 3d Red ring Diameter, 12° 33' The rings will increase in size as the globules become less, and diminish if the globules become larger. The halos round the sun and moon, which have excited most notice, are those which are about 47° and 94° in diame- ter. In order to form a correct idea of them, we shall give accurate descriptions of two ; one a parhelion, and the other a paraselene. The following is the original account of a parhelion, seen by Scheiner in 1630: — (161.) "The diameter of the circle MQN next to the sun, was about 45°, and that of the circle O R P was about 95° 20'; they were colored like the primary rainbow ; but the red was next the sun, and the other colors in the usual order. The breadths of all the arches were equal to one another, and about a third part less than the diameter of the sun, as repre- sented in Jig. 136. ; though I cannot say but the whitish circle CHAP. XXXIII. ON HALOS AND PARHELIA. 229 O G P, parallel to the horizon, was rather broader than the rest. The two parhelia M, N were lively enough, but the Fig. 136. other two at O and P were not so brisk. M and N had a pur- ple redness next the sun, and were white in the opposite parts. O and P were all over white. They all differed in their du- rations ; for P, which shone but seldom and but faintly, van- ished first of all, being covered by a collection of pretty thick clouds. The parhelion O continued constant for a great while, though it was but faint. The two lateral parhelia M, N wen; seen constantly for three hours together. M was in a lan- guishing state, and died first, after several struggles, but N continued an hour after it at least. Though 1 did not see tlte last end of it, yet I was sure it was the only one that accom- panied the true sun for a long time, having escaped those clouds and vapors which extinguished the rest. However, it vanished at last, upon the fall of some small showers. This phenomenon was observed to last 4£ hours at least, and since it appeared in perfection when I first saw it, I am persuaded its whole duration might be above five hours. " The parhelia Q, R were situated in a vertical plane pass- ing through the eye at F, and the sun at G, in which vertical the arches II RC, O R P either crossed or touched one an- other. These parhelia were sometimes brighter, sometimes fainter than the rest, but were not so perfect in their shape and whitish color. They varied their magnitude and color ac- cording to the different temperature of the sun's light at G, and the matter that received it at Q and R; and therefore - U 230 A TKEATISE ON OPTICS. PART III. their light and color were almost always fluctuating, and con- tinued, as it were, in a perpetual conflict. I took particular notice that they appeared almost the first and last of the par- helia, excepting that of N. " The arches which composed the small halo M N next to the sun, seemed to the eye to compose a single circumferpnee, but it was confused, and had unequal breadths; nor did it con- stantly continue like itself, but was perpetually fluctuating. But in reality it consisted of the arches expressed in the figure, as I accurately observed for this very purpose.* These arches cut each other in a point at Q, and there they formed a parhelion ; the parhelia M, N shining from the common inter- sections of the inner halo, and the whitish circle ONM P." (162.) Hevelius observed at Dantzic, on the 30th of March, 1660, at one A. M., the paraselene shown in fig. 137. The moon A was seen surrounded by an entire whitish circle Fig. 137. B C D E, in which there were two mock moons at B and D ; one at each side of the moon, consisting of various colors, and shooting out very long and whitish beams by fits. At about two o'clock a larger circle surrounded the lesser, and reached to the horizon. The tops of both these circles were touched by colored arches, like inverted rainbows. The inferior arch at C was a portion of a large circle, and the superior at F a portion of a lesser. This phenomenon lasted nearly three hours. The outward great circle vanished first. Then the larger in- verted arch at C, and then the lesser ; and last of all the inner * The four intersecting circles which form this inner halo are described from four centres, one at each angle of a small square. CHAP. XXXIII. ON HALOS AND PARHELIA. 231 circle BCDE disappeared. The diameter of this inner circle, and also of the superior arch, was 45°, and that of the exterior circle and inferior arch was 90°. On another occasion Hevelins observed a large white rec- tangular cross passing through the disc of the moon, the moon being in the intersection of the cross, and encircled with a halo exactly like the inner one in the preceding figure. (163.) The frequent occurrence of the halos of 47° and 94° in coll weather, and especially in the northern regions of the globe, led to the belief that they must be formed by crystals of ice and snow floating in the air. Descartes supposed that they were produced by refraction, through flat stars of pellucid ice ; and Huygens, who investigated the subject both experi- mentally and theoretically, has published an elaborate theory of halos, in which he assumes the existence of particles of hail, some of which are globular and others cylindrical, with an opaque nucleus or kernel having a certain proportion to the whole. lie supposes these cylinders to be kept in a vertical position, by ascending currents of air or vapor, and to have their axes at all possible inclinations to the horizon, when they are dispersed by the wind or any other causes. He considers these cylinders to have been at first a globular collection of the softest and purest particles of snow, to the bottom of which other particles adhere, the ascending currents preventing them from adhering to the sides ; they will, therefore, assume a cylindrical shape. Huygens then supposes that the outer part of the cylinders may be melted by the heat of the sun, a small cylinder remaining unmelted in the centre, and that if the melted part is again frozen, it may have sufficient trans- parency to refract and reflect the rays of the sun in a regular manner. By means of this apparatus, the existence of which is not impossible, Huygens has given a beautiful solution of almost all the difficulties which have been encountered in ex- plaining the origin of halos. Sir Isaac Newton regarded the halo of 45° as produced by a different cause from the small prismatic corona? ; and he was of opinion that it arose from refraction " from some sort of hail or snow floating in the air in a horizontal posture, the re- fracting angle being about 58° or 60°." When we consider, however, the great variety of crystal- lino forms which water assumes in freezing ; that these crys- tals really exist in a transparent state in the atmosphere, in the form of crystals of ice, which actually prick the skin like needles ; and that simple and compound crystals of snow, of every conceivable variety of shape, are often falling through the atmosphere, and sometimes melting in passing through its 232 A TREATISE ON OPTICS. TART III. lower and warmer strata, we do not require any hypothetical cylinders to account for the principal phenomena of halos. Mariotte, Young, Cavendish, and others, have agreed in ascribing the halo of 45° or 46° in diameter, to refraction through prisms of ice, with refracting angles of 60° floating in the air, and having their refracting angles in all directions. The cry stals of hoar-frost have actually such angles, and if we compute the deviation of the refracted rays of the sun or moon incident upon such a prism, with the index of refraction for ice, taken at 1-31, we shall rind it to be 21° 50', the double of which is 43° 40'. In order to explain the larger halo, Dr. Young supposes that the rays which have been once refracted by the prism may fall on other prisms, and the effect then be doubled by a second refraction, so as to produce a deviation of 90°. This, however, is by no means probable, and Dr. Young has candidly acknowledged the " great apparent prob- ability" of Mr. Cavendish's suggestion, that the external halo may be produced by the refraction of the rectangular termi- nations of the crystals. With an index of refraction of 1 # 31, this would give a deviation of 45° 44', or a diameter of 91° 28', and the mean of several accurate measures is 91° 40', a very remarkable coincidence. The existence of prisms with such rectangular terminations is still hypothetical ; but I have removed the difficulty on this point, by observing in the hoar-frost upon stones, leaves, and wood, regular quadrangular crystals of ice, both simple and compound. Although halos are generally represented as circles, with the sun or moon in their centres, yet their apparent form is commonly an irregular oval, wider below than above, the sun being nearer their upper than their lower extremity. Dr. Smith has shown that this is an optical deception, arising from the apparent figure of the sky, and he estimates that when the circle touches the horizon, its apparent vertical diameter is divided by the moon, in the proportion of about 2 to 3 or 4 ; and is to the horizontal diameter drawn through the moon as 4 to 3, nearly. With the view of ascertaining if any of the halos are form- ed by reflexion, I have examined them with doubly refracting prisms, and have found that the light which forms them has not suffered reflexion. The production of halos may be illustrated experimentally by crystallizing various salts upon plates of glass, and looking through the plates at the sun or a candle. When the crystals are granular and properly formed, they will produce the finest effects. A few drops of a saturated solution of alum, for ex- CHAP. XXXIH. ON HALOS AND PARHELIA. 233 ample, spread over a plate of glass so as to crystallize quickly, will cover it with an imperfect crust, consisting of flat octo- hedral crystals, scarcely visible to the eye. When the ob- server, with his eye placed close behind the smooth side of the glass plate, looks through it at a luminous body, he will per- ceive three fine halos at different distances, encircling the source of light. The interior halo, which is the whitest of the three, is formed by the refraction of the rays through a pair of faces in the crystals that are least inclined to each other. The second halo, which is blue without and red within, with all the prismatic colors, is formed by a pair of more in- clined faces; and the third halo, which is large and brilliantly colored, from the increased refraction and dispersion, is formed by the most inclined faces. As each crystal of alum has three pairs of each of these included prisms, and as these refracting faces will have every possible direction to the horizon, it is easy to understand how the halos are completed and equally luminous throughout. When the crystals have the property of double refraction, and when their axis is perpendicular to the plates, more beautiful combinations will be produced. (164.) Among the luminous phenomena of the atmosphere, we may here notice that of converging and diverging solar beams. The phenomenon of diverging beams, represented in Jig. 138., is of frequent occurrence in summer, and when the sun is near the horizon ; and arises from a portion of the sun's Fig. 138. rays passing through openings in the clouds, while the adjacent portions are obstructed by the clouds. The phenomenon of converging beams, which is of much rarer occurrence, is U2 ggm* 234 A TREATISE ON OPTICS. PART III. shown in fig. 139., where the rays converge to a point. A, as far below the horizon M N as the sun is above it. This phe- nomenon is always seen opposite to the sun, and generally at Fig. 139. Mr IN the same time with the phenomenon of diverging beams, as if another sun, diametrically opposite to the real one, were below the horizon at A, and throwing out his divergent beams. In a phenomenon of this kind which I saw in 1824, the eastern portion of the horizon where it appeared was occupied with a black cloud, which seems to he necessary as a ground, for rendering visible such feeble radiations. A few minutes after the phenomenon was first seen, the converging lines were black, or very dark ; an effect which seems to have arisen from the luminous beams having become broad and of unequal intensity, so that the eye took up, as it were, the dark spaces between the beams more readily than the luminous beams themselves. This phenomenon is entirely one of perspective. Let us suppose beams inclined to one another like the meridians of a globe to diverge from the sun, as these meridians diverge from the north pole of the globe, and let us suppose that planes pass through all these meridians, and through the line joining the observer and the sun or their common intersection. An eye, therefore, placed in that line, or in the common intersec- tion of all the fifteen planes, will see the fifteen beams con- verging to a point opposite the sun, just as an eye in the axis of a globe would see all the fifteen meridians of the globe converge to its south pole. If we suppose the axis of a globe or of an armillary sphere to be directed to the centres of the diverging and converging beams, and a plane to pass through CHAP. XXXIV. COLORS OP NATURAL BODIES. 235 the globe parallel to the horizon, it would cut off the meri- dians so as to exhibit the precise appearances in Jig. 138. and fig. 139. ; with this difference only, that there would be fifteen beams in the diverging system in the place of the number shown in Jig. 139. CHAP. XXXIV. ON THE COLORS OF NATURAL BODIES. (165.) There is no branch of the application of optical science which possesses a greater interest than that which proposes to determine the cause of the colors of natural bodies. Sir Isaac Newton was the first who entered into an elaborate investigation of this difficult subject; but though his specula- tions are marked with the peculiar genius of their author, yet they will not stand a rigorous examination under the lights of modern science. That the colors of material nature are not the result of any quality inherent in the colored body has been incontrovertibly proved by Sir Isaac. He found that all bodies, of whatever color, exhibit that color only when they are placed in white light. In homogeneous red light they appeared red, in violet light violet, and so on ; their colors being always best displayed when placed in their own daylight colors. A red wafer, for example, appears red in the white light of day, because it re- flects red light more copiously than any of the other colors. [f we place a red wafer in yellow light, it can no longer ap- pear red, because there is not a particle of red light in the yellow light which it could reflect. It reflects, however, a portion of yellow light, because there is some yellow in the red which it does reflect. If the red wafer had reflected no- thing but pure homogeneous red light and not reflected white light from its outer surface, which all colored bodies do, it would in that case have appeared absolutely black when placed in yellow light. The colors, therefore, of bodies arise from their property of reflecting or transmitting to the eye certain rays of white light, while they stifle or stop the re- maining rays. To this point the Newtonian theory is support- ed by infallible experiments; but the principal part of the theory, which has for its object to determine the manner in which particular rays are stopped, while others are reflected or transmitted, is not so well founded. 230 A TREATISE ON OPTICS. TART III As Sir Isaac has stated the principles of his theory with the greatest clearness, we shall give them in his own words. "1st, Those superficies of transparent bodies reflect the greatest quantity of light which have the greatest refracting power ; that is, which separate media that difier most in their refracting power. And in the confines of equally refracting media there is no reflexion. "2d, The least parts of almost all natural bodies are in some measure transparent ; and the opacity of these bodies arises from the multitude of reflexions caused in their internal parts. " 3d, Between the parts of opaque and colored bodies are many spaces, either empty, or replenished with mediums of other densities; as water between the tinging corpuscles wherewith any liquor is impregnated ; air between the aqueous globules that constitute clouds or miste ; and for the most part spaces, void of both air and water, but yet perhaps not wholly void of all substance, between the parts of all bodies. " 4th, The parts of bodies and their interstices must not be less than of some definite bigness, to render them opaque and colored. "5th, The transparent parts of bodies, according to their several sizes, reflect rays of one color, and transmit those of another, on the same grounds that thin plates or bubbles do reflect or transmit these rays; and this I take to be the ground of all their colors." " 6th, The parts of bodies on which their colors depend are denser than the medium which pervades their interstices. " 7th, The bigness of the component parts of natural bodies may be conjectured by their colors." Upon these principles Sir Isaac has endeavored to explain the phenomena of transparency, black and white opacity, and color. He regards the transparency of water, salt, glass, stones, and such like substances, as arising from the smallness of their particles, and the intervals between them ; for though he considers them to be as full of pores or intervals between the particles as other bodies are, yet he reckons the particles and their intervals to be too small to cause reflexion at their common surfaces. Hence it follows, from the table in page 93, that the particles of air and their intervals cannot exceed the half of a millionth part of an inch ; the particles of water the gth of a millionth, and those of glass the |d of a millionth; because at these thicknesses the light reflected is nothing, or the very black of the first order. The opacity of bodies, such as that of white paper, linen, &c, is ascribed by Newton to a CHAP. XXXIV. COLORS OF NATURAL BODIES. 237 greater size of the particles and their intervals, viz. such a size as to reflect the white, which is a mixture of the colors of the different orders. Hence in air they must exceed 77 rnillionths of an inch, in water 57 millionths, and in glass 50 millionths. In like manner all the different colors in Newton's table are supposed to be produced when the particles and their in- tervals have an intermediate size between that which pro- duces transparency and that which produces white opacity. If a film of mica, for example, of an uniform blue color, is cut into the smallest pieces of the same thickness, every piece will keep its color, and a heap of such pieces will constitute a mass of the same color. So far the Newtonian theory is plausible ; but in attempting to explain black opacity, such as that of coal and other bodies absolutely impervious to light, it seems to fail entirely. To produce blackness, " the particles must be less than any of those which exhibit color. For at all greater sizes there is too much light reflected to constitute this color ; but if they be supposed a little less than is requisite to reflect the white and very faint blue of the first order, they will reflect so very littlo light as to appear intensely black." That such bodies will be black when seen by reflexion is evident ; but what becomes of all the transmitted light ? This question seems to have perplexed Sir Isaac. The answer to it is, " it may perhaps be variously refracted to and fro within the body, until it happens to be stifled and lost ; by which means it will appear intensely black." In this theory, therefore, transparency and blackness are supposed to be produced by the very same constitution of the body ; and a refraction to and fro is assumed to extinguish the transmitted light in the one case, while in the other such a refraction is entirely excluded. In the production of colors of every kind, it is assumed that the complementary color, or generally one half of the ljght, is lost by repeated reflexions. Now, as reflexion only changes the direction of light, we should expect that the light thus scattered would show itself in some form or other ; but though many accurate experiments have been made to discover it, it has never yet been seen. For these and other reasons,*! which it would be out of place here to enumerate, I consider the Newtonian theory of * See a more detailed examination of the theory in my Life of Sir Isaac Newton. t For an account of Sir David Brewster's outline of a new theory of t!ia colors of natural bodies, see Note VII. of Am. ed. %>& A TREATISE ON OPTICS. TART III. 238 colors as applicable only to a small class of phenomena, while it leaves unexplained the colors of fluids and transparent solids, and all the beautiful hues of the vegetable kingdom. In numerous experiments on the colors of leaves, and on the juices expressed from them, I have never been able to see the complementary color which disappears, and I have almost in- variably found that the transmitted and the reflected tint is the same. Whenever there was an appearance of two tints, I have found it to arise from there being two differently color- ed juices existing in different sides of the leaf. The New- tonian theory is, we doubt not, applicable to the colors of the wings of insects, the feathers of birds, the scales of fishes, the oxidated films on metal and glass, and certain opalescences. The colors of vegetable life and those of various kinds of solids arise, we are persuaded, from a specific attraction which the particles of these bodies exercise over the differently col- ored rays of light. It is by the light of the sun that the colored juices of plants are elaborated, that the colors of bodies are changed, and that many chemical combinations and decompo- sitions are effected. It is not easy to allow that such effects can be produced by the mere vibration of an ethereal medium; and we are forced, by this class of facts, to reason as if light was material. When a portion of light enters a body, and is never again seen, we are entitled to say that it is detained by some power exerted over the light by the particles of the body. That it is attracted by the particles seems extremely probable, and that it enters into combination with them, and produces various chemical and physical effects, cannot well be doubted ; and without knowing the manner in which this combination takes place, we may say that the light is absorbed, which is an accurate expression of the fact. Now, in the case of water, glass, and other transparent bodies, the light which enters their substance has a certain small portion of its particles absorbed, and the greater part of it which escapes from absorption, and is transmitted, comes out colorless, because the particles have absorbed a propor- tional quantity of all the different rays which compose white light, or, what is the same thing, the body has absorbed white light. In all colored solids and fluids in which the transmitted light has a specific color, the particles of the body have ab- sorbed all the rays which constitute the complementary color, detaining sometimes all the rays of a certain definite refran- gibility, a portion of the rays of other refrangibilities, and al- lowing other rays to escape entirely from absorption ; all the rays thus stopped will form by their union a particular com- CHAP. XXXIV. COLORS OF NATURAL BODIES. 239 pound color, which will be exactly complementary to the color of the transmitted rays. In black bodies, such as coal, &c, all the rays which enter their substance are absorbed ; and hence we see the reason why such bodies are more easily heated and inflamed by the action of the luminous rays. The influence exercised by heat and cooling upon the absorptive power of bodies furnishes an additional support to the preceding views. ^ (166.) Before concluding this chapter, we may mention a few curious facts relative to white opacity, black opacity, and color, as exhibited by some peculiar substances. 1st, Tabasheer, whose refractive power is 1-111, between air and water, is a silicious concretion found in the joints of the bamboo. The finest varieties reflect a delicate azure color, and transmit a straw-yellow tint, which is complement- ary to the azure. When it is slightly wetted with a wet needle or pin, the wet spot instantly becomes milk white and opaque. The application of a greater quantity of water re- stores its transparency. 2dly, The cameleon mineral is a solid substance made by heating the pure oxide of manganese with potash. When it is dissolved in a little warm water, the solution changes its color from green to blue and purple, the last descending in the order of the rings, as if the particles became smaller. 3dly, A mixture of oil of sweet almonds with soap and sul- phuric acid is, according to M. Claubry, first yellow, then orange, red, and violet. In passing from the orange to the red, the mixture appears almost black. 4thly, If, in place of oil of almonds, in the preceding ex- periment, we employ the oily liquid obtained from alcohol heated with chlorine, the colors of the mixture will be pcle yellow, orange, black, red, violet, and beautiful blue. 5thiy, Tincture of turnsole, after having been a consider- able time shut up in a bottle, has an orange color; but when the bottle is opened and the fluid shaken, it becomes in a few minutes red, and then violet-blue. 6thly, A solution of hcematine in water containing some drops of acetic acid is a greenish yellow. When introduced into a tube containing mercury, and heated by surrounding it with a hot iron, it assumes the various colors of yellow, orange, red, and purple, and returns gradually to its primitive tint. ?thly, Several of the metallic oxides exhibit a temporary change of color by heat, and resume their original color by cooling. M. Chevreul observed, that when indigo, spread upon pi'per, is volatilized, its color passes into a very brilliant 240 A TREATISE ON OPTICS. PAET III. poppy-red. The yellow phosphate of lead grows green when hot. 8thly, One of the most remarkable facts, however, is that discovered by M. Thenard. He found that phosphorus, purified by repeated distillations, though naturally of a whitish yellow color when allowed to cool slowly, becomes absolutely black when thrown melted into cold water. Upon touching some little globules that still remained yellow and liquid when he was repeating this experiment, M. Biot found that they in- stantly became solid and black. CHAP. XXXV. ON THE EYE AND VISION. An account of the structure and functions of the human eye, that masterpiece of divine mechanism, forms an interest- ing branch of applied optics. This noble organ, by means of which we acquire so large a portion of our knowledge of the material universe, is represented in Jigs. 140. and 141., the former being a front and external view of it, and the latter a section of it through all its humors. The human eye is of a spherical form, with a slight pro- jection in front. The eyeball or globe of the eye consists of four coats or membranes, which have received the names of the sclerotic coat, the choroid coat, the cornea, and the retina ; and these coats inclose three humors, — the aqueous humor, the vitreous humor, and the crystalline humor, the last of which has the form of a lens. The sclerotic coat, a a a a, or the outermost, is a strong and tough membrane, to which are attached all the muscles which give motion to the eyeball, Fig. 140. CHAP. XXXV. DESCRIPTION OF THE EYE. 241 and it constitutes the white of the eye, a a, Jig. 140. The cornea, b b, is the clear and transparent coat which forms the Fig. 141. front of the cyehall, and is the first optical surface at which the rays of light are refracted. It is firmly united to the sclerotic coat, filling up, as it were, a circular aperture in its front. The cornea is an exceedingly tough membrane, of equal thickness throughout, and composed of several firmly adhering layers, capable of opposing great resistance to ex- ternal injury. The choroid coat is a delicate membrane lining the inner surface of the sclerotic, and covered on its inner surface with a black pigment. Immediately within this pig- ment, and close to it, lies the retina, rrr, which is the inner- most coat of all. It is a delicate reticulated membrane, formed by the expansion of the optic nerve, O O, which enters the eye at a point about y'^ of an inch from the axis on the side next the nose. At the extremity of the axis of the eye, in a line passing through the centre of the cornea, and perpendicular to its surface, there is a small hole with a yellow margin, called the foramen centrale, which, notwithstanding its name, is not a real opening, but only a transparent spot, free from the soft pulpy matter of which the retina is composed. In looking through the cornea from without, we perceive a flat f ircular membrane, ef, Jig. 141., or within, b b, Jig. 140., which is grey, blue, or black, and divides the anterior of the eye into two very unequal parts. In the centre of it there is a circular opening, d, called the pupil, which widens or ex- pands when a small portion of light enters the eye, and closes or contracts when a great quantity of light enters. The two parts into which the iris divides the eye are called the anterior and the posterior chambers. The anterior chamber, which is anterior to the iris, ef, contains the aqueous humor; and the posterior chamber, which is posterior to the iris, contains the crystalline and vitreous humors, the last of which fills a great portion of the eyeball. V 242 A TREATISE ON OPTICS. PART III. The crystalline lens, c c,fig. 141., is a more solid substance than either the aqueous or the vitreous humor. It is suspended in a transparent bag or capsule by the ciliary processes, g g, which are attached to every part of the margin or circumfer- ence of the capsule. This lens is more convex behind than before ; the radius of its anterior surface being 0-30 of an inch, and that of its posterior surface 0-22 of an inch. The lens increases in density from its circumference to its centre, and possesses the doubly refracting structure. It consists of con- centric coats, and these are again composed of fibres. The vitreous humor, V V, is contained in a capsule, which is sap- posed to be divided into several compartments. The total length of the eye from O to b is about 0-91 of an inch; the principal focal distance of the lens, c c, is 1*78; and the range of the moving eyeball, or the diameter of the held of distinct vision, is 110°. The field of vision is 50° above a horizontal line and 70° below it, or altogether 120° in a ver- tical plane. It is 60° inwards and 90° outwards, or altogether in a horizontal plane 150°. I have found the following to be the refractive powers of the different humors of the eye ; the ray of light being inci- dent upon them from air : — Aqueous Crystalline Lens. Humor. Surface. Centre. Mean. 1-33G6. 1-3767. 1-3990. 1-3839. Vitreous Humor. 1-3394. But as the rays refracted by the aqueous humor pass into the crystalline, and those from the crystalline into the vitreous humor, the indices of refraction of the separating surface of each of these humors will be : — From aqueous humor to outer coat of the crystalline 1-0300 From do. to crystalline, using the mean index - - 1-0353 From crystalline outer coat to vitreous 0-9729 From do. to do. using the mean index - - - 0-9 G79 As the cornea and crystalline lens must act upon the rays of light which fall upon the eye exactly like a convex lens, inverted images of external objects will be formed upon the retina r r r in precisely the same manner as if the retina were a piece of white paper in the focus of a single lens placed at d. There is this difference, however, between the two cases, that in the eye the spherical aberration is corrected by means of the variation in the density of the crystalline lens, which, having a greater refractive power near the centre of its mass, refracts the central rays to the same point as the rays which pass through it near its circumference c c. No provision, however, is made in the human eye for the correction of color, CHAP. XXXV. ON THE SEAT OF VISION. 243 because the deviation of the differently colored rays is too small to produce indistinctness of vision. If we shut up all the pupil excepting a portion of its edge, or look past the ringer held near the eye, till the finger almost hides a narrow line of white light, we shall see a distinct prismatic spectrum of this line containing all the different colors; an effect which could not take place if the eye were achromatic. That an inverted image of external objects is formed on the retina has been often proved, and may be ocularly demon- strated by taking the eye of an ox, and paring away with a sharp instrument the sclerotic coat till it becomes thin enough to see the image through it. Beyond this point optical science cannot carry us. In what manner the retina conveys to the brain the impressions which it receives from the rays of light we know not, and perhaps never shall know. On the Phenomena and Laws of Vision (107.) 1. On the seat of vision. — The retina, from its deli- cate structure, and its proximity to the vitreous humor, had always been regarded as the seat of vision, or the surface on which the refracted rays were converged to their foci, for the purpose of conveying the impression to the brain, till M. Mariotte made the curious discovery that the base of the optic nerve, or the circular section of it at O, fig. 141., was in- capable of conveying to the brain the impression of distinct vision. He found that when the image of any external object fell upon the base of the optic nerve, it instantly disappeared. In order to prove this, we have only to place upon the wall, at the height of the eye, three wafers, two feet distant from each other. Shutting one eye, stand opposite to the middle wafer, and while looking at the outside wafer on the same hand as the shut eye, retire gradually from the wall till the middle wafer disappears. This will happen at about five times the distance of the wafers, or ten feet from the wall ; and when the middle wafer vanishes, the two outer ones will be distinctly seen. If candles are substituted for wafers, the middle candle will not disappear, but it will become a cloudy mass of light. If the wafers are placed upon a colored wall, the spot occu- pied by the wafer will be covered by the color of the wall, as if the wafer itself had been removed. According to Daniel Bernoulli, the part of the optic nerve insensible to distinct impressions occupies about the seventh part of the diameter of the eye, or about the eighth of an inch. This unfitness of the base of the optic nerve for giving 244 A TRKATISK ON T OPTICS. PART III. distinct vision, induced Mariotle to believe that the choroid coat, which lies immediately below the retina, performs the functions ascribed to the retina; for where there was no choroid coat there was no distinct vision. The opacity of the choroid coat and the transparency of the retina, which render- ed it an unfit ground for the reception ot" images, were argu- ments in favor of this opinion. Comparative anatomy furnishes us with another argument, perhaps even more conclusive than any of those urged by Marietta. In the eye of the sepia loligo, or cuttle-fish, an opaque membranous pigment is inter- posed between the retina and the vitreous humor;* so that, if the retina is essential to vision, the impressions of the image on this black membrane must be conveyed to the retina by the vibrations of the membrane in front of it .Now, since the human retina is transparent, it will not prevent the-imagea of objects from being formed on the choroid coat ; and the vibrations which they excite in this membrane, being com- municated to the retina, will be conveyed to the brain. These views are strengthened by another fact of some interest. I have observed in young persons, that the choroid coat (which is generally supposed to be black, and to grow fainter by age,) reflects a brilliant crimson color, like that of dogs and other animals. Hence, if the retina is affected by rays which pass through it, this crimson light which must necessarily be trans- mitted by it ought to excite the sensation of crimson, which 1 find not to be the case. A French writer, M. Lehot, has recently written a work, endeavoring to prove that the seat of vision is in the vitreous humor ; and that, in place of seeing a flat picture of the ob- ject, we actually see an image of three dimensions, viz. with length, breadth, and thickness. To produce this effect, lie supposes that the retina sends out a number of small nervous filaments, which extend into the vitreous humor, and convey to the brain the impressions of all parts of the image. If this theory were true, the eye would not require to adjust itself to different distances; and we liesides know for certain, that the eye cannot see with equal distinctness two points of an object at different distances, when it sees one of them perfectly- M. Lehot might indeed reply to the first of these objections, that the nervous filaments may not extend far enough into the vitreous humor to render adjustment unnecessary ; but if we admit this, we would be admitting an imperfection of work- manship, in so far as the Creator would then be employing two Dr. Knox, Edinb. Journal of Science, No. VI. p. J99. CHAP. XXXV. LAW OF VISIBLE DIRECTION. 245 kinds of mechanism to produce an effect which could have boon easily produced by cither of them separately. As difficulties still attach to every opinion respecting the seat of vision, we shall still adhere to the usual expression used by all optical writers, viz. that the images of objects are painted on the retina. (108.) 2. On the law of visible direction. — When a ray of light falls upon the retina, and gives us vision of the point of an object from which it proceeds, it becomes an interesting question to determine in what direction the object will be seen, reckoning from the point where it falls upon the retina. In fig. 142., let F be a point of the retina on which the image of a point of a distant object is formed by means of the crystalline Fig. 1 12. lens, supposed to be at L L. Now, the rays which form the image of the point at F fall upon the retina in all possible di- rections from L F to L F, and we know that the point F is seen in the direction F C R. In the same manner, the points ff are seen somewhere in the directions f S,fT. These lines F R,/' &,fT', which may be called the lines of visible (Unction, may either be those which pass through the centre C of the lens L L, or, in the case of the eye, through the centre of a lens equivalent to all the refractions employed in producing the image; or it may be the resultant of all the directions within the angles L F L, \jf L; or it may be a line perpendicular to the retina at F,f'f. In order to determine this point, let us look over the top of a card at the point of the object whose image is at F till the edge of the card is just about to hide it, or. what is the same thing, let us obstruct, all the rays that pass through the pupil excepting the upper ones, R L, R C ; we shall then find that the point whose image is at F, is seen in the same direction as when it was seen by all the rays L F, F, L F. If we look beneath the card in a similar manner, so as to see the object by the lower rays, R L F, R C F V2 ^^mm 246 A TREATISE ON OPTICS. PAKT III. we shall see it in the same direction. Hence it is manifest that the line of visible direction does not depend on the direc- tion of the ray, bat is always perpendicular to the retina. This important truth in the physiology of vision may be proved in another way. If we look at the sun over the top of a card, as before, so as to impress the eye with a permanent spectrum by means of rays L F falling obliquely on the retina, this spectrum will be seen along the axis of vision F C. In like manner, if we press the eyeball at any part where the retina is, we shall see the luminous impression which is produced, in a direction perpendicular to the point of pressure ; and if we make the pressure with the head of a pin, so as to press either obliquely or perpendicularly, we shall find that the luminous spot has the same direction. Now, as the interior eyeball is as nearly as possible a perfect sphere, lines perpendicular to the surface of the retina must all pass through one single point, namely, the centre of its spherical surface. This one point may be called the centre of visible direction, because every point of a visible object will be seen in the direction of a line drawn from this centre to the visible point. When we move the eyeball by means of its own muscles through its whole range of 110°, every point of an object within the area of the visible field either of distinct or indistinct vision remains absolutely fixed, and this arises from the immobility of the centre of visible direction, and, consequently, of the lines of visible direction joining that centre and every point in the visible field. Had the centre of visible direction been out of the centre of the eyeball, this perfect stability of vision could not have existed. If we press the eye with the finger, we alter the spherical form of the surface of the retina ; we consequently alter the direction of lines perpendicular to it, and also the centre where these lines meet ; so that the directions of visible objects should be changed by pressure, as we find them to be. (169.) 3. On the cause of erect vision from an inverted image. — As the refractions which take place at the surface of the cornea, and at the surfaces of the crystalline lens, act ex- actly like those in a convex lens in forming behind it an in- verted image of an erect object ; and as we know from direct experiment that an inverted image is formed on the retina, it has been long a problem among the learned, to determine how an inverted image produces an erect object. It would be a waste of time to give even an outline of the different opinions which have been entertained on this subject ; but there is one so extraordinary as to merit notice. According to this opinion, all infants see objects upside down, and it is only by comparing t il VI'. XXXV. CAUSE OF EttECT VISION. 247 roneous information acquired by vision with the accurate information acquired by touch, that the young learn to see objects in an erect position ! To refute such an opinion would he an insult to the intelligent reader. The establishment of the true cause of erect vision necessarily overturns all erro- neous hypotheses. The law of visible direction above explained, and deduced from direct experiment, removes at once every difficulty that besets the subject. The lines of visible direction necessarily cross each other at the centre of visible direction, so that those from the lower part of the image go to the upper part of the object, and those from the upper part of the image to the lower part of the object. Hence, in jig. 142. the visible direction of the point/', formed by rays coming from the upper end S of the object, will be/'C S, and the visible direction of the pointy] formed by rays coming from the lower end T of the object, will be/C T; so that an inverted image necessarily produces an erect object. This conclusion may be illustrated in another way. If we hold up against the sun the erect figure of a man, cut out of a piece of black paper, and look at it steadily for a little while; if we then shut both eyes, we shall see an erect spec- trum of the man when the figure of the paper is erect, and an inverted spectrum of him when the figure is held in an inverted position. In this case, there are no rays proceeding from the object to the retina after the eye is shut, and therefore the object is seen in the positions above mentioned, m virtue of the lines of visible direction being in all cases perpendicular to the impressed part of the retina. (170.) 4. On the law of distinct vision. — When the eye is directed to any point of a landscape, it sees with perfect distinctness only that point of it which is directly in the axis of the eye, or the image of which fells upon the central hole of the retina. But, though we do not see any point but the one with that distinctness which is necessary to examine it, we still see the other parts of the landscape with sufficient distinctness to enable us to enjoy ite general effect. The ex- treme mobility of the eye, however, and the duration of the impressions made upon the retina, make up for this apparent defect, and enable us to see the landscape as perfectly as if every part of it were seen with equal distinctness. The indistinctness of vision tor all objects situated out of the axis of the eye increases with their distances from that axis ; so that we are not entitled to ascribe the distinctness of vision in the axis to the circumstance of the image being formed on the central hole of the retina, where there is no ipfp • 248 A TREATISE OiV OPTICS. PART III. nervous matter ; for if this were the case, there would be n precise boundary between distincl and indistinct vision, or the retina would be found to prow thicker and thicker as il re- ceded from the central hole, which is not the case. In making some experiments on the indistinctness of vision at a distance from the axis of the eye, 1 was led to observe a very remarkable peculiarity of oblique vision. If we shut one eye, and direct the other to any fixed point, such as the head of a pin, we shall see indistinctly all other objects within the sphere of vision. Let one of these objects thus seen indis- tinctly be a strip of white paper, or a pen lying upon a green cloth. Then, after a short time, the strip of paper, or the pen, will disappear altogether, as if it. were entirely removed, the impression of the green cloth upon the surrounding parts of the eye extending itself over the part of the retina which the image of the pen occupied. In a short time the vanished image will reappear, and again vanish. When both eyes are open, the very same effect takes place, but not so readily as with one eye. If the object seen indistinctly is a black stripe on a white ground, it will vanish in a similar manner. When the object seen obliquely is luminous, such as a candle, it will never vanish entirely, unless its light is much weakened by being placed at a great distance, but it swells and contracts?, and is encircled with a nebulous halo; so that the luminous impressions must extend themselves to adjacent parts of the retina which are not influenced by the light itself. If, when two candles are placed at the distance of about eight or ten feet from the eye, and about a foot from each other, we view the one directly and the other indirectly, th°! indirect image will swell, as we have already mentioned, and will be surrounded with a bright ring of yellow light, while the bright light within the ring will have a juile blue color. If the candles are viewed through a prism, the red and green light of the indirect imao-e will vanish, and there will bo lej'i only a large mass of yellow terminated with a portion of blue light. In making this experiment, and looking steadily and directly at one of the prismatic images of the candles, I was surprised to find that the red and green rays began to dis- appear, leaving only yellow and a small portion of blue; and when the eye was kept immovably fixed on the same point of the image, the yellow light became almost pure white, so that. the prismatic image was converted into an elongated image of white light. If the strip of white paper which is seen indirectly with both eyes is placed so near the eye as to be seen double, the rays which proceed from it no longer fall upon corresponding CHAP. XXXV LAW OF DISTINCT VISION. 249 points of the retina, anil the two images do not vanish instan- taneously. But when the one begins to disappear, the other begins soon after it, so that they sometimes appear to be ex- tinguished at the same time. From these results it appears that oblique or indirect vision is inferior to direct vision, not only in distinctness, but from its inability to preserve a sustained vision of objects; but though thus defective, it possesses a superiority over direct vision in giving us more perfect vision of minute objects, such as small stars, which cannot be seen by direct vision. This curious fact has been noticed by Mr. Herschel and Sir James South, and some of the French astronomers. " A rather sin- gular method," say Messrs. Herschel and South, "of obtaining a view, and even a rough measure, of the angles of stars of the last degree of faintness, has often "been resorted to, viz. to direct the eye to another part of the field. In this way, a faint star, in the neighborhood of a large one, will often be- come very conspicuous ; so as to bear a certain illumination, which will yet totally disappear, as if suddenly blotted out, when the eye is turned full upon it, and so on, appearing and disappearing alternately as often as you please. The lateral portions of the retina, less fatigued by strong lights, and less exhausted by perpetual attention, are probably more sensible to faint impressions than the central ones; which may serve to account for this phenomenon." The following explanation of this curious phenomenon seems to me more satisfactory : — A luminous point seen by direct vision, or a sharp line of light viewed steadily for a considerable time, throws the retina into a state of agitation highly unfavorable to distinct vision. . If we look through the teeth of a fine comb held close to the eye, or even through a single aperture of the same narrowness, at a sheet of illumi- nated wbitc paper, or even at the sky, the paper or the sky will appear to be covered with an infinite number of broken serpentine lines, parallel to the aperture, and in constant mo- tion; and as the aperture is turned round, these parallel undu- lations will also turn round. These black and white lines are obviously undulations on the retina, which is sensible to the impressions of light in one phase of the undulation, and insen- sible to it in another phase. An annlogous effect is produced by looking stedfastly, and for a considerable time, on the par- allel lines whieli represent the sea in certain maps. These lines will break into portions of serpentine lines, and all the prismatic tints will be seen included between the broken cur- vilinear portions. A sharp point or lino of light is therefore M^M 250 A TREATISE ON OrTICS. TART II* unable to keep up a continued vision of itself upon the retina when seen directly. Now, in the case of indirect vision, we have already seen that a luminous object does not vanish, but is seen indistinctly, and produces an enlarged image on the retina, beside that which is produced by the defect of convergency in the pencils. Hence, a star seen indirectly, will affect a larger portion of the retina from these two causes, and, losing its sharpness, will be more distinct. It is a curious circumstance, too, that in the experiment with the two candles mentioned above, the candles seen indirectly frequently appear more intensely bright than the candle seen directly. (171). 5. On tin' insensibility of the eye to direct impres- sions of faint light. — The insensibility of the retina to indi- rect impressions of objects ordinarily illuminated, has a sin- gular counterpart in its insensibility to the direct impression of very faint light. If we fix the eye steadily on objects in a dark room that are illuminated with the faintest gleam of light, it will be soon thrown into a state of painful agitation ; the objects will appear and disappear according as the retina has recovered or lost its sensibility. These affections arc no doubt the source of many optical deceptions which have been ascribed to a supernatural origin. In a dark night, when objects are feebly illuminated, their disappearance and reappearance must seem very extraordinary to a person whose fear or curiosity calls forth all his powers of observation. This defect of the eye must have been often noticed by the sportsman in attempting to mark, upon the mo- notonous heaths, the particular spots where moor-game liad alighted. Availing himself of the slightest difference of tint in the adjacent heaths, he endeavors to keep his eye steadily upon it as he advances ; but whenever the contrast of illumi- nation is feeble, he almost always loses sight of his mark, or if the retina does take it up a second time, it is only to lose it again.* (172.) 6. On the duration of impressions of light on the retina. — Every person must have observed that the effect of light upon the eye continues for some time. During the twinkling of the eye, or the rapid closing of the eyelid- l.'n the purpose of diffusing the lubricating fluid over the cornea, we never lose sight of the objects we are viewing, in like manner, when we whirl a burning stick with a rapid motion, its burning extremity will produce a complete circle of light * Sue tin 1 Edinburgh Journal nf Science, No. VI p, 288. CHAP. XXXV. SINGLE VISION WITH TWO EYES. 251 although that extremity can only be in one part of the circle at the same instant. The most instructive experiment, however, on this subject, and one which it requires a good deal of practice to make well, is to look for a short time at the window at the end of a long apartment, and then quickly direct the eye to the dark wall. In general, the ordinary observer will see a picture of the window, in which the dark bars are white and the white panes dark ; but the practised observer, who makes the observ- ation with great promptness, will see an accurate representa- tion of the window witli dark bars and bright panes; but this representation is instantly succeeded by the complementary picture, in which the bars are bright, and the panes dark. M. D'Arcy found that the light of a live coal, moving at the dis- tance of 1(55 feet, maintained its impression on the retina during the seventh part of a second.* (173.) 7. On the cause of single vision with two ryes. — Although an image of every visible object is formed on the retina of each eye, yet when the two eyes are capable of di- recting their axes to any given object, it always appears single. There is no doubt that, in one sense, we really see two objects, but these objects appear as one, in consequence of the one oc- cupying exactly the same place as the other. Single vision with two eyes, or with any number of eyes, if we had them, is the necessary consequence of the law of visible direction. By the action of the external muscles of the eyeballs, the axes of each eye can be directed to any point of space at a greater distance than 4 or 6 inches. If we look, for example, at an aperture in a window-shutter, we know that an image of it is formed in each eye ; but, as the line of visible direc- tion from any point in the one image meets the line of visible direction from the same point in the other image, each point will be seen as one point, and, consequently, the whole aper- ture seen by one eye will coincide 'with or cover the whole aperture seen by the other. If the axes of both eyes are di- rected to a point beyond the window, or to a point within the room, the aperture will then appear double, because the lines of visible direction from the same points in each image do not meet at the aperture, [f the muscles of either of the eyes is unable to direct the two axes of the eyes to the same point, the object will in that case also appear double. This inability of one eye to follow the motions of the other is frequently the cause of squinting, as the eye which is, as it were, left behind necessarily looks in a different direction from the other. The * For a farther illustration, see Note VIII. of Am. etl. Wm mm. 252 A TREATISE ON OITICS. PART III. same effect is often produced by the imperfect vision of one eye, in consequence of which the good eye only is used. Hence the imperfect eye will gradually lose the power of fol- lowing the motions of the other, and will therefore look in a different direction. The disease of squinting may be often easily cured. (174.) 8. On the accommodation of the eye to different distances. — When the eye sees objects distinctly at a great distance, it is unable, without some change, to see objects dis- tinctly at any less distance. This will be readiiy seen by looking between the fingers at a distant object. When the distant object is seen distinctly, the fingers will be seen indis- tinctly ; and, if we look at the fingers so as to see them dis- tinctly, the distant object will be quite indistinct. The most distinguished philosophers have maintained different opinions respecting the method by which the eye adjusts itself to dif- ferent distances. Some have ascribed it to the mere enlarge- ment and diminution of the pupil ; some to the elongation of the eye, by which the retina is removed from the crystalline lens; some to the motion of the crystalline lens; and others to a change in the convexity of the lens, on the supposition that it consists of muscular fibres. I have ascertained, by direct experiment, that a variation in the aperture of the pupil, produced artificially, is incapable of producing adjustment, and as an elongation of the eye would alter the curvature of the retina, and consequently the centre of visible direction, and produce a change of place in the image, we consider this hypothesis as quite untenable. In order to discover the cause of the adjustment, I made a series of experiments, from which the following inferences may be drawn : — 1st, The contraction of the pupil, which necessarily takes place when the eye is adjusted to near objects, does not pro- duce distinct vision by the diminution of the aperture, but by some other action which necessarily accompanies it. 2dly, That the eye adjusts itself to near objects by two actions ; one of which is voluntary, depending wholly on the will, and the other involuntary, depending on the stimulus of light falling on the retina. 3dly, That when the voluntary power of adjustment fails, the adjustment may still be effected by the involuntary stimu- lus of light. Reasoning from these inferences, and other results of ex- periment, it seems difficult to avoid the conclusion that the power of adjustment depends on the mechanism which coi> tracts and dilates the pupil ; and as this adjustment is inde- CHAP. XXXV. LONG AND SHORT-SICHTEDNESS. 253 pendent of the variation of its aperture, it must be effected by the parts in immediate contact with the base of the iris. By considering the various ways in which the mechanism at the base of the iris may produce the adjustment, it appears to be almost certain thai the lens is removed from the retina by the contraction of the pupil.* (175.) 9. On the cause of longsightedness and shortsight- edness. — Between the ages of 30 and 50, the eyes of most persons begin to experience a remarkable change, which generally shows itself in a difficulty of reading small type or ill-printed books, particularly by candlelight. This defect of sight, which is called longsightedness, because objects are seen best at a distance, arises from a change in the state of the crystalline lens, by which its density and refractive power, as well as its form, are altered. It frequently begins at the margin of the lens, and takes several months to go round it, and it is often accompanied with a partial separation of the laminee and even of the fibres of the lens. " If the human eye," as I have elsewhere remarked, " is not managed with peculiar care at this period, the change in the condition of the lens often runs into cataract, or terminates in a derangement of fibres, which, though not indicated by white opacity, occa- sions imperfections of vision that are often mistaken for amaurosis and other diseases. A skilful oculist, who thoroughly understands the structure of the eye, and all its optical func- tions, would have no difficulty, by means of nice experiments, in detecting the very portion of the lens where this change has taken place; in determining the nature and magnitude of the change which is going on ; in applying the proper reme- dies for stopping its progress ; and in ascertaining whether it has advanced to such a state that aid can be obtained from convex or concave lenses. In such cases, lenses are often re- sorted to before the crystalline lens has suffered a uniform change of figure or of density, and the use of them cannot fail to aggravate the very evils which they are intended to remedy. In diseases of the lens, where the separation of fibres is confined to small spots, and is yet of such magnitude as to give separate colored images of a luminous object, or irregular halos of light, it is often necessary to limit the aper- ture of the spectacles, so as to allow the vision to be performed by the good part of the crystalline lens." This defect of the eye, when it is not accompanied with disease, may be completely remedied by a convex lens, which * For a fuller account of these experiments, see Edinburgh Journal of Science, No. I. p. 77. w %$m. w$¥??. 254 A TREATISE ON OPTICS. PART III- niakcs up for the flatness and diminished refractive power of the crystalline, and enables the eye to converge the pencil? flowing from near objects to distinct foci on the retina. Shortsightedness shows itself in an inability to see at a dis- tance; and those who experience this defect bring minute ob- jects very near the eye in order to see them distinctly. The rays from remote objects are in this case converged to foci be- fore they reach the retina, and therefore the picture on the retina is indistinct. This imperfection often appears in early life, and arises from an increase of density in the central part:-; of the crystalline lens. By using a suitable concave lens the convergency of the rays is delayed, so that a distinct image can be formed on the retina. CHAP. XXXVI. ON ACCIDENTAL COLORS AND COLORED SHADOWS. (176.) When the eye has been strongly impressed with any particular species of colored light, and when in this state it looks at a sheet of white paper, the paper does not appear to it white, or of the color with which the eye was impressed, but of a different color, which is said to be the accidental color of the color with which the eye was impressed. If we place, for example, a bright red wafer upon a sheet of white paper, and fix the eye steadily upon a mark in the centre of it, then if we turn the eye upon the white paper we shall see a cir- cular spot of bluish green light, of the same size as the wafer. This color, which is called the accidental color of red, will gradually fade away. The bluish green image of the wafer is called an ocular spectrum, because it is impressed on the eye, and may be carried about with it for a short time. If we make the preceding experiment with differently col- ored wafers, we shall obtain ocular spectra whose colors vary with the color of the wafer employed, as in the following table. Color of the Wafer Accidental Color, or Color of thr Ocular Spfctrtim. Red. Bluish green. Orange. Blue. Yellow. Indigo. Green. Reddish violet Blue. Orange red. Indigo. Orange yellow Violet. Yellow green Black. White White Black CHAP. XXXVI. ON ACCIDENTAL COLOR?. 255 In order to find the accidental color of any color in the spec- trum, take half the Length of the spectrum in a pair of com- passes, and setting one root in the color whose accidental color is required, the other will fall upon the accidental color. Hence the law of accidental colors derived from observation may be thus stated : — The accidental color of any color in a prismatic spectrum, is that color which in the same spectrum is distant from the first color half the length of the spectrum ; or, if we arrange all the colors of any prismatic spectrum in a circle, in their due proportions, the accidental color of any particular color will be the color exactly opposite that par- ticular color. Hence the two colors have been called opposite colors. If the primitive color, or that which impresses the eye, is reduced to the same degree of intensity as the accidental color, we shall find that the one is the complement of the other, or what the other wants to make it white light ; that is, the primitive and the accidental colors will, when reduced to the same degree of intensity which they have in the spec- trum, and when mixed together, make white light. On this account accidental colors have been called complementary colors. With the aid of these facts, the theory of accidental colors will be readily understood. When the eye has been for some time fixed on the red wafer, the part of the retina occupied by the red image is strongly excited, or, as it were, deadened by its continued action. The sensibility to red light will therefore be diminished ; and, consequently, when the eye is turned from the red wafer to the white paper, the deadened portion of the retina will be insensible to the red rays which form part of the white light from the paper, and consequently will see the paper of that color which arises from all the rays in the white light of the paper but the red; that is, of a bluish green color, which is therefore the true complementary color of the red. Winn a black wafer is placed on a white ground, the circular portion of the retina, on which the black image tails, in place of being deadened, is refreshed, as it were, by the absence of light, while all the surrounding parts of the retina, beiu^' excited by the white light of the paper, will be deadened by its continued action. Hence, when the eye is directed to the white paper, it will see a white circle corresponding to the black image on the retina; so that the accidental color of black is white. For the same reason, if a white wafer is placed on & black ground, and viewed stedfastly for some time, the eye will afterwards see a black circular space; so that the accidental color of white is black. 256 A tkkatise on optics. Such are the phenomena of accidental colors when weak light is employed; but when the eye is impressed powerfully with a bright white light, the phenomena have quite a different character. The first person who made this experiment with any care was Sir Isaac Newton, who sent an account of the results to Mr. Locke, but they were not published till 1829.* Many years before 1691, Sir Isaac, having shut his left eye, directed the right one to the image of the sun reflected from a looking-glass. In order to see the impression which was made, he turned his eye to a dark coiner of his room, when he observed a bright spot made by the sun, encircled by rings of colors. This " phantom of light and colors," as lie calls it, gradually vanished; but whenever he thought of it, it return- ed, and became as lively and vivid as at first. He rashly re- peated the experiment three times, and his eye was impressed to such a degree, " that whenever I looked upon the clouds, or a book, or a bright object, I saw upon it a round bright spot of light like the sun ; and, which is still stranger, though I looked upon the sun with my right eye only, and not with my left, yet my fancy began to make an impression on my left eye as well as upon my right ; for if I shut my right eye, or looked upon a book or the clouds with my left eye, I could see the spectrum of the sun almost as plain as with my right eye." The effect of this experiment was such, that Sir Isaac durst neither write nor read, but was obliged to shut himself com- pletely up in a dark chamber for three days together, and by keeping in the dark, and employing his mind about other tilings, he began, in about three or four days, to recover the use of his eyes. In these experiments, Sir Isaac's attention was more taken up with the metaphysical than with the op- tical results of them, so that he has not described either the colors which he saw, or the changes which they underwent. Experiments of a similar kind were made by M. ./Epinus. When the sun was near the horizon, he fixed his eye steadily on the solar disc for 15 seconds. Upon shutting his eye he saw an irregular pale sulphur yellow image of the sun, encir- cled with a faint red border. As soon as he opened his eye upon a white ground, the image of the sun was a brow nisi;. red, and its surrounding border sky blue. With his eye again tshut, the image of the sun became green with a red border, different from the last. Turning his eye again upon a white ground, the sun's image was more red, and its border a brighter sky blue. When the eye was shut, the green spectrum be- came a greenish sky blue, and then a fine sky blue, with the * fu Lord King's Life of Locke. CHAP. XXXVl. IMPRESSIONS ON THE RETINA. 257 border growing a finer red ; and when the eye was open, the spectrum became a finer red, and its border a liner blue. M. ^Epinus noticed, that when his eye was fixed upon the white ground, the image of the sun frequently disappeared, returned, and disappeared again. About the year 1808, i was led to repeat the preceding ex- periments of .F.pinus; but, instead of looking- at the sun when of a dingy color, i took advantage of a fine summer's day, when the sun was near the meridian, and 1 formed upon a white ground a brilliant image of his disc by the concave speculum of a reflecting telescope. Tying up my right eye, 1 viewed this luminous disc with my left eye through a tube, and when the retina was highly excited, I turned my left eye to a white ground, and observed the following spectra by al- ternately opening and shutting it: — s ; ectn wi ii; ted ej i , Spectra with left eye »hut. 1. Pink surrounded with green. Green. 2. Orange mixed with pink. Blue. .'■!. Yellowish brown. Bluish pink 4. Yellow. 5. Pure red. Sky blue 6. Orange. Indigo. Upon uncovering my right eye, and turning it to a white ground, I was surprised to observe that it also gave a colored spectrum, exactly the reverse of the first spectrum, which was pink with a green border. The reverse spectrum was a green with a pinkish border. This experiment was repeated three times, and always with the same result; so that it would appear that the impression of the solar image was conveyed by the optic nerve from the left to the right eye. Sir Isaac Newton supposed that it was his fancy that transferred the image from his left to his right eye ; but we are disposed to think that in his experiment no transference took place, be- cause the spectrum which he saw with both eyes was the same, whereas in my experiment it was the reverse one. We cannot however speak decidedly on this point, as Sir Isaac did not observe that, the spectra with the eye shut were the reverse of those seen with tiie eye open. If a spectrum is strongly formed on one eye, it is a very difficult, matter to de- termine on which eye it is formed, and it would be- impossible to do this if the spectrum was the same when the eye was open and shut. The phenomena of accidental colors are often finely seen when the eye has not been strongly impressed with any par- ticular colored object. It was long ago observed by M. Meus- W2 ^p ?&mm mn™ 258 A TREATISE ON OPTICS. rAKT III nier, that when the sun shone through a hole a quarter of an inch in diameter in a red curtain, the image of the luminous, spot was green. In like manner, every person must have ob- served in a brightly painted room, illuminated by the sun, that the parts of any white object on which the colored light does not fall, exhibit the complementary colors. Jn order to see this class of phenomena, I have found the following method the simplest and the best. Having lighted two candles, hold before one of them a piece of colored glass, suppose bright red, and remove the other candle to such a distance that the two shadows of any body formed upon a piece of white paper may- be equally dark. In this case one of the shadows will be red, ami the other green. With blue glass, one of thein will be blue, and the other orange xjellow ; the one being invariably the accidental color of the other. The very same effect may be produced in daylight by two holes in a window-shutter; the one being covered with a colored glass, and the other trans- mitting the white light of the sky. ■ Accidental colors may also be seen by looking at the image of a candle, or any white object seen by reflexion from a plate or surface of colored glass sufficiently thin to throw back its color from the second surface. In this case the reflected image will always have the complementary color of the glass. The same effect may be seen in looking at the image of a candle reflected from the water in a blue finger-glass ; the image of the candle is yel- lowish : but the effect is not so decided in this case, as the retina is not sufficiently impressed with the blue light of the glass. These phenomena are obviously different from those which are produced by colored wafers ; because in the present case the accidental color is seen by a portion of the retina which is not affected, or deadened as it were, by the primitive color. A new theory of accidental colors is therefore requisite, to embrace this class of facts. As in acoustics, where every fundamental sound is actually accompanied with its harmonic sound, so in the impressions of light, the sensation of one color is accompanied by a weaker sensation of its accidental or harmonic color.* W hen we look at the red wafer, we are at the same time, with the same por- tion of the retina, seeing green ; but being much faiy v .er, it seems only to dilute the red, and make it, as it were, whiter, by the combination of the two sensations. When the eye looks from the wafer to the white paper, the permanent sen- * The term harmonic lias been applied to accidental colors ; because the .primitive audits accidental color harmonize with each other in painting. ■i CHAP. XXXVI. IXSKNSIKILITY TO COLORS. 259 cation of the accidental color remains, and we see a green imago. The duration of the primitive impression is only a fraction of a second, as we have already shown ; but the dura- tion of the harmonic impression continues for a time propor- tional to the strength of the impression. In order to apply these views to the second class of facts, we must have re- course to another principle ; namely, that when the whole or a great part of the retina has the sensation of any primitive color, a portion of the retina protected from the impression of the color is actually thrown into that state which gives the accidental or harmonic color. By the vibrations probably communicated from the surrounding portions, the influence of the direct or primitive color is not propagated to parts free from its action, excepting in the particular case of oblique vision formerly mentioned. When the eye, therefore, looks at the white spot of solar light seen in the middle of the red light of the curtain, the whole of the retina, except the por- tion occupied by the image of the white spot, is in the state of seeing every thing green; and as the vibrations which constitute this state spread over the portions of the retina upon which no red light falls, it will, of course, see the white circular spot green. (177.) A very remarkable phenomenon of accidental colors, in which the eye is not excited by any primitive color, was observed by Mr. Smith, surgeon in Fochabers. If we hold a narrow strip of white paper vertically, about a foot from the eye, and fix both eyes upon an object at some distance beyond it, then if we allow the light of the sun, or the light of a can- dle, to act strongly upon the right eye, without affecting the left, which may be easily protected from its influence, the left hand strip of paper will be seen of a bright green color, and the right hand strip of a red color. If the strip of paper is sufficiently broad to make the two images overlap each other, the overlapping parts will be perfectly white, and free from color ; which proves that the red and green are complementary. When equally luminous candles are held near each eye, the two strips of paper will be white. If when the candle is held near the right eye, and the strips of paper are seen red and green, then on bringing the candle suddenly to the left eye, the left hand image of the paper will gradually change ta green, and the right hand image to red. (178.) A singular affection of the retina, in reference to colors, is shown in the inability of some eyes to distinguish certain colors of the spectrum. The persons who experience this defect have their eyes generally in a sound state, and are capable of performing all the most delicate functions of vision. ?p$ spsg IgPfP^ H IIP 260 A TRKATISK ON OPTICS. PART III. Mr. Harris, a shoemaker at Allonby, was unable from his in- fancy to distinguish the cherries of a cherry-tree from its leaves, in so far as color was concerned. Two of his brothers were equally defective in this respect, and always mistook orange for grass green, and light green for yellow. Harris himself could only distinguish black and white. Mr. Scott, who describes his own case in the Philosophical Transactions, mistook pink for a pale blue, and a full red for a full green. All kinds of yellows and blues, except sky blue, he could discern with great nicety. His father, his maternal uncle, one of his sisters, and her two sons, had all the same defect. A tailor at Plymouth, whose case is described by J\lr. Harvey, regarded the solar spectrum as consisting only of yel- low and light blue ; and he could distinguish with certainty only yellow, while, and green. He regarded indigo and Prus- sian blue as black. Mr. R. Tucker describes the colors of the spectrum as fol- lows : — Red mistaken for Brown. Orange .... Green. Yellow sometimes Orange. Green .... Orange. Blue sometimes Pink. Indigo - - - Purple. Violet - - - Purple. A gentleman in the prime of life, whose case I had occasion to examine, saw only two colors in the spectrum, viz. yellow and blue. When the middle of the red space was absorbed by a blue glass, he saw the black space, with what he called the yellow, on each side of it. This defect in the perception of color was experienced by the late Mr. Dugald Stewart, who could not perceive any difference in the color of the scar- let fruit of the Siberian crab and that of its leaves. Mr. Dalton is unable to distinguish blue from pink by daylight, and in the solar spectrum the red is scarcely visible, the rest of it appearing to consist of two colors. Mr. Troughton has the same defect, and is capable of fully appreciating only blue and yellow colors; and when he names colors, the names of blue and yellow correspond to the more and less refrangible rays, all those which belong to the former exciting the sensation of bl ueness, and those which belong to the latter the sensation of yellowness. In almost all these cases, the different prismatic colors have the power of exciting the sensation of light, and giving a dis- tinct vision of objects, excepting in the case of Mr. Dalton, who is said to be scarcely able to see the red extremity of the spectrum. Mr. Dalton has endeavored to explain this peculiarity of CHAP. XXXVII. PLANE AND CURVED MIRRORS. 261 vision by supposing that in his own case the vitreous humor is blue, and, therefore, absorbs a great portion of the red rays and other least refrangible rays ; but this opinion is, we think, not well founded. Mr. Herschel attributes this state of vision to a defect in the sensorium, by which it is rendered incapable of appreciating exactly those differences between rays on which their color depends.* PART IV. ON OPTICAL INSTRUMENTS. All the optical instruments now in use have, with the ex- ception of the burning mirrors of Archimedes, been invented by modern philosophers and opticians. The principles upon which most of them have been constructed have already been explained, in the preceding chapters, and we shall therefore confine ourselves, as much as possible, to a general account of their construction and properties. CHAP. XXXVII. ON PLANE AND CURVED MIRRORS. (179.) One of the simplest optical instruments is the single plane mirror, or looking-glass, which consists of a plate of glass with parallel surfaces, one of which is covered with tin- foil and quicksilver. The glass performs no other part in this kind of plane mirror than that of holding and giving a polished surface to the thin bright film of metal which is extended over it. If the surfaces of the plate of glass are not parallel, we shall see two, three, and four images of all luminous objects seen obliquely ; but even when the surfaces are parallel, two images of an object are formed, one reflected from the first surface of glass, and tiie other from the posterior surface of metal ; and the distance of these images will increase with- the thickness of the glass. The image reflected from the glass is, however, very faint compared with the other ; so that for ordinary purposes a plane glass mirror, is sufficiently ac- curate; but when a plane mirror forms a part of an optical * For the theory recently advanced by Sir David Brewster to explain these cases, see Note IX. of Am. ed. ■ fspp^p^#ppv ? W&$&$ 202 A TREATISE ON OPTICS. PART IV. instrument where accuracy of vision is required, it must, be made of steei, or silver, or of a mixture of copper and tin ; niwl iu this case it is called a speculum. The formation of images by mirrors and specula has been fully described in ( Ihap. II. Kaleidoscope. (180.) When two plane mirrors are combined in a particu- lar manner, and placed in a particular position relative to an o'oject, or series of objects, and the eye, they constitute the kaleidoscope, or instrument for creating and exhibiting beautiful forms. If A C, B C, for example, be sections of two plane mirrors, and M N an object placed between them or in Fig. m;j. front of each, the mirror A C will form behind it an image m n of the object M N, in the manner shown in fig. 1(5. In like manner, the mirror B C will form an image M'N' behind it. But, as we have formerly shown, these images may be con- sidered as new objects, and therefore the mirror A C will form behind it an image, M" N", of the object or image M' N', and B C will form behind it an image, m' n', of the object or image m n. In like manner it will be found that m" n" will be the image of the object or image M" N", formed by B C, and of the object or image m'n', formed by AC. Hence m" n" will actually consist of two images overlapping each other and forming one, provided the angle A C B is exactly 00°, or the sixth part of a circumference of 300°. In this case all the six images (two of the six forming only one, m" n",) will, along with the original object, M N, form a perfect equilateral triangle. The object, ]V1 N, is drawn perpendicular to the mirror B C, in consequence of which M N and M'N' form one straight line ; but if M N is moved, all the images will move, and the figure of all the images combined will form another figure of perfect regularity, and exhibiting the most beautiful variations, all of which may be drawn by the methods already described. In reference to the multiplication and ar- rangement of the images, this is the principle of the kaleido- scope; but the principle of symmetry, which is essential to the instrument, depends on the position of the object and the eye. This principle will be understood from for. 144., where ACE and B C E represent the two mirrors inclined at an angle A C B, and having C E for their line of junction, or common intersection. If the object is placed at a distance, as at M N, then there is no position of the eye at or above E which will CHAP. XXXVII. ON THE KALEIDOSCOPE. 263 give a symmetrical arrangement of the six images shown in fig. 143. ; for the corresponding parts of the one will never join the corresponding parts of the other. As the object is brought nearer and nearer, the symmetry increases, and is most complete when the object M N is quite close to A B C, the ends of tlic reflectors. But even here it will not be per- fect, unless the eye is placed as near as possible to E, the line of junction of the reflectors. The following, therefore, are the three conditions of symmetry in the kaleidoscope : — 1. That the reflectors should be placed at an angle which is an even or an odd aliquot part of a circle, when the object is regular and similarly situated with respect to both the mir- rors ; or an even aliquot part of a circle, when the object is irregular. 2. That out of an infinite number of positions for the object both within and without the reflectors, there is only one posi- tion where perfect symmetry can be obtained, namely, by placing the object in contact with the ends of the reflectors, or between them. 3. That out of an infinite number of positions for the situa- tion of the eye, there is only one where the symmetry is per- fect, namely, as near as possible to the angular point, so that the whole of the circular field can be distinctly seen; and this point is the only one at which the uniformity of the reflected light is greatest. In order to give variety to the figures formed by the instru- ment, the objects, consisting of pieces of colored glass, twisted glass of various curvatures, &c, are placed in a narrow cell between two circular pieces of glass, leaving them just room to move about, while this cell is turned round by the hand. The pictures thus presented to the eye are beyond all descrip- tion splendid and beautiful; an endless variety of symmetrical combinations presenting themselves to view, and never again recurring with the same form and color. For the purpose of extending the power of the instrument, and introducing into symmetrical pictures external objects, 264 A TREATISE ON OPTICS. PART IV whether animate or inanimate, I applied a convex lens, L L : Jig. 144., by means of which an inverted image of a distant object, M N, may be formed at the very extremity of the mir- rors, and therefore brought into a position of greater sym- metry than can be effected in any other way. In this construc- tion the lens is placed in one tube and the reflectors in an- other ; so that by pulling out or pushing in the tube next the eye, the images of objects at any distance can be formed at the place of symmetry. In this way, flowers, trees, animals, pictures, busts, may be introduced into symmetrical combina- tions. When the distance E B is less than that at which the eye sees objects distinctly, it is necessary to place a convex lens at E, to give distinct vision of the objects in the picture. See my Treatise on the Kaleidoscope. Plane burning Mirrors (181.) A combination of plane burning mirrors forms a pow- erful burning instrument ; and it is highly probable that it was with such a combination that Archimedes destroyed the ships of Marcellus. Athanasius Kircher, who first proved the effi- cacy of a union of plane mirrors, went with his pupil Scheiner to Syracuse, to examine the position of the hostile fleet ; and they were both satisfied that the ships of Marcellus could not have been more than thirty paces distant from Archimedes. Buffon constructed a burning apparatus upon this principle, which may be easily explained. If we reflect the light of the sun upon one cheek by a small piece of plane looking-glass, we shall experience a sensation of heat less than if the direct light of the sun fell upon it. If with the other hand we re- flect the sun's light upon the same cheek with another piece of mirror, the warmth will be increased, and so on, till with five or six pieces we can no longer endure the heat. Buffon combined 163 pieces of mirror, 6 inches by 8, so that he could, by a little mechanism connected with each, cause them to reflect the light of the sun upon one 6pot. Those pieces of glass were selected which gave the smallest image of the sun at 250 feet. The following were the effects produced by different num- bers of these mirrors : — Small combustibles inflamed. Beech plank burned. Tarred beech plank inflamed. Pewter flask 61b. weight melted. Tarred and sulphured plank set on fire. f o. of Distance of Object. 12 20 feet 21 20 40 66 45 20 98 126 CHAP. XXXVII. CONVEX AND CONCAVE MIKUORS. 265 No. of Mirrors. Diatsnre Object 112 138 117 20 128 150 148 150 154 150 154 250 274 40 Effect produced. Plank covered with wool set on fire. Some thin pieces of silver melted. Tarred fir plank set on fire. Beech plank sulphured inflamed violently. Tarred plank smoked violently. J Chips of fir deal sulphured and mixed with ( charcoal set on fire. Plates of silver melted. As it is difficult to adjust the mirrors while the sun changes his place, M. Peyrard proposes to produce great effects by mounting each mirror in a separate frame, carrying a tele- scope, by means of which one person can direct the reflected rays to the object which is to be burned. He conceives that with 590 glasses, about 20 inches in diameter, he cuuld reduce a fleet to ashes at the distance of a quarter of a league, and with glasses of double that size at the distance of half a league. Plane glass mirrors have been combined permanently into a parabolic form, for the purpose of burning objects placed in the focus of the parabola, by the sun's rays; and the same combination has been used, and is still in use, for lighthouse reflectors, the light being placed in the focus of the parabola. Convex and Concave Mirrors. (182.) The general properties of convex and concave mir- rors have been already described in Chap. II. Convex mirrors are used principally as household ornaments, and are charac- terized by their property of forming erect and diminished images of all objects placed before them, and these images ap- pear to be situated behind the mirror. Concave mirrors are distinguished by their property of forming in front of them, and in the air, inverted images of erect objects, or erect images of inverted objects, placed at some distance beyond their principal focus. If a fine trans- parent cloud of blue smoke is raised, by means of a chafing- dish, around the focus of a large concave mirror, the image of any highly illuminated object will be depicted, in the middle of it, with great beauty. A skull concealed from the observer is sometimes used, to surprise the ignorant; and when a dish of fruit has been depicted in a similar manner, a spectator, stretching out his hand to seize it, is met with the image of a drawn dagger, which has been quickly substituted for the fruit at ihe other coniuffate focus of the mirror X $m mn 266 A TREATISE ON OPTICS. l'ART IV. Concave mirrors have been used as lighthouse reflectors, and as burning instruments. When used in lighthouses, they are formed of plates of copper plated with silver, and thev are hammered into a parabolic form, and then polished with the hand. A lamp placed in the focus of the parabola will have its divergent light thrown, after reflexion, into something like a parallel beam, which will retain its intensity at. a great distance. When concave mirrors are used for burning, they are gene- rally made spherical, and regularly ground and polished upon a tool, like the specula used in telescopes. The most cele- brated of these were made by M. Villele, of Lyons, who exe- cuted five large ones. One of the best of them, which con- sisted of copper and tin, was very nearly four feet in diameter, and its focal length thirty-eight inches. It melted a piece of Pompey's pillar in fifty seconds, a silver sixpence in seven seconds and a half, a halfpenny in sixteen seconds, cast-iron in sixteen seconds, slate in three seconds, and thin tile in four seconds. Cylindrical Mirrors (183.) All objects seen by reflexion in a cylindrical mirror are necessarily distorted. If an observer looks into such a mirror with its axis standing vertically, he will see the image of his head of the same length as the original, because the surface of the mirror^s a straight line in a vertical direction. The breadth of the face will be greatly contracted in a hori- zontal direction, because the surface is very convex in that CHAP. XXXVIII. ON SPECTACLES. 267 direction, and in intermediate directions the head will have intermediate breadths. If the axis of the mirror is held hori- zontally, the face will be as broad as life, and exceedingly short. If a picture or portrait M N is laid clown horizontally before the mirror A B, fig. 145., the reflected image of it will be highly distorted ; but the picture may be drawn distorted according to regular laws, so that its image shall have the most correct proportions. Cylindrical mirrors, which are now very uncommon, used to be made for this purpose, and were accompanied with a series of distorted figures, which, when seen by the eye, have neither shape nor meaning, but when laid down before a cylin- drical mirror, the reflected image of them has the most per- fect proportions. This effect is shown in fig. 145., where M N is a distorted figure, whose image in the mirror A B has the appearance of a regular portrait. CHAP. XXXVIII. ON SINGLE AND COMPOUND LENSES Spectacles and reading glasses are among the simplest and most useful of optical instruments. In order to enable a per- son who has imperfect vision to see small objects distinctly, when they are not far from the eye, such as small manuscript, or small type, a convex lens of very short focus must be used both by those who are long and short sighted. When a short-sighted person, who cannot see well at a dis- tance, wishes to have distinct vision at any particular distance, he must use a concave lens, whose focal length will be found thus, — Multiply the distance at which he sees objects most distinctly by the distance at which he wishes to see them dis- tinctly with a concave lens, and divide this product by the difference of the above distances. Along-sighted person, who cannot see near objects distinctly, must use a convex lens, whose focal length is found by the preceding rule. In choosing spectacles, however, the best way is to select, out of a number, those which are found to answer best the purposes for which they are particularly intended. Dr. Wollaston introduced a new kind of spectacles, called periscopic, from their property of giving a wider field of dis- tinct vision than the common ones. The lenses used for this purpose, as shown at H and I, fig. 19., are meniscuses, in ''■'"' W7%? 268 A TRKATISK ON OPTICS. PART IV which tlic convexity predominates, for long-sighted persons, and concavo-convex lenses, in which the concavity predomi- nates, for short-sighted persons. Periscopic spectacles de- cidedly give more imperfect vision than common spectacles, because they increase both the aberration of figure and of color ; but they may be of use in a crowded city, in warning us of the oblique approach of objects. Burning and Illuminating Lenses. (184.) Convex lenses possess peculiar advantages for con centrating the sun's rays, and for conveying to an immense distance a condensed and parallel beam of light. M. Bufibn found that a convex lens, with a long focal length, was prefer- able to one of a short focal length for fusing metals by the concentration of the sun's rays. A lens, for example, 32 inches in diameter and 6 inches in focal length, with the di- ameter of its focus 8 lines, melted copper in less than a minute ; while a small lens 32 lines in diameter, with a focal length of 6 lines, and its focus § of a line, was scarcely capa- ble of heating copper. The most perfect burning lens ever constructed was exe- cuted by Mr. Parker, of Fleet Street, at an expense of 700/. It was made of flint glass, was three feet in diameter, and weighed 212 pounds. It was 3^ inches thick at the centre ; the focal distance was G feet 8 inches, and the diameter of the image of the sun in its focus one inch. The rays refracted by the lens were received on a second lens, in whose focus the objects to be fused were placed. This second lens had an ex- posed diameter of 13 inches; its central thickness was ljj of an inch ; the length of its focus was 29 inches. The diameter of the focal image was | of an inch. Its weight was 21 pounds. The combined focal length of the two lenses was 5 feet 3 inches, and the diameter of the focal image \ an inch. By means of this powerful burning lens, platinum, gold, silver, copper, tin, quartz, agate, jasper, flint, topaz, garnet, asbestos, &c. were melted in a few seconds. Various causes have prevented philosophers from construct- ing burning lenses of greater magnitude than that made by Mr. Parker. The impossibility of procuring pure flint glass tolerably free from veins and impurities for a large solid lens ; the trouble and expense of casting it into a lenticular form without flaws and impurities ; the great increase of central thickness which becomes necessary by increasing the diameter of the lens ; the enormous obstruction that is thus opposed to the transmission of the solar rays, and the increased aber- CHAP. XXXVIII. ON POLYZONAL LENSES. 269 ration which dissipates the rays at the focal point, are insuper- able obstacles to the construction of solid lenses of any con- siderable size. (185.) In order to improve a solid lens formed of one piece of glass, whose section isAmpBEDA, Buftbn proposed to cut out all the glass left white in the figure, viz. the portions between m p,fig- 146., and n o, and between n o and the left u;cr iar, hand surface of D E. A lens thus constructed would be incomparably superior to the solid one AmpBE D A ; but such a process we conceive to be imprac- ticable on a large scale, from the extreme difficulty of polishing the surfaces A m, B p, C n, F o, and |D the left hand surface of D E ; and even if it were practicable, the greatest imperfections in the glass might happen to occur in the parts which are left. J-^ In order to remove these imperfections, and to fir construct lenses of any size, I proposed, in 1811, to build them up of separate zones or rings, each of J* which rings was again to be composed of separate segments, as shown in the front view of the lens in fig. 147. This lens is composed of one central lens, A B C D, corre- sponding with its section D E in fig. 146., of a middle ring G E L I corresponding to C D E F in fig. 146., and consisting of five seg- ments ; and another ring, NPRT, corresponding to A C F B, and con- sisting of eight segments. The preceding construction obvi- X ously puts it in our power to execute these compound lenses, to which I have given the name of polyzonal lenses, of pure flint glass free from veins ; but it possesses another great advantage, namely, that of enabling us to correct, very nearly, the spherical aberration, by making the foci of each zone coincide. One of these lenses was constructed, under my direction, for the Commissioners of Northern Lighthouses, by Messrs. W. and P. Gilbert. It was made of pure flint glass, was three feet in diameter, and consisted of many zones and seg- ments. Lenses of this kind have been made in France of crown glass, and have been introduced into the principal French lighthouses; a purpose to which they are infinitely better adapted than the best constructed parabolic reflectors made of metal. A polyzonal lens of at least four feet in diameter will be X2 m ¥$%?> %F® 270 A TREATISE ON OPTICS. TART IV. speedily executed as a burning-glass, and will, no doubt, be the most powerful ever made. The means of executing it have been, to a considerable degree, supplied by the scientific liberality of Mr. Swinton and Mr. Calder, and other gentlemen of Calcutta, chap, xxxix. ON SIMPLE AND COMPOUND PRISMS. Prismatic Lenses. (186.) The general properties of the prism in refracting and decomposing light have already been explained; but its application as an optical instrument, or as an important part of optical instruments, remains to be described. A rectangular prism, ABC, Jig. 148., was first applied by Sir Isaac Newton as a plane mirror for reflecting to a side the rays which form the imago in reflecting telescopes. The angles, B A C, B C A, being each 45°, and B a right angle, rays falling on the face A B will be reflected by the back sur- face B C as if it wcre\ plane metallic mirror; for whatever be the refraction which they suffer at their entrance into the face A B, they will suffer an equal and opposite one at the face B C. The great value of such a mirror is, that as the incident rays fall upon A C at an angle greater than that at which total reflexion commences, they will all suffer total re- flexion, and not a ray will be lost ; whereas in the best me- tallic speeulum nearly half the light is lost. A portion of light, however, is lost by reflexion at the two surfaces A B, B C, and a small portion by the absorption of the glass itself. Sir Isaac Newton also proposed the convex prism, shown at CHAP. XXXIX. ON PRISMATIC LENSES. 271 D E F, the faces D F, F E being ground convex. An analo- gous prism, called the meniscus prism, and shown at G II I, has been used by M. Chevalier, of Paris, tor the camera ob- scura. It differs only from Newton's in the second face, I H, being concave in place of convex. On account of the difficult execution of these prisms, I have proposed to use a hemispherical lens, L M N, the two convex surfaces of which are ground at the same time. When a longer focus is required, a concave lens, R Q, of a longer focus than the hemisphere P R Q, may be placed or cemented on its lower surface, and if the concave lens is formed out of a substance of a different dispersive power, it may be made to correct the color of the convex lens. A single prism is used with peculiar advantage for inverting pencils of light, or lor obtaining an erect image from pencils that would give an inverted one. This effect is shown in Jig. 149., where A B C is a rectangular prism, and R R' R" a par- allel pencil of light, which, after being refracted at the points 1, 2, 3, of the fiace A B, and reflected at the points «, b, c, of the base B C, w ill be again refracted at the points 1, 2, 3, of the face A C, and move on in parallel lines, 3r", 2r', lr; the ray R I, that was uppermost, being now undermost, as at 1 r. Compound and Variable Prisms. (187.) The great difficulty of obtaining glass sufficiently pure for a prism of any size, has rendered it extremely diffi- cult to procure good ones; and they have therefore not been introduced as they would otherwise have been into optical in- struments. The principle upon which polyzonal lenses are constructed is equally applicable to prisms. A prism con- structed like A T),Jig. 150., if properly executed, would have exactly the same properties as A B C, and would be incom- parably superior to it, from the light passing through such a small thickness of glass. It would obviously be difficult to exe- cute such a prism as A D out of a single piece of glass, though %?m s^^w^^ ^^^^^H 272 A TREATISE ON OPTICS. TART IV. it is quite practicable ; but there is no difficulty in combining six small prisms all cut out of one prismatic rod, and therefore necessarily similar. The summit of the rod should have a flat narrow face parallel to its base, which would be easily done if the prismatic rod were cut out of a plate of thick par- allel glass. The separate prisms being 1 cemented to one an- other, as in the figure, will form a compound prism, which will be superior to the common prism for all purposes in which it acts solely by refraction. (188.) A compound prism of a different kind, and having a variable angle, was proposed by Boscovich, as shown in Jig. 151., where A B C is a hemispherical convex lens, moving in a concave lens, DEC, of the same curvature. By turning Fig. 151. one of the lenses round upon the other, the inclination of the faces A B, D E, or A B, C E, may be made to vary from 0° to above 90°. (189.) .As this apparatus is both troublesome Lo execute and difficult to use, I have employed an entirely different principle for the construction of a variable prism, and have used it to a great extent in numerous experiments on the dispersive pow- ers of bodies. If we produce a vertical line of light by nearly closing the window-shutters, and view the line with a flint glass prism whose refracting angle is 60°, the edge of the re- fracting angle being held vertical, or parallel to the line of CHAP. XXXIX. ON PRISMATIC LKXSES. 273 light, the luminous lino will be seen as a brightly colored spectrum, and any small portion of it will resemble almost ex- actly the solar spectrum. If we now turn the prism in the plane of one of its refracting faces, so that the inclination of the edge to the line of light increases gradually from 0° up to 90° when it is perpendicular to the line of light, the spectrum will gradually grow less and less colored, exactly as if it were formed by a prism of a less and less refracting angle, till at an inclination of 90° not a trace of color is left. By this simple process, therefore, namely, by using a line of light instead of a circular disc, we have produced the very same effect as if the refracting angle of the prism had been varied from 90° down to 0°. (190.) Let it now be required to determine the relative dis- persive powers of flint glass and crown glass. Place the crown glass prism so as to produce the largest spectrum from the line of white light, and let the refracting angle of the prism be 40°. Then place the flint glass prism between it and the eye, and turn it round, as before described, till it cor- rects the color produced by the crown glass prism, or till the line of light is perfectly colorless. The inclination of the edge of the flint glass' prism to the line of light being known, we can easily find, by a simple formula the angle of a prism of flint glass which corrects the color of a prism of crown glass with a refracting angle of 40°. See my Treatise on New Philosophical Instruments, p. 291. Multiplying Glass. (191.) This lens is more amusing than useful, and is intend- ed to give a number of images of the same object. Though it has the circular form of a lens, it is nothing more than a mm ggjfg f^B - ■ H 274 A TREATISE ON OPTICS. number of prisms formed by grinding various flat faces on the convex surface of a plano-convex glass, as shown in fig. 152., where A B is the section of a multiplying glass in which only three of the planes are seen. A direct image of the object C will be seen through the face G H, by the eye at E ; an- other image will be seen at D, by the refraction of the face H B, and a third at F, by the refraction of the face A G, an image being seen through every plane face that is cut upon the lens. The image at C will be colorless, and all those formed by planes inclined to A B will be colored in proportion to the angles which the planes form with A B. Natural multiplying glasses may be found among trans- parent minerals which are crossed with veins oppositely crys- tallized, even though they are ground into plates with parallel faces. In some specimens of Iceland spar more than a hun- dred finely colored images may be seen at once. The theory of such multiplying glasses has already been explained in Chap. XXIX. CHAP. XL. ON the camera obscura, magic lantern, and CAMERA LUCIDA. (192.) The camera obscura, or dark chamber, is the name of an amusing and useful optical instrument, invented by the celebrated Baptista Porta. In its original state it is nothing more than a dark room with an opening in the window-shutter, in which is placed a convex lens of one or more feet focal length. If a sheet of white paper is held perpendicularly be- hind the lens, and passing through its focus, there will be painted upon it an accurate picture of all the objects seen from the window, in which the trees and clouds will appear to move in the wind, and all living objects to display the same movements and gestures which they exhibit to the eye. The perfect resemblance of this picture to nature astonishes and delights every person, however often they may have seen it. The image is of course inverted, but if we look over the top of the paper it will be seen as if it were erect. The ground on which the picture is received should be hollow, and part of a sphere whose radius is the focal distance of the convex lens. It is customary, therefore, to make it of the whitest plaster of Paris, with as smooth and accurate a surface as possible. In order to exhibit the picture to several spectators at once, CHAP. XL. ON THE CAMERA OBSCURA. 27ii Fig. 153. and to enable any person to copy it, it is desirable that the image should be formed upon a horizontal table. This may be done by means of a metallic mirror, placed at an angle of 45° to the refracted rays, which will reflect the picture upon the white ground lying horizontally ; or, as in the portable camera obscura, it may be reflected upwards by the mirror, and received on the lower side of a plate of ground glass, with its rough side uppermost, upon which the picture may be copied with a fine sharp-pointed pencil. A very convenient portable camera obscura for drawing landscapes or other objects is shown in fig. 153., where A B is a meniscus lens, with its concave side uppermost, and the radius of its convex surface being to the radius of its concave sur- face as 5 to 8, and C D a plane metallic speculum inclined at an angle of 45° to the horizon, so as to reflect the landscape downwards through the lens A B. The draughtsman introduces his head through an opening in one side, and his hand with the pencil through another opening, made in such a manner as to allow no light to fall upon the picture which is exhibited on the paper at E F. The tube containing the mirror and lens can be turned round by a rod within, and the inclination of the mirror changed, so as to introduce objects in any part of the horizon. When the camera is intended for public exhibition, it con- sists of the same parts similarly arranged ; but they are in this case placed on the top of a building, and the rotation of the mirror, and its motion in a vertical plane, are effected by turning two rods within the reach of the spectator, so that he can introduce any object into the picture from all points of the compass and at all distances. The picture is received on a table, whose surface is made of stucco, and of the same radius as the lens, and this surface is made to rise and fall to accom- modate it to the change of focus produced by objects at dif- ferent distances. A camera obscura which throws the image down upon a horizontal surface may be made without any mirror, by using any of the lenticular prisms D E F, G II I, M L N, when the objects are extremely near, and P R Q,,jig. 148. The convex surfaces of these prisms converge the rays which are reflected to their focus by the flat faces D E, G H, ART IV 290 In order to use object glasses of such great focal lengths with- out the encumbrance of tubes, Huygens placed the object glass in a short tube at the top of a very long pole, so that the tube could be turned in every possible direction upon a ball and socket by means of a string, and brought into the same line with another short tube containing the eye glass, which he held in his hand. As these telescopes were liable to all the imperfections arising from the aberration of refrangibility and that of spher- ical figure, they could not show objects distinctly when the aperture of the object glass was great ; and on this account their magnifying power was limited. Huygens found that the following were the proper proportions : — Magnifying power. 20 88* 44 62 140 197 216 In the astronomical telescope, the object, M N, is always seen inverted. Terrestrial Telescope. (206.) In order to accommodate this telescope to land ob- jects which require to be seen erect, the instrument is con- structed as in Jiff. 165., which is the same as the preceding one, with ^he addition of two lenses E F, G H, which have the Fig. 1C5. Focal length of the Aperture of the Fecal length of th object glass. object glass. eye gloss. 1ft. 0-545 inches. 0-605 3 0-94 1-04 5 1-21 1-33 10 1-71 1-88 50 3-84 4-20 100 5-40 5-95 120 5-90 6-52 same focal length as C D, and are placed at distances equal to double their common focai length. If the focai lengths are not equal, the distance of any two of them must be equal to the sum of their focal lengths. In this telescope the progress of the rays is exactly the same as in the astronomical one, as far as L, where the two pencils of parallel rays CL, DL cross in the anterior focus L of the second eye glass E F. These rays falling on E F form in its principal focus an erect image, m! n', CHAP. XLII. €RECORIAN TF.LF.SCOPK. 291 which is scon erect by the third eye glass G II, as the rays diverging' from m' and n' in the focus of G II enter the eye in parallel pencils at E'. The magnifying power of this tele- scope is the same as that of the former when the eye glasses are equal. Galilean Telescope. (207.) This telescope, which is the one used by Galileo, differs in nothing from the astronomical telescope, excepting in a concave eye glass C D, fig. 166. being substituted for the convex one. The concave lens C D is placed between the Fig. ice. image m n and the object glass, so that the image is in the principal focus of the concave lens. The pencils of rays ABn, A B m fall upon C D, converging to its principal focus, and will therefore be refracted into parallel lines, which will enter the eye at E, and give distinct vision of the object The magnifying power of this telescope is found by the same rule as that for the astronomical telescope : it gives a smaller and less agreeable field of view than the astronomical telescope, but it has the advantage of showing the object erect, and of giving more distinct vision of it. Gregorian Reflecting Telescope. (208.) Father Zucchius seems to have been the first person who magnified objects by means of a lens and a concave spec- ulum ; but there is no evidence that he constructed a reflecting telescope with a small speculum. James Gregory was the first who described the construction of this instrument, but he does not seem to have executed one ; and the honor of doing this with his own hands was re- served for Sir Isaac Newton. The Gregorian telescope is shown in fig. 167., where A E is a concave metallic speculum with a hole in its centre. For very remote objects the curve of the speculum should be a parabola. For nearer ones it should be an ellipse in whose farther focus is the object, and in whose nearer focus is the image; and in both these cases the speculum would be free from spherical aberration. But, as these curves cannot be «l$ w&$; w^^m^mw. ■ 292 A TREATISK ON OPTICS. PART IV. communicated with certainty to specula, opticians are satisfied with giving to them a correct spherical figure. In front of the -wijHNi^ large speculum is placed a small concave one, C D, which can be moved nearer to and farther from the large speculum by means of the screw W at the side of the tube. This spec- ulum should have its curvature elliptical, though it is gene- rally made spherical. An eye-piece consisting of two convex lenses, E, F, placed at a distance equal to half the sum of their focal lengths, is screwed into the tube immediately behind the great speculum A B, and permanently fixed in that position. If rays M A, N B, issuing nearly parallel from the extremities M and N of a distant object, fall upon the speculum A B, they will form an inverted image of it at in n, as more distinctly shown in fig. 14. If this image m n is farther from the small speculum C D than its principal focus, an inverted image of it, in' n', or an erect image of the real object, since in n is itself an inverted one, will be formed somewhere between E and F, the rays passing through the opening in the speculum. This image in' n' might have been viewed and magnified by a convex eye glass at F, but it is preferable to receive the converging rays upon a lens E called the field glass, which hastens their con- vergence, and forms the image of inn in the focus of the lens F, by which they are magnified ; or, what is the same thing, the pencils diverging from the image in' n' are refracted by F, so as to enter the eye parallel, and give distinct vision of the image. If the object M N is brought nearer the speculum A B, the image of it, in ??, will recede from A B and approach to C D ; and, consequently, the other image in' n' in the con- jugate focus of C D will recede from its place in' n', and cease to be seen distinctly. In order to restore it to its place m' n', we have only to turn the screw W, so as to remove C D farther from A B, and consequently farther from in n, which will cause the image in' n' to appear perfectly distinct as be- C1IA1'. XLII. CASSEGH WNIAN TFLKSCOPK. 293 tore. Tlie magnifying power of this telescope may be found by the following rule : — Multiply the focal distance of the great speculum by the distance of the small mirror from the image next the eye, as formed in the anterior focus of the convex eye glass, and mul- tiply also the focal distance of the small speculum by the focal distance of the eye glass. The quotient arising from dividing the former product by the latter will be the magnifying power. This rule supposes the eye-piece to consist of a single lens. The following table, showing the focal lengths, apertures, powers, and prices of some of Short's telescopes, will exhibit the great superiority of reflecting telescopes to refracting ies : — Focal lengths in feet. Aperture in InehM. MagnifyinR powers. Price ingu 1 30 35 to 100 14 2 4-5 90 300 35 3 6-3 100 400 75 4 7-6 120 500 100 7 12-2 200 800 300 12 18-0 300 1200 800 Cassegrainian Telescope. (209.) The Cassegrainian telescope, proposed by M. Cas- segrain, a Frenchman, differs from the Gregorian only in hav- ing its small speculum C T>,fig. 168., convex instead of con- cave. The speculum is therefore placed before the image m n Fig. 168. ^ of the object M N, and an image of M N will be formed at m' n' between E and F as in the Gregorian instrument. The advantage of this form is, that the telescope is shorter than the Gregorian by more than twice the focal length of the small speculum; and it is generally admitted that it^gives more light, and a distincter image, in consequence of the con- vex speculum correcting the aberration of the concave one. Z2 mm mi wm. vmm™ ^M .'*>*,?' 294 A TREATISE ON OPTICS. Newtonian Telescope. PART IV. (210.) The Newtonian telescope, which may be regarded as an improvement upon the Gregorian one, is represented in fig. 169., where A B is a concave speculum, and in n the in- verted image which it forms of the object from which the rays Fig. 169. M, N proceed. As it is impossible to introduce the eye into the tube to view this image without obstructing the light which comes from the object, a small plane speculum C D, in- clined 45° to the axis of the large speculum, and of an oval form, its axes being to one another as 7 to 5, is placed between the speculum and the image m n, in order to reflect it to a side at m' n', so that we can magnify it with an eye glass E, which causes the rays to enter the eye parallel. The small mirror is fixed upon a slender arm, connected with a slide, by which the mirror may be made to approach to or recede from the large speculum A B, according as the image m n approaches to or recedes from it. This adjustment might also be effected by moving the eye lens E to or from the small speculum. The magnifying power of this telescope is equal to the focal length of the great speculum divided by that of the eye glass. As about half of the light is lost in metallic reflexions, Sir Isaac Newton proposed to substitute, in place of the metallic speculum, a rectangular prism ABC, fig. 148., in which the light suffers total reflexion. For this purpose, however, the glass requires to be perfectly colorless and free from veins, and hence such a prism has rarely been used. Sir Isaac also proposed to make the two faces of the prism convex, as D E F, Jig. 148., and by placing it between the image m n and the object, he not only erected the image, but was enabled to vary the magnifying power of the telescope. The original telescope, constructed by Sir Isaac's own hands, is preserved in the library of the Royal Society. The following table shows the dimensions of Newtonian =t2r^s: CHAP. XLII. N KWTON I A N TELESCOPE. 295 telescopes, which we have computed by taking a fine telescope made by Hawksbee as a standard : — Focal Ji-ngth cf great Aperture of speculum. Focal length of .-ye (hi.. Magnifyint power 1ft. 2*23 inches. 0-129 inches. 93 2 379 0-152 158 3 514 0-168 214 4 6-36 0181 265 6 8-64 0-200 360 12 14-50 0-238 604 24 24-41 0-283 1017 (211.) On account of the great loss of light in metallic re- flexions, which, according to the accurate experiments of Mr. R. Potter, amounts to 45 rays in every 100, at an incidence of 45°,* and the imperfections of reflexion, which even with per- fect surfaces make the rays stray five or six times more than the same imperfections in refracting surfaces, I have proposed to construct the Newtonian telescope, as shown in fig. 170., where A B is the concave speculum, m n the image of the object M N, and C D an achromatic prism, which refracts the image m n into an oblique position, so that it can be viewed by the eye at E through a magnifying lens. Nothing more is required by the prism than to turn the rays as much aside as will enable the observer to see the image without obstructing the rays from the object M N. As the prisms of crown and flint glass which compose the achromatic prism may be ce- mented by a substance of intermediate refractive power, no more light will be lost than what is reflected at the two sur- faces. In place of setting the small speculum, C D, of the New- tonian telescope, fig. 169., at 45°, to the incident rays, I have proposed to place it much more obliquely, so as to reflect the image m n, fig. 170., out of the way of the observer, and no farther. This would of course require a plane speculum, C D, * FMnburgh Journal of Science, No. VI., new seriffS, p. 283. W*. ~>k#M 296 A TREATISE ON OPTICS. of much greater length; but the greater obliquity of the re- flexion would more than compensate for this inconvenienced It might be advisable, indeed, to use a small speculum of dark glass, of a high refractive power, which at great incidences reflects a3 much light as metals, and which is capable of being brought to a much finer surface. The fine surfaces of some crystals, such as ruby silver, oxide of tin, or diamond, might be used. A Newtonian reflector, without an eye glass, may be made by using a reflecting glass prism, with one or both of its sur- faces concave, when the prism is placed between the image in n and the great speculum, so as to reflect the rays parallel to the eye. The magnifying power will be equal to the focal length of the great speculum, divided by the radius of the concave surface of the prism if both the surfaces are concave, and of equal concavity, or by twice the radius, if only one surface is concave. Sir William HerscheTs Telescope. (212.) The fine Gregorian telescopes executed by Short were so superior to any other reflectors, that, the Newtonian form of the instrument fell into disuse. It was revived, how- ever, by Sir W. Herschel, whose labors form the most brilliant epoch in optical science. With an ardor never before exhibit- ed, he constructed no fewer than 200 seven feet Newtonian reflectors, 150 ten feet, and 80 twenty feet in focal length. But his zeal did not. stop here. Under the munificent, patron- age of George III., he began, in 1785, to construct a telescope forty feel long, and on the 27th of August, 1780, the day on which it was completed, he discovered with it the sixth satel- lite of Saturn. » The great speculum had a diameter of 49£ inches, but its concave surface was only 48 inches. Its thickness was about 3£ inches, and its weight when cast was 2118 lbs. Its focal length was forty feet, and the length of the sheet iron tube which contained it was 39 feet 6 inches, and its breadth 4 feet 10 inches. By using small convex lenses, Dr. Herschel was enabled to apply a power of 6450 to the fixed stars, but a very much lower power was in general used. In this telescope the observer sat at the mouth of the tube, and observed by what is called the front view, with his back to the object, without using a plane speculum, the eye lens being applied directly to magnify the image formed by the great speculum. In order to prevent the head, &c. from ob- ■ CHAP. XI.III. ON ACHROMATIC TELESCOPES. 297 st meting too much of the incident light, the image was formed out of the axis of the speculum, and must, therefore, have heen slightly distorted. As the frame of this instrument was exposed to the weather, it had greatly decayed. It was, therefore, taken down, and another telescope, of 20 feet focus, with a speculum 18 inches in diameter, was erected in its place, in 1822, by J. F. W. Herschel, Esq., with which many important observations have been made. Mr. Ramage's Telescope. (213.) Mr. Ramage, of Aberdeen, has constructed various Newtonian telescopes, of great lengths and high powers. The largest instrument at present in use in this country, and we believe in Europe, was constructed by him, and erected at the Royal Observatory of Greenwich in 1820. The great specu- lum has a focal length of 25 feet, and a diameter of 15 inches. The image is formed out of the axis of the speculum, which is inclined so as to throw it just to the side of the tube, where the observer can view it without obstructing the incident rays. The tube is a 12-sided prism of deal, and when the instrument is not in use it is lowered into a box, and covered with canvas. The apparatus for moving and directing the telescope is ex- tremely simple, and displays much ingenuity. CHAP. XLIII. ON ACHROMATIC TELESCOPES. (214.) The principle of the achromatic telescope has been briefly explained in Chap. VII., and we have there shown how a convex lens, combined with a concave lens of a longer focus, and having a higher refractive and dispersive power, may pro- duce refraction without color, and consequently form an image free from the primary prismatic colors. It has been demonstrated mathematically, and the reader may convince himself of its truth by actually tracing the rays through the lenses, that a convex and a concave lens will form an achromatic combina- tion, or will give a colorless image, when their focal lengths are in the same proportion as their dispersive powers. That is, if the dispersive powers of crown and flint glass are as - 60 to 1, or 6 to 10; then an achromatic object glass could be formed by combining a convex crown glass lens of 0, or 60, or r - W'VvS StSmSK' M 298 A TREATISE ON OPTICS. PART IV. 600 inches with a concave flint glass lens of 10, or 100, or 1000 inches in focal length. But though such a combination would form an image free from color, it would not be free from spherical aberration, which can only be removed by giving a proper proportion to the cur- vatures of the first and last surface, or the two outer surfaces of the compound lens. Mr. Herschel has found that a double object glass will be nearly free from aberration, provided the radius pf the exterior surface of the crown lens be 6-72, and of the flint 1-120, the focal length of the combination being 10-00, and the radii of the interior surfaces being computed from these data by the formulae given in elementary works on optics, so as to make the focal lengths of the two glasses in Kg. 171. the direct ratio of their dispersive powers. This _a_ c combination is shown in fig. 171., where A B is the convex lens of crown glass, placed on the out- side towards the object, and C D the concavo-con- vex lens of flint glass placed towards the eye. The two inside surfaces that come in contact are so nearly of the same curvature that they may be ground on the same tool, and united together by a cement to prevent the loss of light at the two sur- faces. In the double achromatic object glasses con- structed previous to the publication of Mr. Her- schel's investigations, the surface of the concave lens next the eye was, we believe, always con- cave. Triple achromatic object glasses consist of three lenses A B, fig. 172. C D, E F, fig. 172., A B and E F being convex lenses of crown glass, and C D a double concave lens of flint glass. The object of using three lenses was to ob- tain a better correction of the spherical aberra- tion ; but the greater complexity of their con- struction, the greater risk of imperfect centering, or of the axes of the three lenses not being in the same straight line, together with the loss of light at six surfaces, have beer; considered "V CHAP. XLIII. ON ACHROMATIC TELESCOPES. 299 A B, or first Crown Lens. FIRST OBJECT GLASS. SECOND OBJECT GLASS. Radii of first surface, - - - 28 inches 28 second surface, --40 35"5 C D, or Flint Lens. Radii of first surface, - - - 20-9 21-1 second surface, - - 28 - 25-75 E F, or second Crown Lens. Radii of first surface, - - - 28-4 - - - 28 second surface, - - 28-4 28 Focal length of the compound lens, 46 inches 46 - 3 In consequence of the great difficulty of obtaining flint glass free from veins and imperfections, the largest achromatic object, glasses constructed in England did not greatly exceed 4 or 5 inches in diameter. The neglect into which this im- portant branch of our national manufactures was allowed to fall by the ignorance and supineness of the British government, stimulated foreigners to rival us in the manufacture of achro- matic telescopes. M. Guinand of Brenetz, in Switzerland, and M. Fraunhofer, of Munich, successively devoted their minds to the subject of making large lenses of flint glass, and both of them succeeded. Before his death, M. Fraunhofer executed two telescopes with achromatic object glasses of 9f^ inches, and 12 inches in diameter; and he informed me that he would undertake to execute one 18 inches in diameter. The first of those object glasses was for the magnificent achro- matic telescope ordered by the emperor of Russia, for the ob- servatory at Dorpat. The object glass was a double one, and its focal length was 25 feet; it was mounted on a metallic stand which weighed 5000 Russian pounds. The telescope could be moved by the slightest force in any direction, all the movable parts being balanced by counter weights. It had four eye glasses, the lowest of which magnified 175, and the high- est 700 times. Its price was 1300/., but it was liberally given at prime cost, or 950Z. The object glass, 12 inches in diameter, was made for the king of Bavaria, at the price of 2720Z. ; but as it was not perfectly complete at the time of Fraunhofer's death, we do not know that it is at present in use. In the hands of that able observer, Professor Struve, the telescope of Dorpat has already made many important discoveries in as- tronomy. A French optician, we believe, M. Lereboure, has more m «3*gww«fi«w 300 A TREATISE ON OI'TICS. recently executed two achromatic object glasses of glass made by Guinand. One of them is nearly 12 inches in diameter, and another above 13 inches. The first of these object glasses was mounted as a telescope at the Royal Observatory of Paris ; and the French government had expended 500J. in the pur- chase of a stand for it, but had not the liberality to purchase the object glass, itself. Sir James South, our liberal and active countryman, saw the value of the two object glasses, and ac- quired them for his observatory at Kensington. ON ACHROMATIC EYEPIECES. (215.) Achromatic eyepieces when one lens only is wanted, may be composed of two or three lenses exactly on the same principles as object glasses. Such eyepieces, however, are never used, because the color can be corrected in a superior manner,, by a proper arrangement of single lenses of the same kind of glass. This arrangement is shown in Jig. 173., where A B and C D are two plano-convex lenses, A B being the one Fig. 173. next the object glass, and C D the one next the eye, a ray of white light R A, proceeding from the achromatic object glass, will be refracted by A B at A, so that the red ray A r crosses the axis at r, and the violet ray A v at v. But these rays being intercepted by the second lens C D at the points m, n, at dif- ferent distances from the axis, will suffer different degrees of refraction. The red ray m r suffering a greater refraction than the violet one n v, notwithstanding its inferior refran- gibility, so that the two rays will emerge parallel from the lens C D (and therefore be colorless) as shown at m r', m v'. When these two lenses are made of crown glass, they must be placed at a distance equal to half the sum of their focal lengths, or, what is more accurate, their distance must be equal to half the sum of the focal distance, of the eye glass C D, and the distance at which the field glass A B would form an image of the object glass of the telescope. This eyepiece is called the negative eyepiece. The stop or diaphragm must be placed half-way between the two lenses. The focal length of an equivalent lens, or one that has the same magnifying Fig. 174. CHAP. XLIII. ON ACHROMATIC TELESCOPES. ""* 301 power as the eyepiece, is equal to twice the product of the focal lengths of the two lenses divided by the sum of the same numbers. An eyepiece nearly achromatic, called Ra7nsden , s Eyepiece, and much used in transit instruments and telescopes with mi- crometers, is shown in Jig. 174., where A B, CD, are two plano-convex lenses with their convex sides inwards. They have the same focal length, and are placed at a distance from each other, equal to two-thirds of the focal length of either. The focal length of an equiva- lent lens is equal to three-fourths the focal length of either lens. The use of this eyepiece is to give a flat field, or a distinct view of a system of wires placed at M N. This eyepiece is not quite achromatic, and it might be rendered more so by increasing the distance of the lenses ; but as this would require the wires at M N to be brought nearer A B, any particles of dust or imperfections in the lens A B would be seen magnified by the lens C D. The erecting achromatic eyepiece now in universal use in all achromatic telescopes for land objects is shown in Jig. 175. It consists of four lenses, A, C, D, B, placed as in the figure. Mr. Coddington has shown, that if the focal lengths, reckoning from A, are as the numbers 3, 4, 4 and 3, and the distances between them on the same scale 4, 6, and 5-2, the radii, reckoning from the outer surface of A, should be thus : — C First surface Second surface \ First surface ) Second surface y, J First surface ) Second surface n S First surface ) Second surface nearly plano-convex. a meniscus. 2 . > nearly plano-convex, 2 . v Double convex. 2 A 302 A TREATISE ON OITICS. PART IV. The magnifying power of this eyepiece, as usually made, differs little from what would be produced by using the first or fourth lens alone. I have shown, that the magnifying power of this eyepiece may be increased or diminished by varying the distance between C and D, which even in common eyepieces of this kind may be done, as A and C are placed in one tube A C, and D and B in another tube D B, so that the latter can be drawn out of the general tube. In jig. 175., I have shown the eyepiece constructed in this way, and capable of having its two parts separated by a screw nut E ; and rock. This contrivance for obtaining a variable magnifying power, and consequently of separating optically a pair of wires fixed before the eye glass, I communicated to Mr. Carey in 1805, and had one of the instruments constructed by Mr. Adie in 1806. It is fully described in my Treatise on Philosophical Instruments, and has been more recently brought out as a new invention by Dr. Kitchener, under the name of the Pancratic Eye Tube. Prism Telescope. (216.) In 1812, I showed that colorless refraction may be produced by combining two prisms of the same substance, and the experiments which led to this result were published in my Treatise on New Philosophical Instruments in 1813. The practical purposes to which this singular principle seemed to be applicable were the construction of an achromatic telescope with lenses of the same glass, and the construction of a Teinoscope, for extending or altering the lineal proportions of objects. If we take a prism, and hold its refracting edge downwards and horizontal, so as to see through it one of the panes of glass in a window, there will be found a position, namely, that in which the rays enter the prism and emerge from it at equal angles, as in fig. 20., where the square pane of glass is of its natural size. If we turn the refracting edge towards the window, the pane will be extended or magnified in its length or vertical direction, while its breadth remains the same. If we now take the same prism and hold its refracting edge ver- tically, we shall find, by the same ' process, that the pane of glass is extended or magnified in breadth. If two such prisms, therefore, are combined in these positions, so as to magnify the same both in length and breadth, we have a telescope com- posed of two prisms, but unfortunately the objects are all highly fringed with the prismatic colors. We may correct CHAP. XMI1. ON ACHttOM \TK' TELESCOPES. :?03 these colors in tliree ways : 1st, We may make the prisms of a kind of glass which obstructs all the rays but those of one homogeneous color ; or, we may use a piece of the same glass to absorb the other rays when two common glass prisms are used : 2d, We may use achromatic prisms in place of common prisms : or, 3d, What is best of all lor common purposes, we may place other two prisms exactly similar, but in reverse positions, or they may be placed as shown in Jig. 176., which represents the prism telescope ; A B and A C being two prisms VJT M 1.iim. .U 1 jHl7iillTi. "u.lm,.. 'J of the same kind of glass, and of the same refracting angles, with their planes of refraction vertical, and E D, E F, other two perfectly similar prisms, similarly placed, but with their planes of refraction horizontal. A ray of light, M a, from an object, M, enters the first prism, E F, at a, emerges from the second prism, E D, at b, enters the third prism, A C, at c, emerges from the fourth prism, A B, at d, and enters the eye at O. The object, M, is extended or magnified horizontally by each of the two prisms, E F, E D, and vertically by each of the two prisms, A B, A C ; objects are magnified by look- ing through the prisms. This instrument was made in Scotland by the writer of this Treatise, under the name of a Teinoscope, and also by Dr. Blair, before it was proposed or executed by Professor Amici of Modena. Dr. Blair's model is now before me, being com- posed of four prisms of plate glass with refracting angles of about 15°. It was presented to me two years ago by his son ; but as no account of it was ever published, Mr. Blair could not determine the date of its construction. In constructing this instrument, the perfect equality of the four prisms is not necessary. It. will be sufficient if A B and D E are equal, and A C and 1A F, as the color of the one prism can be made to correct fliat of the other by a change in its position. For the same reason it is not necessary that they ho all made of the same kind of glass. 'mm '&?&mm$ W *>* 304 A TREATISE ON OPTICS. T.ye gift* Magnifying made of power. Flint glass 1* Oil of cassia 2 Flint glass 2 Oil of aniseseed 3 Oil of cassia 3 Oil of cassia 6 Achromatic Opera Glasses with Single Lenses. (217.) M. d'Alembert has long ago shown that an achromatic telescope may be constructed with a single object glass and a single eye glass of different refractive and dispersive powers. To effect this, the eye glass must be concave, and be made of glass of a much higher dispersive power than that of which the object glass is made : but the proposal was quite Utopian at the time it was suggested, as substances with a sufficient difference of dispersive power were not then known. Even now, the principle can be applied only to opera glasses. If we use an object glass of very low dispersive power, the refraction of the violet rays may be corrected by a concave eye lens of a high dispersive power, as will be seen by the following table. Crown glass Water Rock crystal Rock crystal Crown glass Rock crystal Although all the rays are made to enter the eye parallel in these combinations, yet the correction of color is not satis- factory. Mr. Barlow's Achromatic Telescope. (218.) In the year 1813 I discovered the remarkable dis- persive power of sulphuret of carbon, having found that it " exceeds all fluid bodies in refractive power, surpassing even flint glass, topaz, and tourmaline; and that in dispersive power it exceeds every fluid substance except oil of cassia, holding an intermediate place between phosphorus and balsam of tolu. * * * Although oil of cassia surpasses the sulphuret of car- bon in its power of dispersion, yet, from the yellow color with which it is tinged, it is greatly inferior to the latter as an op- tical fluid, unless in cases where a very thin concave lens is required. The extreme volatility of the sulphuret is undoubt- edly a disadvantage ; but as this volatility may be restrained, we have no hesitation in considering the sulphuret of carbon as a fluid of great value in Optical researches, and which may be of incalculable service in the construction of optical instruments ."* This anticipation has been realized by Mr. * On the Optical Properties of Sulphuret of Carbon, in Edinburgh Trans. vol. viii. p. 285. Feb. 7. 18)4. cit.vr. XLiir. ox achromatic telescopes. 305 Barlow, who has employed sulphuret of carbon as a substitute for flint cflass, in correcting the dispersion of the convex lens. It had been proposed, and the experiment even tried, to place the concave lens between the convex one and its focus, lor the purpose of correcting the dispersion of the convex lens, with a lens of less diameter, but Mr. Barlow has the merit of hav- ing first carried this into effect. The telescope which he lias made on this principle, consists of a single object lens of plate glass, 7 - 8 inches in clear aper- ture, with a focal length of 78 inches. At the distance of 40 inches from this lens was placed a concave lens of sulphuret of carbon, with a focal length of 59 - 8 inches, so that parallel rays falling on the convex plate lens, and converging to its focus, would, when refracted by the fluid concave lens, have their focus at the distance of 104 inches from the fluid lens, and 141 inches, or 12 feet, from the plate glass lens. The flu id is contained between two meniscus cheeks, and a glass ring, so that the radius of the concave fluid lens is 144 inches towards the eye, and 56-4 towards the object lens. The fluid is put in at a high temperature, and the contraction which it experiences in cooling is said to keep every thing perfectly tight. No decomposition of the fluid has yet been observed. The great secondary spectrum which I found to exist in sul- phuret of carbon is approximately corrected by the distance of the fluid lens from the object glass ; but We are persuaded that it is not free from secondary color. Mr. Coddington remarks, that the general course of an oblique pencil is bent outward by the fluid Ions;, and the violet rays more than the red, so as to produce indistinctness ; but we are not aware that this defect was observed in the instrument. The tube of the tele- scope is 11 feet, and the eyepieces one foot. " The telescope," says Mr. Barlow, " bears a power of 700 on the closest double stars in South'a and Herschel's catalogue, although the field is not then so bright as I could desire. Venus is beautifully white and well defined with a power of 120, but shows some color with 300. Saturn, with the 120 power, is a very bril- liant object, the double ring and belts being well and satisfac- torily defined, and with the 360 power it is still very fine." Mr. Barlow remarks, also, that the telescope is not so com- petent to the opening of the close stars, as it is powerful in bringing to light the more minute luminous points. Achromatic Solar Telescopes with single Lenses. (219.) An achromatic telescope for viewing the sun or any highly luminous object may be constructed by using a single 2A2 r*a^:#"». wmm 306 A THEATISK ON OPTICS. object glass of plate glass ; and by making any one of the eye glasses out of a piece of glass which transmits only homage neous light: or the same thing may be effected by a piece of plane glass of the same color; but this introduces the errors of other two surfaces. In such a construction it would be preferable to absorb ail the rays but the red; and there are various substances by which this may be readily effected. The object glass of' this telescope, though thus rendered monochro- matic, will still be liable to spherical aberration. Hut if the radii of the lens are properly adjusted, the excess of solar light will permit us to diminish the aperture, so as to render the spherical aberration almost imperceptible. Such a telescope, when made of a great length, would, we are persuaded, be equal to any instrument that has yet been directed to the sun. If we could obtain a solid or a fluid which would absorb all the other rays of the spectrum but the yellow, with as little loss as there is in red glasses, a telescope of* the preceding construction would answer for day objects, and for all the pur- poses of astronomy. If the art of giving lenses a hyperbolic form shall be brought to perfection, which we have no doubt will yet be done, the spherical aberration would disappear; and a telescope upon this principle would be the most perfect of all instruments. Even by using red light only, a great improvement might be effected in the common telescopes for day objects and for astronomical purposes. If the red rays, for example, form f '„th of white light, we have only to increase the area of the aper- ture 10 times to make up completely for this defect of light. The spherical aberration is, no doubt, greatly increased also: but if we consider that, when compared to the aberration of color, it is only as 1 to 1200, we can afford to increase it in order to gain so great an advantage. Common telescopes, indeed, may be considerably improved by applying colored glasses, which absorb only the extreme rays of the spectrum, even though they do not produce an achromatic or homoge- neous image. These observations are made for the benefit of those who cannot afford expensive instruments, but who may yet wish to devote themselves to astronomical observations, with the ordi- nary instruments which they may happen to possess. On the Improvement of imperfectly achromatic Telescopes. (220.) There are many achromatic telescopes of consider- able size, in which the flint lens either over corrects or under corrects the colors of the crown glass lens. This defect may CHAP. XLIir. ON ACHROMATIC TELESCOPES. 307 be easily removed by altering' slightly the curvature of one or other of the lenses. But all achromatic telescopes whatever, when made of crown and flint glass, exhibit the secondary colors, viz. the wine-colored and the green fringes. These colors are not very strong ; and in many, if not in all cases, we may destroy them by absorption through glasses that will not weaken greatly the intensity of the light. The glasses requisite for this purpose must be found by actual experiment; as the secondary tints, though generally of the colors we have mentioned, are variously composed, according to the nature of ihe glass of which the two lenses are made. ^p^ppg ■ $m?m m&em qm ®<^#^m^-$% ■^y'.^.'^m 309 APPENDIX OF THE AUTHOR, . CONTAINING TABLES OF REFRACTIVE AND DISPERSIVE POWERS, &c. OF DIFFERENT MEDIA. TABLE I. (Referred to from Page 30.) Table of the Refractive Powers of Solid and Fluid Bodies. Judex of Refraction. Realgar artificial Octohedrite Diamond Nitrite of lead Blende Phosphorus Sulphur melted Zircon Glass — lead 2 parts, flint ) 1 part S Garnet Ruby G lass— lead 3 parts, flint ) 1 part S Sapphire Spinelle Cinnamon stone Sulphuret of carbon Oil of cassia Balsam of Tolu Guaiacum Oil of anisieseed Quartz Rock salt Sugar melted Canada balsam 2-549 2-500 2139 2-322 2-2(50 2-224 2-148 1-961 1-830 1-815 1-779 2028 1-794 1-764 1-759 1-768 1-641 1-628 1-619 1-601 1-548 1-557 1-554 1-549 Index of Refraction. Amber 1-547 Plate glass, from 1-514 to . . . 1-542 Crown glass, from 1-525 to . . 1-534 Oil of cloves 1-535 Balsam capivi 1-528 Gain arable 1-502 Oil of beech nut l-5t>0 Castor oil 1-490 Cajeputoil 1-483 Oil of turpentine 1-475 Oil of olives 1-470 Alum 1-457 Fluor Spar 1-434 Sulphuric acid 1-434 Nitric acid 1-410 Muriatic acid 1-410 Alcohol 1-372 Cryolite 1349 Water 1-336 Ice 1-309 Fluids in minerals 1-294 to . 1-131 Tabasheer Fill Ether expanded to thrice ) , nc -*, its volume < luo/ Air 1-000294 Table of the Refractive Powers of Gases. Index cf Refraction. Vapor of sulphuret of) 1.001530 carbon ) Phosgene gas 1 Cyanogen 1 Chlorine Olefianl gas Sulphurous acid Sulphuretted hydrogen Nitrous oxide 1 1 Hydrocyanic acid . Muriatic acid 1 001159 ■000834 ■000772 000678 •000665 000644 000503 •000451 000149 Index of Rt-frnction. Carbonic acid v 1-000449 Carburetted hydrogen . .* 1-000443 Ammonia 1-000385 Carbonic oxide 1 000340 Nitrous gas 1-000303 Azote 1-000300 Atmospheric air 1 000294 Oxygen 1000272 Hydrogen 1-000138 Vacuum 1000000 ■ |p^ $$fflmwm ^M 4^>v;v/vU'- 310 A TREATISE ON OPTICS. TABLE II. (Referred to from Page 31.) Table of the Absolute Refractive Powers of Bodies. Tabasheer Cryolite Fluor spar Oxygen Sulphate of baryta Sulphurous ccid gas Nitrous gas Air Carbonic acid Azote Chlorine Nitrous oxide Phosgene Selenite Carbonic oxide Quartz Glass Muriatic acid Sulphuric acid Calcareous spar Alum Borax Kefraetion. 0-0976 0-2742 0-3426 0-3799 0-3829 0-4455 0-4491 0-4528 0-4537 0-4734 0-4813 0-5078 0-5188 0-5386 0-5387 0-5415 0-5436 0-5514 0-6124 0-6424 0-6570 0-6716 ; Index of Refraction, Nitre 0-7079 Rain water 0-7815 Flint glass 0-7986 Cyanogen 0-8021 Sulphuretted hydrogen ... 0-8419 Vapcr of sulphuret of > i.q~ A i carbon..... j ° -8 ' 43 Ammonia 1 -0032 Alcohol rectified 1-0121 Camphor 1-2551 Olive oil 1-2607 Amber 1-3654 Octohedrite 13816 Sulphuret of carbon 1-4200 Diamond 1-4566 Realgar 1-6666 Ambergris 1-7000 Oil of cassia* 1-7634 Sulphur 2-2000 Phosphorus 2-8857 Hydrogen 30953 No. I. (Referred to from Page 72.) In order to convey to the reader some idea of the variety of dispersive powers which exist in solid and fluid bodies, I have given the following table, selected from a much Larger one, founded on observations which I made in 1811 and 1812-t The first column contains the difference of the indices of refraction for the extreme red and violet rays, or the part of the whole refraction to which the dispersion is equal; and the second column contains the dispersive power. Table of the Disjiersive Powers of Bodies. Dlroen.lT. x>i "- ° r In< "«" l«>»er. for nUtm Kuy> Oil of cassia 0139 0089 Sulphur after fusion 0-130 0149 Phosphorus 0-128 0-156 Sulphuret of carbon 0-115 0077 Balsam of Tolu 0103 0-065 Balsam of Peru 0-093 0058 * See Edinburgh Journal of Science, No. XX. p. 308. t See my Treatise on New Philosophical Instruments, p. 315. ■^ TABLE OF DISPERSIVE POWERS. DliperalTe Barbadoes aloes ' 0085 Oil of bitter almonds 0-079 Oil of aniscseed 0077 Acetate of lead melted 0069 Balsam of Styrax 0067 Guaiacum 0-066 Oil of cumin 0-065 Oil of tobacco 0064 Gum ammoniac 0-063 Oil of Barbadoes tar 0-062 Oil of cloves 0-062 Oil of sassafras 0060 Rosin 0057 Oil of sweet fennel seeds 0055 Oil of spearmint 0-054 Rock salt 0-053 Caoutchouc 0-052 Oil of pimento 0-052 Flint glass 0-052 Oil of angelica 0-051 Oil of thyme 0-050 Oil of caraway s«eds 0-049 Flint glass 0-048 Gum thus 0-048 Oil of juniper 0047 Nitric acid 0.045 Canada balsam 0-045 Cajeput oil 0-044 Oil of rhodium 0-044 Oil of poppy 0-044 Zircon, greatest ref. 0-044 Muriatic acid 0043 Gum copal 0043 Nut oil 0-043 Oil of turpentine 0042 Feldspar 0042 Balsam capi vi 0-041 Amber 0041 Calcareous spar — greatest 0-040 Oil of rape-seed 0-040 Diamond 0038 Oil of olives 0038 Gum mastic 0038 Oil of rue 0037 Beryl 0037 Fther 0037 Selenite 0-037 Alum 0-036 Castor oil 0036 Crown glass, green 0036 Gum arabic 0-036 Water 0035 Citric acid 0-035 Claw of Borax 0034 311 Diff. of Imlicra of Refraction for extreme Rays. 0058 0-048 0044 0040 0039 0041 0033 0035 0-037 0032 0-033 0032 0-032 0028 0026 0029 0028 0.020 0-026 0-025 0-024 0.024 0029 0028 0-022 0-019 0021 0021 0-022 0-022 0045 0016 0024 0022 0020 0022 0021 0O23 0027 0019 0056 0018 0-022 0016 0022 0-012 0020 0-017 0-018 0-020 0-018 0012 0-019 0018 %$ w$m. m^^^im^mmm^ mmzm 312 A TREATISE ON OPTICS. Dispersive power. Garnet 0034 Chrysolite 0033 Crown glass 0-033 Oil of wine 0032 Glass of phosphorus 0-031 Plate glass 0-032 Sulphuric acid 0031 Tartaric ocid 030 Nitre, least ref. 0030 Borax 0030 Alcohol 0029 Sulphate of baryta 0029 Rock crystal 0026 Borax glass (1 bor. 2 silex) 0026 Blue sapphire 0026 Bluish topaz 0025 Chrysoberyl 0025 Blue topaz 0024 Sulphate of strontia 0024 Prussic acid 0027 Fluor spar 0022 Cryolite 0022 Dlff. of Indict. of Refraction for extreme iUye. 0018 0022 0018 0012 0017 0017 0014 0016 0-009 0-014 001 i 0011 0014 0014 0021 0016 0019 0016 0015 0-008 0010 0007 No. II. (Referred to from Page 73.) The following table contains the results of several experiments whicli I made in the manner described in pp. 72, 73. The bodies at the top of the table have the least action upon green light, and those at the bottom of it the greatest. The relative position of some of the substances is empirical ; but, by referring to the original experiments in my Treatise on New Philosophical Instruments, p. 354., it will be seen whether or not the relative action of any two bodies upon green light has been determined. Table of Transparent Bodies, in the order in which tiiey exercise the least action upon Green Light. Oil of cassia. Sulphur. Sulphuret of carbon. Balsam of Tolu. Oil of bitter almonds. Oil of aniseseed. Oil of cumin. Oil of sassafras. Oil of 6weet fennel seeds. Oil of cloves. Canada balsam. Oil of turpentine. Oil of poppy. Oil of spearmint. Oil of caraway seeds. Oil of nutmeg. Oil of peppermint. Oil of castor. Gum copal. Diamond. Nitrate of potash. Nut oil. Balsam of capivi. Oil of rhodium. Flint glass Zircon. TABLE OF ACTION OF MEDIA ON GREEN LIGHT, &C. 313 Table of Transparent Bodies, fyc. — continued. Oil of olives. Calcareous spar. Rock salt Gum juniper. Oil of almonds. Crown glass. Gum arabic. Alcohol. Ether. Glass of borax. Selenite. Beryl. Topaz. Fluor spar. Citric acid. Acetic acid. Muriatic acid. Nitric acid. Rock crystal. Ice. Water. Phosphorous acid. Sulphuric acid. No. III. (Referred to from Page 80.) Table of the Indices of Refraction of several Glasses and Fluids. Refracting Media. Sj« Indices of Refraction for the Seven Ktivs in the Spectrum marked 1 iufif. 65. with the fallowing Letter*. 1! Red ray. c Red ray. D Orange. B Green. F Blue. O Indigo. H Violet. Water - - • Solution of? Potash 5 Oil of Tur- ) pentine ) Crown Glass Crown Glass Flint Glass Flint Glass M«..)( I- 111. 0-885 2-535 2-761 3-723 3-612 1-330935 1 -39962a 1-470496 1-525832 1-554774 1-627749 1-602042 1-331712 1-400515 1-471530 1-526S49 1-555933 1-629681 I-6O3SO0 1-333577 1-402805 1-474434 1-529587 1-559075 1-635036 1-608494 1-335851 1-405632 1-478353 1-533005 1-563150 1-612021 1-614532 1-337818 1-408082 1-4S1736 1-5360'. 1 1-566741 1-648266 1-620042 1-341293 1-412579 1-4S819S 1-641667 1-573535 I-6602K-, 1-630772 1-344177 1-416368 I-493S74 1-546566 1-579470 1-671062 1-640373 ... 211 mm ppg|p^ipi ip pi^^pg^^ m tm NOTES THE AMERICAN EDITOR. No. I. (Referred to from article 32.) If the remark of Mr. Herschel be admitted, the consequence may be drawn in relation to all the simple gases, except oxygen, that their ab- solute refractive powers will be expressed by the square of the index of refraction, diminished by unity: for in them, the specific gravity is directly proportional to the weight of the atom. The same remark applies to the vapors of simple bodies, and to many compound gases. Jf the specific gravity, and weight of the atom, of hydrogen be called unity, the specific gravity of nitrogen, chlorine, &c. will be expressed by the weight of the atom of each : hence the square of the index of refraction diminished by unity, will be, by the process directed in article 32, multiplied and divided by the same quantity. The inflammable substance hydrogen, instead of presenting a high intrinsic refractive power, would occupy a low place on the scale, while chlorine would rank high upon it. This consequence was observed by Mr. Herschel himself. No. II. (Referred to from article 66, page 67.) This remark in relation to the absorptive power of water, though true for moderate thicknesses, in relation to the colored rays of the spec- trum, appears, by a recent discovery of Signor Melloni, not to be true in regard to the heating rays. An account of the interesting experi- ments which have established this fact, will be more in place, in connexion with the article which treats of the heating power of the spectrum. No. III. (Referred to from article 66, page 70.) This interesting analysis of the solar spectrum, by Sir David Brewster, will, probably, have its value to the reader increased by a brief state- ment of the experiments from which the results, given in pages 69 and 70, were deduced. The matter of this note is taken from a paper, by Sir David Brewster, in the Edinburgh Journal of Science for October, 1831. * 1. The first position is that, " red, yellow, and blue light exist at every point of the solar spectrum." The eye gives evidence of the existence of red light, in the red, orange, and violet spaces, which, together, con- stitute more than half the length of the spectrum. If the blue and * And Edinburgh Trans. Vol. XII. Part. I. §m -mm %m», mm^ W^vf^ 316 A TREATISE ON OrTICS. indigo spaces be transmitted through olive oil, the light becomes of a violet tint, rendering evident the red, previously existing in the blue and indigo, by absorbing rays which had neutralized it. In the yellow and green spaces the existence of red is proved by showing that white light may be detected in them. Yellow light is recognized by the eye in rather more than one-fifth of the length of the spectrum, namely in the orange, yellow, and green spaces. It may be proved to be present in the blue and indigo spaces by many experiments, among which is the one already described, in which a violet tint is developed, bypassing the spaces through olive oil : the tint absorbed by the oil cannot be red, because violet, reddish blue, is made to appear by the transmission; it cannot be blue, for blue taken from blue will not leave triolel : ii is, then, yeilow, which mixed wilh the red and blue had formed white light, at ibis part of the spec- trum. Farther, the spectrum examined through a deep blue glass, show's green in the blue space, and through a transparent Avafer of gelatine, produces a whitish band in the same space. Yellow is shown to exist in the red space by examining it through a prism of ]xirt wine, the re- fracting angle of which is 90°, and the whole of the red space assumes a yellowish tint by the absorption of the blue rays, by certain thicknesses oi pitch, balsam of Peru, &c. In the violet space, owing to the extreme facility with which that color is absorbed, and the extreme faintness of the rays, yellow light has not yet been detected. Blue light is perceptible to ihe eye through more than two-thirds of the length of the spectrum, that is in the green, blue, indigo, and violet spaces. The absorptive powers of pilch, balsam of Peru &C, show green light extending considerably within the red space ; and the blue is farther proved to be spread throughout that space by the yellow tinge which it assumes, when viewed through the media already alluded to; a tint which could only result from the absorption of blue rays. 2. Wliile light exists, at every point of the spectrum, and may be in- sulated by absorbing the excess of the colored rays at any point. By a particular thickness of smalt-blue glass, the ycllcnv space, the brightest of the spectrum, becomes greenish while, and, with a different blue, reddish white. A mixture of red ink and sulphate of copper re- duces the yellow space to nearly a white, the tint being slightly red when the ink is in excess, and green when there is too much of the solution of sulphate of copper. By particular methods, not described, Sir David Brewster states that he has succeeded in insulating white light in both the orange and green spaces. The curious property possessed by this white light of not being de- composable by refraction, is a powerful support of the new theory of the spectrum. By a principle of absorption applied to the heating rays of the solar spectrum, and which will be described in a subsequent note, the exist- ence of a spectrum of heating rays, exceeding in length the three colored spectra, is proved, bringing a new analogy to bear upon this question. No. IV. (Referred to from article 71, page 62.) Comparative experiments on the heat in different parts of the solar spectrum, require the most delicate instruments. The new branch of science, thenno-magnetism, furnished Signor Melloni with a much more sensible means of measuring temperature than the common thermom- NOTES BY THE AMERICAN EDITOR. 317 eter, namely, by the magnetic currents developed by heat, in a battery composed of bars of bismuth and antimony. By the aid of this instrument, he found that the heat accompanying the violet space of the spectrum, from a crown glass prism, was not at all absorbed by pure water, while a small portion of the heat in the indigo was absorbed, a greater portion of that in the blue, and so on through the colored spaces and into the invisible heating rays beyond the spec- trum, the extreme rays of which were entirely absorbed. The relative degrees of heat in the spectrum formed by a crown glass prism, which, however, it must be recollected, has absorbed the rays unequally, is represented by the annexed diagram, in which the °-5 | r |o| y | g | T, | j | v ordinates, 2 v, 5 1, &c, of the upper curve represent, nearly, the relative temperatures; the lengths of the ordinates, and of course the relative degrees of heat, being expressed by the numbers written above them : thus the amount of heat in the middle of the space v, the violet space, compared with that in the middle of the blue space b, is as 2 to 9 ; with the middle of the red space r, as 2 to 32. The points marked 25, 12, &c. beyond the colored space r, correspond to the bands, in the spectrum, having, re- spectively, the same temperatures as the middle of the yellow, of the green, of the blue, &c. By passing the spectrum through a thickness of less than a twelfth of an inch of water, contained between plates of thin glass with parallel surfaces and free from defects, the heating powers of the several rays became as represented in the lower curve ; none of the heat accompanying the violet rays having been absorbed, a little of that accompanying the blue, and so on increasing as the re- frangibility diminished, until in the band, 2, 0, having the same tem- perature as the violet, all the healing rays were absorbed. Different media stopped the heating rays in different degrees, those of higher refractive powers permitting them to pass more readily than those of lower powers. Although this subject cannot be considered as fully developed, we are able to understand by it, why Seebeck found the point of greatest heat to vary, according to the material of the prism used to form the spectrum, being in the red when a prism of crown glass was used, and in the yellow when water was the refracting material. Melloni ibund the greatest heat in the orange after passing the spectrum formed by the crown glass prism through water, as appears by the diagram, in which the greatest heat in the red and yellow are 20, and in the orange is 21. Seebeck found the point of greatest heat in the yellow, when the prism contained water; a sufficiently near coincidence with the ob- servation of Melloni, if we consider that in the experiments of Seebeck the rays were exposed to absorption by the glass forming the hollow prism in which the water was contained, and in those of Signor 2B2 &&p HH| ■ 3&&*;j\ 318 A TREATISE ON OPTICS. Melloni to the absorption by the crown glass prism first, and then to that by two plates of glass and the water contained between them. The power of transmitting the heating rays without absorption being greater as the refractive power is greater, according to the law before referred to, we should expect that in (lint glass the greatest heat would lie farthest from the violet end of the spectrum: in plate and crown glass, that it should be at a less dislance from that end ; in sul- phuric acid and oil of turpentine slill less; in alcohol and water yet nearer to the violet end: and these deductions we find, by consulting the table on page 82, to be correct. Minute differences, which could not be detected by the instruments used by Seebeck and others, in the points of greatest heat as given by that tabic, will probably hereafter appear. The experiments discussed in ibis note authorize the addition of a fourth spectrum to the three colored spectra represented in fig. 51., namely, a healing spectrum containing rays which are less refrangible than the extreme red rays of the spectrum. No. V. (Referred to from page 116.) The undulatory hypothesis represents in so simple a manner the phe- nomena to which Dr. Voung applied his principle of interference, thai I have been induced to reler to it here, with a view- to a general expla- nation of the hypothesis. The reader will be better satisfied if he lake up the subject, as briefly referred to in the 84th article of the text, be- fore entering upon the account to be given in this note. As stated in that article, the hypothesis of undulations supposes all space, the planet- ary spaces as well as the interstices between the particles of bodies, to be occupied by an elastic medium, or ether, which is put in a state of vibratory motion by luminous bodies, and in which impulses are propagated according to the same mechanical laws, as the impulses which, communicated to air, produce sound. If we suppose a luminous point surrounded by this elastic medium, the particles immediately about the point have a vibratory motion im- pressed upon them, or a motion to and fro ; this they communicate to the adjacent particles, and thus a wave is formed, which spreads about. the point as a centre, just as the waves formed by a stone, thrown into still water, spread around the point at which the stone struck the sur- face. As these waves would communicate to a floating body which they might meet, an impulse in a direction radiating from the point where they originated, so luminous waves striking the retina, give the sensa- tion of light in a similar direction. In the annexed diagram, let AB, No. 1, represent one of the directions in which the impulse given by a luminous body, is propagated : we shall, lincl, according to the hypothesis, along that, line particles of the elastic medium, or ether, having all rates of motion, from rest, or when the mo- tion is nothing, to the greatest rapidity of the vibration ; and in the two opposite directions, from A towards R, and from B towards A. For example, let the particles at A, D, and C, be at rest, then if from A lo D we find particles moving towards B, their velocities, or rates of motion, will be found increasing between A add a', mid-way from A to D, and then de- creasing between a' and D; between D and C the vibration will be in the contrary direction, namely, from B towards A ; and the velocities after NOTES BY THE AMERICAN EDITOR. 319 increasing tof, will diminish toC. The distance between A and C includes all velocities from nothing to the greatest, and in the two opposite direc- tions A B and B A ; the same would he true of C B, if made equal to A C : this distance AC, or C B, is called the Length of a wave, and it is this which in red light is 256 ten millionths, and in violet light 174 ten millionths, of art inch (see page 119, text.) To represent to the eye the velocities of the different particles of ether between A and D, the curve A a D is described, in which the ordinates, or lines in the same direction as a! a, represent these velocities, as, for example, a 1 a the velocity at a'; the particles between D and C vibrating in a contrary direction, the curve D b C, representing their velocities, is traced on the side of the line A B op|xjsite to A at). In the same way the velocities between C and E, and between E and B, are shown by similar curves, on opposite sides of A B. Let No. 2. represent a system of waves in which the lengths M P and P O are equal to AD" and DC, and let the motion between M and P coincide in direction with that between A and D, the curves of velocities M m P and A a I) coinciding, and the motion be- tween P and O with that between D and C, the curves P p C and D b C coinciding; then it is plain that the motion of the particles between OQ, C E, and Q N, EB, &c. will be the same, or that the undulations will coincide throughout; one undulation will, therefore, add to the effect of the other, and the light will he the united light produced by the two undulations. The same will be true whether the point M coincides with A, with C, or with B, &c. ; that is, whether the lengths of the paths of the two rays A B and M N are exactly equal, or differ by one, two, or more undulations. If the rays, instead of moving in the same direction, meet under a small angle, the remarks will still apply ; and the first result, stated on page 115, text, is in accordance with the hypothesis, under consideration, namely, that bright spots, illuminated by the sum of the two lights, will be formed when the; differences in the lengths of the paths of the rays, are d, (A C,) 2 d, (A B,) 3 d, &c. Next let the curves described in No. 3 represent the velocities in another system of undulations, S V and V U being equal to A Dand BC in No. 1, the particles of the ether between S and V, as shown by the curve, being in a state of vibration from T towards S, those between V and U from S towards T, and so on. If the ray S T in No. 3 were brought to coincide with A B in No. 1, the point S being placed at A, the particles between S and V in No. 3 moving in opposite directions from 320 A TREATISE ON OPTICS. those between A and D, and those between V and U in opposite dircc tions from those between D and C, and their velocities being supposed equal, their motions would destroy each other, and the wave would be destroyed, or darkness would result. The path S T differs from A B by the distance S V, or half an undulation. The same would be the result if No. 3 began one undulation to the left hand of S, or two or more un- dulations, that is if the path ST, No. 3, differed from A B, No. 1, by one and a half, two and a half, &c undulations. As the Bame consequences would follow, if the rays ST and A B met under a small angle, we infer (page 115, text) that when the difference in the lengths of the paths, of the two pencils of rays, is hd (A D), lid (A E), 2k d, &c, " instead of adding to one another's intensity, they destroy each other and produce a dark spot." * « No. VI. (Referred to from page 120.) Professor Hare lias observed, in relation to the translucency of gold leaf; — "Gold leaf transmits a greenish light, but it is questionable, if it be truly translucent. Placed on glass, and viewed by transmitted light, it appears like a retina. It is erroneously spoken of as a continuous superficies." The nature of the process by which gold is reduced to leaves, strengthens this conclusion. On examining gold leaf by the solar microscope, I find in it innumer- able rents, and also various gradations of thickness ; the rents have their edges colored, a blue fringe appearing on one side, and a reddish brown on the opposite side; the thickest parts transmit no light, and through the very thin parts a dclicategrcen light is transmitted. The surface thus exhibited is very beautiful, The" rents are visible to the naked eye, when the leaf is very strongly illuminated. No. VII. (Referred to from page 237.) The following classification of colored bodies is alluded to in the text, * as given by Sir David Brewster, in the life of Newton. The colors of each of the classes require, in his view, to be explained upon different principles. 1. " Transparent colored fluids, transparent colored gems, transparent colored glasses, colored powders, and the colors of the leaves and flowers of plants." The colors of these bodies are derived from the absorption of particu- lar colored rays: thus, water at great depths appears red by transmitted light, owing to the absorption of the blue and yellow rays which with :!:r red constituted white light; certain thicknesses of smalt-blue glass appear intensely blue by transmitted light, while at greater thicknesses the glass appears red. In the case of opaque and colored todies, as in the leaves of plants, we are to suppose certain rays to be absorbed by the indefinitely thin film through which the rays reflected from any surface may be supposed to pass, the complementary tint being reflected; as all the incident fight is not reflected, the transmitted tint will be com- plementary to the colors ateorbed, and thus the body will appear of the same color by both reflected and transmitted light. Colored powders NOTES BV THE AMERICAN EDITOR. 321 would seem not always lo belong lo this class, since many of them i- tange their hues with a change in the size of the particles. 2. " Oxidations on metals, colore of Labrador feldspar, colors of precious a id hydrophanous opal and other opalescences, the colors of the feathers of birds, of the wings of insects, and of the scales of fishes." To these the Newtonian theory is strictly applicable. 3. " Superficial colors, as those of mother-of-pearl and striated sur- faces." 4. " Opalescences and colors in composite crystals having double re- fraction." 5. " Colors from the alisorption of common and polarized light, by doubly refracting crystals." 6. "Colors at the surfaces of media of different dispersive powers." 7. " Colors at the surfaces of media in which the reflecting forces extend to different distances, or follow different laws." No. VIII. (Referred to from page 251.) The philosophic toy called by its inventor, Dr. Paris, the thaumatrope, or wonder-turner, illustrates very perfectly the fact of the duration of impressions on the retina. On one side of the card, represented in the diagram, is drawn a chariot and horses, and- on the opposite side the charioteer; on causing the card to revolve by turning the strings C and D between the thumb and fore- finger of each hand, the charioteer appears in the act of driving die chariot, as in the figure. It requires but little skill to give to the card, exactly the motion, which shall perfectly unite the two objects on tho opposite sides into one picture*, and yet not render it confused by tho rapidity of the turning. Many amUiing illustrations accompany the toy; for example, Harlequin and Columbine are painted upon opposite sides, and by a turn of the card are seen to join in a dance: royalty, stripped of its robes, occupies one side of the card, and the robes the opposite, the robes are donned by a turn : a potter seated at his wheel moulding the unformed clay, occupies one side of the card, and an urn is grasped by an arm on the reverse; on turning, the urn appears grasped by the potter's arm, the foot of the vase being yet unfinished. ■ mm 822 A TREATISE ON OPTICS. No. IX. (Referred to from page 261.) Referring to his now analysis of the solar spectrum, Sir David Brews- ter advances the following hypothesis to account for certain of the cases just detailed* "By means of this analysis we are now ahle to explain the phe- nomenon observed by those who are insensible to particular colors. (Edin. Journ. Sc. No. XIX. old series. No. IX, new series.) The eyes of such persons are blind to red light ; and when we abstract all the red rays from a spectrum constituted as already)" described, there will be left two colors, blue and yellow, the only colors which are recognized by those who have this delect of vision. To such eyes, light is always seen in the red space ; but this arises from the eye being sensible to the yellow and blue rays, which are mixed with the red light. Hence blue light will be seen in the place of the violet, and a greenish yellow will appear in the orange and red spaces, or, which is the same ihing, the spectrum will consist only of the yellow and the blue spectra. The physiological fact, and the optical principle, are therefore in perfect accordance; and while the latter gives a precise explanation of the former, the former yields to the latter a new and an unexpected sup- port." The details of the cases referred to, fully sustain this conclusion. There are other circumstances connected with them, and with others described in the text, page 2G0, not unworthy of notice. In the second of the cases described in the Journal of Science, No. XIX, although the individual never failed to detect a full blue or a lull yellow, he seems to have had very imperfect ideas of those colors when presented in a state of mixture ; green, as such, he did not know, and when blue was diluted with yellow, forming what to a good eye would appear yellowish green, the blue tint escaped him, and the mix- ture appeared yellow. In like manner, his discrimination of yellow, when mixed with blue, was very defective ; he called grass green yellow, and yet yellowish green appeared to be "yellow with a good deal of blue in it." This remark may serve to explain why the same white seen at different times, appeared to him to vary in its tint, at one time being white, at another " white with a dash of yellow and blue," at another " white with yellow and blue in it." When requested to arrange colors so as to produce the strongest contrasts, he divided them into two classes, to one of which he gave the name of blue, and to the other i/e!low. In these contrasts he invariably placed white among the blues, and was never perplexed, as in the preceding examinations, when tasking himself as to the precise shades. That white should be classed by him as blue, appears consistent with the other observations, for being blind to red light the tint of white should be that which appears when red is removed from the spectrum or a bluish green, which tint he saw as a hhie. In examining other cases we shall find reason to be satisfied that this blindness extends to light of other colors than red, and that in those eases also there is a want of discrimination between shades in mixtures of the colors to which the eye is sensible. The Plymouth tailor, whose * Edinburgh Trans. Vol. XII. Part. I., or Edin. Journalof Science, No. X. new series, f See text, p. 69. NOTES BY THE AMERICAN EDITOR. 323 case is described by Mr. Harvey (see page 260, text), seems not to have been entirely blind to red light, and to have been in a measure blind to blue; thus the prismatic spectrum appeared to consist entirely of yellow, and light blue ; the red, orange, and yellow spaces appearing as if red had been withdrawn from them, while the full blue, the in- digo and violet were light blue, and dark blue and indigo stuffs ap- E eared to be black, and crimson was either blue or black. A dark green e regarded as brown, by which, since he was blind lo red light, he must have meant a shade of black, and light green as orange, by wjiich, for the reason just slated, he meant a variety of yellow. The blue in both these mixtures escaped his perception. An extreme case seems to have occurred in the vision of Mr. Harris, of Allonby, who, according to the statement of Mr. Huddart, could only distinguish black from white, or was entirely blind to colors. It is much to be desired, for the elucidation of this curious subject, that more well examined cases were on record. The colors of the spectrum afford rigid tests, not to be found in colored stuffs ; and by such tests only, the minutiae of peculiarities of vision can be satisfac- torily determined. ?"'.'>ii- ■ ' -'14'! ■■■■ m APPENDIX TO SIR DAVID BREWSTER'S TREATISE ON OPTICS; INTENDED TO ADAPT THE WORK TO USE, AS A TEXT-BOOK IN THE COLLEGES OF THE UNITED STATES. BY A. D. BACHE, PROFESSOR OF NAT. PHILOS. AND CHEMISTRY IN THE UNIVERSITY OF PENNSYLVANIA. 3$*iS&i&§!$§fcy Wm^$Ml?Q$S%& &$£ttt^ ■P ■I ADVERTISEMENT. The object of the following Appendix is to place in the hands of the students of our Colleges, a text-book which will furnish them with some of the analytical methods of the most recent writers, upon the elements of Optics. The work of Sir David Brewster is from the pen of a master, and presents, in a popular form, the results which have flowed from experiment and from theory, applied to the investigation of the different branches of Optics. The Appendix merely aims at supplying to the student the mode of determining the results given in the text, more particularly in what relates to Reflexion and Refraction. It may not be amiss to state, that I do not present, to the notice of instructers, an untried course, but that most of the propositions in the following pages have entered into the mathematical portion of the course, taught to the Senior Class of the University of Pennsylvania. The works in which the full development of these subjects may be found, and which have been consulted in the composition of the Appendix, are Coddington's Optics, Coddington on Reflexion and Refraction, Lloyd's Treatise on Light and Vision. The more advanced student will find the subject treated by the most general methods in Herschel's Treatise on Light. A. D. BACHE. Philadelphia, March, 1833. ^$f|i$ ^M $0$$ $Z&$M^;^>&$MjM CONTENTS OF THE APPENDIX BY THE THE AMERICAN EDITOR. Art 1 •I 3. •1. 5. 6. 7. 8, REFLEXION. CHAPTER I. REFLEXION BY SPHERICAL AND PLANE MIRRORS. Page Direct and oblique pencil defined 9 Reflexion of a small direct pencil by a Spherical Mirror. — Case 1. Diverging Rays falling upon a Concave Mirror ... ib. Case 2. Converging Rays upon a Concave Mirror 10 Comparison of the Formulae for the two Cases ib. Case 3. Diverging Rays upon a Convex Mirror 11 Case 4. Converging Rays upon a Convex Mirror ib. Comparison of Formulae. — General Formula for the Reflected Pencil 12 Rule. — The sum of the vergencies of incident and reflected Pencils constant ib. Application of the general Formula for the reflected Pencil, to the Plane Mirror 13 Parallel Rays falling upon a Plane Mirror ib. 12. Diverging and Converging Rays ib. Application of the general Formula to Reflexion by a Concave Mirror ib. Principal Focal distance found 14 Case of diverging Rays; Reflected Pencil considered for dif- ferent values of the distance of the Radiant Point ib. Converging Rays upon a Concave Mirror. — Rule for Focal length 16 Reflexion of a small direct Pencil by a Convex Mirror ib. 19, 20. Parallel, diverging, and Converging Rays ib. 22. Incident tuid reflected Pencils referred to the centre of the Mirror 17 24. Distance of the principal Focus from centre, a mean propor- tional between the Distance of the Radiant Point and Focus, of any small Pencil, from the same centre 18 Cases of the oblique Pencil 19 28. Reflexion of a small oblique Pencil which meets a Mirror near its vertex ib. Focus of reflected Pencil in primary plane. General Formula 20 Geometrical interpretation of general Formula. — Rule ib. Focus of reflected Pencil in secondary plane. — Rule 21 Reflexion of a small oblique Pencil which crosses the Axis of a Mirror belbre meeting its Surface 22 A2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^M H ■ ■ VI CONTENTS OP THE APPENDIX. CHAPTER II. FORMATION OF IMAGES BY REFLEXION. Art. Page 33. Image of a Plane, formed by a Concave Mirror 23 34. Curvature of Image at Vertex 24 35. Change in the form of Image, by change in distance of object from Mirror ib. 36. Hypothesis that Image and Mirror are similarly curved 20 37. Image formed from Section ib. 38. Case of Convex Mirror. — Case of Plane Mirror ib. REFRACTION. CHAPTER m. REFRACTION BY PRISMS AND LENSES. 39, 40. Refraction at a plane Surface. — Law expressed. — Angle of deviation, general Formula 26 41. Case of total Reflexion 27 42. Formulae for Reflexion deduced from those for Refraction ib. 43. Refraction by a Prism, general Formula 28 44. 45. Case in which the angle of Incidence is small. — Other cases in which the general Formula is simplified ib 46. Case of Rays incident perpendicularly upon the first Surface of a Prism 29 47. Case in which the angles of Incidence and emergence are equal ib. 48. Proof thai, in this last case, the deviation is a minimum 30 49. Refraction by Lenses. — Suppositions made in first approxi- mations. — Correct ions 31 50. A small Pencil of Rays refracted by a spherical Surface. — Formulas ib. 51. Signs of the algebraic quantities in the general Formula .... 32 52. Refraction of a direct Pencil by a Lens. — Approximate re- lations 33 53. 54. Formula, neglecting thickness of Leas. — Rule. — Cases in ' practice 34 55. Mode of applying Correction for thickness in any case 35 56. Refraction by a plate with parallel Surfaces ib. 57. Particular case of thick plate 36 58. 59, 60. Parallel, diverging, and converging Rays falling upon a thick Plate ib. 61. Refraction by a thin double Convex Lens 37 62, 63. Parallel Kays. Principal focal distance 38 64,65. Diverging Rays. — Rulf.s, for focal distance, in different forms of double Convex Lens, investigated 39 66. Converging Rays. — Rules for Focal Length 40 67, 68. Correction for thickness applied to Case of parallel Rays falling upon a double Convex Lens. — Nature of results ... 41 69. Principal Focal Length of a Refracting Sphere deduced from the general Formula 42 70. Refraction by a thin plano-convex Lens 43 m CONTEXTS OF THE APPENDIX. Vll Art. Page 71. Plane Side fumed to Incident Light. — No correction for thick- ness 44 72,73. Convex side to Incident light. — Correction for thickness .. ib. 74. Refraction by a double Concave Lens 45 75, 76, 77. Cases of parallel, diverging, and converging Rays. — Effect of thickness in a practical point of view 46 78. Plano-concave Lens. — Concave side to incident Rays. — Plane side to incident Rays 47 79. Refraction by a Meniscus. — Power of the Meniscus with Con- vex side to incident Rays 48 80. 81. Meniscus with Concavity towards Incident Light — Focal length ■. ib. 82. Concavo-convex Lens ib. 83. Spherical Surfaces of the same Curvature 49 CHAPTER IV. FORMATION OF IMAGES BY REFRACTION. 84. Formation of Images by Lenses. — Property of Centre of a Lens 50 85. Centres of different kinds of Lenses found 51 86. Mode of using this Point in the formation of Images ib. 87. Image of a plane, by a Convex Lens 52 88. Object considered as a portion of a circular arc 53 89. Image by a Concave Lens 54 90. Theory of the magnifying power of a Lens used as a Telescope ib. CHAFrER V. SPHERICAL ABERRATION OF MIRRORS AND LENSES. 91. Spherical aberration of Mirrors. — Longitudinal aberration de- fined. — Lateral aberration 55 92,93. Formula for longitudinal aberration. — Approximate rela- tioas introduced ib. 94. Aberration for parallel Rays 58 95. Lateral Aberration ib. 96. Physical Focus of a Mirror, or circle of least Aberration, de- termined in position and magnitude ib. 97. Introduction to Aberration of Lenses. — Aberration produced by a single spherical Refracting Surface 61 98. Approximate values introduced into general Formula 63 99. Aberration by a Convex Surface. — Cases of no Aberration . . 64 100. Spherical Aberration of Mirrors found from Formula for Aber- ration by a Single refracting Surface ib. 101. Spherical Aberration of a Lens 65 102, 103. Case of parallel Rays.— Double Convex Lens 66 104. Spherical Aberration of an equi-convex Leas of glass 68 105. Case of Lens of which the Radii of the Surfaces are in the ratio of two to five ib 106. Aberration by plano-convex Lens. — Plane side towards Inci- dent Light.— Convex side towards Incident Light . . , 69 107. Lens of least Spherical Aberration with a given power ib Vlll CONTENTS OF THE APPENDIX. Art. Page 108. Equations of Curves producing Surfaces of accurate Refrac- tion 71 109. Convex Surface of accurate Refraction for parallel Riys pass- ing from a rarer to a denser Medium. — Lens formed 72 110. Concave Surface of accurate Refraction for parallel Rays pass- ing from a denser to a rarer Medium. — Lens corresponding to this Case 73 112. Surfaces of accurate Reflexion determined » 74 113. Caustic for parallel Rays falling upon a concave spherical Mirror ib. For Rays diverging from the Extremity of a Diameter 75 114. Equation of caustic Curve for any reflecting Curve 76 115. Example. — Caustic by Reflexion from a portion of a Logarith- mic Spiral 78 CHAPTER VI. DOUBLE REFRACTION AND POLARIZATION OF LIGHT. 117. Formula for extraordinary Refraction in the Plane of Section of a doubly refracting Crystal 79 118. Law of exlraordinary Refraction in Plane of principal Section 80 In Plane perpendicular to principal Section ib. 119. Formula for relative Brightness of Images, after transmission through a second doubly refracting Crystal ib. 120. Law of Polarization of Light by Reflexion at the Maximum Polarizing Angle. — Consequence of the Law 81 121. Relation of polarized Ray to refracted Ray ib. 122. Polarization at the second Surface of a Plate 82 123. Law of partial Polarization of Light, by Reflexion ib. 124. Effect of successive Reflexions of Common Light, at Angles differing from the Maximum Polarizing Angle 83 125. Law of Polarization of Light by ordinary Refraction ib. 126. Polarization of Light by Refraction and Reflexion at the two Surfaces of a Plate ib. Consequences of the Law 84 CHAPTER VII. ON THE RAINBOW. 127. To determine the Deviation, produced by two Refractions and any number of Reflexions, by a sphere, when the deviation is a Minimum S5 128. Theory of the Primary Rainbow 86 129. Theory of the Secondary Rainbow 87 APPENDIX. REFLEXION. CHAP. I. REFLEXION BY SPHERICAL AND PLANE MIRRORS* (1.) In considering the cases of reflexion from spherical or plane surfaces, two divisions will be made : in the first, the axis of the incident pencil will be supposed perpendicular to the surface of the mirror, in the second, oblique to it ; in the first case the pencil is termed direct, in the second, oblique. (2.) Prop. I. To determine the form given by reflexion to a small direct pencil of light, proceeding from a point in the axis of a mirror. Case 1. In fig. 8., p. 18 of the text, let A represent the radiant point of a pencil of diverging rays, F its focus, and C the centre of a concave mirror. Call AD = u, FD = v, and CD = r. The angle of reflexion FMC being equal to the angle of incidence AMC, the line CM bisects the vertical angle of the triangle AMF: whence (Legendre's Geom., Book III., Art. 201., or Euclid, VI. 3.) AC : AM : : FC : FM, or AC _ FC_ AM ~ FM The pencil being, by supposition, very small, the point M is very near to D, hence for the approximate value of AM we may take AD, and for that of FM, FD. The equation just found becomes AC FC AD FD By the notation adopted above AC = AD — CD = u — r, and FC = CD — FD = r — v. We have therefore 1 = 1 , whence ♦Throughout the Appendix, the student is supposed to be acquainted with the corresponding chapters in the body of the work. ■ ^H ■ ^M H 10 CONCAVE MIRRORS. APPENDIX. dividing by r and transposing 4- + 4— W (3?) Case 2. We next proceed to the case in which a converging pencil falls upon a concave mirror. Fig. 10., p. 20 of the text. AM, AN representing the two extreme rays of the pencii, con- verging to A', the imaginary radiant point, MF and NF are the reflected rays. The radius CM therefore bisects the angle AMF, the outward angle of the triangle A'MF, and (Euc. VI. A.) A'C _ FC_ AM ~ FM ' but for A'M and FM we may substitute A'D and FD, their ap- proximate values, whence A'C _ FC_ AD ~ FD ' Using the notation before employed, A'D = u, FD = v, and DC = r, whence A'C = u + r, and FC = r — u; these values substituted in the equation just found give, u 4- r r — v i^ — or. 1 -f — = 1 and u v _i + -L.i o» (4.) Comparing equation (b) with (a) we find that it differs from it only in the sign of — which is positive in (a) and negative in u (b) ; botli these cases may, therefore, be represented by the same equation, if we agree to give the positive sign to the distance of the radiant point for diverging rays, negative for converging rays ; that is, if we consider the distance (u) positive, when the radiant point is in front of the mirror, negative when it is behind the mirror. If, then, dLu represents the distance of the radiant point from the mirror, will denote the degree of divergency or convergency of the incident rays. In like manner, — will represent the con- v vergency of the reflected rays. From equations (a) and (b) 1 J_ _ 2 ^am REFLEXION. 11 where r is a constant quantity ; a result which may be thus ex- pressed : the divergency (or convergency) of the incident rays to- gether with the convergency of the reflected rays is a constant quantity for the same mirror. The curvature of the mirror is measured by — , the reciprocal of the radius. r (5.) Case 3. Diverging rays falling upon a convex mirror. Fig. 12., p. 21, text. The line CM, bisects the outward angle of the triangle AMF t whence (Euc. VI. A.) AC __ FC AM ~~ ~FM ' substituting their approximate values for AM and FM, AC _ FC AD ~ FD' and by the notation adopted in the cases already considered, u 4- r r — v , — J- — = , whence u v J 1_ __ JJ_ ,. u v r On comparing this equation, in which we have made — posi- u tive as it corresponds to a real radiant, with (a), we perceive that 1 2 the signs of both and are different. From the figure we v r observe that v corresponds to an imaginary focus, and that the ra- dius is now behind the mirror. Equation (a) may, then, be used to represent this case if the sign of the radius be changed ; the re- sulting negative value of the focal distance corresponds to a focus behind the mirror. (6.) Case 4. Converging rays falling upon a convex mirror. The formula for this case may be deduced from Jig. 12., if B M and BNbe made to represent the incident, and MA, NA the reflected rays. We should have by proceeding as in the last case, FC AC FM AM FC _ FD AC AD and since FD — u, F being the imaginary radiant point, and AD = v, A being the focus ; FC = r — u, and AC = r -{- » t whence 12 CONVEX MIRRORS. GENERAL FORMULA. APPENDIX. r — u r 4- v = -JL- ,. or u v 2 + ^=- w the first term being made negative to correspond to the case of converging rays. This formula differs from (c) in the sign of _ , which was negative in (c), and in that of — . The figure v u shows that v in this latter case corresponds to a real focus, while in the former — v denoted the distance to an imaginary fbcus. The change of sign in _ conforms to the remarks made in W article (4.) (7.) Comparing the four equations (a) (b) (c) and (d), we per- ceive that the formula i + — a) r may be made to include them all, by attributing to u, and r, respec- tively, the positive sign when the radiant point or the centre is in front of the mirror, the negative sign when either of these points is behind the mirror, and by considering the positive value of v as corresponding to a focus in front of the mirror, its negative value to one behind it. (8.) We might have commenced by giving to the student this conventional mode of considering the quantities used in the an- alysis, and then have deduced the general equation by reference to a single diagram : we have preferred in the outset to show him that the variations in the algebraic signs are not arbitrary, but required by the geometrical relations of the quantities. From the formula J- + J- = A a, u v r we deduce the general rule, that the sum of the vergencies* of the incident and reflected pencils is a constant quantity. (9.) Having obtained an equation (1) expressing the relation be- tween the distances of the radiant point and focus of a small pencil, by means of the radius of the mirror, we shall proceed to interpret it in its application to different kinds of mirrors, and under different circumstances of the incident pencil. * This convenient term, expressing, as the case may be, either divergency or convergency, is proposed and used by Lloyd in his Treatise oh Light and Vision. CHAP. I. REFLEXION.— PLANK MIRRORS. 13 Prop. II. To determine the form given to a small pencil of rays by rejlexion from a plane mirror. 2 In the plane mirror the radius is infinite, or — = o , whence from (1), — + — = o (2), or, The focus and radiant point are at equal distances from the mirror, but on opposite sides of it. (10.) If parallel rays fall upon the mirror u = oo , and v = — oo , or the reflected rays are parallel. This corresponds to the case represented in Jig. 4., p. 15, text. (11.) If the rays diverge before reflexion, the formula _L — J_ V u shows that they will be equally divergent after reflexion ; and that the focus is as far behind the mirror as the radiant point is in front of it. Fig. 5., p. 15, text. (12.) For converging rays (Jig. 6., p. 16, text,) u takes the nega- tive sign, and (2) becomes ~— + 1 = o (3), whence, 1 1 V =s U. The sign of v being positive the focus is real, Hs distance v m front of the mirror is equal to the distance of the imaginary ra- diant point behind it. (13.) Prop. III. Rejlexion of a small pencil of light by a concave mirror. The formula which applies to this case is, i=-I <■). u v r By transposition J_ _ _2_ l_ _ 2u — r v r u ur i + 2u—r 14 COIVCAVE MIRRORS. APPENDIX This value of v gives the rule on page 19 of the text. (14.) When the rays are parallel — = o . And = , or, The sign of v being positive shows that the focus is in front of the mirror, and its value JL , that the focal distance is half the radius. This is represented in fig. 7., p. 17, of the text, where FD = 4 CD. The focus of parallel rays is called the principal focus. As it serves as a point of comparison for the foci of other rays, we shall 2 represent its distance by a symbol, /. Placing for — its value r 1 7 , formula (1) becomes i + i- . • ■ (4). (15.) The next case to be considered is that of diverging rays. In this, u is positive and the formula is (- = (4), whence, u v f v f u As long as u >/, or the point A in fig. 8., p. 17, is farther than the principal focus from D, we have < — , and — » f f is a positive quantity, or the rays converge after re- u flexion. Since — <« , we have «-- , and » > /, or the focus is farther from the mirror than the principal focus. If we suppose, besides, that A is beyond the centre C, we have a 1 1 k« 1 1 2 12 u> r, and <; — , , whence — , or — > - u r f u r u r , or >. , and — > — , or v < r, the focal distance r r v r is less than the radius. • •HAP. I. REFLEXION. 15 We infer, then, that rays diverging from a point beyond the centre of the mirror, are reflected to a point between the principal focus and the centre. As diminishes in value, — evidently must increase, or, u v as u increases, v diminishes, and vice versa : that is, as the radiant point recedes from the mirror the focus approaches it, and vice versa. When the radiant point becomes infinitely distant, the rays are parallel, and their focus is the principal focus. We have seen that as long as the radiant point is farther from the mirror than its centre (that is, u > r) the focus cannot coincide with that centre (or v < r) which it approaches. If u = r, or the radiant point coincides with the centre, then _L= _L_-L = _l_i- = J-,or, v f r r r r v = r, and the rays are reflected to the centre. Let the radiant point now pass the centre towards the principal focus, that is, let u < r, and at the same time u > f, or — > - and Since ^ , is still a positive 1 1 1 quantity; the rays are, therefore, still brought to a focus: but 1 _J_ or 1 , whence ! < _L and v > r ; that is, the focus lies beyond the centre. v r When the radiant point coincides with the principal focus u =f whence, 1 1 == o, or v = oo 1 ,1 1 , and = — / v f the rays are rendered parallel by reflexion If we suppose the radiant point to approach still nearer to the 1 1 ;f, we have 1 , whence - — , is / / has a negative sign, or the focus is mirror, so that u ■> negative ; that is, V imaginary, the rays diverging after reflexion. The divergency of the reflected rays is less than that of the incident rays, i) \ u f / vergency before reflexion. ami J_ < _L , the di- ■ H ■ 16 CONVEX MIRRORS. ArPEIS'DIX. (16.) For tlio case of con writing rays falling upon a concave mir- >r (y/g-. 'J., p. 19, text,) we make, in formula .(4), « negative ; whence, _1_JL ' (5), or, The signs of the quantities on the right-hand side of this equa- tion being both positive, — is always positive. Converging rays v 1 1 are, therefore, always brought to a focus. Moreover, since _ -|- ~ >. _ , _L > __ , or the convergency is greater after, than be- lt v u fore, reflexion. Since 1 > — , > and t> their di- flexion, and since -\- — > / v u v u vcrgency is increased by tlie reflexion. (20.) Figure 12. will, as has already been stated, represent the case of converging rays falling upon a convex mirror, if we sup- pose DM, and DN to represent the incident rays, meeting in the imaginary radiant point F. To express this case analytically, u must be made negative, and the formula is = r- (8), or, v II f i_ — JL J_ V u f The position of the focus, as shown by this equation, passes through variations, corresponding to those in tlie case of diverging rays falling upon a concave minor (Art. 15.). (21.) It is sometimes convenient to refer the distance of the ra- diant point and focus, to the centre of the mirror, instead of to the vertex. Formula (1) may be readily transformed into one which shall refer to the centre. Prop. IV. To determine the relation of the distance, of the focus and radiant point, of a small pencil of rays, from the centre of a mirror. By Jig. 8., p. 18, text, it appears that AD == AC + CD, and FD = CD— CF. Calling AC = u' and FC = v, CD, as be- fore, = r, AD = u, and FD = v ; we have u — u' -f- r, and v = r — v'. Substituting these values of u and v in the equation (1), it becomes 9 Wm HI 18 REFERENCE TO CENTRE. APPENDIX. Bringing to a common denominator the quantities on both sides of the equation, and reducing, we have, u -\- r u' 4 r -T -I 1 + , or, 1 - -V = — (9)- (22.) This equation is the same in form with the one found when the distances were estimated from the vertex, except that the sign of u' is negative, the distances v' and u' being now reckoned in opposite directions. Equation (9) might have been deduced directly from the relation of the lines AM, FM, AC, CF, Jig. 8., in the triangle AMF. The solution of the question by that method, would have been more simple. 2 1 (23.) If we substitute for , in equation (fl), its value _ , we r f have (10). From this equation, may be deduced the relation expressed in the following proposition. (24.) Prop. V. The distance of the principal focus of a mirror from the centre, is a mean proportional between the distances of the. radiant point and focus of any small pencil, from the principal focus. Equation (10) gives, «/ / u' /+«' /- / — , whence, /+ »' f 4. u ' f + v! f-v':f::f:f+u', in which proportion / — v' represents FO {Jig. 8., p. 18, text,) or CO — CF, and u' 4. /, AO or AC 4 CO. (25.) The subject of reflexion at curved surfaces in general, will be treated briefly in a subsequent chapter. The properties of the CHAP. I. REFLEXION. 19 surfaces formed by tlic revolution of the paraliola and ellipse about their axes, are readily understood, from very simple geometrical considerations, and gain nothing by being presented analytically. They will, however, be referred to in another part of this 'Appendix. (26.) We pass next to the reflexion of a small oblique pencil by a spherical mirror, on which subject two propositions will be given; in the firs! will be considered the ease in which the axis of the pencil does not cross that of the mirror, and the pencil falls upon the mirror near to the vertex, and in the second, the case in which the axis of the pencil crosses that of the mirror. (27.) Pnop. VI. A small pencil having its radiant point out of the axis of a mirror, meets the surface near the vertex, required the form of the rejleeted pencil. Fig. A. Let LMN represent a section of the mirror, made by a plane passing through the lines MR and MC, or through the axis of the pencil and the centre of the mirror. RL is an extreme ray of the pencil incident very near to M, LF is the corresponding reflected ray, meeting the reflected ray MF, which corresponds to the axis of the pencil, in the point F. F is the focus of the pencil LRN, in the plane of the section RMC. (28.) To determine the focus of rays which meet the mirror in a plane perpendicular to RMC; suppose a plane to pass through RM at right angles to that of the figure, this plane will cut from the pencil, RLN, two rays, which reflected will meet the axis, MF, of the reflected pencil, in the focus required. If, now, a plane bo passed throng' 1 one of the incident rays, just described, and the corresponding reflected ray, it will pass through the centre of the mirror; the line RC, joining the radiant point and centre, will be, therefore, its intersection with the plane RMC containing the axis of the incident and of the reflected pencil; and the point, /•", in which RC produced meets MF, will be the point in which the sup- posed plane meets MF, that is, the focus of the reflected pencil. The focus, of the reflected pencil, in any plane between RMC and the one at right angles to it,* will be found between F and F' *Codnington tr-rins the former of tliese planes the primary plane, t lie atttjr, tile secondury plane. ——————————————————— I — — — — — I 20 OBLIQTTE PENCIL. APPENDIX. (29.) To determine, analytically, the position of the point F; draw from the vertex M, MX and MZ perpendicular, respectively, to RL and LF; also from C, CF and CQ perpendicular, respec- tively, to KM and MF. As the arc I,M is very small, it may be con- sidered a straight line perpendicular to the radius CL ; whence the incident and reflected rays making equal angles with CL, also make equal angles with LM, and the triangles LMX and LMZ are similar. But they have the side LM common, hence they are equal, and MX is equal to MZ. The two triangles CFM and CQM being also equal, CF is equal to CQ. And since CT and CS may be considered as perpendicular to RL and LF, they may be taken as equal. Whence PT= QS. By the similar triangles RFT and RMX, RP : RM :: FT : MX; and by the similar triangles FMZ and FQS, QS : MZ : : FQ : FM ; whence, since PT=QS, and MX = MZ, RP : RM : : FQ : FM (e). Let RM = u, MF = v, CM = r, and the angle RMC =

r. cos o. Dividing by r, _1 cos

V u — (11), 2« cos (i, (30.) To interpret equation (11) geometrically, we should ob- serve that the ratio of MX to RM, that is, ( __\ , measures the \RM/ divergency of the incident pencil LRN. But in the triangle MXL, MX= ML . cos LMX, and since the angles XMR and LMC CHAP. I. KEFLEXION. 21 arc nearly right angles, LMX is nearly equal to RMC, or MX = Ihcrefore varies with 1 , which, conse- ML. cos 2 or, X) = — . cos , r : u : sin (0 — 0) : sin 0, r sin (9 — ' sin Adding together the values found for — , and — _ , and re- u v during, wc find, — 4- — r = 2 . cos

f, or d , or c A (1 — e 2 ) AQ. — e 2 ) < 1, and the image is a portion of an ellipse. When d =/, c = 1, and the curve is a parabola. For d 1, and the curve becomes a branch of a hyperbola. If d = o, or the object passes through the centre of the mirror, _ — oo , and e is. infinite, or the image is a straight line, coin- d ciding with the object. When the object passes the centre, towards the mirror, d be- comes negative, and the equation (14) changes, if we reckon from GC, to 1 1 cos 9 .. cn v' " f d This equation gives a hyperbola while d f. Each of these cases presents curious circumstances. For ex- ample, in the case, d < /, if a point be taken in the object, so that v! = /, that is, = /, the equation for the focal distance of cos the pencil proceeding from that point, is 1 o, v == oo the image is infinitely remote from the mirror. If we suppose the object to be sufficiently extended to cut the mirror, the point com- mon to the object and mirror is its own image, and for that point u' = r, and v' = r ; between the points for which u' = r and u' = /, the distance of the image has varied from r to infinity, and, therefore, that portion of the object which is between these limits, has a virtual image, the part of each branch of which, between v' = r and the vertex, is wantitig. C ^W 26 REFRACTION AT A SURFACE. APPENDIX The part of the object between u' = d and u' =f, has its imago beyond the centre; it is the branch of a hyperbola conjugate to the first. (36.) If the section of the object be an arc concentric witli tho mirror, u' is constant, and 1 JL + J, is constant, or the image is also a circular arc concentric with the mirror. In this case, the relative magnitudes of the object and image are as their distances from the centre of the mirror. (37.) We have considered a section of the object, of the image, and of the mirror, made by a plane passing through the axis of the mirror ; if these sections be supposed to revolveabout the com- mon axis, tiie section of the object will generate a plane, and that of the mirror, and of the image, surfaces of revolution correspond, ing to the sections. (38.) The case of a convex mirror is embraced by equation (14), if r be made negative. 2 For the plane mirror, r = oo , and = o, whence, = ^L! (16), cos ~d and the image is similar to the object REFRACTION. CHAP. III. REFRACTION BY PRISMS AND LENSES. (39.) The most simple case of the refraction of light, is that in which it takes place at a plane surface. The perpendicular being drawn, the refracted ray is connected with the incident, by the law (p. 29, text,) that the sine of the angle of refraction bears a constant ratio, for a given medium, to the sine of the angle of incidence. To represent this law analytically, suppose a ray passing from a rarer to a denser medium, call the angle of incidence : sin = m . sin (17). CHAP. III. REFRACTION. When the ray passes from a denser to a rarer medium, ' repre- sents the angle of incidence, and

\ (40.) The difference between the angles of incidence and refrac- tion is termed the deviation of the ray, for a single surface. When the angles are very small, we have, for the deviation, nvp' — can never exceed unity, m sin ' cannot exceed unity, whence sin $' cannot exceed , (in which .m case, m . sin . The ray then ceases to be refracted ; it is wholly Te- rn fleeted. The angle, at which, light, passing through a denser medium, and meeting the separating surface of the denser and of a rarer medium, ceases to be refracted, is found from the equation sin = , whence, .,,'; ."-.y.,; ,.;"■•,,»-■" 28 EKFRACTXpN BY A PRISM. APPENDIX. (43.) Prop. IX. To determine the course of a single ray, or of a small pencil of rays, refracted bij a prism. {Fig. 20., p. 3:2, text.) Let the angle, HRM, of incidence, upon the first surface, be called ft ^'i ant ^ $1 as given by the following equations : sin (p = in . sin — ' 4 «//) ; but (19) gives 0' 4 V*' = *i whence, 5 = (m — 1) .« (21), a value depending only upon the refractive power of the materia! of the prism, and upon its refracting angle. (45.) There are two other cases in which the deviation, as given by equations (20), (17), (17'), and (19), assumes a somewhat simple form. These we shall consider, in order. < HAP. III. HV-FKACTION. 29 (4G.) Pltop. X. To find the deviation of a ray, or of a small pencil of rays, incident perpendicularly on one of the surfaces of a prism. When the incidence upon the first surface is perpendicular, = o, and ' = o, and equation (20) becomes, i = $ — a, and (19), i^' = a, whence, sin t/.'= m . sin ip' = m . sin a ; but from the value just found for <5, we have, ip = a -\- 6, whence, sin (a _|_ <5) = ?n . sin a (22) : from which equation, 5 becomes known when a and m are given ; or, sin (a 4- <5) m = ; — ! — - , sin a may be found by determining 5 and a. This is one method of determining the refractive power of a substance. Another method is furnished by the next proposition. (47.) Frop. XI. To determine the deviation of a ray, or of a small pencil of rays, when the angles of incidence and emergence are equal. (Fig. 20., p. 32, text.) By the condition of the question

*s 1 sin — (a -(- <5) = m . sin -— a an equation from which osition which follows. (48.) Prop. XII. The angles of incidence and emergence, of a ray passing through a prism, are equal, when the deviation is a minimum. The proposition requires the deviation to be a minimum. We therefore find its value, differentiate it, and put the differential co- efficient equal to zero. The value of the deviation, from equation (20), is, 8 =

— a, the differential of which, a being constant, is, d& = d

J/, by the differentiation of which, we obtain ~d$ From equation (17), by a similar process, we have, d . sin = m . d sin £', or, cos

In like manner, we obtain, by differentiating (IT), cos iJ/ d

= m dV. COS "J; Dividing the second of these equations by the first, d^ cos <(> . cos •J/ d-vj/ ~d4 ~ cos _ d

cos . cos '4' = 1 (25); an equation which is satisfied if

-. The equation for the refraction at the second surface may be in- ferred from (26),' or may be obtained directly, by proceeding in the triangles FMC and F'MC, as was done, in the last proposition, in ERC and FRC. Thus, FC _ sin FMC FM "~ sin FCM ' and, by division, FC sin F'MC FC FM ' F'M F'C and ss F'M sin FMC sin F'MC Bin FCM Call VV, the thickness of the lens, t ; F'V, v; CV\ r ; then FV = u' 4. t, FC == u -f t — r', and FC =v — r'. These values being substituted, in the equation just found, instead of FM, F'M, &.C., we have. u' -}- t — r u' -f t = , whence, ■■■■ ,-_.-■- KaKKiK3K3liaK3 34 JEN'ERAL rOR?.IULA. APPENDIX V — r' u' 4- t — r' — in . AL v! -f t f — ( »' + .) dividing by r', and collecting the terms, 1 m 1 v m'-|- t r' m being greater than unity, the sign of the second member is really negative, and as it will be convenient to show this, we change the form of that member, and the equation becomes, _L _ m _- _ m ~ * (27). v u' -\- t r' This is the general equation between w', v, r', t, and m, which, combined with (26)> will determine v in terms of u, r, r\ t, and m, all of which arc given quantities. (53.) If the thickness of the lens is so small that it may be neglected, equation (27) becomes, but, from (26), whence, in — 1 i* 1 • + m — 1 u r m - -1 _ or, 1 L = (m - 1) (1 L\ (28).' v u \ r r / Since represents the divergency, or convergency, of the u' incident pencil, and — that of the refracted pencil, we deduce, v from (28), that the difference of the vergencies of the refracted and incident pencils is a constant quantity for the same lens. This formula applies to the different cases of the incident pencil and lens, as in the single surface (art. 51), if we consider the dis- tances of the radiant point, locus, and centre, positive when in front of the lens, and negative in the contrary case. The same remark applies to the general equations (26) and (87). (54.) In most of the cases which occur in practice, the thick- ness of the lens may be neglected, and, therefore, equation (28) is applicable to them; in all cases this equation determines an ap- proximate value of the focal distance, to which a correction for the thickness of the lens may be, conveniently, applied. CHAP. III. REFRACTION. (55.) To obtain this correction for thickness, expand iin equation (27), into a series, by division, 35 m u' -f t mt 2 — w' -f- t */' it' 2 «/3 &.C. u' v! \u' u' 2 / If the thickness of the lens is not very great compared with the dlistance of the point F, the powers of — T , higher than the first, u imay be neglected, and we have, m m mt u' -f t Substituting this value for — -^- + u -\- t mt in (27), it becomes, m — 1 on- substituting for 1 1 its value from (26), vi — 1 + mt by transposing and collecting the terms, JL= J_ + (m-l) (_L _ \\ __^. (29),' » u \ r r / u' 2 in which we perceive the approximate value of JL given by v equation (28), and a correction JH for the thickness of the u' J lens. (56.) Prop. XV. To determine the form of a small pencil refracted by a medium, bounded by parallel planes. For plane surfaces, r and r' are infinite, whence = o, and r Equation (28) becomes, 1 1 =. o, or. = — (30). JtBUBllMMM MHH HSBBjBHUUUMHkKmSVSK 9E9E9 36 REFRACTION BY THICK PLATES. AN>ENDIX. The vergency of the refracted pencil is tlie same with that of the incident pencil, when the plate is indefinitely thin. (57.) Applying the correction from equation (2'J), we obtain, 1 1 mt but, from (26), by making — = o, r= , whence, u' = mu ; Substituting this value of u' in that last given, for 1 1 t i = -L(.-^y v u \ mu / (31). Equation (31) contains the theory of refracting plates of con- siderable- thickness. (58.) If the incident rays are parallel, — o, and — o, u v or, t) = oo, or parallel rays are unchanged by the refraction (fig. 23, p. 36, text.) (59.) The value of — for diverging rays, given by (31), is posi- v tive, zero, or negative, according as we have 1 > as t < mu, t For ordinary cases of relation between u, and t, (t < mu) — , 1 = — , 1 < , that is, mu mu mu mu, t > mu, in which m is greater than unity. 1 . positive, and therefore the focus imaginary, or the rays still diverge after refraction. As u is measured from the first surface, and v from the second, the effect of refraction in bringing the object nearer to, or removing it farther from the plate, is not expressed by the relation of?) and u, but bv that of v — t, and u. To ascertain the effect of refraction in this point of view, we take the value of v in (31),_or, CHAP. III. INFRACTION. 37 this, by dividing and neglecting the powers of t above the first, gives V = u -f- -, whence v — t = u -f t (1 1 ) • • \m / 1 (32). In which since m > 1, — - < 1, and 1 is subtractive, m m or) x> — t < u ; the point of divergence, therefore, is brought nearer the first surface by the distance, t II V 3 11 In a glass plate, m — _, and 1 = — Th e point of divergence is, therefore, brought nearer the first surface by one third the thickness of the plate. For water, m = — , and 1 1 (60.) If the incident rays converge, u is negative, whence from (31), (33), J- = _ i. (1 + ±) v u \ muz in which — is always negative, and, therefore, the rays still con- verge. Proceeding to find the value of v — 1, as before, we have mu~ I v — t = t = — u -\ t, mil -\- t v — t = — u — t (l V v — t = —(u + t(l— — )) (34,) from which it appears that the focus is farther from the first surface than the imaginary radiant point, by the distance til 1. The reverse of the result for diverging rays. (61.) Prop. XVI. To determine the form of a small pencil of rays refracted by a double convex lens. We will first consider the case in which the thickness of the lens may be neglected. To this, equation (28) will be adapted by D 38 DOUBLE COKVEX LENS. APPENDIX. making r, the radius of the first surface, negative, since its centre is turned from incident light. This gives — - — = ~ (»» ~ 1) (— + X\ (35), v u \ r r / since — represents the vcrgency of the refracted pencil, and — that of the incident pencil, and is negative, m,r,r', being v u constant, for the same lens, we infer that the divergency destroyed, or convergency produced, by this lens, is a constant quantity. (62.) For parallel rays, — — o, and (36). In tlie refracting media of which lenses are made, m > 1, and this value of — is negative : hence the focus lies behind the lens, v and is real. For the distance of the principal focus we have by taking the value of v from (36), 1 rr' *>= ^--T-^ (37). m — 1 r -\- r 3 1 If the lens is of glass, m = -— - , and — = 2, whence, 2 m — 1 2rr' r -j- r' corresponding to the rule given in the text (page 42). If the glass is equally convex, r = r', and — 7^— r. The principal focal distance is equal to the radius of the surfaces of the lens. (63.) The principal focal distance may serve as a convenient term of comparison for the focal distances of diverging and con- verging rays. Denoting it by/, we have the value of/ given by (37), and of — by (36), substituting in (35) — for its equal ; and J f transposing, we have, i.~I > (38). v u f CHAP. III. REFRACTION. 39 (G4.) Diverging rays. Equation (38) applies to this case. From that equation it appears that for the same lens, the vcrgency of the refracted rays / — -\ is less than the divergency of the incident rays I — J by a constant quantity ( , ] de- pending upon the index of refraction of the material of the lens, and upon the curvature of its surfaces. If u > / {Jig. 29., p. 43), — < -T- 1 and _ is negative, or u f v the rays are brought to a focus. The reciprocal of the focal dis- tance is, from (38), V \ f u / Since <; — / — < — . , or v > /, and the focus » / is farther from the lens than the principal focus. Aa the radiant point approaches the lens, — increases, and, of course, , or u v , diminishes, or v increases : that is, as theradiant point f u , approaches the lens, the focus recedes, and vice versa. When u «= 2/, JL = _ ( — L\ = L , and v = v \f 2// 2/' — 2/. The focus is as far from the lens as the radiant point. If the rays proceed from a point as far from the lens as the prin- cipal focus (as from O', Jig. 29.), u = f, and — ~ ; the re- v fractcd rays are parallel. The radiant point being still supposed to approach the lens, we 1 J_ and the rays are no longer brought to a focus. (65.) Equation (35) will give the value of the focal distance for diverging rays : to determine this, transpose — and bring the u terms on the right-hand side of the equation to a common denomi- nator, whence, 1 _ rr'—u(m—l) (r + /) d 1 t 1 1 have u — - « / — , or — , is then positive, ^^^^^^^^^^^^^^^^Q^g 40 DOUBLE CONVEX LENS. APPENDIX. rr' — u (m — ■ 1) (;• _|_ r') or changing the sign to correspond to a real focus, to which we have found the rays to be brought as long as u> f, urr' For a glass lens, m u (m — 1) (r -\- r) — - rr J 3 i i = — , and m — 1 2 _ 1 2 2urr' . • (40) ; whence, u (r + r') — 2;V This value of v gives the rule found in art. 45, of the text. The arithmetical operation there directed is changed for the subtraction 2rr' of u (r -f- r') from 2>r', when 2rr > u (r -f- r'), or u < , r -f- r' or u u • f - , the convergency of the ravs is increased by refraction. u . . 3 Taking the value of v from (41), and making, in it, m = — — , as was done in the case of diverging rays in the last article, we find for a glass lens, cttAin nr. REFRACTION. 41 u (r -\- r') -j- 2/r' For a glass lens equally convex, we have, Mr (43). . . . (44). u -j- r These values for the focal distance give the rules on p. 44, of the text. (67). In what precedes, we have neglected the thickness of the lens, and next proceed to show how a correction, for the eflectof the thickness, may be introduced, Prop. XVII. To skoio the method of applying, to the approximate focal length found for a double convex lens, a correction for the effect of the thickness. As an example, let us take the case of parallel rays falling upon the lens. Equation (29) is applied to this case by making r nega- tive, whence, -L=-L_ Cm _ 1) (i + _L)_ v u \ r r / And for parallel rays, for which = o, u 1 mt (45). _1)(2- + J\_^ \ r r J u J (46). Equation (26) adapted to the case of a convex surface, gives, m 1 tn — 1 and for parallel rays, m — 1 _ (m , whence, l) a in (46), we have, (m — I) 2 t Substituting this value of — = — (« - 1) (— + v \ r 1 _ 1 _ (771 V f 7 To determine from this equation the correction tc be applied to the focal length, », we reduce the terms of the secccd member to u common denominator, whence,. D2 • 1)H mr- • (47). MM^«MMlfiMiSMMS BBS ERfi ^^^^^^^S 42 CORRECTION FOR THICKNESS. SPHERE. 1 mr 2 -f- (in — l)"ft APPENDIX V mr-f mr''f Z_ and •«»•- _|_ (m — l)-/i mr*f mr~ -f (m — l) 2 /t + / = w»r a + (mi — l) 2 /< Dividing, and neglecting the terms containing powers of t higher than the first, (m - l)T-t »+/ (48), the correction which is to be applied to the focal distance obtained by equation (37). When the lens is equiconvex and of glass, we find (art. 62) that f = — r, to which a correction, f r- is to be applied. The sign of the correction is contrary to that of the focal distance, and the effect is therefore subtractive. The corrected focal length is v = — r+lt. (68.) The method which has just been shown gives, at last, only an approximate value of the focal distance, which, however, is sufficiently accurate for all cases in which the thickness does not bear a considerable relation to the focal distance. In the case of a sphere used as a lens, the thickness is too considerable to use the method of correction already exhibited. (69.) Prop. XVIII. To find the focal length of a sphere for parallel rays. The supposition that the rays are parallel simplifies the question, without deducting much from its utility. Since — = o, equations (26) and (27) become, by making r negative, * =-?L=} (49), and in — 1 For the sphere t 1 u' 4. t 2r, r = r (50). v! 4- 2r and (50) gives _ r ZLZli , but from (49), CHAP. III. RETRACTION 7iir 43 2r — 2r — u' -f 2/- = , whence 2mr — 2r 1 to — 1 Substituting this value of «' 4. 2r in (50), to (to — 1) to — 1 r (m- -2) 711 — -1 ' (50), to — 1 or , or, 1 v and v r (in — 2) m (m — 1) — (m — 1) (m— 2) _ (to — 1) (to — to 4. 2) r(m — 2) 1 2 (to- r (?n — 2) r (to — 2) _ r(m— 2) , whence 2 (to — 1) The value just found is the distance of the focus from the second surface ; call / the distance from the centre, then r(m— 2) / = t '- r= 27^TI)- r ' or, bringing to a common denominator and reducing, mr f = ■ (51). 2 (/« — 1) The rule on page 40 of the text, is given by the value of / just found. If the refracting sphere be of to6asAeer,m =1.11145, and/= — 5r, of course FQ (fig. 26, text,) = — 4/-. If the sphere be of water, to = 1.3358 and/ = — 2r nearly, or FQ (fig. 2C>) = — r. For a sphere of glass, m = 1.5, / = — ljr, and FQ = — Jr. For a sphere of zircon, to = 2, / = — r, and FQ = 0. (70.) Returning to the discussion of the formula for the refracted pencil when the lens is indefinitely thin, we take up the case next in order. Pit op. XIX. To determine the form of a small pencil after re- fraction by a plano-convex lens. As in other discussions, the refractive power of the substance of the lens is assumed to exceed that of the medium traversed by the incident pencil, or m > 1. The question obviously includes two cases ; in the one, the •'-.-■■■■ ■.-.■■ ^^^^^^^^g 44 PLANO-CONVEX LENS. APPENDIX. plane side is turned towards incident light, in the other the curved side is thus directed. (71.) First: when the plane side is turned to incident rays, — — o, whence from (28), r 1 1 m — 1 — = r~ (52). From this we infer, that the divergency destroyed, or convergency produced, by this lens, is a constant quantity, as in the double convex lens (art. CI), but the effect is less than in that lens by , the power of the first surface ; it is not necessary, therc- r fore, to carry out the discussion of the properties of this lens. There will be no correction for thickness, for parallel rays, no refraction being produced by the first surface. This is shown by the analysis, the term —^ (in 29) vanishing, since from (26), — 7 — o . The principal focal distance given by making — — o in (52), and inverting, is r' m — 1 For a glass lens, / = - 2r' (72.) Second : when the convex side is turned to incident light, — T = o, and r is negative ; from (28), r ±_± = _^LZ1. 1 (53 ). v u r The effect is, as in the first case, to destroy divergency in the incident pencil, or to produce, or increase, convergency ; and if we suppose r = r' , that is, the same lens to be used in both cases, the effect produced is the same. For the principal focal distance, / 1 ' and for a glass lens, /=-2r. (73.) The thickness of the lens produces in this case an effect on the principal focal length, since the rays refracted by the first surface fall obliquely upon the second. RKl'KACTIOJf. 45 To introduce tlic correction into the value of the principal focal iistance, we recur to equations (26) and (27) ; making, in these, 1 ,1 r negative, — r = o, and — = o, we obtain, m — 1 v u' + t The value of u' from (54), is , . (54), (55). , and u ' -f t = m — 1 - -i- t = —m ( — r -— 1, but from (55), 1 \m — 1 m / v! -f t and substituting for u' -\. t the value just found, v = r —r +— (56). m — 1 m The correction, therefore, shortens the approximate focal length of the lens by — th part of its thickness ; if the lens be of glass, v = — 2r -|- It, or, „ = _ (2r — §t). (74.) Prop. XX. To determine the form of a small pencil, after refraction by a douhle concave lens. In this form the radius of the first surface is positive, that of the second negative. When the thickness of the lens may be neglect- ed, we have from (28), — - — = (m - 1) (— + -L) (57), v u \ r r / and where an approximate value of the thickness may be used, from (29,) = (»» — 1) I h — I ~ (58), V u \ r r / u - in which u is determined from equation (2G), or J!L = J. + t=i (59). ^^^^^^^^^^^^^^^^^v^gg ^^^^^^^^^^^^^^^^q^^^^^q ^^b^rk^d^br 46 DOUBLE CONCAVE LENS. APPENDIX. (75.) When the incident rays are parallel, (fig. 31, p. 44, text,) (57) gives v \ r i / or calling / the principal focal distance, and determining it from the equation just given, /=_ 2 " (GO). J - (m _ 1) (r + ,.') This value being positive, the locus is imaginary, and at a distance expressed by the product of the radii divided by the index of refraction, less one, into the sum of the radii. The rule cor- responds to that for a double convex lens ; in fact, equations (60) and (37), differ only in their sign. (7G.) Diverging rays. In this case — is positive, and, there- in fore, as long as m > 1, the value of v, from equation (57), will always be positive and the focus imaginary ; since v u \ r r / (57), it appears that — > — , or the divergency of the rays is increas- ed by the refraction, (fig. 32, text.) In a glass lens, 1 l , 1 ( l . l \ v u 2 \ r r / 2urr' ~ 2rr' -f u (r -f 7) ' whence the rule on page 45 of the text. (77.) When the rays converge, — is negative, and (57) becomes u ~ = - — + (m - 1) (— + 4) (61-)- v u \ r r / The pencil still converges, is rendered parallel, or diverges, ae- 1 ' I 1 \ cording to the relation between — and (m — 1) ( — -4 — 1, or u \ r r / its equal — - . If <- — - or u > f , — is positive, and the / « / » rays stnl converge ; if — 1 t l —=■ , or v =f, — o, and the CHAP. III. REFRACTION. 47 refracted rays arc parallel; if — > — , or u when we shall have — < —r- and -I — -< — , or u f u f f — •< — and v > — / : the revorse will of course be true if » / If, as was first supposed, u > f or — < — , though the rays " / still converge after refraction, they converge less than before it, for 1 1 1 We shall not introduce the correction for thickness, as it would be determined by the same method with that for the double convex lens. Practically the thickness of double concave lenses is of little importance, since it is least at the central parts. (78.) Proi\ XXI. To determine the form of a small pencil of rays, after refraction by a plano-concave lens. First. When the concave side is turned to incident rays, = o, and r is positive ; equation (28) gives 1 1 _ (m — 1) (G2). The divergency produced by this lens is, therefore, less than that produced by a double concave lens, by . — ' , the effect of r' the second surface. Second. When the plane side is towards incident light. Then = o, and r' is negative, whence, r JL _ JL = fr"- 1 ) (63), v u r' agreeing with the expression found above, if the lens is the same in each case, or r = r'. (79.) The next form to be considered is (he meniscus. ' E^aEl^aE^EqB^gWB»E^g9gWgJBBqg^HWfWflgygHpj|g^ 48 MENISCUS. CONCAVO-CONVEX LENS. Al'I'ENDIX Prop. XXII. To determine the form of a small pencil of light after refraction by a meniscus. When the convex side of the meniscus is turned towards inci- dent light, the signs of both r and r' are negative. The general formula (28) gives J^ _ J_ = _ , _ ! / JL 1\ v u V r r' / (64), in which, by the nature of this lens, r' > r, or <* — . r' r From this relation of — and it follows, that . — - r' r r r is a positive quantity, and therefore the sign of the right hand side of this equation is negative. The equation corresponds to that for the double convex lens (35), but the divergency destroyed by the me- niscus is (m — 1) / r J , while that by the double convex lens was (m — 1) I — -(- — J . The power of the meniscus \ r r' / is the difference between the powers of its two surfaces. (80.) When the concavity of the meniscus is turned to incident light, r and r' are positive, and r > r', or — <- — . r r Equation (28) applies directly to this case, and v u \ r r / Since < , is a positive quantity, hence r " r' r' r v is negative. Equations (65) and (64) are identical, the u surface which first received the incident rays in the case of (64), being now the second surface. (81.) For the focus of parallel rays, we have, from (64), / = i-r • -^~ (66). m — 1 r — r (82.) The formulas just found for the meniscus apply to the con- cavo-convex lens, recollecting that when the convexity is turned to incident light r > r', and the reverse, r' > r, when the concavity is thus turned. ■ CHAP. III. REFKACTION. For the first of these cases \vc have (64), 1 1 49 (m "(v-f) (64), which, since < . , will be more expressive if written (m »CW) (67). The second member of this equation is positive, and by referring to the case of the double concave lens (art. 74), we shall find that the convergency destroyed by the concavo-convex lens is the differ- ence of the effects of its two surfaces, while in the double concave lens it was the sum of the same effects. It is obvious that turning the concave side of this lens to inci- dent light does not alter the effect of the lens, as was shown in the case of the meniscus. The virtual focal length of the concavo-convex lens for parallel rays, when the convex side of the lens is turned to the incident pencil, is / = —L, . JzL (68). m — 1 r — r (83). For two spherical surfaces of the same curvature, we have r = r\ and (28) gives 1 1 The effect is that of a plane glass. E 2E9 WMpngNHiuung ^^^g ■ 50 IMAGES BY LENSES. APPENDIX CHAP. IV. FORMATION OF IMAGES BY REFRACTION. (84.) The subject of the formation of images by lenses becomes simple, by introducing' the consideration of the ray which passes through the two surfaces of the lens, at points where the tangents are parallel. Prop. XXIII. All the rays which suffer no deviation by refraction, by a lens, pass through a single point. Fig.F. In the figure, let RL be a ray, refracted by the first surface of the lens MN into LL', and finally emerging in the direction L'R', parallel to RL. Produce LL' until it intersects the axis of the lens in O. Since, by hypothesis, the tangents at the points L and L' are parallel, the radii CL and C'L' are also parallel, and the tri- angles COL and COL' are similar. Whence, CO : CO i: C'L' : CL, or, CO = CO . < Hi , and CL C'C = CO — CO = CO Calling CL = r, CL' = r', the thickness of the lens = t, and CO = u', we have CC= CV—CV= CV— VV'—CV= r' — t — r, and CHAP. IV. REFRACTION. r' 51 or, and >'— — «— r -(-f- 1 ) u'=r — t. —H— (69) This value of CO is made up of quantities, constant for the same lens ; from which we infer, that all the rays which experience no deviation in passing through a lens, icould, if produced after the first refraction, meet in a single point in the axis of the lens. This point is called the centre of the lens. (85.) The distance of the centre of a lens from the vertex of the first surface is found, readily, from equation (69) ; for, since VO = CV — CO = r — u', we have, by taking the value of u' from (6.9) and calling r — u\ x. u' = t (70). The distance from the centre of a lens to the vertex of its first surface, is equal to the thickness of the lens, multiplied by the ra- dius of that surface, and divided by the difference of the radii of the two surfaces. In the double convex lens, r is negative and r positive, whence, rt x = . r' -f r The sign of (x) VO shows that, in this case, it lies on the right- hand side of the vertex. Since is a fraction, x < t, the r' -|- r tentrt is therefore between the two surfaces. In the equiconvex lens r' = r, and _ t x _. The centre is midway between the vertices. It is from this cir- cumstance that the point, which we have defined, derives its name. The same remarks apply to the double concave lens, since for that lens r is positive and r' negative, whence, rt X r= » r' -\- r the same expression which we have above. (86.) To use the position of the centre of the lens, in de- termining the image formed by an object, we observe, that one 52 IMAfiES BY A DOEELE COXVEX LEXS. ArPF.xnix ray of the pencil, which proceeds from every point of the object to the lens, passes through this centre. This ray is called the principal ray or axis of the pencil, and when the lens is thin may be regarded as suffering no refraction. It docs not fall per- pendicularly, nor nearly so, upon the surface which it meets, and, therefore, in strictness, the refraction of an oblique pencil should be investigated and applied to this case. Approximate results may, however, be' obtained, by taking the focal distance already deter- mined tor a direct pencil ; this distance being found, for the pencil proceeding from each point of the object, we have a series of points, the assemblage of which constitutes the image. An application of this method is given in the following proposition. (87.) Prop. XXIV. The object, of which the image by a convex lens is required, is a plane perpendicular to the axis of the lens. Fig. G. Let AB represent a section of the object, MM that of a double convex lens, PC a line drawn from any point in the object through the centre, C, of the lens ; this line may be regarded as the axis of a pencil of rays proceeding from P, and may, farther, be con- sidered to suffer no refraction. Call a the distance DC ; u, PC ; and the angle DCP : we have from the triangle DCP, a = u . cos 9 , whence cos but from equation (38), article 63, -T- 7 m ' or substituting for — its value just found, 1 cos 1 (71). The polar equation of a conic section referred to the fbcus. CHAP. IV. REFRACTION. 53 From this equation, inferences might be drawn similar to those, found in the chapter on the formation of images by mirrors. (88.) When the object subtends a small angle, we may consider its section as a circular arc ; the image will be, also, a circular arc, since if u is constant (38), v will be so ; and the arcs will, evidently, be similar. If the distance of the object and image, respectively, from the centre of the lens, be called a and v, their magnitudes d and d\ we shall have £--=- (72); a a being assumed very small, equation (71) gives, making cos 0=1, 1 f—a — = J — r - , or v af a f »«= ~— : f~ a this value of v substituted in (72), gives £. / d ~ /- As long asa>/,/ — a is negative, and the image is real. To show the results of this case more clearly, put equation (73) under the form *-__ J— d (73). If a > 2/, <*—/>/ and than the object When a = 2/, d is a fraction, or the image is less d' — = 1, the image is equal to the object. — f ' L 2»' V t>' r /J But F'C = r — »', and LC = u — r ; and substituting the values of F'C, F'M, LC and LM in the ratio found in the be- ginning of this article, v L 2» \ »' r /J = iiz! fi + _2l (J L\l . m L 2u V r m /J . Dividing by r and performing the multiplications by the quanti- ties outside of the vinculum, in each member of the equation just found, J L _ vL /_! _ JL\ a = v r 2v' \ »' r / = J._i + |L(±_J.y («,. r u 2u \ r u / CHAP. V. SPHERICAL ARBITRATION. 57 Wc have tlius a general relation between v' and u in terms of the radius and semi-aperture of the mirror. (J3.) If in (75) we use for , in the multiplier of X_ , v' r 2v' its approximate value, derived from equation (1), art. (7), namely 11 11 , f . . — = , we obtain ■ v r r u J_ _ J_ _ j£ /J 1_V_ «' r 2o \ r u / r u 2u \ r u / _1__L = J L+ vl(± + ±-\(±- J_V, »' r r u 2 \ u v' / V r u/' fartlicr, by using for I + — ) the value given by (1), \ u v' / 1 1 _ 2 DMT and substituting this in the equation last found, J___L = J l + _y!(J__ «' r r « r V r J_ = A _ i_ + ^i c± _ j_y »' r u r\r w/ In this equation, which gives the value of — , corresponding u / or, (7G). to a point of incidence distant from the vertex, we find the recip- rocal of the approximate focal length, obtained when the rays were 2 supposed to meet the mirror near the vertex, namely, r , and a correction for aberration. This correction contains u JL — , a quantity proportional to the versed sine of the semi-angle r of the pencil, and therefore depending upon this angle, or the semi- aperture of the mirror ; and also r, and u, the radius of the mirror and distance of the radiant point. If these latter quantities are constant, the aberration is a function of the semi-aperture of the mirror. The correction for aberration is additive, showing that the reciprocal of the focal length, for rays distant from the vertex, is greater than the reciprocal focal length of those near the vertex, or that the point F' is nearer to the mirror than F. awgwwwgn j MeH e ww gg^«wwwsBi ^^^^^^^^^^^^^^^^n 59 CIRCLE OF LEAST ABERRATION. APPENDIX. (94.) For parallel rays — = o, and from equation (7G), u 1 2 . y* + , whence, 2r* + y» performing the division, and neglecting the powers of y higher than the second, 2 4r »' = -- — 2— (77). Tlic correction for aberration is, therefore, — i or — - , or is subtractive, and equal to the square of the semi-aperture of the mirror divided by eight titnes the principal focal length. (95.) Second : To find the lateral aberration of the extreme ray. The value of FI, which measures the lateral aberration of the extreme ray, may be obtained as follows. In the similar triangles F'FI and F'MN , H=^L,zndFI=FF>.m.. FF' F'N F'N MN To approximate to the ratio — — . i F'D may be taken instead of F'N, and the value of the aberration is MAT FI = FF'- F'D (78), in which all the terms are known when FF' has been determined. (96.) We propose in this article to determine the position and magnitude of the physical focus of a mirror, or of the circle which includes all the rays of a reflected pencil, when they are spread over the least space. Puor. XXVI. To determine the position and magnitude of the cir- cle of least aberration, in a pencil of rays reflected by a concave mirror. In the figure let LM be the extreme ray of a pencil, incident upon the mirror, MF' the corresponding reflected ray, F the focus of rays very near the vertex. Farther, let LP be any incident ray, in the lower portion of the pencil ; FR the corresponding re- flected ray intersecting MF' produced in c : draw cb perpendicular to the axis, from the point c. If we suppose the arc DP very small, the reflected ray PR will coincide very nearly with the axis, and the distance cb will be indefinitely small ; as the arc DP increases, E^K &#%&$* ^r^rowtSy^iS?^^^^ SPHERICAL ABERRATION. 59 flic reflected rays being removed farther from the axis, cb, at first, increases ; it afterwards diminishes as the jx>int of intersection of PR with the axis approaches to F', and when DP = DM, PR coincides with PF', and cb vanishes. Between the case, then, in which DP is very small and DP = DM, there is a position of the incident ray LP, for which the reflected ray PR gives a maxi- mum value for cb When cb is a maximum, all the rays of the Fig. I. reflected pencil pass through the circle of which that line is the radius, which is, thus, the physical focus of the mirror. The question resolves itself into determining the values of F'b and cb when the latter is a maximum. Call MN, y; PT,y'; DF,v; DF', v' ; FF', the longitudinal aberration, a ; F'b = x ; be = z. Since (art. 93) the aberration of a ray is proportional to the square of the distance of its point of incidence, from the axis of the mirror, FR : FF' : : PT 2 : MN 2 ::y' 3 :y 2 , or FF'— FR : FF' : : y 2 — y' 2 : y 2 , whence F'R = a.y-Z-JLl, y- But from the similar triangles F'bc and F'NM, bc-.bF': :MN:F'N, or MN FN' And from the similar triangles Rbc and RTP Rb : be : : RT : TP , or RT_ TP' .substituting for be in this expression the value found above, be = bF' Rb = bc — ^^^— ^^JWI 60 CIRCLE OF LEAST ABERRATION. AI>PENDIX. MN RT F'N' TP' and by the notation, FN y" If we approximate, by considering RT and FN to be equal. Rb = IP . Rb = x . -^- , and y FR = Rb + bF'=x .^L + x V + y' y" y Equating this value of F'R with the one before found, x . y , " 1= a . ± — _£_ whence j, y- 7* W 3 — v' 2 i = a . _ . J - _ ,or y'~ y 2 y + yf iy-y') (79). As we have supposed the ray LM to remain fixed, and LP to take different positions, and have found, z = bc = bF' . , F'N be (or z) will be a maximum when bF (or x) is a maximum ; but from (79), x is a maximum, since a and y are constant, when y' (y — y') is a maximum, or when y' (.y — y') = y 12 , °r In that case, from (79), ay 3 4y 3 JVflV F'JV (80), and, MN If a perpendicular, FI, be drawn from the focus F, of rays in- cident near the vertex, to the axis, meeting the extreme ray MF' in /, by article (95), MN FI F'N whence the value of z, or FF a MN 4 FN FI 4 — » , becomes (81). CHAP. V. SPHERICAL ABERRATION. 61 From these values, (80) and (81), of a and z, it appears, that the distance of the circle of least aberration, from the focus of rays near the vertex, is three fourths of the longitudinal abei-ration of the extreme ray, and that the radius of the same circle is one fourth of the lateral aberration of the extreme ray. Spherical Aberration of Lenses. (97.) In this investigation we begin by determining the aberra- tion produced by a single surface. We shall assume the light to pass from a rarer into a denser medium, as when it enters a lens through its first surface. Trop. XXVII. To determine the aberration produced by a single refracting surface. Fig. K. Let the ray RL fall upon the spherical surface ZFat any point L, and be refracted into the direction LM. Continue LM until it intersects the axis of the surface, at F. Draw the radius CL. Call m the ratio of the sine of incidence to that of refraction, in the passage of the ray from the rarer to the denser medium, the sine of refraction being unity ; then, by proceeding as in article (50), Chap. III., we find RC _ FC ~RL ~~ m ' FL From the centres R and F with the radii RL and FL, respec- tively, describe the ares LS and LT cutting the axis in £ and T; SV will be the difference between R V and RL, and TV that between FV and FL. If the perpendicular LN be let fall, from /„ upon the axis RV, SV = NV — NS, and TV = NV — NT. As in the notation of Chap. III., let RV = u, FV = m', CV == r ; and call LN, y. NS is the versed sine of the arc LS, NT of LT, and NV of LV ; and if for the chord of each of those arcs we substitute, as an approximate value, the sine, we have F ^^B !— — — — — IB^M 62 SINGLE SPHERICAL SURFACE. 2RL nt= y2 , « 2 JVF= JL_ , 2CF APPENDIX. or subsl ituting for RL and JX, the approxinn ite values R V and FV, ns= i—, 2u NT JtL, w = _y: 2u' 2, , whence, SF = NV— NS = ll (— L\ , and 2 \ r u / = ivr_ ivr = -11 /J i-\ . 2 \ r a' / and TV From these values of SV and TF we obtain, RL = RV — SV= u — lLLl - — -L \ 2 \ r u / FL = FV—TV=u' — JLL- (— L\ . 2 V r w'/ Taking the reciprocals of RL and FL, that is dividing 1 unity by the values just found for those lines, and neglecting the terms which involve the quotients of the powers of y- by those of u, after the first term, (-1. — \ , we have J_= JLT1 + JL/J__JL\»I1. RL u L « V r «/ 2 J FL it' L u ' V r u' / 2 J From the figure we have RC = RV — CV = u — r, and FC = FV — CF= v! — r, and the equation lor the relation of RC, RL, FC and FL, becomes lzzL.h+±(±-±\2>L] = u L u \ r u / 2 j u' L * V t «'/ 2 J Performing the divisions by u and u', indicated by the terms of the equation, and dividing both sides of the equation by r, (1 LUi +±(± -\^il=» V r m / L u\r «/ 2 J v&m CHAP. V. SPHERICAL ABERRATION. 63 = m (1 L\ fl + _L (± _ _1\ 111 , \ r m'/L u \ r «/ 2 J performing the multiplications required by the expressions, i___L + i(J._±) s i = r u u \ r u / 2 - m (± - ±-\ + -™ (1 LVll , whence, \ r u' / u' \ r «'/ 2 II. ==JL + <&-!) J- + _£ rjl(J___L) 2 __L(JL_J_\ 2 l (82). \_ u \ r u / u \ r m/J In this value of , the first two terms correspond to the u' value found, (26) art. 50, on the supposition that the pencil is small, and the third contains the correction for the aberration, pro- duced by the single spherical surface. (98.) The expression just found, may be simplified by substi- tuting in the terms of the second member for u', its approximate value from equation (26), art. 50. From that equation 1 1 -f (m — 1) — , whence, u u r i_ - _L (J_ + (Mi-l) 1\ , and u m \ u r / ±-±= _L_ WA. +(,*-!) J_\, or, r u r in \ u r / J__JL=2_(jL__L\, and r u in \ r u / l---Y=-(---) a \ r u' / m 2 \ r u / In order to reduce, with greater convenience, the coefficient of H- in (82) to its simplest form, call that coefficient k, then sub- 2 stituting in it the approximate values of and I . 1 , u' \ r u' / just obtained, we have, * /!_+(,„_!) n._L (2. _I\ 2 \u r / in- \r u / __L(i_±)*,.r, IBBB SHi HSjoBSS EsESEaHBi GB SBB 64 CASES OF SO ABERRATION - . APPENDIX. i= (i__JLV.M (! + (,»-,) 1)_J_] V r u / L"*" \ u »" / it J t^/i--J-\ 2 .r 1 - 7 " 3 + ]5z^ii_, \ r m / L « r j in- k = (L_L\ 2 .(L- m + l ) \ r u / \ r u / whence (82) becomes m — 1 in — 1 + \r u ) \r u / 2 (83). (99.) When the surface is convex, r is negative, and (83) takes the form, mi 1 in — 1 in — 1 (f+^)(l+^)' (84). And for converging rays, u being negative, in 1 m — 1 ii (L _ wt + 1 \ . (1_ _ 2\ 2 . vL \r u / V r u/ 2 (85). The term in (85) which contains the correction for aberration, will vanish if either of the factors composing it should be equal to zero. First, let 1 in 4- 1 ., m 4- 1 1 -I- — = o, then — IE — — — , or, r u . u r 1 : r : : i» -{- \ : «. There is, therefore, no aberration for converging rays, falling upon a convex spherical surface, when the distance of the radiant point is a fourth proportional to 1, r, and m -{- 1. From whence the result on page 56, of the text, is easily deduced. Next, let = o, and u = r, or the incident rays converge to the centre of the spherical surface. (100.) In art. 42, it was remark'ed that making in = — 1 in the formuke for refraction, the cases would represent the cor- responding ones in reflexion. Making m = — 1 in (83) we have, CHAP. V. SPHERICAL ABERRATION. 1 1 2 65 2/1 1 \2 «2 _ ( ) . ±_, or, r \ r u / 2 J_ = JL_-L + i!(i__±y, w' r ?t r V r w / a result which agrees with equation (76), art. 93. (101.) Prop. XXVIII. To determine the aberration in a pencil of rays, after refraction by a spherical lens. Fig.L. R being the radiant point of a pencil of rays falling upon the lens LVV'L', let RL be the extreme ray of the pencil, and R' the virtual focus of the extreme rays, after refraction by the first sur- face of the lens. If now we suppose a pencil to proceed from R', considered as in the denser medium, the extreme ray of this pencil, R'U, will be refracted into the direction, L'M, which, if continued backward to F, will give the virtual focus of the extreme rays. As before, represent RV by u, R' V by u', and CL by r ; fartlacr, let R V = »', FV = v, CL' = r', and VV = t. By the preceding proposition (equation 83,) we have rn — 1 m — 1 i + ,/i " + i\ (L-±-V. V r u / \r u / (83). The case of the second surface will correspond to that of the first, if we consider F the radiant point, and R the virtual focus ; v must be written in (83), for u, v' for u\ and r' for r ; we then obtain m __ 1 m — 1 , m — 1 (T-^MT—r)' F2 Mgg^HgB^BBWiWWBI ^HHHBHBWBMMWHWHBWI 66 SPHERICAL ABERRATION OF LENSES. 1 m in — 1 m — 1 /J m + 1 \ /J i_y V" v /' \/ ' v / ' APPENDIX. (86); but v' = m' -f- <, whence — : , performing the division v' u' -(- t and neglecting the powers of t above the first, m m mi We may farther approximate to this value of _ '— , by substi- v' tuting for , in the second member of the equation, its ap- u' 2 proximate value, from (26), art. 50, namely, i_ = J_ (J- + m ~ l Y; whence, u' 2 m' z \ u r / m m t / 1 m — 1\ 2 . v u m V u r / in which the value of — , from (83), being written, u' J_(J_ + 'JL=i!) 2 + m \ u r / ?n 1 m — 1 m — 1 ! \ r u / \ r u / V 2 By substituting for — , in equation (86), its value just found, we have .L= L + (m -i)(J__JL)_i(i- + «i=i) 3 + v u \ r r / m\ u r / m — \ T/ 1 m -|- 1\ /J_ J_\ 2 / 1 m -f K j/i j LV r u / \r u / \r' v ) ■%-m- v i < e7 »- We see, in this formula, first, the two terms which denote the reciprocal of the focal distance of an indefinitely small pencil ; second, the correction for thickness ; and, in the last term, the correction for aberration. (102.) The general formula, (87), becomes less complex, and gives results of considerable practical importance, when applied to the case of parallel rays. CHAP. V. SPHERICAL ABERRATION. 67 Prop. XXIX. The incident rays being parallel, to determine the aberration of the pencil after refraction by a spherical lens. In this case, = o, and (87) becomes, m — 1 i_ = («-i)/_L_J r \_ JL'.Siz v \ r r' ' m r L r 3 V r' v / \ r' v / J I) 2 + The correction for thickness, contained in the second term, has already been separately considered, articles 55, 67, &c. ; we may therefore leave it out of the question here, making in (88) t = o. Farther, to approximate to the value of », we may substitute for in the second member of the same equation, the approximate v value , or (m — 1) ( ) , obtained by making / V r r / « = o in (28), art. 53. We have, then, from (88), 1 1 . m — 1 r 1 f + VI' / or, taking the value of v, dividing by the denominator thus found, and neglecting the powers of/ higher than the second, ^j rj_ __/j !* + 1 \ >i 2 I. r •' V r> f ) the aberration in length, therefore, is represented by a = _ !!Lz! H _ (L - !L±i\ m 3 Ir 3 V /•' J / / / J 2 (103.) To apply the formula just obtained, to a double convex icns, r and / (art. G2,) must he made negative, whence m -f 1) in- L r J \ r + ') 68 ABERRATION OF GLASS LENSES. APPENDIX. •(wn-^ •* This value of the aberration having the positive sign, while the approximate focal length has the negative sign, its effect on the focal length, for rays not near the vertex, is subtractive ; showing that the focus of such rays is nearer the lens, than the focus of rays incident near the vertex. 3 (104.) For an equi-convex, glass lens, r = r', m = , and / = r, disregarding the sign, since / has already been made nega- tive in (90) ; and from (00), 9 Lr 3 T \r T 2r/ raj 2 „ = i_|- l + (1 + i-).4].f, 5 y 2 3 r If we suppose the beam of light to occupy the whole aperture of the lens, y becomes the semi-breadth, and y 2 — — . 2r nearly, or y 3 = rt, and t = £—.; writing t for JL_ in the value r r of a, just found, a = 1§<, the result stated in paragraph 3, page 53, of the text. 3 5 (105.) If m = — , and r : r' : : 2 : 5, or r' — - — r, we have ; 2 2 from (3G), *— •—>(t-+v)— i4«- '— ¥» Substituting these values of m, r', and / in (90), recollecting thai / bus been already made negative in that equation, and that now its value is to be placed there without regard to the sign, it gives, 9 Lr J ^ \5r 2 lOr / HI &^^&'M?it$^&»i*i 5*wi V. SPHERICAL ABERRATION. / 2 7 yi 10 a r 3 ya ' \~5r" + 107/ J ' "~ 73" ' 2/ 9 L V 5 ^ 20 / V 5 10/ J 7 69 10 3 / This is the case of an unequally convex lens, in which the more convex side is turned to incident light (106.) In the plano-convex lens, if the plane side be turned towards parallel rays, — o, and / = 2r / ; if the material be r 3 glass, m = — , and from (90) we obtain = m+i)-(i + i-)\H->'-f •=^t (, +-f)4] 8 f- or, a = 4.5 JL / = 4 . 5 t. The result given in paragraph 1, page 53, of the text. In the same lens, with the convex side turned to parallel rays, ==o, and / ss 2r, whence from (90), r and / having already r' been made negative, a = -L \l + ± • J-l 8 . ll , or, 9 L T 4 4 J / a m 1.17 J£. = 1 .17.*. / The result stated in paragraph 2, page 53, of the text. (107.) Prop. XXX. To determine the ratio of the radilFbfthc sur- faces of a double convex lens, which shall produce the least aher ration, with a given focal length and aperture. To solve this question we must determine the ratio of r and r\ when a is a minimum, / and y being constant. Differentiating the value of a given in (90), considering r and r' as variable, and disregarding the constant multipliers, we obtain, after changing all the signs, ^^B Bl — W — WWWWHWI 70 LENS OF LEAST ABERRATION. APPENDIX. rill 3dr / 1 , m 4. 1 \ / 1 . 1 \ 2tfr' - — + \1^ + -T-) VV + 77*7* + Substituting in (91) this value of — _ , and dividing by dr da lr~ (v + ^ J : ' V7" + T) * ~ (~ + T) ~^ ' which, by the question, is equal to zero. Multiplying by r 2 we obtain ^_,.(i r+ -±«).(^ + i) -(^ + iV= (92). From equation (36), disregarding the sign of/, r in — 1 / r 1 Substituting this value in (92), and arranging the terms _3 6 _,_ 3 _2 2(m + 2) r' 3 (m - l)/r' 2(m + 1) - + (m - I)*/* 1 A' i =r o, or, ''* fr' P - (—% + 2m + 6 ) • -h + \m — 1 / fr ( - 2m — 3\ . J- = o, V(TO — l) a / f 2 CHAP. V. SPHERICAIi ABERRATION. and by transposition and multiplication, 71 = ( («. + a. + e\ . » = \m — 1 / r (m — lf 2m - 3 )t (93). If the lens is of glass, orm=: — = — or r' — _?L but from (36) 1 2 12 / 21 / 12 ^ and Comparing togetlier the values obtained for r and r', r : r' : : 1 : 6. This lens is known to opticians as the crossed lens. With the 15 t/ 2 more convex side turned to parallel rays the aberration is — . i_ , which is less than that for the plano-convex lens with the convex side turned to parallel rays. (108.) It would carry us beyond the limits of this Appendix, to go into the investigation of the aberration of combined lenses. Before leaving this subject we purpose to show a method by which the surfaces which refract rays accurately to a point, may be determined. Prop. XXXI. To determine the curvature of the surface of a me~ dium, so that rays passing into it, from a rarer medium, may be refracted to a point. Fig. M. As we have found a concave surface to give only a virtual focus, »vc proceed, at once, to examine the case in which the surface of the denser medium is convex. Let K be the radiant point, RL a ray meeting the surface at L and refracted to F : let L' be a point farther from the vertex V than L, RL' being the incident and Ump B BBPBBBI BWB S IW B MHBHWlMBHWBiB I t ^^Q 72 LENSES WITHOUT AEEKRATION. APPENDIX L'F the refracted ray for this point. Draw the perpendiculars LR! and L'S upon the incident and refracted rays RL' and L'F, respectively. LR' will be nearly equal to the increment of the incident ray, and LS to the decrement of the refracted ray, in- passing from the point L to L'. Call RL, u', and LF, — v'. Then if L' be supposed very near to L, LR = dv', and LS = dv'. LR' In the triangle L'LR', _,__ — cos. RLL' — sin. incidence, LL and in L'LS T whence, LL' "Zs - = LR' LS cos. SLL' sin. incidence sin. refraction — m, or, sin. refraction du' dv' du' — mdv' — o (94), the differential equation of the curve which, by a revolution about the axis RF, will produce the surface required. To integrate, let RV = if, and FV = — v, the complete integral of (94) will be «' — u = m (»' — v) (95). (109.) If the incident rays be parallel, u' — u = VM, Jig. N. Fig. N. V M If we put VM = A — x, (95) becomes A — x = mi (t)' — »), whence, A — x V = V -f — (96). The equation for the distance of any point in an ellipse from the farther focus is, (Young's Analyt. Geom. art. 47, p. 72), v = A -\- ex, in which e < 1 ; with this (96) agrees in form, and will be identical if 1 _ c , / A . m A m CHAP. V. LENSES WITHOUT ABERRATION. 73 Substituting for to in the second of these equations, its value from the first, w' — c = A, or v' = A -f- c. We find, then, that an ellipsoid of which the semi-transverse axis is to the excentricily as the index of refraction is to unity I = to | will refract parallel rays, accurately, to the farther focus. If a lens be formed, of which the first surface is a portion of the ellipsoid just determined, the second surface should be (art. 99.) a portion of a sphere, having the farther focus of the ellipsoid as its centre {Jig- 38, text). (110.) Equation (95) may be applied to the case in which the inci. dent pencil passes from a denser to a rarer medium, through a con- cave surface. Then FL, FL', Jig. M, would represent the incident rays, and LR, L'R the refracted rays, and the ratio of the sine of incidence to the sine of refraction would be represented by the fraction ; substituting this for tn in (95) we have 1 , , ^ u u = (v V) (97). For the case of parallel rays, (Jig. 40., p. 55, text,) by proceed- ing as in the last article, making u' — u = A — x, v' — v = m {A — x), and v = v' — in A -f- mx ; an equation of the same form with that before obtained, and re- presenting the distance of a point in a conic section from the farther focus ; in it m = e — — , and v' — mA = A. A Since m > 1, e > 1, and the equation belongs to a hyperbola, (Young's Analyt. Geom., article 79, p. 104,) the equation of which is v' = A -f- mA = A -j- c. If, then, we form a lens witli the first surface plane, and the second that of a hyperholoid of which the excentricity is to the semi- transverse as the index of refraction, of the material of the lens, is to unity, parallel rays, incident perpendicularly upon the first surface of the lens, icill he refracted to the farther focus of the hyperholoid which forms the second surface (Jig. 40, text). (111.) The cases in which the aberration of converging raya upon a spherical surface is zero, (art. 99,) are contained in (.95) ; it is unnecessary, however, to discub-s it farther. G ^^^9 m^BW ^gg^Big^^^p ^^| MMMI 74 CAUSTIC FOR PARALLFX RAYS. APPENDIX. (112.) The forms of mirrors without aberration may also be inferred from the equations just discussed. The convex mirror will be given by making m = — 1 in (1)4), whence, da! -\- dv = o, and integrating »' — t/ (-») = C (98). By this property we recognize the hyperbola, the distances u and — e' being those of the point, from tiie two foci. For a concave mirror, u' and v' have the same sign, in equation (94), and du -f- dv = o, or, u' -{- v' — C (99). The mirror is an ellipsoid, the radiant point coinciding with one focus, and the rays being collected at the opposite focus. If one focus remove to an infinite distance, the ellipsoid becomes a paraboloid, into the focus of which the rays which have been supposed parallel are collected. Caustics by Reflexion. (113.) It is not intended to enter fully into this subject in rela. tion to both reflexion and refraction, but to confine the discussion to examples of the caustics produced by reflexion. The formula for the oblique pencil, art. 29, &c, gives, in certain cases, an elegant and easy method of determining the form of a section of the caustic surface, produced by reflexion from a spherical mirror. Prof. XXXII. To determine the form of the caustic produced by the reflexion of a pencil of rays from a spherical mirror, when the rays are parallel; and also when the radiant point is at a diameter's distance from the vertex of the viirror. First. When the radiant point is infinitely distant, oi Jhe rays parallel. LDM representing a section of the mirror, let RL be a ray inci- dent upon it and reflected into LB ; then, the focus of a small pencil meeting the mirror near to L will be the point F found from the value of v in the equation which concludes art. 30, namely, r v — . cos. , the ra- dius CB = CB' = r. Join Fi£ and let fall the perpendicular RQ, upon BF. F being a point in the caustic, FR is the radius vector of that point and RQ a perpendicular upon the tangent ; call RF, u\ and RQ, p'. An equation between u' and p' will be that of the caustic curve. In the acute angled triangle RFB, since the segment BQ = RB . cos. RBQ, u' 2 = u 2 -f v 2 — 2«« . cos. 2tf> (100) ; and in the right angled triangle RBQ p' = m . sin. 2

the value just given, cos. 2 = Bil 1. u 2 We have also, by trigonometry, sin. 2

may be applied to this case by taking r to represent the radius of the osculating circle, which is, udu dp Substituting this value for r, 1 Qdp udu dp . cos.

d u If this value of v be substituted in equation (103), we shall obtain a new equation, which, in conjunction with (102), will give the relation of u' and p' in terms of u, p, du, and dp. The relation of the last four quantities mentioned will be "given by the equation of the reflecting curve and by its differential ; eliminating these quantities, there will result a single equation between u' and p', the equation of the caustic curve. (115.) To give an example of this method of proceeding, let the reflecting curve be any portion of a logarithmic spiral, of which the equation is, p = mu. The general value of v (104), is first to be applied to this par- ticular case. Differentiating the equation of the curve, , dp = mdu, whence (104) becomes _ pudu pu pu "zmudu — pdu 2mu — p 2p — p This value of v substituted in equation (103), gives w' 3 — 4a 2 __ -J. = 4m 2 4p 2 , or, 4m 2 — 4m 2 u 2 = 4u 2 (1 — m 2 ), and DOUBLE REFRACTION. 79 u' = 2w V 1 — m*. Also, from (102), p' = 2mu J \ — ? -^|- — 2mu jL—m2 , or, since we have just found 2« n / 1 — /« 2 = u' j>' = mu'. The section of the caustic surface is, therefore, a logarithmic spiral differing only in position from the reflecting curve. CHAP. VI. ON THE DOUBLE REFRACTION AND POLARIZATION OF LIGHT. (116.) Although it docs not enter into the design of this Appen- dix to show the method of deducing, from theoretical considera- tions, any of the general laws of Optics, I have thought that it may assist the student to give the formula? to which these consid- erations lead, or which have been deduced from experiment, in certain particular cases, discussed in the text. The formula, or general law, once remembered, the details of the phenomena flow naturally from it, and the memory is not tasked to recollect indi- vidual results. Double Refraction of Light. (117.) The formula which represents the law of extraordinary retraction in doubly refracting crystals, becomes, when the inci- dent ray is in a plane passing through the axis of the crystals, m'- — m' 2 — (?n- — m"-') . sin. 2

m\ or — > — , that is a > b, ( , \ b a \ b a a' J / WWW WMPBBBWlPBWWW WBBIMMMMBBMMWWMMW1 80 INCIDENCE ON A SECOND CRYSTAL, APPENDIX. (fig.Tl), will be negative, whence the term crystals with a nega- tive axis which applies to this class. When a m, and ( . — 1 becomes additive, and b ^ a Vfc 2 a 3 / the crystals are said to be crystals with a positive axis of double refraction. (118.) In the plane of principal section the tangents of the angles of extraordinary and of ordinary refraction arc in a constant ru'.io to each other. In the plane perpendicular to this, the law of the fines applies equally to the extraordinary and to the ordinary ray, but the value of the constant quantity is different for the two rays. These are the only two cases, in which the extraordinarily refracted ray is contained in the plane of incidence. ► (119.) When light which has been polarized by double refraction, in the plane of principal section of a crystal Iceland spar (Jigs. 84. and 85., text), passes through a second crystal, the relative brightness of each image, supposing that no light is lost by re- flexion or absorption, may be expressed by the following formula; in which Oo, Ee, Oe, and Eo represent the images formed as de- scribed on page 140 of the text, a is the angle which the plane of principal section of the second rhomb makes with the same plane in the first, and A is the brightness of the incident ray. Oo = ± A 2 1 . 2 a — Ee Oe — A . sin. 2 a = Eo 2 (106). (107). The sum of the brightness of the four images, Oo -[- Ee -(- Oe -f- Eo — A (cos. 3 a -f. sin.- a) — A. From the foregoing formulfe (106. and 107.) we may trace the changes of brightness in the several images, as described in pages 140, 141, of the text (Jig. 86.) When the principal sections are parallel, a — 0, cos. a sin. a = 0, therefore 1, and Oo = Ee= —A 2 Oe = Eo = 0. By turning the lower crystal, a assumes a finite value and the images Oe, Eo appear. As a increases, sin. a increases and cos. a diminishes ; Oe and Eo, therefore, increase in brightness, and Oo, Ee decrease. When a = 45°, cos. a — sin. a, and the four im- ages are equally bright. The angle a increasing farther, Oo and Ee become more and more faint, and disappear when a — 90° ; at CHAP. VI. POLARIZATION OF LIGHT 1 81 this angle Oe = Eo A. The rotation of the lower crystal being continued beyond 90°, cos. a takes the negative sign and increases negatively, while sin. a again diminishes ; when a = 180°, cos. a = — 1, sin. a = 0, and Oe, Eo again disappear. At this angle the two images Oo, Ec coalesce, the two extraordinary refractions taking place in opposite directions. Polarization of Light by Reflexion. (120.) When light has been polarized by reflexion from a sur- face, upon which it falls at the maximum polarizing angle, the following ernpyrical formula, determined by Malus, will represent the intensity of the light reflected from another surface, upon which the pencil is incident at the polarizing angle : (Jig. 87, page 143, text.) 7= A. cos. 2 a (108), in which J is the intensity of the reflected light, A that of the inci- dent light, and a the angle between the plane of incidence and that of the second reflexion, or the azimuth of the plane of the second re- flexion. When a = 0, or 180°, J is a maximum, and when a = 90°, or 270°, 1=0, and no light is reflected. As a consequence of this law, a beam of common light, as far as brightness is concerned, may be represented by two beams of polarized light, having their planes of polarization at right angles to each other : for, the angle between the planes of polarization and of reflexion of the one being called a, that of the other will be 90° — a, and from (108) we shall have, for the brightness of the two reflected pencils, I = A. cos. 2 a I' = A. cos. 2 (90 — a) = A. sin. ' J a ; whence, I -\- I' = A (cos. 2 a -j- sin. 2 a) = A, the sum of the intensities, of the two supposed pencils, remaining the same whatever be the angle a, which is characteristic of com- mon light. Equation (108) applies to the case of light polarized by refrac- tion, and incident upon a reflecting surface at the angle of com- plete polarization, a being the angle between the plane of polariza- tion of the incident ray and the plane of reflexion. (121.) The law, deduced by Sir David Brewster, as expressing the relation between the phenomena of refraction and polarization by reflexion, when light falls upon the first surface of a body, is tan. P = m (109). P being the polarizing angle, and m the index of refraction of the material used. iBI^B^BiBWB^BBB^BJJBJtBKBI^P^w^BiBBPtBWPCBWW '.■'■•.■•■■■■.■.■■'. 82 POLARIZATION BY REFLEXION. APPENDIX. From this formula, if wc suppose the light to be incident at the polarizing angle, and call It the angle of refraction at this inci- dence, tan. P = sin. It ; but tan. P sin. R = cos. P, sin. P cos. P • (HO), , whence or the maximum polarizing angle is the complement of the cor- responding angle of refraction, and the reflected ray is perpen- dicular to the refracted ray. (122.) If the light which hah passed through the first surface fall upon a second, parallel to the first, the angle of incidence upon the first surface being P, that on the second is R, and R = 90 — P (110); whence, tan. R = cot. P : hut cot. P = _i tan. P 1 , and therefore, tan. R = .— or the tangent of the incidence upon the second surface is the index of the refraction from the denser to the rarer medium. R is, therefore, the angle of polarization for the second surface, and the light reflected from that surface, as well as that from the first, will be polarized. Law of Partial Polarization of Light by Reflexion. (123.) Sir David Brewster has verified by an extensive series of experiments a law, which is due to Fresnel, by which the effect of any number of reflexions, on the inclination of the planes of polarization of a beam of light, may be determined. The effect of a single reflexion at an angle differing from the polarizing angle, is given by the equation tan.

= cot x . cos. (» — i'), or tan. = ! . cos. (i — i') From formula (111), for partial polarization by reflexion, ,, , cos. (i 4- i' . _ cos. (i _L i') tan. w — tan. a : — _L_ . = tan. x . cos. (i — i') cos. (i — i) - Equation (115), applied to the second surface of the plate, gives cot. (p" = cot. ' the reciprocal of the value just found for tan. cos. (i -\- i'), which would occur by dimin- ishing i, cot. $" > 1, and " = cos. (i — »'), or the light is repolarized at the second refraction, and the effect of the plate is that of a single surface.* CHAP. VII. OF THE RAINBOW. (127.) To explain the theory of the rainbow, wc begin by tho following proposition. Prop. XXXIV. A ray of light enters a refracting sphere, is re- fleeted any numher of times, and emerges ; to determine the devia- tion when it is a maximum, or minimum. Fig.*. ll>^ Let RL be a ray of light, meeting the refracting sphere LMNP at L, and refracted into LM: LM meeting the second surface of the sphere at M, is in part reflected into MN, which farther suffers reflexion at NP, taking the direction NP ; that part of NP which is not reflected, passes out of the sphere, being refracted into the direction PF. By the law of reflexion tiie angles CML, CMN, &c, are all equal to CLM the angle of refraction at the first surface; the angle of emergence IPF is, therefore, equal to the angle of incidence RLK. Call the angle of incidence ' ; the angle of deviation of the refracted ray LM, or the angle HLM = (p — tp' ; the angle of deviation at emergence, or the angle NPG —

= J7i. sin. ' (17). When 5 is a maximum or minimum, dS = o, and by differen- tiating (120,) considering

\ and (n -|- 1) cos.

= 59° 21', and sin. — .8G03, whence from (17) sin. = . 5199, = 58° 41 \', and sin.

and a .3118,

'=; . 712t;, = . 3184, = 71° 27. 1 ., sin. ' = .7046, 1, must subsist ; but from the investigation it appears, that sin.

4. Camera obscura, an optical instru- ment, invented by the celebrated Baptista Porta, 274. Camera lucida, invented bv Dr. Wol- laston, 277. Cameleon mineral, 239. Carbon, sulphuret of, of great use in optical researches ; employed as a substitute for flint glass by Mr. Barlow, 305. Carpa, M.. and M. Ridolfi, repeat Dr. Mi richini's experiment with success, 84. * The Appendix referred to is that of the American editor, unless when the contrary is expressly stated. H2 teSfe; EBMBE fflBMWBW CBlBraB«B 3E8 »8 » ™M B^B^E^B^^S 90 INB Cassia, oil of, 31—72. Catoptrics, 13. App. 9. Caustics formed by reflexion, 58. App. 74. Formed by refraction, 62. Charcoal the most absorptive of all bodies, 120. Chevalier, M., of Paris, makes use of a meniscus prism for the ca- mera obscura, 271. Christie, Mr., of Woolwich, his ex- periment confirmed by those of M. Barlocci and M. Zantedeschi, 84. Coddington, Mr., his observations on the compound microscope, 284. Colors, accidental, and colored shadows, 204. Phenomena of, il- lustrated by various experiments, 259. Compression and dilatation, their optical influence, 203. Crossed lens, App. 69. Crystals with one axis of double re- fraction, 128. Whether mineral bodies or chemical substances have two axes of double refraction, 133. A list of the primitive forms of, according to Uaiiy, 134. With one axis; system of colored rings in, 172. The influence of uniform heat and cold on, 202. Composite exhibited in the bi pyramidal sul- phate of potash, 20(5. I»apophvl- lite, ib. In Iceland spar, 20S. The multiplication of images by the crystals of calcareous spar with one axis, 210. Different colors of the two images produced by dou- ble refraction in crystals with one axis, 211. Cubes of glass with double refrac- tion, 199. Curves, caustic, formed by reflexion and refraction, 58. App. 74. Cylinders of glass with one positive axis of double refraction, 197. With a negative axis of double refraction, 198. D. D'Alembert,304. Davy, Sir Humphry, repeats Berard's experiments on the heating power of the spectrum in Italy and at Geneva ; the result of these ex- periments a confirmation of those of Dr. Herschel, 32. De Chaulnes, duke, 98. Descartes, 54. Deviation, angle of, 33. App. 27. Diamond, 31. Dichroism, or the double color of bodies, 210. Dioptrics, 26. App. 26. Dispersion, irrationality of, 73. Dispersive powers, table of, Author'} App. 310. Di Torre, father, of Naples, his im- provement on Dr. Hooke's spheres for microscopes, 280. Dollond, Mr., the achromatic tele- scope brought to a high degree of perfection by, 70. E. Ellipsoid, 54. App. 72. Englefleld, Sir Henry, 81. Eriometer, an instrument proposed by Dr. Young, a description of it ; and the manner in whicb it is to be used, 101. Eye, the human; the structure and functions of, 240. The refractive powers of humors of, 212. '!",>. in- sensibility of, to direct impressions of faint light ; duration of the im- pressions of light on the retina, 250 and 321. The cause of sin- gle vision with two eyes, 251. The accommodation of, to different distances, proved by various ex- periments, 252. Long-sightedness and short-sightedness accounted for, 253. Insensibility to particu- lar colors, 259 and 322. Eye-pieces, achromatic, Ramsden's ; in universal use in all achromatic telescopes for land objects, 301. Faraday, Mr., his observations on glass tinged purple with manga- nese ; its absorptive power altered by the transmission of the solar rays, 124. Fata Morgana, seen in the straits of Messina, accounted for. 218. Fibres, minute, colors of. 101. Fits, the theory of, superseded by the doctrine of interference. 111. Flui. Is, circular polarization in, dis- covered by M. Biot and Dr. See- beck, 188. Focal point, 18. Foci, conjugate, 18. App. 15. Focus, principal, for parallel rays, 17. rules for finding the principal ; for convex lenses, 41. App. 38 ; for mirrors, 17. App. 14. Distance from centre, a mean proportional, &c, App. 18. Physical, of mirrors, App. 59. Fraunhofer, M., of Munich, I'lis ob- servations on the lines in the spec- trum, 78; perceives similar bands in the light of planets, and fixed stars, 79. Illuminating power of the spectrum. 80. Fresnel, M., explains the phenomena or inflexion or diffraction of light, 80. Formula for polarization of light by reflexions, not at maxi- mum polarizing angle, App. 82. His experiment on the interference of polarized light, 180. Discoveries of, on circular polarization, 169. O. Classes, plane, 31. Multiplying, 273. Refraction of light through plane, 30. App. 35. Achromatic opera, with single lenses, 304. Gorilon, the duchess of, 91. Goring, Dr., his improvements in all kinds of microscopes; introduces the use of test objects, 287; a work published by him and Mr. Pritch- ard on the microscope, 281. Cray, Mr. Stephen, 2ti0. Gregory, James, the first who de- scribed the construction of the re- flecting telescope, 291. Grimaldi, his discovery of the in- flexion or diffraction of light, SO. H. Hall, Mr., inventor of the achro- matic telescope, 70. Halley, Dr., his observations on the rainbow, 226. Halos, 227. The colors of, described : the origin of, and how produced, 232. Hare, Dr., observation on translu- cencyof gold leaf, 320. Haiiy, the abbe, discovers the want of electricity by friction in anal- cime, 184. Heat, the influence of, on the ab- sorbing power of colored media ; analogous phenomena in mineral bodies, 123. Heat and cold, tran- sient influence of, 197. Herschel, Mr., his discovery of an- other pair of prismatic images in thin plates of mother-of-pearl, 105. The principal data of the uiidula- tory theory given by, 119. The results of many authors on the subject of colored Haines, given by, l-M. His discovery that, in crystals with two axes the axes change their position according to the color of the lifeht employed, 131. Herschel, Sir W., his experiment of the heating power of the spectrum confirmed by Sir Henry Englefield, 81. Ink applied by him for obtain- ing a white image of the sun. 121 Constructs a telescope, 40 feet long, with which he discovers the sixth satellite of Saturn, 296. ;x. 91 Heveliin>, his observations on a pa- raselene, 230. Home, Sir Everard, his description of tlie pearl, 10.5. Hooke, Dr., constructs small spheres for microscopes. 280. Huygens, his discovery of the law of double refraction in crystals. 131 ; determines the extraordinary re- fraction of any point of the sphere, 132. Publishes an elaborate history of halos, 231 ; his discov- ery of the ring and the fourth sat- ellite of Saturn, 289. Huddart, Mr., several cases described by him of unusual refraction, 210. Iceland spar; of what composed; found in almost all countries, 120. Image by mirrors, curvature at ver- tex of, App. 24. Change of form by change of distance of object, App. 24; formed from section, App. 20 ; by a convex lens, App. 52 ; by a concave lens, App. 54. Incidence, angle of, equal to angle of emergence, 14. App. 29. Plane of, 14. Induration, the influence of, 205. Ink, diluted, absorbs all the colored rays of the sun in equal propor- tion ; applied by Sir William Her- schel, as a darkening substance, for obtaining a white image of the sun, 121. Interference, the law of, 115. lolite, properties of, 211. Iris ornaments invented by John Barton, Esq. 107. Jansen, his invention of the single microscope, 279. Jit ri no and Soret, observation of an unusual refraction, 218. K. Kaleidoscope, formation and prin- ciple of the, 202. Kircher, the inventor of the magic lantern, 270. Kitchener's, Dr., pancratic eye-tube, 302. Landriani, 81. Lantern, magic, invented by Kir- cher, 270. Latham, Mr., 221. Lerebours, M., has lately executed two achromatic object-glasses, which are in Sir James South'-, i itory at Kensington, 300. HI 92 ' ixr Lehnt, hi* work on the seat of vis ion, '.'44. Le Maire, 51. Lens, spherical, concavo-convex, double-convex, plano-convex, dou- ble-concave, plano-concave, 31 ; of least aberration, App. ti9. Lens, a plano-convex, (he principal focus of, 42. App. 43. Plano-con- cave, refraction by, App. 47. achromatic. 76. Lenses, the formation of images by, App 50. Their magnifying power, 4ii. Convex and concave, 267. App. 37,45, and 54. Burning and illu- minating, 2e8. Lenses, polyzonal, constructed for the Commiisioners of Northern Lighthouses? introduced into the principal French lighthouses, 269. Light, the velocity with which it moves; moves in straight lines, 12. Falling upon any surface, the an- gle of its reflexion equal to the angle of its incidence. 14. The to- tal reflexion of, 34. App. 27. Re- fraction of, through curved sur- faces. 37. Refraction of, through spheres, 38. A pp. 42. Refraction of, through concave And convex sur- faces, 40. App. 31. Refraction of, through convex lenses, 41. App. 37. Refraction of, through concave lenses, 44. App. 45. Refraction of, through meniscus and concavo- convex lenses, 45. App. 48. On the colors and decomposition of white light, the composition of, discover- ed by Sir I. Newton, (53. Different refrangibilities of the rays of; re- composition of white light, 65. De- composition of, by absorption, 67, and 31-5. The inflexion or diffrac- tion of, 86. Several curious prop- erties of. 107. The interference of, 111. The absorption of, 120. A new method proposed of analyzing while liulit, 124. Double refraction of, firs! discovered ill Iceland spar, 126 Polarization of. by double re- fraction, 138. Partial polarization of, by reflexion, 149. App. 82; and by ordinary refraction, 152 App. Kt. Polarized, the colors of crys- tallized plates in, 1H2. The action of metals upon ; absorptive powers of. 210. Loadstone, various experiments by professor Rarlocci and Znnte- deschi on the magnetizing power of light on, 85. M. Magnetism, experiments illustrative of, as developed by light, 85. Mains, M., discovers the polariza- tion of light by reflexion, 142. Law of, App 81. Marietta, his curious discovery tliat the base of the optic nerve was incapable of conveying to the brain the impression of di