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HANDBOOK OF OPTICS 
 
^^^yi>^ 
 
HANDBOOK OF OPTICS 
 
 FOR 
 
 STUDENTS OF OPHTHALMOLOGY 
 
 <7ti/Ul^ 
 
 BY 
 
 J". 
 
 WILLIAM NOEWOOD S^lfTBKr^.A., M.D. 
 
 FEOITBSSOR OF OPHTHALMOLOGY, NATIONAL UNIVBESITY, AND 
 
 ASSISTANT SURGEON, EPISCOPAL EYE, EAR, AND 
 
 THROAT HOSPITAL, WASHINGTON, D.C. 
 
 THE MACMILLAN COMPANY 
 
 LONDON: MACMILLAN & CO., Ltd. 
 1899 
 
 f 
 All rights reserved 
 
Copyright, 1899, 
 bt the MACMILLAN COMPANT. 
 
 Nortoooli ^r«BB 
 
 J. S. Cufihing & Co. — Berwick & Smith 
 
 Norwood Mass. U.S.A. 
 
PREFACE 
 
 In the following work there is presented so much 
 of the science of optics as pertains directly to oph- 
 thalmology. Simplicity has been sought so far as 
 this is not incompatible with thoroughness ; for 
 whoever would become versed in ophthalmology as 
 a science must in the beginning make the mental 
 effort necessary to acquire a clear understanding 
 of the refraction of light through a compound opti- 
 cal system such as the eye. 
 
 The demonstrations, some of which may appear 
 formidable to the student, require no knowledge 
 of mathematics beyond that of simple algebraic 
 equations and the elementary truths of geometry. 
 For those who may not be familiar with the trigo- 
 nometrical ratios, a brief synopsis has been fur- 
 nished in an appendix. 
 
 In demonstrating refraction by prisms and by 
 spherical surfaces. Heath's " Geometrical Optics " 
 has been used as a basis, but many modifications 
 have been made. 
 
VI PREFACE 
 
 A uniform notation, with which the student will 
 easily become familiar, has been preserved through- 
 out the book ; by this means those who may be 
 indisposed to follow the algebraic processes in 
 detail will be aided in understanding the methods 
 of demonstration. 
 
 Washington, June, 1899. 
 
CONTENTS 
 
 OHAPTBR PAGB 
 
 1 
 
 Introduction 
 
 I. Rbfkaction at Plane Surfaces 
 n. Refraction at Spherical Surfaces 
 III. Refraction through Lenses . 
 rV. The Eye as an Optical System 
 
 26 
 40 
 62 
 
 V. The Determination op the Cardinal Points 
 
 OF THE Eye in Combination with a Lens 90 
 
 VI. Errors of Refraction — Lenses used as 
 
 Spectacles 94 
 
 VII. The Effect of Spherical Lenses upon the 
 
 Size of Retinal Images .... 109 
 
 Vin. Cylindrical Lenses 124 
 
 IX. The Twisting Property of Cylindrical 
 
 Lenses 138 
 
 X. The Sphero-Cylindrical Equivalence op Bi- 
 
 CYLINDRICAL LeNSES 148 
 
 XL Oblique Refraction through Lenses . . 163 
 
 Xn. The Effect of Prismatic Glasses upon 
 
 Retinal Images 169 
 
viii CONTENTS 
 
 CHAPTER PAGE 
 
 XIII. The Reflexion op Light 177 
 
 XIV. The Optical Principles of Ophthalmometry 
 
 AND of Ophthalmoscopy .... 184 
 
 Appendix I 198 
 
 Appendix II 200 
 
 Index 205 
 
HANDBOOK OF OPTICS 
 
 INTRODUCTION 
 
 That branch of physical science which treats of 
 light and vision is called Optics. It may be sub- 
 divided into Geometrical, Physical, and Physiological 
 optics. Geometrical optics deals with the theory of 
 light ; it is a " mathematical development " of the 
 experimental laws by which light is supposed to be 
 controlled, — the laws of reflexion and refraction, 
 and the supposition that light travels through homo- 
 geneous media in straight lines. Physical optics 
 investigates the causes and nature of light ; while 
 Physiological optics treats of the phenomena of vision 
 or the sensation produced by the action of light 
 falling upon the retina. 
 
 Catoptrics and Dioptrics, terms less used now than 
 formerly, refer respectively to the phenomena of 
 reflexion and refraction of light. The science of 
 optics was practically unknown to the ancients. 
 They had observed that light travels in a homogene- 
 ous medium in straight lines, and they also knew the 
 simple law of reflexion and the focusing property of 
 
 1 
 
2 HANDBOOK OF OPTICS 
 
 lenses and mirrors ; but their ideas of vision were 
 most crude, it being commonly supposed that light 
 was something given out from the eye. Strange to 
 say, this theory did not entirely disappear for many 
 centuries. 
 
 Spectacles of spherical lenses were probably intro- 
 duced in the thirteenth century. To a spectacle 
 maker, Hans Lippershey, is ascribed the first tele- 
 scope in 1608, and in the following year Galileo 
 independently constructed his telescope ; but with 
 the astronomer Kepler, who died in 1630, begins the 
 true science of optics. Willebrod Snellius, professor 
 of mathematics in Leyden, discovered the law of 
 refraction ; and to Sir Isaac Newton is due the dis- 
 covery that white light is composed of various 
 colors, capable of separation by the action of a 
 prism. 
 
 Up to this time the propagation of light was 
 thought to be instantaneous. Romer, a Danish 
 astronomer, discovered in 1676 that time is re- 
 quired for the transmission of light. This he 
 inferred from discrepancies between the calculated 
 and actual time in the observation of eclipses of 
 Jupiter's satellites ; and he rightly attributed these 
 discrepancies to the unequal distances through 
 which light had to travel, owing to the varying 
 distance between the planet and the earth. Ter- 
 restrial measurement of the rate of propagation of 
 
INTRODUCTION 3 
 
 light was not accomplished until the middle of the 
 present century. This was done with instruments 
 of great precision by Fizeau and Foucault, and more 
 recently by Professor Newcomb. The velocity as 
 thus determined is 300,000,000 metres or 186,000 
 miles per second. 
 
 The question as to the method of transmission of 
 light has been the subject of much controversy. 
 Leaving aside the speculations of the ancients, the 
 two theories are the Corpuscular or Emission The- 
 ory and the Wave Theory. The first supposef a 
 luminous body to give off certain particles, which, 
 striking the eye, produce vision. The wave theory 
 supposes that all space is pervaded by a substance 
 called ether ; and that by means of this substance, 
 waves, excited in the luminous body, are transmitted 
 to the eye. While some of the phenomena of light 
 can be explained by either theory, experiments by 
 Huyghens, Young, and Fresnel have rendered the 
 emission theory untenable. While we are forced to 
 the belief that light is propagated in waves, we are 
 ignorant as to the nature of these waves. It was 
 formerly supposed that the ether was a highly elastic 
 body, transmitting vibrations of its particles through 
 space just as a rod of steel, if struck near one end, 
 will convey the vibrations to the other end. 
 
 To explain the phenomena of light it is necessary 
 to suppose the vibrations transverse to the direction 
 
4 HANDBOOK OF OPTICS 
 
 of propagation, as is the case in the illustration 
 cited. The waves of sound travelling through air 
 differ from these in that in the case of the sound 
 waves the particles of air vibrate to and fro in the 
 direction of propagation. 
 
 The modern study of electricity has changed our 
 conception of ether waves ; for according to the 
 electro-magnetic theory of waves, each particle of 
 ether is "polarized" or charged with energy, which 
 in turn is transmitted to the next particle, and so on. 
 These waves of energy produce various effects de- 
 pending upon the rapidity of vibration. Those of 
 least rapidity are manifested as electricity, and next 
 in order come heat-producing waves. As the rapid- 
 ity increases we have light of different colors, red 
 being the color of least rapidity of vibration. Be- 
 low this color in the spectrum are found heat 
 waves. Violet is the color of greatest rapidity, 
 and beyond this no light will be seen, but certain 
 chemical effects produced there indicate the presence 
 of waves capable of causing chemical action. It is 
 also thought probable that gravitation is exercised 
 by the transmission of energy through ether waves. 
 For the further study of the wave theory, which is 
 necessary for the proper understanding of interfer- 
 ence phenomena and of polarized light, the student 
 is referred to complete treatises on the theory of 
 light. 
 
INTRODUCTION 
 
 The investigations in the following chapters are 
 based upon the experimental laws of reflexion and 
 refraction, and upon the supposition that light 
 travels in a homogeneous medium in straight lines. 
 
CHAPTER I 
 
 EEFKACTION AT PLANE SURFACES 
 
 A luminous body emits light in all directions. 
 That portion of light which travels along a partic- 
 ular line is called a ray. A collection of rays which 
 do not deviate far from a central fixed ray is called 
 a pencil. When a ray of light passing through a 
 medium meets another medium of different den- 
 sity, it is divided into two portions; a part of 
 the light is reflected back into the first medium, 
 while the remaining portion passes into the second 
 medium, and is generally altered in direction. 
 Of the reflected light, a part is said to be regu- 
 larly reflected and a part scattered. Strictly speak- 
 ing, all reflexion is regular; but owing to the un- 
 evenness of the surface, the light is reflected in 
 various directions. It is by this means that we see 
 a non-luminous object ; for if the light were all 
 regularly reflected, we should see only the image 
 of the illuminating source. The more even the sur- 
 face, the greater is the regularly reflected and the 
 less the scattered light. If the substance is opaque, 
 no light passes into it ; the incident light is either 
 
REFRACTION AT PLANE SURFACES ^ 
 
 reflected or absorbed. When a ray passes from one 
 medium to another, the two portions of the ray 
 before and after entering the new medium are 
 called, respectively, the incident and the refracted 
 ray ; and the acute angles which they make with the 
 normal to the surface are called, respectively, the 
 angle of incidence and the angle of refraction. In 
 
 Fig. 1. 
 
 Fig. 1 i20 is the incident ray, OR' is the refracted 
 ray, COR is the angle of incidence, and DOR' is 
 the angle of refraction. It is found by experiment 
 that the incident and refracted rays always lie on 
 opposite sides of the normal to the refracting sur- 
 face ; that the angles of incidence and refraction 
 always lie in the same plane ; and that the sine of 
 the angle of incidence always bears a fixed ratio to 
 the sine of the angle of refraction. This ratio, while 
 fixed for the same two media, varies with the nature 
 of the refracting substances. It is called the refrac- 
 
8 
 
 HANDBOOK OF OPTICS 
 
 tive index for the two media. This law of refrac- 
 tion is called Snail's law ; it is sometimes called 
 Descartes' law, since he first published it in its 
 present form. For many years prior to Snell's dis- 
 covery, investigators had constructed tables express- 
 ing the relation between angles of incidence and 
 refraction, but even Kepler was unable to deduce 
 from these the law governing this relation. This 
 
 Fro. 2. 
 
 law, discovered by experiment, acquires new interest 
 from its corroboration of the wave theory of light. 
 It was supposed — and modern experiments have 
 proved it to be true — that light travels with dif- 
 ferent velocities in media of different densities. Let 
 
REFRACTION AT PLANE SURFACES 9 
 
 AB, Fig. 2, be a small portion of the front of a 
 wave of light which proceeds from a distant point ; 
 then AB, being an indefinitely small arc of a circle, 
 is indistinguishable from a straight line, and may- 
 be regarded as such. AA' represents the surface 
 of refraction ; v the velocity of light in the first 
 medium ; v' the velocity in the second medium. 
 Then if t is the time required by the light to trav- 
 erse the distance BA', we have BA' = v ■ t, which is 
 also equal to AB. If the wave had been unob- 
 structed by the second medium, it would occupy the 
 position A'NB at the end of the time t. But the 
 portion of the wave front at A, meeting the more 
 dense refracting substance, does not travel so fast as 
 the portion BA'. The point A becomes the centre 
 of a wave disturbance, which in the time t has 
 reached the point O ; and, consequently, AC=:v' • t. 
 Similarly, the portion of the wave at P travels in the 
 second medium the distance BM', while it would 
 have travelled in the first medium the distance PIf. 
 Therefore, AO : AB = v' : v, and FM' ■.PN=v':v; 
 from which 
 
 AQ v' PM' AO AB A A' 
 
 - or, - 
 
 AB V PN ' ' PM' PN PA' 
 
 From this equation it follows that PA'M' and 
 AA' O are similar triangles, and consequently M 
 lies on the line A'G. Since in like manner any 
 
10 HANDBOOK OE OPTICS 
 
 other point of the wave front will at the end of" the 
 time t lie on the line A' (7, which is perpendicular to 
 AO, then A' O will represent the wave front after 
 refraction, and AC will represent the direction in 
 which the refracted light travels. 
 
 If * be the angle of incidence of the ray, then 
 BA'A = 90 - i, and AA'D = i. If r be the angle of 
 refraction, then A'AC= 90 — r, and AA' C= r. 
 
 From the triangle A' AD we have 
 
 AD 
 
 
 
 
 sin I = 
 
 A A' 
 
 and from 
 
 AA'O 
 
 we have 
 
 
 
 
 
 sin?' = 
 
 AC 
 AA' 
 
 from which 
 
 sini 
 sinr 
 
 AD 
 
 AC 
 
 V 
 
 ''v'~ 
 
 n. 
 
 From this we see that the constant ratio between 
 the sines of the angles of incidence and refraction is 
 that of the velocity of light in the first medium to 
 the velocity in the second medium. We learn from 
 experiment that the deviation is toward the normal 
 to the surface when the ray passes from a rarer to a 
 denser medium, and away from the normal when the 
 ray passes from a denser to a rarer medium. If the 
 deviation is toward the normal, then i is greater 
 than r, and consequently v is greater than v' . Thus, 
 
REFRACTION AT PLANE SURFACES 
 
 11 
 
 according to the wave theory, the velocity of light 
 must be greater in air than in a dense medium, 
 such as water or glass. On the other hand, accord- 
 ing to the emission theory, the velocity must be 
 greater in a dense medium than in a rare one.* But 
 experiments have proved that the velocity is less in 
 dense than in rare media ; and the evidence is in 
 favor of the wave theory. 
 
 Fjg. 3. 
 
 When light passes from a rarer to a denser me- 
 dium, n is greater than unity, and since sinr = ^^^ — , 
 
 sin r is never greater than unity whatever may be the 
 angle i ; but when light passes from a denser to a 
 rarer medium, n is less than unity, and, for certain 
 values of i, sin r may become greater than unity. As 
 
 * Preston's "Theory of Light," 2d ed., p. 17. 
 
12 HANDBOOK OF OPTICS 
 
 the sine of an angle cannot be greater than unity, 
 this would be an impossible value for r. It is found 
 by experiment that when i has such value as to make 
 sin r greater than unity, light does not pass out of 
 the denser medium, but is reflected back into this 
 medium. This is called the total internal reflexion, 
 and the angle of incidence which makes sin r equal 
 to unity is called the critical angle. A glance at 
 Fig. 3 will show the meaning of this. It is also 
 found, as we might expect from this phenomenon, 
 that as the angle of incidence increases the propor- 
 tion of reflected light increases, while that of re- 
 fracted light diminishes. Advantage is taken of 
 the phenomenon of total internal reflexion in the 
 construction of certain optical instruments. 
 
 Another experimental fact is the reversibility of 
 the path of light, that is, if the direction of a ray be 
 reversed so that the angle of refraction becomes the 
 angle of incidence, then the original angle of inci- 
 dence will become the new angle of refraction. This 
 being so, it is evident that a ray after refraction 
 through a medium with parallel surfaces will, upon 
 reentering the original medium, be parallel to its 
 direction before refraction. It will, however, un- 
 dergo a lateral displacement varying with the thick- 
 ness of the medium. See Fig. 4. 
 
 We have learned that = w, where n is the 
 
 sinr 
 
EEFEACTION AT PLANE SURFACES 
 
 13 
 
 ratio of the velocity of light in the first medium to 
 its velocity in the second medium. If the velocity 
 in a vacuum is taken as the standard, the ratio of 
 the velocity in any medium to the velocity in a 
 vacuum is called the absolute refractive index. The 
 ratio of the velocities for any two media may there- 
 fore be expressed in terms of the absolute indices of 
 the two media. Thus, if n be the absolute index 
 
 Fig. 4. 
 
 of the first medium and n' that of the second, the 
 
 relative index for the two media will be 
 
 n' 
 
 Snell's 
 
 law thus becomes n • sin i=n' ■ sin r, and it is usually 
 written in this form. 
 
J4 HANDBOOK OF OPTICS 
 
 In any refraction the greater the angle of inci- 
 dence, the greater will be the deviation ; and the 
 greater the angle of incidence, the greater will be 
 the increase in deviation for a fixed increase in the 
 angle of incidence. This follows directly from the 
 
 equation =n. Reference to Appendix I. will 
 
 sinr 
 
 render this clear, for it is there shown that the sine 
 of an angle increases less rapidly than the angle ; 
 and the greater the angle, the less will be the change 
 effected in its sine by a fixed increase of the angle. 
 Hence, if r be smaller than i, a smaller increase in 
 r will be required to maintain the constant ratio 
 between sin i and sin r than in the greater angle i; 
 and this is true in a greater degree as i approaches 
 90 degrees. Since the deviation is expressed by 
 i — r, it follows that this increases when i increases. 
 As the path of light is reversible, the same holds 
 true when i is less than r, that is, when the ray 
 passes from a denser to a rarer medium. 
 
 A medium bounded by two plane faces meeting in 
 an edge is called a prism. At present we shall con- 
 sider only the refraction of rays through prisms, 
 reserving for a future chapter the more difficult sub- 
 ject of refraction of pencils of light. We shall sup- 
 pose the rays to lie in a principal plane of the prism, 
 that is, in a plane which is perpendicular to the edge 
 of the prism, and consequently to the plane of each 
 
REFRACTION AT PLANE SURFACES 
 
 15 
 
 face of the prism. We shall also confine our atten- 
 tion to prisms whose refractive index is greater than 
 that of air. When a ray of light passes through 
 such a prism, the deviation is in all cases from the 
 apex toward the thicker part of the prism. We 
 have seen that a ray, passing through a plane 
 
 Fig. 5. 
 
 bounded by parallel surfaces, emerges without devia- 
 tion. Let PQRS (Fig. 5) represent a ray passing 
 through the medium ABCB, AB and CB being 
 parallel. BN is normal to the face OB. Now sup- 
 pose the face CB be turned into the position AB' , 
 making the prism BAB' ; then the normal iJiV" must 
 turn into the position RN'; and by this change the 
 angle of incidence is increased from NRQ to N'RQ. 
 
16 
 
 HANDBOOK OF OPTICS 
 
 We have learned that with an increase of the angle 
 of incidence there is also an increase of deviation. 
 KRS represents the deviation at the second surface 
 when the face has the position GB, and as the devia- 
 tion is increased by turning the face into the position 
 AD\ then KRS' , greater than KRS, will represent 
 the deviation at the second face of the prism BAB'. 
 RS is parallel to PQ, the direction of the ray be- 
 
 FiG. 6. 
 
 fore entering the prism ; hence 8RS' represents 
 the deviation of the ray in its passage through the 
 prism, and this deviation is away from the apex of 
 the prism. 
 
 Light from an object at P (Fig. 6) would be so 
 deviated as to enter an observer's eye at E, and the 
 object would appear to be at P' . Hence an object 
 seen through a prism is displaced toward the apex 
 of the prism. 
 
 All light is not equally deviated by prisms. If a 
 
EEFEACTION AT PLANE SUEFACES 17 
 
 narrow beam of sunlight be passed through a prism 
 in a darkened room, and the refracted light be inter- 
 cepted by a screen, it will be found that the beam 
 has been decomposed into bands of colored light. 
 These are violet, indigo, blue, green, yellow, orange, 
 and red. Of these, violet is most and red least 
 deviated. They are called the colors of the spec- 
 trum. Since the deviation of a ray by refraction is 
 explained by the theory that the passage of light is 
 retarded upon entrance into a substance of greater 
 density, it is necessary to suppose that this retard- 
 ing power is different for different colors; violet, 
 which is most deviated by the prism, must suffer the 
 greatest retardation, and red, the color of least 
 deviation, must be least retarded by the prism. It 
 is supposed that only dense substances oifer this un- 
 equal resistance to the passage of light of different 
 colors, and that in space and in air all light travels 
 with the same velocity.* 
 
 The property which prisms possess of separating 
 colors is called dispersion. It is a most important 
 property, but only incidentally concerns the student 
 of ophthalmology. The chief use of prisms in oph- 
 thalmological practice is for the purpose of changing 
 the apparent position of objects. In the weaker 
 prisms dispersion is not noticeable ; but if an opaque 
 
 * Preston's "Theory of Light," 2d ed., p. 97. 
 
18 
 
 HANDBOOK OF OPTICS 
 
 object be viewed through a prism of considerable 
 deviating power, it will be tinged with red toward 
 the apex and with violet toward the base of the 
 prism. 
 
 Let PQRS (Fig. 7) represent a ray passing 
 through the prism whose apex is at 0, and whose 
 faces are inclined at an angle a. This angle is 
 called the refracting angle of the prism. At Q and 
 
 Fig. 7. 
 
 R draw the normals i^ilf and LRN. Let the angle 
 of incidence PQM be called i, and the angle of 
 refraction LQR be called r. Also let LRQ be r\ 
 and NRS be e ; then from the law of refraction we 
 have sin« = W'sinr, and sine = w • sin r'. The 
 angle ORQ is equal to 90 — r' and OQR is equal to 
 90 — r. Since the sum of the three angles of the 
 triangle ROQ must be equal to 180 degrees, we 
 have 
 
 a + 90 - r + 90 - r' = 180 ; or, r + r' = a. 
 
 The deviation of the ray at the first surface is 
 
REFRACTION AT PLANE SURFACES 19 
 
 represented by i — r, and at the second surface by 
 e —r'. The total deviation is denoted by i + e — 
 (r + r'), or t + e — a. 
 
 Hence the deviation produced by a prism is equal 
 to the sum of the angles of incidence and emer- 
 gence, minus the refracting angle of the prism. 
 
 Let us suppose that the ray passes symmetrically 
 through the prism, that is, that the angles of inci- 
 dence and emergence are equal. The angle of 
 incidence i is greater than the angle of refraction r, 
 since the index of the prism is greater than that 
 of air. Hence, as was shown on page 14, when i 
 increases, r also increases, but less rapidly than i. 
 Since in the triangle LM Q the angle L remains con- 
 stant, then when r increases, r' must undergo a cor- 
 responding decrease, for the sum of the three angles 
 of the triangle must be 180 degrees. From the 
 equation sin e = n • sin r', it follows that if r' de- 
 crease, e must decrease more rapidly than r' . There- 
 fore, if we start with the ray which passes sym- 
 metrically through the prism, and increase the angle 
 of incidence, the effect will be to increase the devia- 
 tion at the first face of the prism and to diminish it 
 at the second face ; but as r is now greater than ?•', 
 the increase at the first face outbalances the de- 
 crease at the second face, and the total deviation is 
 increased. If we trace this ray backward, we see 
 the effect of making the angle of incidence smaller 
 
20 HANDBOOK OF OPTICS 
 
 than that of the symmetrical ray, that is, in this case 
 also the deviation is increased. Hence the symmet- 
 rical ray is the ray which undergoes the least devia- 
 tion ; it is called the ray of minimum deviation. 
 
 If 1) denote the deviation of a ray in passing 
 through a prism, D = i + e — a, from which e = a + 
 D-i. 
 
 We have seen also that a = r + r', from which 
 r' = a — r. 
 
 Substituting these values of e and r' in the equa- 
 tion 
 
 sin e 
 
 -. — j = n, 
 
 sin r 
 
 , sin(a + D — i) 
 
 we have t^— — ^ = n, 
 
 sin (a — r) 
 
 or sin \_(a + B} —i'] =n ■ sin (a — r), or sin (a -|- D) 
 cos i — cos(a + -D) sin i=n (sin a cos r — cos a sin r). 
 When the angle of the prism is small, sin (a + IT) 
 and sin a do not differ materially from the angles 
 a + D and a. * Likewise, it is easily seen that the 
 cosines of these small angles do not differ materially 
 from unity. Making these substitutions, we have 
 
 * The measurement of an angle is expressed by the subtending 
 arc divided by the radius of this arc ; it is readily seen that this is 
 practically equivalent to the sine of the angle when the angle is very 
 small. 
 
REFRACTION AT PLANE SURFACES 21 
 
 (a + D) COS i — &uii =-n • a cos r — n ■ sinr; 
 or, 
 
 D • cos i = aQn ■ cos r — cos i), since sin i = n ■ sin »•. 
 
 Hence D = a 
 
 n • cos r 
 cos I 
 
 When tlie ray passes nearly perpendicularly through 
 the prism, as does the ray of minimum deviation 
 in a prism of slight deflecting power, then cos r and 
 cost are both very nearly equal to unity. In this 
 case the deviation is approximately equal to a(n—V). 
 If the index of refraction of the mateiial of which 
 the prism is made is 1.5, as is approximately true 
 of spectacle glass, the deviation becomes equal to -. 
 
 The exact index for glass is greater than 1.5; its 
 average index may be regarded as 1.53, and there- 
 fore the deviation even in weak prisms is more than 
 one-half the refracting angle of the prism, but for 
 practical, purposes the two may be considered equal. 
 In prisms of high deviating power the deviation 
 is perceptibly greater than one-half the refracting 
 angle. 
 
 Prisms in the oculists' trial case are usually num- 
 bered in degrees of the refracting angle, or accord- 
 ing to the deviating power in the position of minimum 
 deviation, the latter method having been first sug- 
 
22 HANDBOOK OF OPTICS 
 
 gested by Dr. Edward Jackson,* of Philadelphia, 
 as being more scientific than the old notation in 
 degrees of the refracting angle. Other systems of 
 numbering prisms have also been advocated and are 
 to some extent used. The units in these systems 
 are : the centrad, introduced by Dennett ; f the 
 prism-dioptre, introduced by Prentice ; | and the 
 metre-angle, introduced as a measure of convergence 
 by Nagel, and as a prism unit by Maddox.§ 
 
 It is sometimes desirable to know the result of 
 combining two prisms whose edges are not parallel. 
 We know that prisms deviate light in a direction 
 at right angles to the edge of the prism. Hence 
 when two prisms are combined so that their edges 
 are not parallel, the deviating power of the second 
 prism must be added to that of the first ; but the 
 direction in which the power is exerted is not the 
 same in the two prisms. 
 
 In Fig. 8, let AB and AD represent the direc- 
 tions in which light is deviated by the first and 
 second prisms respectively ; also let the length AB 
 represent the displacement of a ray of light which 
 the first prism produces, as measured on a screen 
 at a fixed distance from the prism, and let AD 
 
 * Trans. American Ophth. Soc, 1887 and 1888. 
 
 t Ibid., 1889. 
 
 t Archives of Ophthalmology, 1890. 
 
 § Wood's "Med. and Surg. Monographs," Vol. IX., No. 2. 
 
EEFRACTION AT PLANE SURFACES 23 
 
 represent the displacement which the second prism 
 produces at the same distance. Then to find the 
 combined effect of the two prisms, it is only neces- 
 sary to construct the parallelogram ABCB, and 
 the diagonal AO will represent the direction and 
 the length of displacement produced by the two 
 prisms acting together ; for the problem is the same 
 
 
 ,^' 
 
 / 
 
 I 
 
 Fig. 8. 
 
 as that in which an object at A is acted upon by a 
 force which would move it from A to _B, and at the 
 same time by a force which would move it from A 
 to jD, the result being that the object is moved from 
 ^to C. 
 
 It would, however, be inconvenient to find the 
 linear displacements AB and AD ; and to avoid this 
 we must use a relation which exists between the 
 angular deviation and the linear displacement. 
 
 Let d represent the angle of deviation of the first 
 prism, and d' that of the second prism ; then, since 
 
24 HANDBOOK OF OPTICS 
 
 an angle is measured by its subtending arc divided 
 by the radius of this arc, we have 
 
 J arc 1 ji arc' 
 a = — and a' = . 
 
 If the distance of the screen from the prism be the 
 radius, then the arc described with this radius and 
 terminating in A and B will be the subtending arc 
 of the angle d. When d is small, the arc and the 
 straight line AB will be so nearly equal that we may 
 consider them identical. Similarly the arc of the 
 angle d', when this angle is small, may be replaced 
 by the line AB. Thus for small angles we may with 
 any unit lay off AB so that it contains as many units 
 as the arc of the angle d contains degrees, and with 
 the same unit lay ofi AD so that it contains as many 
 units as the arc of the angle d' contains degrees ; 
 then AO as measured by the same unit will give the 
 number of degrees of deviation produced by the 
 equivalent prism, and AO will represent the direc- 
 tion of deviation of this prism. The angle BAB is 
 the angle of inclination of the two prisms, for BA 
 and AD are perpendicular to the edges of the first 
 and second prisms respectively. The angle ABC is 
 equal to 180 — DAB. Hence in the triangle J. J5C 
 we know the two sides AB and BG and the included 
 angle ABO. We can find the third side ^(7 and the 
 
REFRACTION AT PLANE SURFACES 25 
 
 angle CAB, which the equivalent prism must make 
 with the first prism. 
 
 The side AC is obtained from the equation 
 
 iAOy = iABy -{-liBOy - 2 AB X BO COB ABC. 
 
 Having found A (7, the angle CAB is found from the 
 equation 
 
 sin CAB ^BC 
 
 sin ABO AC' 
 
 since in any triangle the sines of the angles are pro- 
 portional to the opposite sides. If the two prisms 
 are at right angles, then 
 
 iAOy = (ABy + (B (7)2, and tan CAB = ^. 
 
 AB 
 
CHAPTER II 
 
 KEFRACTION AT SPHERICAL SURFACES 
 
 Let (Fig. 9) be the centre of a spherical surface, 
 separating two media whose refractive indices are 
 respectively n and n' . 
 
 A ray of light, PR, meets the surface at B, and is 
 refracted so as to assume the direction RQ. NRP 
 is the angle of incidence, and ORQ is the angle of 
 refraction. Then we have n • sin NRP = n'- sin ORQ. 
 
 From the triangles ORP and ORQ we have 
 
 sin (im-NRP) _ sin NRP ^ OP 
 sin RO A sin ROA RP 
 
 sin ORQ _ sin ORQ OQ 
 
 sinQiHO-ROA) sinROA RQ 
 Dividing (1) by (2), we have 
 
 (1) 
 (2) 
 
 sin JSTRP ^ OP RQ ,„. 
 
 sin ORQ RP ^ OQ ^ ^ 
 
 The first member of (3) is equal to — 
 
 Therefore n • -—— = n' • ——• 
 
 RP RQ 
 
 26 
 
REFRACTION AT SPHERICAL SURFACES 27 
 
 The ray FA is normal to the spherical surface at 
 A ; hence it will undergo no deviation, but will con- 
 tinue in the same straight line, PAQ. 
 
 This line is called the axis of the refracting sur- 
 face. The ray which meets the surface at R meets 
 the axis at Q. If every other ray of the pencil which 
 is refracted at the spherical surface should meet the 
 axis at Q, this would be the focus of the pencil after 
 refraction. A study of Fig. 9, however, will show 
 
 that this will not be so. If the arc AB be revolved 
 around the axis, a portion of a sphere will be gener- 
 ated, and all rays meeting this surface at the distance 
 AB from the axis will meet at Q. If we take the rays 
 which meet the spherical surface at a distance AW, 
 they will not meet the axis at Q. The relations 
 between BOA and the angles of incidence and re- 
 fraction are such that as BOA increases, OQ must 
 diminish; and consequently those rays which meet 
 the refracting surface more remotely from A inter-. 
 
28 HANDBOOK OP OPTICS 
 
 sect tlie axis at a less distance from A than do those 
 rays which lie near the axis.* 
 
 The rays RQ and E' Q' meet at q. The curve 
 formed by the locus of all such intersecting points is 
 called the caustic of the refracting surface. An illus- 
 tration of a caustic can be seen by placing a glass of 
 water on a table so that sunlight strikes the glass. 
 The brightly illuminated curve which will be noticed 
 represents the intersection of rays after refraction by 
 the water. The property of spherical refracting sur- 
 faces, by which rays of a pencil proceeding from a 
 point meet the axis at different points after refrac- 
 tion, is called spherical aberration. The nearer that 
 p 7? 
 
 -— — approaches a constant quantity, the less is the 
 jRQ 
 
 spherical aberration. It is easily seen that this condi- 
 tion is best fulfilled when li is near the axis, and when 
 the curvature of the surface is slight as compared with 
 the distances PA and AQ. The greater the lengths 
 PA and AQ, the farther may H be removed from the 
 axis without appreciable spherical aberration. Spec- 
 tacle lenses, being lenses of long focal length, do not 
 cause much aberration. In the eye the focal distances 
 are short ; but, owing to the cutting off by the iris 
 of all peripheral rays, and to the peculiar construction 
 of the crystalline lens, aberration is not appreciable. 
 Since the more peripheral rays are too strongly 
 
 * Heath's " Geometrical Optics," 2d ed., p. 144. 
 
REFRACTION AT SPHERICAL SURFACES 29 
 
 refracted in comparison with those near the axis, it 
 is evident that the curvature of the spherical surface 
 is too great for peripheral rays. A surface with 
 diminishing curvature, such as an ellipsoid, would 
 consequently produce less aberration than the spheri- 
 cal surface. The curve which produces no aberra- 
 tion is known to mathematicians as the Cartesian 
 oval. Such surfaces, however, are not in practical 
 use ; for, by suitable combinations of lenses, aber- 
 ration can be almost entirely overcome, even in the 
 lenses of very short focal distance and wide aperture 
 used in microscopes. 
 
 Besides spherical aberration, we have also chro- 
 matic aberration, which is due to the unequal deviating 
 power of the refracting surface for different colors. 
 In the construction of microscopes and other delicate 
 optical instruments, the annulment of this defect is 
 a matter of the utmost importance, but owing to the 
 long focal length of spectacle lenses and to the pre- 
 vention by the iris of all peripheral rays from enter- 
 ing the eye, chromatic aberration does not attract 
 attention in ordinary vision. 
 
 All our formulae will be based upon the assump- 
 tion that PA, BF, AQ, and RQ are great in com- 
 parison with the curvature of the surface, and that 
 only rays near the axis are allowed to pass into the 
 refracting medium. With this understanding, our 
 equation. 
 
30 HANDBOOK OF OPTICS 
 
 OP , OQ 
 
 becomes 
 
 (PA + AO) ^ . {AQ-AO^ 
 PA AQ 
 
 OP 
 
 For, upon our assumption, — — will not differ 
 
 JL Ja» 
 
 materially from — — , and — ^ will not differ mate- 
 
 rially from --^• 
 ■^ AQ 
 
 If PA be denoted by /, and AQ by/', and V be 
 
 the radius of the spherical surface, the equation will 
 
 become 
 
 / /' 
 
 This equation may be reduced to the form, 
 
 This is the relation between a point P and its 
 focus Q. P and Q are called conjugate foci, and / 
 and /' are conjugate focal distances. A pencil of 
 light from P will be brought to a focus at Q, and, 
 conversely, a pencil proceeding from Q will be 
 brought to a focus at P.* In our demonstration we 
 
 * When Q Is virtual, we must modify this clause so as to read : 
 "A pencil directed towards Q will be focused at P." 
 
REFRACTION AT SPHERICAL SURFACES 31 
 
 have considered all the quantities as positive. This 
 is the convention of signs most convenient in dealing 
 with lenses ; but in working out formulse for several 
 refracting surfaces, it would lead to confusion. In 
 these cases it is better to consider all quantities posi- 
 tive when measured from left to right, and negative 
 when measured from right to left. Thus, AP, 
 measured from A, would be negative ; while AQ^ as 
 in the former convention, would be positive. We 
 should have to replace / in our equation (a) by — /. 
 Doing this, we have 
 
 / /' 
 
 (&) 
 
 We shall use (a) in demonstrating the properties 
 of single lenses, but (6) will be more suitable in 
 tracing the path of light through the several re- 
 fracting media of the eye. 
 
 n n' __n' —n 
 
 is the equation between conjugate foci for all posi- 
 tions of / and /'. If, in this equation,, we make / 
 infinite, we have 
 
 ■n 
 
 This means that if the rays proceed from a point at 
 an infinite distance, that is, if the rays are parallel 
 
32 HANDBOOK OF OPTICS 
 
 to the axis in the first medium, the value for /' is 
 ^ *" * This is called the posterior or second princi- 
 
 K — H 
 
 pal focal distance, and the point where the rays meet 
 the axis is called the posterior or second principal 
 focus. Similarly, if /' is infinite, that is, if rays pro- 
 ceeding from a point, P, are parallel after refraction, 
 
 we have, as the corresponding value of /, — 
 
 n —n 
 
 The point P thus becomes the first or anterior prin- 
 cipal focus, and the distance — is the first princi- 
 
 n' —n 
 
 pal focal distance. The anterior focal distance is 
 denoted by the letter F^ and the posterior focal 
 distance is denoted by I" . 
 
 If we divide equation (a) by , we have 
 
 nr n'r _ ., 
 
 (n' — 71) f (n' — 71) f 
 
 This equation gives us the relation between two 
 conjugate points and the two principal foci. 
 
 F and / are positive when to the left of the refract- 
 ing surface, and F' and /' are positive when to the 
 
 * In the small pencils which enter the eye, the rays may be con- 
 sidered parallel when they proceed from a point situated 6 metres 
 from the eye ; hence / is infinite when it has not less than this 
 length. 
 
REFRACTION AT SPHERICAL SURFACES 33 
 
 right of this surface. When n' is greater than w, F 
 and F' are both positive or both negative according 
 as r is positive or negative ; and when n' is less than 
 w, F and F' are both positive or both negative ac- 
 cording as r is negative or positive. In other words, 
 F and F' are both positive when the convex surface 
 is turned toward the medium of less refractive in- 
 dex, and both negative when the convex surface is 
 turned toward the medium of greater index. 
 
 We have seen that rays which are parallel to the 
 axis in the first medium will, after refraction into 
 a denser medium at a convex surface, meet in an 
 actual focus in the second medium ; and we have 
 also seen that F' in this case is positive. On the 
 other hand, if the refracting surface be concave, the 
 rays will not meet in a point in the second medium ; 
 they will be rendered divergent by the refraction, 
 and their direction will be such that, if prolonged 
 backward, they will meet in a point in the first 
 medium. Such a focus is only imaginary, not a real 
 meeting point of the rays, as in the former case ; it 
 is called a virtual focus. It is, as we have seen, nega- 
 tive. Hence, with the convention of signs which we 
 are using, real foci are positive, and virtual foci are 
 negative. Figure 10 illustrates the virtual focus. 
 
 A study of equation (c?) will show that if n' be 
 greater than n, and r be positive, then, so long as/ is 
 greater than F, f will be positive ; and if / be less 
 
34 HANDBOOK OF OPTICS 
 
 than F, f will be negative. This indicates that if 
 rays proceeding from a point on the axis are re- 
 fracted into a denser medium at a convex surface, 
 they will after refraction meet the axis in a real 
 point so long as the point from which they proceed 
 is farther from the surface than the first principal 
 focus. If the point is within this focus they will 
 
 \ 
 
 Fig. 10. 
 
 not meet the axis after refraction, but if prolonged 
 backward they will meet in a virtual focus, and this 
 will be farther from the surface than the point from 
 which the light proceeds. 
 
 The following equations will also be found useful 
 in our studies : 
 
 Since F=^!^ and ^'=4^, 
 
 we have F : F' =n : n' ; or, —^ = —^. 
 
 Also, if P (Fig. 11) be the position of a point 
 and Q its conjugate, F and F' the principal foci, then 
 
REFllACTION AT Sl'HEHICAL SURFACES 35 
 
 FF = f-F&nd QF'=f' - F'. Let PF be denoted 
 by u and QF' by u'; then/= u + F and/' = m' + F' . 
 
 Hence, 
 
 F 
 
 F 
 
 . u + F u' + F' 
 From which we deduce 
 
 uu'=FF'. 
 
 Fig. 11. 
 
 Let us next take a point P not on the axis AA'. 
 Draw POF' (Fig. 12) through the centre, 0, of the 
 refracting surface. This line is the new axis and F' 
 is the focus conjugate to F. If OF be equal to OA, 
 
 then OF' and OA' will be equal, and FA and F'A' 
 will be arcs of circles. As RRB' is revolved on the 
 axis, producing the spherical surface, so the arcs FA 
 and F'A' also produce portions of spherical surfaces. 
 
36 HANDBOOK OF OPTICS 
 
 If the radii OP and OP' be great as compared with 
 the length of these arcs, the portions of spherical 
 surface which they generate will not differ materially 
 from plane surfaces. 
 
 Therefore the planes at A and A' perpendicular to 
 the axis AA' are called conjugate focal planes. The 
 image of an object lying in the plane at A will lie in 
 the plane at ^'. A plane tangent to the refracting 
 surface at H is called the principal plane of the 
 refracting 'surface. If the lines AR and PR' were 
 terminated in such a plane, and from their points of 
 intersection with the plane lines were drawn to A' 
 and P', respectively, the resulting lines would very 
 nearly coincide with those as drawn in Fig. 12 ; 
 and, for practical purposes, the two results might be 
 considered identical. 
 
 The point H, where the principal plane and axis 
 intersect, is called the principal point. 
 
 We shall now show how we may construct the 
 image of an object if we know the position of certain 
 points. Let PA (Fig. 13) be the linear dimension 
 of an object. If we can determine the point of inter- 
 section after refraction of two rays from P, this point 
 will evidently be the image of P. First we take a 
 ray PFR', which passes through the anterior focus 
 F ; then we know that this ray must after refraction 
 be parallel to the axis AA'. It is represented by 
 R'P' . Next we take a ray PR., which before refrac- 
 
REFRACTION AT SPHERICAL SURFACES 
 
 37 
 
 tion is parallel to the axis ; then, after refraction it 
 must pass through the second principal focus F' . It 
 is represented by RP' . Then P', the point of inter- 
 section of the two rays from P, is the image of P. 
 The image of all points in the line AP must lie in 
 the line A'P'; hence, A'P' is the image of AP. 
 
 We can now appreciate the importance of these 
 points and planes. They are called the cardinal 
 
 
 R 
 
 
 , 
 
 
 ^~~^~-^__F 
 
 H 
 
 A' 
 
 ~~~~~^ 
 
 R' 
 
 — -^L- 
 
 
 \ 
 
 
 r 
 
 Fig. 13. 
 
 points and planes of the refracting surface. The 
 centre of the refracting surface also possesses a dis- 
 tinctive property, in that rays which pass through 
 it undergo no deviation, since such rays are normal 
 to the refracting surface. It is called the nodal 
 point. The cardinal points of a single refracting 
 surface are four ; namely, the principal point, the 
 two principal foci, and the nodal point. The cardinal 
 planes are the principal plane and the two principal 
 focal planes. 
 
 If be the linear dimension of the object, and i 
 
38 
 
 HANDBOOK OF OPTICS 
 
 that of the image, then from the similar triangles 
 PAF and EFR' (Fig. 13) we shall have 
 
 PA _ AF _u £__^* 
 
 HB/~ FH' """"'^rF' ""^'T F' 
 
 Also from the triangles RSF' and P'A'F', 
 
 F' 
 
 F' 
 
 — : = — ; or, -. = -—• 
 — I w I w 
 
 From either of these equations we can determine 
 the size of the image. We may also determine the 
 size of the image in terms of the divergence of the 
 
 N _ 
 
 M 
 
 Fig. 14. 
 
 rays before and after refraction. Let MP (Fig. 14) 
 be the linear dimension of an object. A ray from 
 P, meeting the refracting surface at R, would be 
 refracted so as to assume the direction RS; and if 
 prolonged backward, it would meet the axis at Q, 
 which is conjugate to P. Likewise, iVis conjugate 
 to M, and QN is the image of PM. The angle 
 
 * Since the image, when real, as in the figure, lies on the oppo- 
 site side of the axis to the object, we must consider it negative. 
 
EEFEACTION AT Sl^HERICAL SURFACES 39 
 
 MPA, which, we call a, expresses the divergence of 
 the pencil before refraction, and HQA or a^ is the 
 divergence after refraction. 
 
 Then tan a = — — ■, and tan a^ = — — - ; 
 
 J? 1.1 tan a QA f 
 
 from which = -^— - = ^. 
 
 tan aj PA f 
 
 We have also the equation 
 
 PO_PM 
 QO QN 
 
 From the law of refraction we have, as on page 26, 
 
 PO , QO 
 RP RQ 
 
 or, since RP may for rays near the axis be replaced 
 by/andi?^by/, 
 
 PO , QO PO n'f 
 n--— = n'-^, or ■—- = -- ; 
 / / QO nf 
 
 1 PM n'f J n . n' .-.^ 
 
 ^^^"^ -QN"'~r¥^' ~r"r ^^ 
 
 From this equation and from ^ = we deduce 
 
 / tan a^ 
 
 -n- tan a = i-n' • tan asj. (2) 
 
 This is known as Helmholtz' formula. 
 
CHAPTER III 
 
 REERACTION THROUGH LENSES 
 
 Having investigated the refraction of light at one 
 spherical surface, we are now prepared to study re- 
 fraction through lenses. A lens is defined as a por- 
 tion of a refracting substance- bounded by two curved 
 surfaces centred on the same axis. If the radius of 
 curvature of one surface is infinite, then the corre- 
 sponding surface is plane, and the lens is bounded by 
 one curved and one plane surface. Ordinarily, the 
 surfaces of lenses are spherical ; but lenses have 
 been constructed whose surfaces were ellipsoidal or 
 paraboloid. The only lenses in practical use, how- 
 ever, are those whose generating curve is a circle ; 
 and we shall confine our attention entirely to lenses 
 of this nature. 
 
 The thickness of a lens is the distance between the 
 bounding surfaces as measured along the axis. 
 
 A lens bounded by two convex surfaces is called a 
 double convex or bi-convex lens ; one bounded by two 
 concave surfaces is called a bi-concave lens. 
 
 A lens, of which one surface is convex and one 
 concave, is called a convexo-concave lens, or a menis- 
 
 40 
 
REFRACTION THROUGH LENSES 41 
 
 cus. Lenses of this form are used as spectacles, and 
 are known as periscopic lenses. 
 
 The terms plano-convex and plano-concave need no 
 explanation. 
 
 In our demonstration we shall take as the typical 
 case the double convex lens ; and we shall consider 
 the refractive index of the material of which the lens 
 is composed to be greater than that of the air by 
 which it is surrounded. The index of the lens will 
 be indicated by the letter n, and that of the air by 
 unity. 
 
 We shall first show that there are two points on 
 the axis of the lens which are useful in the deter- 
 mination of the positions of conjugate foci. These 
 points are a pair of conjugate foci, such that any 
 incident ray directed toward one of them will, after 
 refraction, appear to come from the other, and in a 
 direction parE(,llel to that before refraction. These 
 points are called the nodal points of the lens. 
 
 To find the position of these points, we draw any 
 radius of the first surface, as OQ (Fig. 15). Next 
 we draw a radius, O'Q', of the second surface, so that 
 OQ and 0' Q' are parallel. Connect the points Q 
 and Q' by the straight line QQ', which meets tlie 
 axis at C. Then from the similar triangles OCQ 
 and O'C^', wehave 00: 0' = OQ : O'Q' =r:r', 
 
 from which -zrr-, = —:• 
 
 0' C r 
 
42 
 
 HANDBOOK OF OPTICS 
 
 OQ 
 
 Therefore, — -- is a constant quantity, irrespective 
 
 of the position of §, from which the first radius is 
 drawn ; and consequently Q must be a fixed point. 
 The ray of light, TiR\ which passes through C, is, 
 after refraction, parallel to its direction before re- 
 
 FiG. 15. 
 
 fraction ; because, the radii OQ and 0' Q' being par- 
 allel, the planes perpendicular to the curved surfaces 
 at Q and Q' are parallel, and the lens will, for this 
 ray, act as would a piece of plane glass. 
 
 Let N be the point on the axis toward which the 
 ray is directed before refraction. The ray is so re- 
 
REFRACTION THROUGH LENSES 43 
 
 fracted at the first surface as to pass through ; 
 then G is conjugate to N in the first refraction. 
 After refraction at the second surface, the ray ap- 
 pears to pass through N' ; then N' is conjugate to Q 
 in the second refraction. Since N is the virtual in- 
 tersecting point of the ray and axis before refraction, 
 and W is the virtual intersecting point of the ray 
 and axis after refraction, then N and N' are conju- 
 gate points with respect to both refractions. 
 
 The point Q is called the optical centre of the lens. 
 All rays which pass through it are, after refraction 
 through the lens, parallel to their direction before 
 refraction. 
 
 To find the position of the points N, If', and we 
 have the following relations : 
 
 00' = r + r'-e; 
 
 where r and r' are the radii of the surfaces, and e is 
 the thickness of the lens. Also, 
 
 00 _r , 
 
 O'C r'' 
 
 and consequently, 
 
 00 r 00 r 
 
 - or, - 
 
 00+ O'C r + r' r + r'-e r + r' 
 
 0(7= — ^(r + r' - e)= ?• - -^, 
 r + r r+ r 
 
44 HANDBOOK OF OPTICS 
 
 AO=OA-OQ=r-r + ^'^ ^'* 
 
 r + r' r + r' 
 Similarly, A' 0= ■ '"' 
 
 er' 
 
 But since in the first refraction If and are con- 
 jugate foci, we have from formula (a), page 30, 
 
 1 n _ w — 1 
 
 AN AC" r 
 
 Substituting the value of AO, this becomes 
 
 1 n(r + r') _n — 1 
 AN er r 
 
 If rays proceeding from F-^ are parallel to the axis 
 after refraction at the first surface, and these parallel 
 rays after refraction at the second surface meet at 
 I'a, then 
 
 AF^ = -^ = if\, and A'F^ = -^ = F^. 
 w — 1 n — 1 
 
 Making these substitutions, -we have 
 
 1 _ nCF, + JP,) 1 . 
 
 AN eF^ F^ ' 
 
 IT 
 
 or, AN= ^ V, X e. 
 
 n(:F, + F,}~e 
 
REFRACTION THROUGH LENSES 45 
 
 If e be replaced hj n • c, this equation will become 
 
 Since AJV is negative for tbe convex lens, iV" lies 
 to the right of A, and since in the lenses which are 
 in practical use, the numerator Fj^ is less than the 
 denominator n(^F^ + F^}— e, the length ^iV is less 
 than e, and the nodal point JV lies within the lens. 
 Similarly, we find that the nodal point iV' lies within 
 the convex lens, and that its distance from A' is 
 
 F,+F^-c 
 
 Using the same equation, but changing the sign 
 of r, we should find that in bi-concave lenses, also, 
 the nodal points lie within the lens. 
 
 To repeat, the nodal points are two conjugate 
 points such that a ray of light directed toward one 
 of them is so refracted as to pass through the optical 
 centre of the lens, and emerges in a direction parallel 
 to that before refraction. Such a ray undergoes only 
 a lateral displacement due to the thickness of the 
 lens. 
 
 We shall now demonstrate the method of finding 
 the focus Q, after refraction by a lens, of a pencil 
 of light from a point P on the axis of the lens. 
 The ray FH (Fig. 16) meets the first refracting 
 
46 
 
 HANDBOOK OF OPTICS 
 
 surface at -R ; it is refracted toward R', and, if 
 prolonged, it would meet the axis at Q'. But, after 
 travelling the distance HR', it meets the second 
 
 Fig. 16. 
 
 surface of the lens and is refracted to Q. Then in 
 the first refraction Q' is conjugate to P, and we 
 have from equation (a), page 30, 
 
 In 
 
 n-1 
 
 PA AQ< 
 
 or, substituting / for PJ., /' for AQ', and F^ for 
 r 
 
 (1) 
 
 1 w^ J_ 
 / ./' Fx 
 
 Also in the second refraction Q is conjugate to 
 Q' ; therefore, 
 
 n 1 _ w — 1 
 
 A^' A^^~V~' 
 
 or, substituting /j' for A'Q', /j for A' Q, and F^ for 
 
 n-1 
 
 ■, we have 
 
 i?L + l = i- 
 /i' /i < 
 
 (2) 
 
REFRACTION THROUGH LENSES 47 
 
 Reference to the figure will indicate that AA!^ 
 the thickness of the lens, is equal to the difference 
 in length between /' and /j' ; but we also notice 
 that /' and f-l have opposite signs, that when /' is 
 positive, f-l is negative. Hence the difference be- 
 tween /' and /j' will be expressed algebraically by 
 f +f-l ; and if e represent the thickness of the lens, 
 /' +/i' = e, , or nc, n being the refractive index of 
 the lens, and c being of such value that e is equal 
 to nc. From these equations we can determine the 
 relation between P and Q. From (1) we obtain 
 
 JL_1 
 
 F, f 
 
 From (2), // = -^ j- 
 
 Substituting these values in the equation /'-h/j' 
 = nc, we have 
 
 n n 
 
 This, by reduction, becomes 
 
 //iCJ'i + #2 - - ^1/1(^2 - 
 
 -FJiF^-e')=cF^F^-. (4) 
 
48 HANDBOOK OF OPTICS 
 
 Tliis equation is true for all values of/ and /j. If 
 we make / infinite, the corresponding value of/^ will 
 give us the focus for raj's parallel to the axis before 
 refraction. 
 
 Dividing equation (4) by / and making / infinite, 
 we derive 
 
 /i 
 
 F^+F^-c 
 
 Similarly, if /j be infinite, that is, if the rays be 
 parallel to the axis after refraction, we shall obtain 
 
 F,(F,-c-) 
 / F^ + F^-c 
 
 The points determined by these equations are the 
 principal focal points of the lens. They are not 
 usually measured from the surfaces of the lens, how- 
 ever, but from the points iVand N' . It will be seen 
 that as thus measured the two focal distances are 
 equal. 
 
 ■' + ^"■'^ - F, + F^- c^ F^ + F^- c F, + F,-c 
 
 f I A,^r,_F 2(F^-e^) . cF, _ F,F, 
 
 ^^^ F^ + F^-c'^ F^ + F^-c F^ + F^-G 
 
 Hence in a lens the two principal focal distances are 
 equal, and this distance is found from the equation 
 
 F= ^1^2 
 
 F^ + F^-c' 
 
REFRACTION THROUGH LENSES 49 
 
 where F-^ and F^ represent the same quantities as on 
 page 44. 
 
 The nodal points N and N' possess another im- 
 portant property in addition to that already demon- 
 strated. If planes be drawn through these points 
 perpendicular to the axis of the lens, an object in 
 the first plane will have its image in the second 
 plane, and the image and object will be of the same 
 size and on the same side of the axis ; in other words, 
 the line joining the points where the incident and 
 refracted rays meet, respectively, the first and second 
 planes, is parallel to the axis. The two planes are 
 called the principal planes, and the points where 
 they meet the axis are called the principal points. 
 In lenses, the principal points and nodal points coin- 
 cide ; but this is not so in all optical systems, as we 
 shall hereafter learn. 
 
 To prove this property of the principal planes, let 
 be the linear magnitude of an object, i its image after 
 one refraction, and i' its image after two refractions. 
 
 Then from equation (1), page 39, we have 
 
 X l_»*'*_n ^ • n _ i^ _ 
 
 ^T~ ~7~~ ' /i' /r ' 
 
 "_»*/ and --'^i. 
 
 XT ff' 
 
 Hence, .,=^. 
 
50 HANDBOOK OF OPTICS 
 
 But in demonstrating the properties of nodal 
 points we have seen that AQ : A!G =r -.r' . Since 
 in the first refraction C is conjugate to iV, then AN 
 is represented by / and AC by /'; and since in the 
 second refraction W is conjugate to C, then A' Q is 
 represented by/j' and A!N' by/^. Hence, 
 
 /i' ~ r'- 
 
 Applying formula (a), page 30, to the first refrac- 
 tion, we have 
 
 1 n _n-^\ f'+nf_n—l ^^^ 
 
 Similarly, applying this formula to the second 
 refraction, 
 
 Dividing (2) by (1), we derive 
 
 AfiV+nn r' //' 
 from which 
 
 or, #/ + nff^ =fj' + nff^, and /// =fj>. 
 
REFRACTION THROUGH LENSES 51 
 
 ff 
 Therefore the equation — = ^^ becomes o = i', 
 
 »' //i 
 
 which shows that object and image are equal when 
 
 they lie in the planes perpendicular to the axis at the 
 nodal points of the lens. The significance of the 
 principal planes is rendered clear by reference to 
 Fig. 17. A thorough understanding of the proper- 
 ties of these planes is necessary to the further study 
 
 Fig. 17. 
 
 of refraction. A ray of light BR' meets the first 
 surface at R'. It is directed toward P, but before 
 reaching P, it is refracted at R', so as to assume the 
 direction B'R" ; at R" it is again refracted so that 
 its direction on leaving the lens is R"R"'. If R"R"' 
 be prolonged backward, it will meet the plane P'N' 
 at P', and FN^ and P'N' will be equal. Any other 
 ray directed toward P will after refraction appear to 
 come from P' ; in other words, P' is the image of P- 
 It must not be supposed, however, that a real object 
 placed in the lens substance at PN would have for 
 
52 
 
 HANDBOOK OF OPTICS 
 
 its image P'W ; for such an object, as seen by an 
 eye, would have undergone refraction at one surface 
 of the lens only. PN and P'N' are both virtual. 
 
 By making use of the properties which belong to 
 the principal planes, we can construct the image of 
 an object after refraction through a lens. Let PA 
 (Fig. 18) be the linear dimension of an object, F 
 and F' the principal foci of the lens, H and H' the 
 
 F 
 
 R 
 
 
 R 
 
 A 
 
 ^~\.F H 
 
 
 h'^\f' 
 
 A' 
 
 
 
 ^\ 
 
 S 
 
 
 S' 
 
 P 
 
 Fig. 18. 
 
 principal points, RS and R'S' the principal planes. 
 From the point P draw a ray PFS ; then since it 
 passes through the anterior focus, it must after re- 
 fraction be parallel to the axis, and from the prop- 
 erty of principal planes just demonstrated HS and 
 H'S' must be equal. Hence S'P' represents the ray. 
 Next, take a ray PR from P, parallel to the axis ; 
 then it must after refraction pass through the pos- 
 terior focus F', and SR and SR' are equal. 
 
REFKACTION THROUGH LENSES 53 
 
 R'FP will represent the ray after refraction, and 
 P', the point of intersection of the two rays from P, 
 will be the image of P. Similarly, we can show that 
 any other point of PA has a corresponding image in 
 P'A', and P'A' is the image of PA. To find the 
 size of the image we use the similar triangles PAP 
 and PES, or P' A'P' and H'R'P'. When the image 
 is real, as in our figure, it lies on the opposite side of 
 the axis to PA, and it must be considered negative. 
 As on page 35, the distance of the object from the 
 first principal focus is denoted by the letter m, and 
 the distance of the image from the second focus is 
 denoted by u' . Hence, we have the following equa- 
 tions : 
 
 PA^AF_ nr -^= " 
 
 HS 
 
 Also, 
 
 P'A' 
 H'R' 
 
 PR' 
 
 or,- 
 
 - * 
 
 P 
 
 P'A! 
 H'F 
 
 ; or. 
 
 
 — i 
 
 P 
 
 u' 
 
 By referring to page 38, we see that these equa- 
 tions are the same as those which we found to deter- 
 mine the size of the image after one refraction ; but, 
 instead of one principal plane from which all dis- 
 tances are measured, we now have two such planes. 
 
 Having solved the problem for the double convex 
 lens, we may, without repeating the investigation, 
 apply the same formulae to other lenses by making 
 suitable changes in the signs of radii and foci. 
 
54 HANDBOOK OF OPTICS 
 
 The position of the principal or nodal points may 
 be found from the equations, 
 
 cF 
 
 AN (which we call h) = — -J , 
 
 ^ -^ F^+F^ — c 
 
 and A'N' ior A') = ^-^ 
 
 F, + F,-c 
 
 The focal distance of the lens is found from the 
 equation 
 
 F = ^^ 
 
 F^ + F^- c 
 
 If, in these equations, we replace the value of F^ 
 and F^ in terms of n and r, we obtain 
 
 7 ^' 
 
 ~ (r + r'^-Qn- l)c 
 
 (1) 
 
 ?' '''■' 
 
 (2) 
 
 (r + /) — (w— l)e 
 
 F- '■'■' 
 
 (3) 
 
 (n — 1) [r + r' — (w — l)c] 
 
 In spectacle lenses the thickness is so slight, as 
 compared with the radius of curvature, that it may 
 be ignored. Such lenses are spoken of as thin lenses. 
 In thin lenses the nodal points coincide with the 
 centre of the lens ; and the equation which deter- 
 mines F reduces to the form 
 
REFRACTION THROUGH LENSES 55 
 
 (w -!)(»• + >•')" 
 
 This equation serves to determine the focal length 
 in all kinds of spectacle lenses. If both r and r' 
 are positive and equal, -we have 
 
 2(w - 1) ' 
 
 and, if we reckon n as 1.5 for the glass of which 
 spectacles are made, we have F = r. If r' is infinite, 
 that is, if the lens is plano-convex, then 
 
 n— 1 
 
 when n is 1.5. If »• and r' are both negative, the 
 lens is bi-concave, and we have 
 
 F = 
 
 -(n- V)(r + r') 
 
 Hence the principal focal distances are negative. 
 This means that rays which are parallel to the axis, 
 meeting the lens, do not intersect the axis after 
 refraction ; but they would, if prolonged backward, 
 meet the axis on the same side of the lens as that 
 from which the rays proceed. Similarly, rays which 
 are parallel to the axis after refraction, do not come 
 from a real point ; but before refraction they are 
 
56 
 
 HANDBOOK OF OPTICS 
 
 directed toward a point on the axis and on the 
 opposite side of the lens to that from which the light 
 proceeds. Thus we see that in the concave lens both 
 foci are virtual ; also that the first focus lies behind 
 the lens, and the second focus lies in front of theT 
 lens, considering, as we do, that the side of the lens 
 
 O' 
 
 a) 
 
 Pig. 19. 
 
 which is turned toward incident light is the front 
 of the lens. 
 
 In' Fig. 19, (1) represents the action of a convex 
 lens, (2) represents the action of a concave lens. 
 All rays from 0, the anterior focus of the convex 
 lens, are after refraction parallel to the axis ; all 
 rays, which before refraction are parallel to the axis, 
 meet at the posterior focus 0' . Convex lenses are 
 
EEFRACTION THROUGH LENSES 57 
 
 called collective lenses. In the concave lens rays 
 whicli are parallel to the axis before refraction are 
 rendered divergent after refraction. If prolonged 
 backvt^ard, they would meet at 0' . 0' is then the 
 posterior focus of the lens, for it is the virtual inter- 
 secting point with the axis of rays which before 
 refraction are parallel to the axis. Similarly, rays 
 converging to the point before refraction are 
 parallel to the axis after refraction. is therefore 
 the anterior focus of the lens, since it is the virtual 
 intersecting point with the axis of rays which are 
 parallel to the axis after refraction. Concave lenses 
 are called dispersive lenses. 
 
 If, in equation (4), page 47, we disregard the 
 thickness of the lens, making o equal to zero, and if 
 we substitute F, the focal distance of the lens, for 
 its equivalent, the equation reduces to the form, 
 
 f A 
 
 Equation (4) can be reduced to the same form, 
 irrespective of the value of <?, but not so simply as 
 when c is equal to zero ; and, as we need apply this 
 formula to thin lenses only, we shall not make the 
 substitution for thick lenses. From the expression 
 thus obtained, we see that after refraction by convex 
 lenses, conjugate foci lie on opposite sides of the 
 lens and are real so long as both conjugate points 
 
58 HANDBOOK OF OPTICS 
 
 are without the principal foci, but if one point is 
 within the principal focus, and real, the other lies 
 without the principal focus, on the same side of the 
 axis as the first point, and is virtual. We also ob- 
 serve from a study of this equation that as a point 
 approaches the lens, its conjugate moves in the same 
 direction, that is, it recedes from the lens when it is 
 real, and approaches the lens when it is virtual. 
 Since an object lying in one conjugate plane has its 
 image in the other, we may apply the foregoing con- 
 clusions to the images of objects. Since, as we have 
 seen, real images are inverted, a real and inverted 
 image of an object will be formed by a convex lens, 
 provided the object be not within the principal focus 
 of the lens. If the object be placed at the principal 
 focus of the lens, the rays from every point of the 
 object will be rendered parallel, and no image will 
 be formed. If the object be placed within the prin- 
 cipal focus, the rays from every point of the object 
 will, after refraction, be divergent, and if prolonged 
 backward will meet in a virtual focal plane, thus 
 forming a virtual image. 
 
 If, in the equation which we have been studying, 
 we make F negative, as in the concave lens, we 
 shall see that rays from an object will not, after 
 refraction, meet in a real focus. 
 
 Since real images are formed by the actual inter- 
 sections of rays of light, they may be depicted upon 
 
REFRACTION THROUGH LENSES 69 
 
 a screen, or upon a sensitive photographic plate 
 which is capable of retaining the impression; or, 
 again, upon the retina of the eye, a nervous mechan- 
 ism through which the impression is transmitted to 
 the brain, where it is manifested as vision. Virtual 
 images, not being actual intersecting points of rays, 
 cannot be so depicted. 
 
 Plano-convex lenses act in the same manner as 
 bi-convex lenses; and plano-concave act as bi-con- 
 cave lenses. Periscopic lenses act as convex or 
 concave lenses according as the convex or concave 
 surface has the greater curvature. Reference to 
 the equation 
 
 j7_ rr' 
 
 (w — 1) (r -f- /•') 
 
 renders this apparent. 
 
 The power of a lens is inversely proportional to 
 the focal length. If F represents the focal length, — 
 
 is the power of the lens. Lenses may be numbered 
 according to their focal length or according to their 
 power. A lens which has a focal length of ten 
 inches is called a ten-inch lens. Its power is ex- 
 pressed by -jig-. Since, in lenses which have equal 
 curvature at the two. surfaces and whose index is 
 1.5, the focal length is equal to the radius of curva- 
 ture, the power of the lens is expressed by -. 
 
60 HANDBOOK OF OPTICS 
 
 We obtain the power of two or more lenses used 
 in combination by adding the expressions denoting 
 the power of each lens ; but this simple method 
 applies only when the distance between the centres 
 of the lenses is so slight that it may without error 
 be neglected. That the power of two or more lenses 
 used in combination is equal to the sum of the 
 powers of the lenses, may be proved from the rela- 
 tion between bi-curved and piano-curved lenses. 
 Since the principal or nodal points and the optical 
 centre of a thin lens coincide, any bi-curved thin 
 lens is equivalent in all respects to a piano-curved 
 lens of the same focal length. Hence if we wish to 
 combine two lenses, we may first replace them by 
 two piano-curved lenses with their plane faces in 
 contact. We then have as the result of this com- 
 bination a bi-curved lens ; and the focal length of 
 this lens is found from the equation 
 
 F = 
 
 1^ 
 
 F^ + F^-c 
 
 When the thickness of the lens is neglected, this 
 becomes 
 
 jPj being the focal length of the first, and F^ that of 
 the second of the component lenses. 
 
REFEACTION THROUGH LENSES 61 
 
 From this we find 
 
 1 = 1 + 1. 
 
 To find the power obtained by combining a lens 
 of ten inches focal length with one of twenty inches, 
 we have 
 
 1=^ + ^ = 1' «^'^=^^^°''^'^- 
 
 In order to avoid fractions in the addition of 
 lenses, another method of numbering them is em- 
 ployed. This method, which possesses many advan- 
 tages, has almost entirely displaced the old method 
 in ophthalmology. The unit of power is that of a 
 lens whose focal length is one metre. This unit is 
 called a dioptre. A lens which has a focal length 
 of one-half metre has therefore a power of two 
 dioptres. A lens whose focal length is two metres 
 has a power of .5 dioptre, and so on. Hence to find 
 the power of any number of lenses used in combina- 
 tion, we have only to add their dioptric values. 
 Thus a lens of two dioptres in combination with 
 one of three dioptres is equivalent to a lens of five 
 dioptres. 
 
CHAPTER IV 
 
 THE EYE AS AN OPTICAL SYSTEM 
 
 The eye as an optical system consists of three 
 refracting surfaces and three media. The first sur- 
 face is the cornea. Strictly speaking, this is not a 
 spherical surface; it conforms more closely to the 
 small end of an ellipsoid of three unequal axes, but 
 we may without appreciable error replace in our cal- 
 culations the normal cornea by a spherical surface. 
 The anterior and posterior surfaces of the cornea 
 being very nearly parallel, and the* refractive index 
 of the cornea and aqueous being practically identi- 
 cal, the cornea may be disregarded and the aqueous 
 humor may be considered the first refractive me- 
 dium. After traversing the aqueous, light enters 
 the crystalline lens. This is not composed of a 
 homogeneous medium, but of numerous layers, the 
 density of which increases from the outer to the 
 central part of the lens. To trace the path of light 
 through each of these layers would be an impossible 
 task. Helmholtz, accordingly, divides the lens into 
 three portions with increasing index, — the cortical 
 
 62 
 
THE EYE AS AN OPTICAL SYSTEM 63 
 
 or outer portion, the intermediate, and the nuclear 
 portion. From these he has determined the mean 
 refractive index of a lens having the same curvature 
 and refractive power as the crystalline lens. Refer- 
 ence to Fig. 20 will show that this does not mean 
 that the refractive index of Helmholtz' 
 equivalent is equal to the mean of the 
 indices of the three portions into which 
 the lens is divided. The index of the 
 equivalent is greater than the greatest 
 index of the component portions of the ^^ 
 lens ; for if the index of the entire lens 
 were equal to that of the nucleus, its refractive 
 power would be less than in the lens as constituted. 
 This will be seen from the figure. The letter n 
 indicates the nuclear portion, which has a small 
 •radius of curvature ; and therefore, acting alone, 
 it would have a greater refractive power than the 
 entire lens of the same index, for its curvature is 
 greater. The outer portion has the same effect as 
 if two divergent menisci were added to the nucleus. 
 The index of the outer portion being less than that 
 of the nucleus, the addition of the two menisci has 
 a less divergent effect than if they had the higher 
 index of the nucleus. Thus we see that the refrac- 
 tive power of the lens with increasing index is 
 greater than if it were composed of homogeneous 
 material with the index of the most highly refract- 
 
64 HANDBOOK OF OPTICS 
 
 ing part of the lens ; and consequently the index of 
 a homogeneous lens having the curvature and power 
 of the crystalline lens must be greater than that of 
 the nucleus of the lens. This equivalent, as deter- 
 mined by Helmholtz, is 1.4371. 
 
 Another effect of this increasing index is to 
 diminish spherical aberration. We have seen that 
 a spherical lens has a greater refractive power for 
 rays that meet it at a distance from the axis than for 
 those which pass near the axis. By means of the 
 physiological arrangement of layers vrith increasing 
 index, those rays which meet the lens near the axis 
 are acted upon by the more highly refracting nucleus, 
 while those which are remote from the axis escape 
 this portion and are refracted only by the cortical 
 layers of the lens. 
 
 Finally, after passing through the lens, light enters . 
 the vitreous, whose refractive index is the same as 
 that of the aqueous. 
 
 Gauss, the eminent mathematician, has by his 
 researches rendered it possible for us to trace the 
 path of light from an external object through the 
 media of the eye to its focus on the retina.* Prior 
 to his work other mathematicians, notably Moebius, 
 had investigated refraction through a number of 
 media, but they neglected the thickness of the lenses, 
 
 * Gauss, " Dioptrisclie Untersuchiungen," Werke, Band V. 
 Gbttingen, 1867. 
 
THE EYE AS AN OPTICAL SYSTEM 65 
 
 thereby causing an appreciable error in dealing with 
 lenses of considerable thickness in comparison with 
 their focal length, as in the case of the lens of the 
 eye. Furthermore, Gauss demonstrated that an 
 optical system of any number of spherical surfaces 
 and media, the surfaces all being centred on the 
 same axis, has certain points and planes which are 
 very useful in determining the optical effect of 
 the system. These are called cardinal points and 
 planes, and are similar to the cardinal points and 
 planes of single lenses. As in single lenses we 
 have : 
 
 The first and second principal points and the first 
 and second principal planes, and 
 
 The first and second principal foci and the first 
 and second principal focal planes. 
 
 When in any system the positions of these points 
 and planes have been determined, the solution of the 
 system is complete. 
 
 There are also two other points, the first and 
 second nodal points, which are similar to the 
 nodal points of a lens. These points, though 
 useful, are not necessary to the solution of the 
 system. Their properties were first demonstrated 
 by Listing. 
 
 Let P (Fig. 21) represent a point on the axis of 
 the system ; and let HB represent the first surface 
 or cornea of the eye ; H'W the anterior surface and 
 
HANDBOOK OP OPTICS 
 
 C 
 
 H'^R" the posterior surface of the 
 crystalline lens. A ray of light passing 
 through P meets the cornea at i2 ; it 
 will be refracted so as to assume .the 
 direction RR', and if there were no 
 further refraction, it would meet the 
 axis at Q' . But after travelling the 
 distance RR', it meets the anterior 
 surface of the crystalline lens and is 
 refracted so as to assume the direc- 
 tion R'R", and if it continued in this 
 direction, it would meet the axis at 
 Q". But at the posterior surface of 
 the lens it is again refracted so that 
 its final direction is R" Q. Hence we 
 see that in the first refraction Q' is 
 conjugate to P ; in the second refrac- 
 tion Q" is conjugate to Q' ; and in 
 the third and last refraction Q is con- 
 jugate to Q" . The point Q being 
 the point of intersection of the ray 
 with the axis after the three refrac- 
 tions, is conjugate to P with reference 
 to the entire system. 
 
 To find the relation between the 
 position of a point and its conjugate 
 after refraction through the eye, we 
 make use of the following constants : 
 
 Pig. 21. 
 
n' 
 
 n 
 
 THE EYE AS AN OPTICAL SYSTEM 67 
 
 The refractive index of air, whicli is denoted by 1 
 The refractive index of the cornea and aqueous n 
 The refractive index of the crystalline lens . 
 The refractive index of the vitreous . . . 
 The radius of curvature of the cornea . . . r 
 The radius of curvature of the anterior surface 
 
 of the lens r' 
 
 The radius of curvature of the posterior surface 
 
 of the lens r" 
 
 The distance of anterior surface of the cornea 
 
 from the anterior surface of the lens . . . nt-^ 
 The thickness of the crystalline lens .... n't^ 
 
 All distances measured from left to right are posi- 
 tive, and those measured from right to left are nega- 
 tive. In accordance with this convention of signs, 
 we use formula (6), page 31. Then r and r' are 
 positive and r" is negative. The distance from H 
 to H' (Fig. 21) is denoted by w^j, and that from H' 
 to H" by n't^. Let the distance of P from the first 
 surface be denoted by/; and the distance of Q' from 
 the same surface by j-t/j. The distance of the second 
 image Q" from the second surface W is denoted by 
 n'f^, and the distance of the third image Q from the 
 third surface H" by nf^ * 
 
 * The reason for this notation will hecome apparent in the 
 course of the demonstration. 
 
68 HANDBOOK OF OPTICS 
 
 Hence 
 
 Pn=f; HQ' = nf^ ; H> Q" = n% ; and E" Q = nf^. 
 
 Then at the first refraction, applying formula (6), 
 page 31, we have 
 
 1 n 1 — n 
 
 At the second refraction, 
 
 (1) 
 
 S'Q' n% r 
 
 But H' Q' = HQ' - HE' = nf^ - nt^ ; hence 
 1 _ 1 _n — n' 
 
 At the third refraction, 
 
 (2) 
 
 II 
 
 E"Q" nf^ -r 
 
 But E" Q" = E' Q" - E'E" = n% - n't^ ; there- 
 fore 
 
 (3) 
 
 /2 ~ *2 /s *" 
 
 For convenience we make 
 1 — w_, w — »i'_, ■, n' — n 7 
 
 ""O ' i — ""1 ' ^^"- M *2' 
 
THE EYE AS AN OPTICAL SYSTEM 69 
 
 From (1) we have 
 
 From (2), /^ = t^ 
 
 h + ~ 
 
 From (3), /^ = ,^ + 1_. 
 
 Jz 
 Substituting these values of /^ and f^, we have 
 
 i=K+ — ^ — (4) 
 
 ^ k + 4 — 
 
 *2 + — 
 
 h + j 
 
 This is an equation expressing the relation be- 
 tween / and /g, from which we can find the conju- 
 gate focus of a point, P, in any case. The same 
 method may be applied to any number of surfaces, 
 but as the equation becomes much more cumbersome 
 with each additional surface, it is convenient to ob- 
 tain from the known properties of such an expres- 
 sion a simpler relation between / and /g. The 
 second member of equation (4) is called a continuous 
 
70 HANDBOOK OF OPTICS 
 
 fraction. If we should neglect all the terms to the 
 right of ^Q, the expression would become 
 
 Kq 
 
 If we neglect all except Jcq + — , we have 
 
 h 
 
 h^t-^ + 1 
 
 h 
 
 (a) 
 
 (J) 
 
 Neglecting now all after h^ + — , we have 
 
 k^h + 1 " 
 
 Continuing this process, we finally embrace all the 
 terms and get the true value of the second member 
 of the equation. The expressions (a), (5), and (c) 
 are called convergents. If we examine the conver- 
 gent (c), we shall see that its numerator is obtained 
 by multiplying the new letter A;j by the numerator 
 of the preceding convergent and adding the numera- 
 tor of the second preceding convergent; and that 
 the denominator is obtained by multiplying the new 
 letter k^ by the denominator of the preceding con- 
 vergent and adding the denominator of the second 
 preceding convergent. This relation is true for all 
 subsequent terms of a continuous fraction such as 
 the second member of equation (4); we can there- 
 
THE EYE AS AN OPTICAL SYSTEM 71 
 
 fore write out the value of this fraction without 
 further calculation. The successive convergents, as 
 thus obtained, are : 
 
 T' 
 
 Vi+1 
 
 h 
 
 
 ^a|fei(Vi+l) + ^oJ + Vi+l (9) 
 
 t^(k^t^ + \-) + t^ ' (K) 
 
 ^2fe^^i(Vi + l) + ^oi+Vi + l] + ^i(Vi + l) + ^o W 
 
 For convenience we call the last two of these con- 
 vergents ^ and - ; then the next expression, which 
 will be the true value of the fraction, may be written 
 -~j — J-, and equation (4) becomes 
 
 f U + h' 
 
 If we multiply numerator and denominator of the 
 second member of this equation by w, we shall have 
 
 1 _ nfjc + ng 
 f ~ nfj. + nh' 
 
 or w/s ■ ^ + nh =f -nf^-h+f- ng. (5) 
 
72 HANDBOOK OF OPTICS 
 
 Since nf^ is the distance of the last image, §, from 
 tlie last surface, equation (5) expresses the relation 
 between P and its conjugate Q. This is a general 
 equation, true for all positions of P ; we may there- 
 fore, by making nf^ and / respectively equal to in- 
 finity, find the positions of the principal foci. 
 
 Equation (5) may be written 
 
 (Jk -V)nf^ = nh-f- ng. 
 
 When 91/3 = 00, ^^T; 
 
 Similarly we find when /= 00, 
 
 •'^ k 
 
 These values determine the first and second prin- 
 cipal foci as measured respectively from the anterior 
 surface of the cornea and the posterior surface of the 
 crystalline lens ; but we cannot construct the image 
 of an object until we find the position of the prin- 
 cipal planes. These planes are, as we know, conju- 
 gate focal planes such that an object in one of them 
 will have its image in the other, the object and 
 image being equal and on the same side of the axis. 
 Hence to find these planes in a system of refracting 
 
THE EYE AS AN OPTICAL SYSTEM 73 
 
 surfaces, it is evident that we may use equation (5), 
 whicli expresses the relation between conjugate foci. 
 If, in this equation, we impose the condition that 
 object and image be of the same size and sign, the 
 resulting values of / and nf^ will be the distances of 
 the first and second principal planes from the ante- 
 rior surface of the cornea and posterior surface of 
 the crystalline lens, respectively. 
 
 Fig. 22. 
 
 In Fig. 22 let PR represent an imaginary ray 
 directed toward E. After refraction through the 
 system it appears to pass through E' ; then E and 
 E' are conjugate foci. Let F mark the position of 
 the anterior, and F' the position of the posterior 
 focus of the system ; then 
 
 FH=-; mdH"F'=-'!f. 
 k k 
 
74 HANDBOOK OF OPTICS 
 
 Since E is the point of intersection of the entering 
 ray with the axis, -H!S=/;'and likewise E', being 
 the point of intersection of the emergent ray with 
 the axis, H"E' is equal to w/g. Equation (5) may 
 be written, 
 
 or, 
 
 kj\-^ k j k F P 
 
 But by a property of continuous fractions, gl — hk 
 = 1.* 
 
 „e„„e, (/4)(»/.-t^)._|. (6) 
 
 We have also (Fig. 22) FE = HE + HF, but HF 
 is in the figure negative ; if it were positive, we 
 
 should have FE = EE - HF ; or, FE=f-L 
 
 k 
 
 Similarly, FE' = nf, - LzM). 
 
 k 
 
 Let the distance FE be denoted by u. As in the 
 case of single lenses, it will be most convenient to 
 consider u positive when E lies to left of F, and 
 negative when it lies to the right of F, as in our 
 
 * Reference to the simpler convergents on page 71 will make 
 this apparent. 
 
THE EYE AS AN OPTICAL SYSTEM 
 
 75 
 
 figure.* Likewise if the distance F'M be denoted 
 by m', then u' will be positive when E' lies to the 
 right of F', and negative when it lies to the left of 
 F', as in the figure. Then 
 
 / — - becomes — u, and nf^ — '^ — — ^ becomes — u' ; 
 k k 
 
 and equation (6) thus becomes 
 
 n 
 
 1 n 
 k'k 
 
 This equation is true for any two conjugate points 
 E and E' ; and to find the values of u and u' which 
 
 Tig. 23 
 
 give to E and E' the properties of principal points, 
 we make use of Helmholtz' formula for obtaining 
 the .relation between the size of an object and its 
 image. In F'ig. 23 let H, H\ and H" represent the 
 points of intersection of the refracting surfaces with 
 the axis ; then RH, R'R', and R"H" are the princi- 
 pal planes of these surfaces. A ray of light passing 
 
 * See Chap. II., p. 35, 
 
76 HANDBOOK OF OPTICS 
 
 through P meets the principal plane RH at Z2 ; it is 
 refracted to R', where it is again refracted to R". 
 RPH, the angle which the incident ray makes with 
 the axis, is called a, and the angle RP'H, which the 
 refracted ray makes with the axis, is called a-^. Simi- 
 larly, a^ and a^ are the angles which the ray makes 
 with the axis after the second and third refractions. 
 Let 6, Sj, 62, represent the distances from the axis at 
 which the ray meets the successive principal planes. 
 As in our previous demonstrations, the refractive in- 
 dex of air is unity; that of the aqueous and vitreous 
 are each w, that of the crystalline lens n' ; the dis- 
 tance HE' is nt^, and H'H" is n't^. From Fig. 23 
 we obtain the following equations : 
 
 HP = 
 
 h 
 
 HP' 
 
 b 
 
 R'P" 
 
 h 
 
 tana 
 
 tan a^ ' 
 
 tan«2 
 
 Since in the first refraction HP and HP' are con- 
 jugate foci, and both negative, we have formula (J), 
 page 31, 
 
 1 , n 1— n tan a , n . tan a-, ,, 
 
 from which n • tan a-^ = tan a + k^h. 
 Referring again to Fig. 23, we see that 
 Jj = 5 + nij • tan ay 
 
THE EYE AS AN OPTICAL SYSTEM 77 
 
 In exactly the same way it is proved that 
 
 n' tan a^ = n • tan aj + Jc-J)^, and h^ = 1-^^ + n't^ • tan a^, 
 
 and so on. By these equations all the quantities 
 n • tan a^, 6j, n' tan a^, b^, n • tan a^, may be expressed 
 in terms of tan a and b. Their values become 
 
 n tan a^ = kffi + tan a, 
 
 ftj = (^Q^j + 1)6 + ij tan a, 
 
 w' tan ^2 = l^iC^o^i + 1) +^ol ^ + C^^^i^i + l) *^'^ *' 
 
 and so on. 
 
 The coefficients of b and tan a in these equations 
 are seen to be respectively the numerators and de- 
 nominators of the successive convergents on page 71. 
 
 Hence we may write out the value of b^ thus : 
 b^ = gb + h tan a ; and n ■ tan a^ = lcb + 1- tan a, where 
 g. A, k, and I have the same significance as on page 71. 
 
 If represent the linear dimension of an object 
 and ^^, i^, and i^ the coi-responding dimensions of the 
 successive images, then from Helmholtz' formula, 
 page 39, we shall have 
 
 • tan a = w • ij • tan a^ = w' • i^ • tan a2 = n ■ i^- tan a^. 
 
 The relation between the object and its final 
 
 image is 
 
 • tan a = ig- n ■ tan a^. (7) 
 
78 HANDBOOK OF OPTICS 
 
 But n • tan ag = kb + 1 ■ tan a, and from the figure, 
 b = HP tan a, or since HP = — /, b = — / • tan a. 
 Substituting tliis value, we have 
 
 n ■ tan a^= I • tan a — kf- tan a, 
 
 or n • tan ag = k • tan a ( - — /), 
 
 and equation (7) becomes 
 
 • tan a = ijc ■ tan «(-—/). 
 
 Referring to page 75, we see that 
 
 (i-f)- 
 
 u. 
 
 Therefore o = ijcu, and if object and image are 
 equal, which is the condition to be imposed in order 
 that M and u' may determine the principal points, 
 
 = 1*3 and M = -. 
 From the equation 
 
 mm' = — — , we find u'= . 
 
 k^ k 
 
 Thus we have determined the values of u and u' 
 which represent the distances of the first and second 
 principal points from the first and second principal 
 foci, respectively; and since these distances are the 
 
THE EYE AS AN OPTICAL SYSTEM 
 
 79 
 
 first and second principal focal distances, we have 
 only to find the numerical value of k in order that 
 the solution of the system be complete. The value 
 of k can be obtained if we know the radii of the 
 refracting surfaces, the distances between these sur- 
 faces, and the refractive indices of the media. In 
 the human eye these quantities have all been meas- 
 
 p R R 
 
 aI ^^^F E E' ^^F- A- 
 
 S S' '^ 
 
 Fig. 24. 
 
 ured by careful and scientific investigators. Of 
 these the name of Helmholtz is most conspicuous. 
 
 Before substituting these values it will be well to 
 show the construction of the image after refraction 
 through a number of media, and to demonstrate the 
 properties of the nodal points. The geometrical 
 construction of the image is almost a repetition of 
 that for single lenses. 
 
 In Fig. 24, ES and R'S' represent the principal 
 planes, F and F' the principal foci, AP the linear 
 
80 HANDBOOK OF OPTICS 
 
 dimension of the object, and A'P' that of the image. 
 The anterior focal distance HF is denoted by F, and 
 the posterior focal distance F'F' by F'. If, as in 
 our former notation, u denote the distance of the 
 point A from the anterior focus, and u' the distance 
 of its image from the posterior focus, then FA = u 
 and F'A' = u' . From the similar triangles PAF and 
 
 EFS we have 
 
 PA 
 
 FA 
 
 
 ES 
 
 EF' 
 
 and as ES — A'P' 
 
 PA 
 
 FA 
 
 (vlIKa. Ojij JjJ KJ jCL. jL ■ 
 
 P'A' 
 
 EF' 
 
 Since in our construction the object and image 
 are on opposite sides of the axis, the image i must 
 be considered negative, and we have this equation 
 to determine the size of the image : 
 
 u . F 
 
 = — , or i = — — 
 
 — iF u 
 
 Also from the triangles E'F'R' and F'A'P', we 
 obtain the relation 
 
 __F[ 
 
 u' 
 
 
 or i = — 0—- 
 
 - i m' ' 
 
 F' 
 
 This construction differs from that for single 
 lenses only in the respect that in lenses F and F' 
 are equal, while in the present case F is to F' as 
 
THE EYE AS AN OPTICAL SYSTEM 
 
 81 
 
 1 to n, that is, as the index of air is to that of 
 the final medium or vitreous. If, after refraction 
 through any number of media, light enters a medium 
 of the same index as that of the first medium, the 
 system acts toward light as a single lens; if the 
 final medium has not the same index as the first, 
 the system is analogous to a single refracting surface. 
 
 
 
 S 
 
 
 S' J 
 
 P 
 
 F^^ 
 
 E 
 
 N 
 
 t 
 
 E 
 
 ^^ 
 
 f' 
 
 / 
 
 R 
 
 .'' 
 
 ^ 
 
 
 
 
 
 
 Py^ 
 
 
 
 
 
 
 Fig. 25. 
 
 We have seen that in refraction at one surface 
 there is one nodal point at the centre of curvature, 
 and in refraction through lenses there are two nodal 
 points. The characteristic property of nodal points, 
 we remember, is that a ray directed toward one of 
 them appears after refraction to come from the other, 
 and in a direction parallel to that before refraction. 
 Let us then take a ray PR (Fig. 25) directed 
 toward the point N on the axis, so that after refrac- 
 tion through the system it appears to pass through 
 
82 
 
 HANDBOOK OF OPTICS 
 
 N' in a direction If'P' parallel to PB ; then iV and 
 iV' are nodal points. To find the position of these 
 points we construct the principal planes MS and 
 B'S' and the principal focal plane F'P'. The ray 
 Pit meets the first principal plane at M. It must 
 after refraction meet the second principal plane at 
 B' so that UB = WB'. P' is its point of intersec- 
 tion with the focal plane P'F'. If we draw any 
 two rays from P', passing through the system, they 
 
 Fig. 26. 
 
 must after refraction be parallel. Let P'JV' repre- 
 sent one of these rays, and P'S', parallel to the axis, 
 the other. P'S' must after refraction pass through 
 the anterior focus P. SP will represent this ray 
 and BP will represent the ray P'JV' after refraction. 
 jPi\r, the distance from the first focus to the first 
 nodal point, is FE + EN. The triangle. K^iV" is 
 equal to B'P'N', and FES is equal to N'F'P' ; 
 hence FE + EN= E'N' -{- N'F', or FN=F'. In 
 the same way we find iV^'^' = F. When, as in lenses, 
 
THE EYE AS AN OPTICAL SYSTEM 83 
 
 F and F are equal, it is clear that the nodal and 
 principal points coincide. It is also apparent that 
 the distance between nodal points is the same as 
 that between principal points. The size of the 
 image may be determined if we know the distance 
 of object and image from the first and second nodal 
 points respectively. In Fig. 26 AP= o, and A'P' = i ; 
 then from the similar triangles APN a,\\A. A1P'N\ 
 
 AP AN AN 
 
 or — - = 
 
 AlP' A'N' -i A'N' 
 
 The following is a table of measurements of the 
 constants which we shall require in the determina- 
 tion of the refractive power of the eye : 
 
 Radius of curvature of the cornea (r) . 7.829 mm. 
 
 Radius of curvature of the anterior 
 surface of the lens in a state of 
 rest (/) 10 mm. 
 
 Radius of curvature of the posterior 
 
 face of the lens (r") 6 mm. 
 
 Distance of the anterior surface of the 
 cornea from the anterior pole of the 
 lens (nt{) 3.6 mm. 
 
 Thickness of the lens (n't^^ .... 3.6 mm. 
 
 Index of refraction of tlie cornea, aque- 
 ous and vitreous (n) 1.3365 mm. 
 
 Equivalent index of refraction of the 
 
 crystalline lens (w') 1.4371 mm. 
 
84 HANDBOOK OF OPTICS 
 
 Substituting these values in the successive conver- 
 gents on page 71, we deduce the following : 
 
 i..l^ 
 
 = -.043 
 
 n 
 
 = 2.6936 
 
 *'- n' 
 
 = 2.505 
 
 J n-n' 
 
 — — 01 
 
 "l ^: 
 
 ^— * \J J. 
 
 7 _n'-n 
 *2 _^n 
 
 = -.016! 
 
 le^t^ + 1 = . 8842, \t^ + 1 = . 9731, 
 
 hihh + 1) + ^0 = - -0^18, 
 h\hihh + 1) + hi + hh + 1=5- = -7544, 
 hihh + 1) + «i = A = 5.1312, 
 
 *2[*2l*l(Vl + 1) + K\ + Vi + 1] 
 
 + /ci(/c/i + 1) + i„ = A; = - .0645, 
 
 K^hihh + V) + h\ + hh + 1 = ? = .8869. 
 
THE EYE AS AN OPTICAL SYSTEM 85 
 
 The first principal focal distance EF, which is 
 
 equal to -, thus becomes 
 k 
 
 - 15.5038 mm.; and ^^ or -= - 13.7504 mm. 
 
 k 
 
 Therefore EH, which is equal to EF — HF, is 
 — 1.7534 mm.; thus the cornea lies 1.7534 mm. in 
 front of the first principal point. The second focal 
 
 distance E'F', being represented by — -, is 
 
 k 
 
 20.721 mm.; and WF' or -^ is 15.6326 mm. 
 
 Therefore E', the second principal point, lies 5.0884 
 mm. in front of the posterior surface of the lens, 
 and since this surface is 7.2 mm. behind the anterior 
 surface of the cornea, the second principal point lies 
 2.1116 mm. behind the cornea. The distance be- 
 tween the two principal points is .3582 mm. The 
 position of the first nodal point of the eye is, as we 
 know, found by laying off the distance FN equal to 
 E'F', the second focal distance; as thus found, N 
 lies 6.9706 mm. behind the anterior surface of the 
 cornea ; and since the distance between nodal points 
 is the same as that between principal points, the sec- 
 ond nodal point is 7.3288 mm. behind the anterior 
 surface of the cornea. With these points determined 
 our knowledge of the eye as an optical system is 
 
86 HANDBOOK OF OPTICS 
 
 complete. The following table giving the positions 
 of these points will be found useful for reference : 
 
 Distance of the anterior surface of the 
 
 cornea from the first principal point 1.7534 mm. 
 Distance of the anterior surface of the 
 
 cornea from the second principal 
 
 point 2.1116 mm. 
 
 Distance of the anterior surface of the 
 
 cornea from the first nodal point . 6.9706 mm. 
 Distance of the anterior surface of the 
 
 cornea from the second nodal point 7.3288 mm. 
 Distance of the first principal focus 
 
 from the anterior surface of the 
 
 cornea 13.7504 mm. 
 
 Distance of the second principal focus 
 
 from the anterior surface of the. 
 
 cornea 22.8326 mm. 
 
 The hypothetical eye which possesses these car- 
 dinal points is called the schematic eye. 
 
 We have seen that the method of determining the 
 size and position of the image is the same in refrac- 
 tion at one surface as in refraction at any number of 
 surfaces, with the exception that there are two prin- 
 cipal and two nodal points in the latter case and only 
 one principal point in the former, with one nodal 
 point at the centre of curvature. We also know 
 that the size of the image is proportional to the 
 
THE EYE AS AN OPTICAL SYSTEM 87 
 
 anterior focal distance ; and that this distance is 
 measured from the first principal point, while the 
 second focal distance, which determines the position 
 of the image, is measured from the second principal 
 point. In the eye the two principal points are so 
 near to each other that they may with very slight 
 error be merged into one. In this way the optical 
 effect of the eye may be represented by one refract- 
 ing surface whose summit coincides with the merged 
 principal points. Listing first proposed this simpli- 
 fication, and called the resulting equivalent tlie re- 
 duced eye.* In this reduction the single principal 
 point is between the two principal points of the 
 schematic eye as determined by Listing. f The 
 ratio of F to F' remains unchanged, and as F is to 
 F' as 1 to n, we can determine what value n must 
 have in the reduced eye. Listing takes as the an- 
 terior focal distance 15.036 mm., and as the second 
 focal distance 20.133 mm., from which we find n is 
 equal to 1.3365, and this, as a reference to the table 
 on page 83 will show, is the index of the vitreous. 
 
 From the equation F = -, we can determine the 
 
 * Dioptrilc des Auges, Wagner's " Handworterbuch der Phy- 
 siologie." 
 
 t Listing's schematic eye differs slightly from that given in the 
 text, since he takes the radius of curvature of the cornea as 8 mm., 
 and the distance from cornea to lens and the thickness of the lens 
 as each equal to 4 mm. 
 
HANDBOOK OF OPTICS 
 
 radius of curvature of the reduced eye. Using 
 Listing's measurements, this is 5.1248 mm. Figure 
 27 illustrates in (1) the schematic eye and in (2) 
 the reduced eye. 
 
 F 
 
 4 
 
 E'fh ^ 
 
 f' 
 
 F 
 
 \ 
 
 H 
 
 !:Kj 
 
 F' 
 
 
 
 W 
 
 
 Fig. 27. 
 
 Bonders has furnished a less accurate but a more 
 useful reduced eye than that of Listing. He neg- 
 lects fractions, making the anterior focal distance 
 1.5 mm., the posterior focal distance 20 mm., and the 
 
 radius of curvature 5 ram. From F - 
 
 we find 
 
 n = 1.333, which is the index of water. Calculations 
 
THE EYE AS AN OPTICAL SYSTEM 89 
 
 as to the size of images with this eye are extremely 
 simple. If it be remembered that the size of the 
 image is proportional to the anterior focal distance, 
 and that in Bonders' eye this is 15 mm., while in 
 the schematic eye it is 15.5038 mm., it will be ap- 
 parent that the image obtained from calculation with 
 Bonders' eye is to the actual image in a normal eye 
 as 15 to 15.5038. 
 
 Based upon these data, artificial eyes have been 
 constructed for the study of the refraction of the 
 eye. 
 
CHAPTER V 
 
 THE DETERMINATION OF THE CARDINAL POINTS OF 
 THE EYE IN COMBINATION WITH A LENS 
 
 We must now carry our investigations one step 
 further in order to appreciate tlie effect upon vision 
 of placing a lens in front of the eye. A lens has two 
 refracting surfaces, and the eye three such surfaces ; 
 hence, if we wish to use the method given in the 
 preceding chapter, we must write out the convergents 
 for two additional surfaces ; but simpler than this 
 would be an independent geometrical construction, 
 from which we could find the effect of combining 
 two optical systems. But for our purpose it suffices 
 to make use of the reduced .eye of one surface, since 
 we do Jiot care to know the exact size of the retinal 
 image. What we wish to know is the relative size 
 of images with lenses and without them ; and the 
 investigation with the reduced eye furnishes this 
 information. 
 
 In Fig. 28 let A represent the centre of a spheri- 
 cal lens ; and since we neglect the thickness of the 
 lens, the letter A also marks the position of the 
 merged principal points of the lens. Let A! repre- 
 
 90 
 
THE EYE IN COMBINATION WITH A LENS 91 
 
 sent tlie position of the principal point of the reduced 
 eye. We know that in the lens A the proportion of 
 curvature at the two faces is immaterial, in other 
 words, any thin bi-spherical lens may be replaced by 
 a plauo-spherical lens of the same focal length ; and 
 for the sake of simplicity we shall make this substi- 
 tution. To find the cardinal points of the combina- 
 tion of eye and lens, we use the formulae deduced in 
 the preceding chapter. If F denote the anterior and 
 
 P P Fi^ A ?, f f!f FY f; 
 
 Fig. 28. 
 
 F' the posterior focal length of the combination, then 
 
 -?= -, and F' = — -, in which n represents the index 
 
 of the reduced eye, that is, if we use Listing's re- 
 duced eye, it is equal to the index of the vitreous, 
 and if we use Bonders' reduction, n is equal to the 
 index of water. The thickness of the lens, which 
 in our formula is represented by nt^, is equal to zero ; 
 and the distance of the lens from the eye, which is 
 represented by n't^, becomes t^, since n', being now 
 the index of the air, is unity. This distance between 
 the eye and lens may be conveniently denoted by the 
 letter e ; then t^ in the formula is replaced by e. 
 
92 HANDBOOK OF OPTICS 
 
 Thus we have hr. = , in which r represents the 
 
 r 
 
 radius of curvature of the first surface of the lens and 
 n the index of the lens ; in the piano-spherical lens 
 
 is equal to the focal length of the lens. If we 
 
 1— w 
 
 denote this by F-^, we have ]cq = —• The radius of 
 curvature of the second face of the lens is infinite, 
 
 F^ being the anterior focal length of the eye. Mak- 
 ing these substitutions in the expressions (^), (^), 
 and (Z), page 71, we derive 
 
 hence ^j = 0. In the same way we find lc^ = — , 
 
 '- F 
 
 e + F, 
 
 f ' 
 
 Hence, F = \= — ^^^ 
 
 k F,^ + F^ + e 
 
 p_ n_ nF,F,, _ F,F^, _ 
 
 k F^ + F^+e F-^ + F^+e 
 
 With the convention of signs whicli we are using, 
 F^ and F^ , being measured to the left of A and A' 
 
THE EYE IN COMBINATION WITH A LENS 93 
 
 respectively, are both negative. In order that the 
 formulae may be applicable in any case without pre- 
 fixing the minus sign to #i and #2> we must change 
 the sign of these quantities. Making this alteration, 
 the equations become 
 
 i^ = l= ^ x^-i . p- ^- ^^^^ 
 
 k Fi + F^-e k F^ + F^-e 
 
 It is in this form that these equations are usually 
 written. To find the distance from the lens to the 
 anterior focus F, we have 
 
 AF=i=-^:ii^:^^^; 
 
 h F^ + F^-e 
 
 and to find the distance from the principal point -4' 
 of the eye to the posterior focus F , we have 
 
 k F^ + F^-e 
 
 Having found the positions of the two foci and the 
 two focal distances, we know also the positions of the 
 two principal points, and the solution of the system 
 is complete. 
 
CHAPTER VI 
 
 ERRORS OF REFRACTION — LENSES USED AS 
 SPECTACLES 
 
 We have studied the eye as an optical system, 
 taking as our measurements those found to exist 
 with close approximation to uniformity in a large 
 number of eyes. The posterior focal distance of 
 this system we have found to lie 22.8326 mm. behind 
 the anterior surface of the cornea. When the retina 
 lies at the same distance from the cornea, the image 
 of a distant object will be accurately formed on the 
 retina. When the retina lies in front of the focus 
 of the eye, the image of a distant object will be 
 blurred. This condition of the eye is called hyper- 
 opia ; and when the retina lies behind the focus, the 
 resulting condition is called myopia. 
 
 Eyes so constituted that retina and principal focus 
 do not coincide are said to be ametropic, or affected 
 with errors of refraction. When retina and focus 
 coincide, as in the normal eye, the condition is called 
 emmetropia. A distant object will be clearly seen 
 by a healthy emmetropic eye, but the image of a 
 near object will fall behind the retina. In order 
 
 94 
 
ERRORS OF REFRACTION 95 
 
 that a near object be clearly seen, either the eye 
 must be elongated or the focus must be brought 
 forward. The latter change is the one which occurs 
 by an increase in curvature of the crystalline lens 
 (principally of the anterior surface of the lens) 
 under the influence of the ciliary muscle. This 
 change in curvature is called accommodation. By it 
 we are enabled to adapt the eye for varying distances. 
 
 Since the curvature of the lens is increased during 
 accommodation, the optical system of the eye in this 
 state differs from that when the eye is adapted for 
 a distant object ; the focal distance has been short- 
 ened by the exercise of accommodation, and the retina 
 now lies behind the principal focus ; in other words, 
 the eye has become myopic. Thus we see that the 
 emmetropic eye can render itself myopic in order 
 to see near objects. Similarly, a hyperopic eye, 
 possessing sufficient accommodative power, may be- 
 come emmetropic to see distant objects, and by a 
 still further increase in curvature of the lens may 
 even become myopic and thus see clearly near objects.* 
 The myopic eye is unable to increase its focal dis- 
 tance, thus bringing the principal focus back to the 
 retina ; it cannot therefore see distant objects clearly. 
 
 Since an eye may be myopic either from increase 
 in curvature or from increase in the antero-posterior 
 
 * These conditions, however, are not Included in the usual 
 acceptation of the words emmetropia and myopia. 
 
96 HANDBOOK OF OPTICS 
 
 diameter, we have curvature myopia and axial 
 myopia. Similarly, we have curvature hyperopia 
 and axial hyperopia. In axial hyperopia and myopia 
 the eye as an optical system is normal, the defect 
 being in the position of the retina; in curvature 
 ametropia the deviation from the normal is in the 
 optical system.* 
 
 Without using the formulae for the cardinal points 
 of the eye and lens in combination, we can easily 
 
 Fig. 29. 
 
 understand how lenses can in hyperopia and myopia 
 bring the retina and principal focus into coincidence. 
 Let A'R' (Fig. 29) represent the cornea of a hyper- 
 opic eye whose principal focus is at F^ behind the 
 retina. Rays of light parallel to the axis before 
 entering the eye will after refraction meet at F^. 
 In order to bring these rays to a focus at F on the 
 retina, we introduce a convex lens A. Let Q be 
 the principal focus of this lens; then a ray PR 
 parallel to the axis will, after passing through the 
 
 * Index ametropia, which in optical effect resembles curvature 
 ametropia, occurs in exceptional cases. 
 
EREOKS OF REFRACTION 97 
 
 lens, take the direction BQ; but at i2' it is refracted 
 by tlie eye and assumes the direction R'F. 
 
 Let A'R' (Fig. 30) represent the cornea of a 
 myopic eye wliose foous F^ lies in front of the 
 retina ; then rays of light parallel to the axis before 
 entering the eye will, after refraction, meet at F^. 
 A ray passing through the point Q will meet the 
 axis at its conjugate focus behind the principal focus. 
 Let Q be so taken that its conjugate lies on the 
 retina; then Q is the far point of the eye, since light 
 
 Fig. 30. 
 
 from any more distant point will meet the axis in 
 front of the retina. If we place at A a concave 
 lens whose focal length is A Q, then any ray paral- 
 lel to the axis will, after refraction through the lens, 
 appear to pass through Q, and its direction will be 
 QR'. But the ray QJR' will, after refraction by the 
 eye, meet the axis at F. Hence a concave lens 
 whose focal length is AQ will bring parallel rays 
 to a focus on the retina. 
 
 It is clear from these diagrams that the farther 
 from the eye the convex lens is placed the weaker 
 
98 HANDBOOK OF OPTICS 
 
 is the lens required to bring the retina and focus 
 into coincidence, since AQ is the focal length of the 
 required lens. In the case of myopia the opposite 
 is true, for AQ, the focal length of the lens, dimin- 
 ishes as the lens is removed from the eye. 
 
 We can determine the amount of shortening of 
 the eye in hyperopia, or of lengthening in myopia, 
 from the equation 
 
 1+^=1. 
 f f 
 
 In this equation I' and F' are the focal distances 
 of the eye, and / and /' are the conjugate distances 
 A'Q -And A' F. 
 
 This equation is true not only for refraction at 
 one surface and for lenses, as we have seen; it applies 
 equally to a system of refracting surfaces. That 
 this is so, follows readily from the equation 
 
 , 1 n 
 
 uu' = • -I 
 
 k k 
 
 since u=f-F, u' =/' -F', and - 1 . ^ = FF'. 
 
 k k 
 
 Applying this equation in myopia, if Q, the far 
 point of the eye, is 100 mm. from the first principal 
 point of the eye, then / is 100 mm. Substituting for 
 F, the anterior focal distance, its value 15.5038 mm., 
 and for F' its value 20.721 mm., we find the corre- 
 
ERRORS OF REFRACTION 99 
 
 spending value of /' to be 24.5219 mm. That is, 
 the retina lies 24.5219 mm. behind tlie second prin- 
 cipal point of the eye. Subtracting 20.721mm., 
 whicli is the distance of the retina from the second 
 principal point in the normal eye, we have 3.8009 mm. 
 as the amount of lengthening. 
 
 In hyperopia if rays directed toward Q (Fig. 29) 
 would be focused on the retina, and if the distance 
 of Q from the first principal point of the eye is 
 100 mm., tlien / in our equation becomes — 100 mm. 
 Making the proper substitutions as before, we find 
 /' to be 17.9402 mm., and the amount of shortening 
 is 2.7808 mm. From this we see that the shortening 
 in hyperopia is less than the lengthening in the same 
 degree of myopia. 
 
 Since the strength of the lens required to correct 
 an error of refraction varies with the position of the 
 lens, it is clear that we cannot measure the error 
 by the correcting lens unless we adopt some fixed 
 position at which the lens should be placed. To be 
 strictly correct, this should be at the first principal 
 point, from which Q is measured, but as this is im- 
 possible, the position at which spectacle lenses are 
 ordinarily worn, about 15 mm., in front of the cornea, 
 is taken as the standard. In our example of the 
 myopic eye, 100 mm., as measured from the first 
 principal point E (Fig. 22) indicates the position of 
 the far point in a myopia of 10 dioptres; but the 
 
100 HANDBOOK OF OPTICS 
 
 focal length of the correcting lens, placed 15 mm. in 
 front of the cornea and consequently nearly 17 mm. 
 in front of E, must be a trifle more than 83 mm. 
 Taking this as 83 mm., the dioptric power of the 
 lens is slightly more than 12 D. In the second ex- 
 ample we have a hyperopia of 10 dioptres as meas- 
 ured from E, while the degree, as indicated by the 
 correcting glass placed 15 mm. in front of the 
 cornea, is about 8.5 D. 
 
 If the crystalline lens has been extracted from the 
 eye, there is only one refracting surface and one 
 medium, for the index of the aqueous and vitreous 
 are identical. The second principal focal distance 
 of the aphakic eye is therefore derived from the 
 equation 
 
 n — \ 
 
 and from this the focus is found to lie 31.095 mm. 
 behind the anterior surface of the cornea. The diop- 
 tric power of the aphakic eye is therefore 32.16 D. 
 The principal focus of the normal eye lies 22.8326 mm. 
 behind the anterior surface of the cornea ; hence its 
 dioptric power is 43.8 D. Subtracting the power 
 of the aphakic from that of the normal eye, we find 
 that the action of the crystalline lens, when adapted 
 for distant vision, is nearly equivalent to that of a 
 lens of 11.5 D. placed in contact with the cornea. 
 
ERRORS or REFRACTION IQl 
 
 After extraction of the lens from an emmetropic eye, 
 a convex lens of 11.5 D. would be required to bring 
 the image of a distant object to an accurate focus on 
 the retina, if the lens were worn in contact with the 
 cornea ; but as this is impracticable, a weaker lens 
 is required. If the distance of the lens from the 
 cornea be 15 mm., then its focal length must be 
 87 mm. (the focal length of a lens of 11.5 D.) 
 + 15 mm. A lens having this focal length repre- 
 sents a dioptric power slightly in excess of 10 D. 
 Hence after extraction of the lens from an emme- 
 tropic eye, we should expect a lens of 10 D. to rec- 
 tify the refractive condition for distant objects, and 
 this corresponds very closely with what we find in 
 practice. 
 
 Let us now examine the condition which exists 
 after extraction of the lens from a myopic eye. If 
 the myopia is caused by excessive curvature of the 
 cornea, the amount of myopia relieved by the ex- 
 traction is the same as in emmetropia, that is, if 
 there existed prior to extraction 11.5 D. of myopia 
 as measured from the cornea, the eye would be 
 rendered emmetropic by the extraction. This, 
 however, is not so when, as is usually the case, the 
 myopia is due to lengthening of the antero-posterior 
 diameter of the eye, As previously shown, the pos- 
 terior focus of the aphakic eye of, normal curvature 
 lies 31.095 mm. behind the cornea. Hence if an eye 
 
102 HANDBOOK OF OPTICS 
 
 is emmetropic after extraction of the lens, the retina 
 also must lie at this distance from the cornea. To 
 find what degree of myopia exists in an eye whose 
 retina lies at this distance from the cornea, we must 
 use the equation 
 
 / /' 
 
 As we can find from this equation the amount of 
 lengthening, if we know the distance (/) of the far 
 point from the eye ; so if we know the distance (/') 
 of the retina from the second principal point, we can 
 find the distance (/) of the far point from the eye, 
 and this distance measures the myopia. Our result 
 will he sufficiently accurate if we consider F as 
 15| mm., F' as 21 mm., and /' as 29 mm. We find 
 this last value by subtracting 2.1116 mm., the dis- 
 tance of the second principal point from the cornea, 
 from 31.095 mm., the distance of the retina from 
 the cornea, the result being the distance (/') of 
 the retina from the second principal point. Making 
 these substitutions, we find the corresponding value 
 of / to be 56.2 mm. This is the distance of the far 
 point of the eye from the first principal point ; its 
 distance from the cornea is therefore approximately 
 54.4 mm. ; or, expressed in dioptres, the myopia, as 
 measured from the cornea, is 18.3 D. 
 
 To find what is the required power of the correcting 
 
ERRORS OF REFRACTION 103 
 
 lens for this amount of myopia we shall consider the 
 lens placed not, as before, 15 mm. from the cornea, 
 but 15 mm. from the first principal point, or about 13 
 mm. from the cornea. The position at which spec- 
 tacle glasses are usually worn is more remote from 
 the eye than as indicated in the standard (15 mm.) 
 which we have taken, but this is not so when very 
 strong concave lenses are worn. These lenses, when 
 tolerated, ■ are worn very near the eye ; hence we 
 make the change in distance so as to be more in 
 accord with the actual conditions with which we 
 meet. It will be noticed that a slight change in 
 position of these strong lenses makes perceptible 
 change in dioptric power. The focal length of the 
 correcting lens in the case which we are considering 
 thus becomes 56.2 mm. — 15 mm., or 41.2 mm. The 
 power of the lens is accordingly 24 D. From this 
 we see that the eye must have a myopia of 24 D. as 
 measured by its correcting lens in order to become 
 emmetropic after extraction of the lens from the eye. 
 Let us now deduce from calculation the lens re- 
 quired for distant vision in an aphakic eye, in which, 
 before extraction, a concave spherical lens of 20 D. 
 was found to correct the ametropia. Making the 
 proper substitutions, we first find that in this case 
 the retina lies 29.7 mm. behind the cornea, and since 
 the posterior focus of the aphakic eye is 31 mm. 
 behind the cornea, the eye will be hyperopic. 
 
104 HANDBOOK OF OPTICS 
 
 In the equation :^ + -7^ = li -^ now represents 
 
 the anterior focal length of the aphakic eye ; it is 
 therefore approximately 23 mm. ; F' represents the 
 posterior focal length, approximately 31 mm.; /' 
 represents the distance A'F (Fig. 29), and in this 
 case is 29.7 mm. From these data we can find / 
 (A'Q, Fig. 29), the focal length of the required lens 
 if placed in contact with the cornea. We thus de- 
 rive the value/ = — 512 mm. Supposing that the 
 lens will be worn in the same position as the con- 
 cave lens previous to extraction, we add 16 mm. to 
 512 mm. The dioptric power of the lens required 
 to correct the existing hyperopia is therefore 1.9 D. 
 This result agrees very closely with the condition as 
 found to exist in a particular case after the removal 
 of the transparent lens for the cure of myopia. So 
 close an approximation could not be obtained in 
 every case, however, for in high myopia there is 
 frequently defect in curvature and in position of 
 the lens as well as in length of the eye. 
 
 From the foregoing studies we also learn the 
 effect of a change in the position of the crystalline 
 lens upon the refractive power of the eye ; if the 
 lens move toward the cornea the eye will be rendered 
 myopic, and if it move away from the cornea the eye 
 will become hyperopic. 
 
 We shall next study the effect of changing the 
 
ERRORS OF REFRACTION 105 
 
 position of spectacle lenses when used for near 
 vision. We have seen that convex lenses used to 
 aid the distant vision of hyperopes increase in power 
 as they are withdrawn from the eye. When convex 
 lenses are worn by presbyopes to replace the accom- 
 modation which has failed, the effect of withdrawing 
 the lens varies under different circumstances. Sup- 
 pose first that we are dealing with an emmetropic 
 eye whose accommodative power has entirely failed ; 
 this eye can focus only parallel rays upon the retina. 
 
 Fig. 31. 
 
 Let P (Fig. 31) represent the point where a near 
 object intersects the axis ; then, in order that the 
 object be clearly seen by the eye without accommo- 
 dation, a convex lens A, whose focal length is AP, 
 must be placed before the eye. - As the lens is re- 
 moved from the eye the focal length AP diminishes, 
 and the strength of the lens necessary to produce 
 distinct vision must be increased. 
 
 We shall next take the case of a hyperopic eye 
 whose accommodative power has been lost. Let P 
 (Fig. 32) represent the intersection of a near object 
 with the axis ; then the conjugate of the point P in 
 
106 HANDBOOK OF OPTICS 
 
 the refraction by the lens is Q, which lies behind the 
 retina. If the position of Q be such that it is con- 
 jugate to R in the refraction by the eye, then the 
 object at P will be accurately focused on the retina 
 
 with the aid of the lens. If P remain stationary 
 while the lens is moved to J.", the effect upon the 
 power of the lens will not be the same in all cases. 
 "We have the equation 
 
 J-4-J_ = l 1 + 1 = 1 
 
 AP AQ F' °^ f f F' 
 
 When the lens is moved from A to A"., the dis- 
 tance/is diminished by AA" , and/' is increased by 
 AA" . To study the effect of this change upon the 
 equation, we take the case when / and /' are equal, 
 each being then equal to 2F. The equation then 
 
 When the lens is at A" we have the similar equa- 
 tion 
 
 2F - AA" 2F + AA" F^ 
 
ERRORS OF REFRACTION 107 
 
 from which 
 
 1 4F 1 
 
 ^1 4:F^-{AA"y J, (AA"y 
 
 4F 
 
 From this we have F.=F- (AA^. Therefore, 
 
 4F 
 
 J\ is less than F, and the dioptric power of the lens 
 must be increased in order that Q may be conjugate 
 to P. In other words, for a fixed lens the line PQ 
 is shorter when AP and AQ are equal than for any 
 other position of the leus ; conversely, if the line PQ 
 remains constant, a stronger lens is required when it 
 is moved from its central position at A, whether it is 
 moved toward P or toward Q. Hence we have this 
 rule : If the distance between object and lens is less 
 than twice the focal length of the lens, the power of 
 the lens is diminished by movingit from the eye and 
 toward the object ; if, on the other hand, the distance 
 between object and lens is more than twice the focal 
 length of the lens, the power of the latter is increased 
 by moving it away from the eye. It will be observed 
 that the distance at which a book is held for reading 
 is ordinarily less than twice the focal length of the 
 lens worn as a reading glass ; consequently the power 
 of such a glass is weakened by moving it from the 
 eye. 
 
 In Fig. 33 we have a representation of an eye 
 myopic either from abnormal length of axis or from 
 
108 
 
 HANDBOOK OF OPTICS 
 
 act of accommodation. Without the aid of a lens 
 the conjugate of § is at J? on the retina ; but in 
 order that an object at P, nearer than Q, may be 
 seen, a convex lens is required ; then in the refrac- 
 tion by the lens, P and Q are conjugate. It is seen 
 that AP is positive and AQ is negative ; therefore 
 AP must be less than 2P, for it must be less than F 
 in order that A Q may be negative. Hence, as in the 
 
 Fig. 33. 
 
 previous case, the power of the lens vi^ill be dimin- 
 ished by moving it away from the eye. 
 
 From this we see that the statement that pres- 
 byopes increase the power of their glasses by moving 
 them to th.e tip of the nose is inaccurate. The power 
 of the lens can be increased in this way only on con- 
 dition that the object is moved in the same direction 
 and to the same extent as the lens ; but as the object 
 is removed from the eye, the retinal image becomes 
 smaller — a manifest disadvantage. 
 
CHAPTER VII 
 
 THE EFFECT OF SPHERICAL LENSES UPON THE 
 SIZE OF RETINAL IMAGES 
 
 Spherical lenses used for the purpose of bringing 
 the image of an object to an accurate focus on the 
 retina have also an effect upon the size of retinal 
 images. In many cases this is so slight as to escape 
 notice ; but when strong glasses are worn the effect 
 is considerable. If the glass before one eye differs 
 much in strength from that before the other eye, 
 confusion results from the unequal images, and such 
 glasses are usually rejected. The size of the image 
 after refraction through any optical system is ob- 
 tained, as we have learned, from the equation 
 
 t = — 2, 
 
 U 
 
 where u is the distance of the object from the an- 
 terior focus of the system, and F^ is the anterior 
 focal distance. This equation gives a negative value 
 for the image when it is real, since such an image 
 lies on the opposite side of the axis to the object, 
 and is consequently inverted. As we shall have in 
 
 109 
 
110 HANDBOOK OF OPTICS 
 
 our further studies to deal only with retinal images, 
 we shall neglect the minus sign, because the image 
 is seen by the observer as erect, the transformation 
 being made by mental act. Hence we shall consider 
 as positive those images which appear to be erect ; 
 and we shall consider as negative those which appear 
 to be inverted. Our equation thus becomes 
 
 « = • — ^• 
 u 
 
 If now a lens be introduced before the eye, we 
 shall have a new optical system ; and, as before, the 
 size of the image formed by this system will be 
 found from the equation 
 
 where Mj represents the distance of the object from 
 the new anterior focus, and F represents the new 
 focal distance. Hence the relation of the size of 
 the image with the lens to that without the lens 
 is expressed by the equation 
 
 -V 
 
 u X F 
 
 i u-y X F,^ 
 
 2 
 
 When the distance of the object is great in com- 
 parison with the change in position of the anterior 
 
EETINAL IMAGES HI 
 
 focus caused by adding the lens, then u and Mj may- 
 be considered identical. As this is true in the case 
 of lenses used as spectacles, we have the simplified 
 expression 
 
 But on page 93 we learned that the anterior focal 
 distance of the eye combined with a lens is derived 
 from the equation 
 
 ^= ^\^->. 
 
 F^ + F^-e 
 
 Making this substitution, we have 
 
 Ii_ 
 
 i F^ + F^-e 
 
 In this equation F^ is the focal length of the lens, 
 F^ is the anterior focal length of the eye, and e is 
 the distance between eye and lens. This formula 
 is sufficiently accurate to determine the effect of 
 spectacles upon retinal images except when the 
 object is Yerj near the eye ; but as we wish to show 
 the magnifying power of lenses in all positions in 
 front of the eye, we shall examine the condition 
 w^hen u and Mj cannot be considered identical. Re- 
 ferring to Fig. 28 we see that u = PF^, and m^ = PF. 
 
112 HANDBOOK OF OPTICS 
 
 Hence Wj =. w + FF^ = u + (^AF — AF^ ; 
 
 4 
 
 but on page 93 we found that AF is equal to 
 
 F^^F^-e 
 We see also that 
 
 AF^ = e-F^. 
 
 From this we find 
 
 uF^ + u{F^ - e) + (J; - g)^ . 
 
 F^ + F^-e 
 
 or, 
 
 and i^= ?^^^^ ; 
 
 i uF^ + uiF^~e-) + iF^-ey' 
 
 or, 
 
 ^1 
 
 ' ^i+(J'2_e) + I:5i-=liI'. 
 
 This is the general expression for the magnifying 
 power of any lens in combination with the eye, or 
 with any other optical system. Examining this 
 expression, we see that if F^ = e, that is, if the lens 
 be placed at the anterior focus of the eye, 
 
RETINAL IMAGES 113 
 
 Hence a lens so placed has no effect upon the size 
 of retinal images. This is true whether the lens is 
 convex or concave, and irrespective of the distance 
 of the object. 
 
 If the lens be convex and F^ be less than e, that 
 is, if the lens be vrithout the anterior focus of the 
 eye, F^ — e will be negative, but (^F^ — e)^ will be 
 positive. The least_ value which u can have is equal 
 to e — F^, since the object cannot lie between the 
 lens and eye. When u = e — F^, i' and i are equal, 
 which indicates that there is no effect on the size of 
 the image when object and lens are in contact. But 
 as the distance of the object is increased and u 
 assumes a greater value, the lens exerts a magnify- 
 ing power upon the image, and this magnification 
 increases as the distance between lens and object 
 increases. We observe also from a further study of 
 the equation that as e, the distance between eye and 
 lens, varies, the magnifying power varies. When 
 this distance becomes such that 
 
 <ie-F^^=F^ + ^^^-'^\ 
 u 
 
 the denominator of the expression which denotes the 
 
 i' 
 magnifying power becomes zero, and — becomes 
 
 infinite. 
 
 If (e - #2) is greater than J\ + (-^g ~ ^)^ , then ^ 
 
 becomes negative. 
 
114 HANDBOOK OF OPTICS 
 
 To render the meaning of this clear, we shall first 
 suppose the object to be so distant in comparison 
 with the distance between lens and eye that the 
 
 fraction ^^-2 ^ may be neglected as inappreciable. 
 
 In this case — becomes — ^ , and this expres- 
 
 i F^ + F^-e ^ 
 
 sion is negative when e — F^is greater than F-^, that 
 
 is, the image is inverted when the distance of the 
 
 lens from the anterior focus of the eye is greater 
 
 than the focal length of the lens. An aerial image 
 
 of the object will then be formed by the lens in 
 
 front of the eye, and a second image will be formed 
 
 by the eye.* 
 
 As the object approaches the lens, the term 
 
 ^^—2 ^ can no longer be ignored, and e — F^, the 
 
 It 
 
 distance of the lens from the anterior focus of the 
 
 eye, must be greater than F-, + - — 2 L in order 
 
 u 
 that the inverted image be formed. 
 
 The meaning of this is that as the object ap- 
 proaches the lens, its image recedes behind the pos- 
 terior principal focus, F-^, and in order that the 
 aerial image be formed in front of the anterior focus 
 of the eye, the distance of the lens from the eye 
 
 * When the aerial image is near the anterior focus, the final 
 image will lie far behind the retina, and no distinct image will be 
 perceived by the eye. 
 
RETINAL IMAGES 115 
 
 must be greater than when the object is more re- 
 mote. If we solve the equation 
 
 :-^.=i'.+a^'. 
 
 u 
 
 we obtain the relation between e — F,^ and u, which 
 exists when — is infinite. This relation thus becomes 
 
 i(i±V' 
 
 «-«,=?(i±\;i-i^ 
 
 Since we cannot extract the square root of a nega- 
 
 4:F 
 
 tive quantity, the fraction • 1 must be less than 1, 
 
 u 
 
 or equal to it, in order to give a real value to the ex- 
 pression. Hence we see that the least value which u 
 
 i' 
 can have and satisfy the condition —=ao is m = 4 ^j; 
 
 and when u = 4iF^ we find the corresponding value 
 
 ioY e — F^to be -. In other words, the image can- 
 
 not become infinite, and consequently cannot become 
 negative, when the distance of the object from the 
 anterior focus of the eye is less than four times the 
 focal length of the lens. When this distance is 
 equal to four times the focal length of the lens, the 
 image becomes infinite provided the lens is placed 
 midway between the object and anterior focus of the 
 eye. All this follows from what has been said in 
 the preceding chapter in regard to the effect of 
 
116 HANDBOOK OS" OPTICS 
 
 changing the position of a lens in near vision, and 
 we need not have deduced this result algebraically. 
 We have seen that PQ (Fig. 32) is shorter when the 
 lens, A, is half way between P and Q than in any 
 other position. Similarly, in Fig. 28, the least dis- 
 tance from #2 3't which P may be situated and still 
 form an image at this point is four times the focal 
 length of the lens, and the lens must be at the mid- 
 way point of the line PF^. Whether the lens be 
 moved toward the object or toward the eye, the 
 image will be thrown to the right of F^, and the 
 final image will be reduced in size. Thus the image 
 attains its greatest size when the lens is in this mid- 
 way position, and we have this rule for determining 
 the effect of changing the position of the lens : As 
 the convex lens is removed from the eye, the magni- 
 fying power increases so long as the distance of the 
 object from the lens is more than twice the focal 
 length of the lens, and when the distance between 
 object and lens is less than twice the focal length of 
 the lens, the magnifying power is diminished by 
 further removal of the lens from the eye. 
 
 If, while the lens is convex, F^ — e be positive, 
 that is, if the lens be placed within the anterior 
 focus of the eye, then i' will always be less than i. 
 
 We next suppose the lens to be concave. In this 
 case F-^ is negative. Our equation shows that when 
 F^ is less than e, or when the lens is without the an- 
 
RETINAL IMAGES 117 
 
 terior focus, i' is less than i, except when u= e — Fg,; 
 in this case, as with convex lenses, i' and i are equal. 
 When F^ is greater than e, i' is greater than i. 
 
 To summarize : A convex or concave lens placed 
 at the anterior focus of the eye or of any optical sys- 
 tem, though it alters the position of the image of an 
 object, has. no effect upon the size of the image. 
 
 A convex lens placed without the anterior focus 
 of an optical system magnifies the image, the degree 
 of magnification varying with the distance between 
 the lens and anterior focus, and with the distance 
 between the object and the lens. 
 
 A convex lens placed within the anterior focus of 
 an optical system minifies the image, the degree of 
 minification varying with the distance between the 
 lens and anterior focus and with the distance between 
 the object and lens. 
 
 A concave lens placed within the anterior focus 
 magnifies the image ; if placed without the anterior 
 focus it minifies the image, the degree of magnifi- 
 cation or minification varying with the distance be- 
 tween lens and anterior focus and with that between 
 object and lens.* 
 
 * The student should verify these phenomena by taking a con- 
 vex lens of 20 dioptres, which may represent the eye ; if another 
 lens be held before this and be moved to and fro while an object is 
 viewed through the combination, the effect of the second lens upon 
 the size of the image can be easily noted. 
 
118 HANDBOOK OF OPTICS 
 
 The anterior focus of the eye we have found to 
 be 13.7504 mm. in front of the cornea. Spectacle 
 glasses, being usually worn farther from the eye 
 than this on account of the projecting eyelashes, 
 must affect the size of retinal images ; convex lenses 
 magnify and concave lenses minify these images. 
 Thus in the hyperopic eye, which sees with the aid 
 of a convex lens, the image is larger than it would 
 be in the emmetropic eye, because the optical sys- 
 tem is the same in the two eyes, and the addition 
 of a convex lens placed without the anterior focus 
 magnifies the image. Similarly, in a myopic eye, 
 which sees with the aid of a concave lens placed 
 without the anterior focus, images will be smaller 
 than in an emmetropic eye. 
 
 We must now compare the image as formed by 
 the unaided hyperopic eye with that formed by the 
 emmetropic eye. The hyperopic eye is enabled to 
 see objects by an increase of curvature of the crys- 
 talline lens, by which means its dioptric power is 
 increased and its focal distances are diminished. 
 Hence the size of the retinal image is diminished 
 by the change, since this is proportional to the an- 
 terior focal distance. The hyperopic eye without 
 a correcting lens has smaller images than the emme- 
 tropic eye ; and as the same eye with a lens has 
 larger images than the emmetropic eye, it is evident 
 that in hyperopia images are increased in size by the 
 
RETINAL IMAGES 119 
 
 correcting lens. The same holds true in near vision, 
 since a greater amount of increase in curvature is 
 necessary for the hyperopic eye to see near objects 
 than for the emmetropic eye to see the same objects, 
 and in consequence the image in the hyperopic eye 
 is smaller than in the normal eye ; but with the aid 
 of the correcting lens the image is larger in the 
 hyperopic than in the normal eye. Likewise, pres- 
 byopes, who wear convex lenses to enable them to 
 see near objects, have larger images than those who 
 see near objects by act of accommodation. 
 
 The myopic eye cannot adapt itself to distant 
 vision, and hence without a lens all images of dis- 
 tant objects are blurred. Such an eye, however, 
 sees near objects either without any increase of 
 curvature or with less increase than would be re- 
 quired in an emmetropic eye. Thus we see that 
 the myopic eye has the advantage of larger images 
 of near objects than the normal , eye has. If the 
 myopia is corrected by a concave lens, the eye must 
 now make the same effort of accommodation that the 
 normal eye makes. The image is accordingly mini- 
 fied by the increase of curvature and by the concave 
 lens if this is worn without the anterior focus of the 
 eye. In the high degrees of myopia, in which strong 
 glasses are required, the minifying effect is a serious 
 obstacle to their use. In these cases the lenses 
 should be worn as near the eyes as possible. 
 
120 HANDBOOK OF OPTICS 
 
 We shall now investigate the aphakic eye. The 
 second principal focal distance of this eye is 
 31.095 mm. As the retina is only 22.8326 mm. 
 behind the cornea, we have in the aphakic eye a high 
 degree of curvature hyperopia. The anterior focal 
 distance of the aphakic eye is 23.2659 mm. Hence 
 the eye which has been deprived of its crystal- 
 line lens has larger images than the normal eye. 
 The image, however, will be formed far behind the 
 retina ; and in order to bring it forward to the 
 retina a strong convex lens is required. This lens 
 worn as a spectacle glass will be within the anterior 
 focus of the aphakic eye. It will, accordingly, 
 reduce the size of images so that they more nearly 
 correspond in size with those as formed in the nor- 
 mal eye. If the eye be hyperopic, prior to the 
 extraction of the lens, the images will be still more 
 nearly approximated . in size to those formed in 
 the normal eye, since a stronger convex lens will 
 be required to focus rays on the retina than if the 
 eye had been normal. But even in high degrees 
 of hyperopia the image will not be reduced by the 
 correcting lens to the size of the normal image. 
 
 In the myopic eye images will be larger than if 
 the eye had been emmetropic before extraction, 
 because a less convergent lens will be required than 
 if the eye had been normal. Thus we see that in 
 all aphakic eyes the retinal images, as formed with 
 
RETINAL IMAGES 121 
 
 the aid of correcting lenses, are larger than in nor- 
 mal eyes, and that of aphakic eyes, those which 
 were hyperopic prior to the extraction are least 
 favorably situated as regards the size of images, 
 and those which were myopic prior to the extraction 
 are most favorably situated in this respect. To this 
 enlargement of images is due the remarkable increase . 
 of visual acuity after extraction of the lens for the 
 cure or improvement of myopia. We have seen 
 that the excessively myopic eye which requires a 
 strong concave lens to produce clear images is very 
 much hampered by the minifying effect of the cor- 
 recting lens. If, after the extraction of the cr3'stal- 
 line lens, parallel rays of light are focused on the 
 retina without the aid of a lens, the size of the image 
 . of a distant object, as formed in this ej-e, will be 
 to the size of the corresponding image m a nor- 
 mal eye as the anterior focal distance of the aphakic 
 eye is to that of the normal eye. Omitting fractions 
 this ratio is as 23 to 15.* When we consider that, 
 before the extraction, images were reduced in size, 
 we are prepared to expect great improvement in 
 visual acuity in those cases in which the operation 
 has been successfully accomplished. 
 
 To the enlargement of images, though in a less 
 
 * This ratio refers to the linear dimensions of the images. The 
 relative amount of retinal surface covered by the images is pro- 
 portional to the squares of these numbers. 
 
122 HANDBOOK OF OPTICS 
 
 degree, is also due the excellent visual acuity which 
 sometimes follows cataract extraction in emmetropic 
 eyes. In the most successful cases a visual acuity 
 of -1^ may be obtained in spite of the fact that the 
 pupil is not entirely free from particles of opaque 
 matter. 
 
 Finally, we shall consider the effect upon retinal 
 images of changing the position of the convex lens 
 in near vision. We have seen, as we might infer, 
 that the size of the image varies according to the 
 same rule as does the dioptric power of the lens 
 in any position, that is, if the distance between the 
 object and lens is less than twice the focal length 
 of the lens, the power of the lens is weakened 
 and the size of the image is diminished by removal 
 of the lens from the eye ; and if the distance of 
 the object is more than twice the focal length of 
 the lens, the power of the lens and the size of the 
 image are increased by removal of the lens from 
 the eye. Thus, as the prevalent 'belief that pres- 
 byopes increase the power of their glasses by placing 
 them on the tip of the nose was shown to be not 
 generally true, so the statement made in some text- 
 books that by so placing them larger images are 
 obtained, is also erroneous. 
 
 The condition when the eye is emmetropic and 
 without accommodative power is shown in Fig. 31. 
 
 When one looks through an ordinary hand magni- 
 
EETINAL IMAGES 123 
 
 fying glass, the same conditions are present. If, as 
 is usually the case, the distance of the object from 
 the anterior focus of the eye is less than four times 
 the focal length of the lens, the image is never in- 
 verted. If this distance is less than twice the focal 
 length of the lens, then as the lens is removed from 
 the eye the image is diminished in size. But when 
 the lens A is farther from the object P than the 
 focal length of the lens, the pencils of light after 
 passing through the lens are convergent, and cannot 
 be focused by a normal eye ; hence the nearest point 
 to the eye at which the lens can be placed and afford 
 distinct vision is such that the object is at the focus 
 of the lens, and this is for the emmetropic eye the 
 position of greatest magnifying power. The hyper-' 
 opic eye can focus the convergent pencils and receive 
 a clear image when the lens is to the right of ^ ; on 
 the other hand, the myopic eye can only focus diver- 
 gent pencils such as are formed when the lens is on 
 the left of A and nearer to the object. 
 
CHAPTER VIII 
 
 CYLINDRICAL LENSES 
 
 We have assumed that the refracting surfaces of 
 the eye are spherical in form. This is permissible 
 in normal cases, but in a large proportion of eyes the 
 refracting surfaces — more especially the cornea — 
 do not have this regularit)' of curvature ; the curva- 
 ture is found to vary appreciably in different meridi- 
 ans, the meridians of greatest and least curvature 
 being usually at right angles to each other. These 
 are called the principal meridians. Such a surface 
 is called a torus or toric surface. The outer convex 
 surface of a ring is an example of a toric surface. 
 In an eye whose cornea is of this form the image of 
 a point will not be another point, and from this fact 
 the defect is called astigmatism. This asymmetry of 
 refraction was first noticed by Dr. Thomas Young, 
 the celebrated physicist, but Sir George Airy, who 
 had the defect in his own eyes, was, in 1827, the 
 iirst to correct it by means of suitable lenses. 
 
 Astigmatism may be produced by faulty curvature 
 or oblique position of the crystalline lens, but its 
 most common source is asymmetrical curvature of 
 
 124 
 
CYLINDRICAL LENSES 125 
 
 the cornea. If the curyature of the cornea be nor- 
 mal in one meridian and too little or too great in the 
 meridian at right angles to this one, the defect may 
 evidently be corrected by a lens having no curvature 
 in the normal meridian, and having in the meridian 
 at right angles to this such curvature as will counter- 
 act the defective curvature of the cornea in this me- 
 ridian. A lens of this nature would be cylindrical 
 in form. If, with the equalization of curvature, the 
 eye is still ametropic, a spherical lens may be com- 
 bined with the cylindrical lens. The spherical 
 surface may be ground on one side of the glass 
 and the cylindrical on the other ; or a toric curva- 
 ture may be ground on one side of the glass, leav- 
 ing the other side plane ; and, if desired, this may 
 then be made concave and the lens periscopic. 
 Toric lenses are not much used, as they are more 
 difficult to grind than spherical and cylindrical 
 surfaces. 
 
 In a cylindrical -lens the line drawn through the 
 summit of curvature and parallel to the axis of the 
 cylinder is called the axis of the lens.* Reference 
 to Fig. 34 renders it clear that in the direction at 
 right angles to the axis AAl of the lens a cylindrical 
 lens has the same action that a spherical lens of like 
 radius of curvature and index would have in this 
 
 *This must not be confounded with the axis of the optical 
 
126 
 
 HANDBOOK OF OPTICS 
 
 direction, and that the deviating power of the lens 
 is confined entirely to this meridian. Rays of light 
 parallel to the axis 00' passing through such a lens 
 would not be united in a focus. They would, how- 
 ever, all meet a line which is parallel to the axis of 
 the lens, and whose distance from the lens is equal 
 to the focal length of a spherical lens of the same 
 
 Fig. 34. 
 
 radius of curvature and index. This distance is 
 called the focal distance of the cylindrical lens, and 
 the line is called the focal line. Similarly, rays 
 diverging from a point will, after refraction by the 
 lens, all pass through a line conjugate to the point 
 from which the rays proceed. 
 
 If we take another cylindrical lens whose axis is at 
 right angles to that of the first lens, it will bring all 
 
CYLINDRICAL LENSES 
 
 127 
 
 the rays from a point into a line at right angles to 
 the first line.* 
 
 In Fig. 35 the rays from the point P all pass 
 through the line AB ; and rays from P' all pass 
 through the line A'B' at right angles to AB. 
 Hence, if these two lenses have equal refractive 
 
 Fig. 35, 
 
 power, and if we combine them so that the rays after 
 deviation by the first lens pass immediately through 
 the second lens, the effect of both lenses will be to 
 cause the rays to meet in the point where the lines 
 intersect ; for the rays must all pass through both 
 lines. In other words, two equal cylindrical lenses 
 placed with their axes at right angles are equivalent 
 
 * It will be understood that we are bound by the same restric- 
 tions as in the case of spherical lenses, that is, we must suppose 
 that only rays near the axis of the pencil pass through the lens. 
 
128 
 
 HANDBOOK OF OPTICS 
 
 to a spherical lens of the same index and radius of 
 curvature. In Fig. 36 let P represent the point at 
 which a ray parallel to the axis of the refracting 
 system meets a spherical lens ; then when this ray 
 reaches the focus of the lens it will have been so 
 deviated as to pass through the axis at F. The mo- 
 tion of a point from P to F may be considered as the 
 
 ?/ 
 
 Fig. 36. -"-V 
 
 resultant of three motions at right angles to each 
 other : 1st, from P to M; 2d, from M to ; and 
 3d, from to F. In the case of the spherical lens 
 the deviation in the plane PMOHwill be represented 
 by the line PO ; if instead of a spherical lens we 
 take two equal cylindrical lenses with axes at right 
 angles, then in this plane the ray will be deviated by 
 one lens from P to M, and by the other from M to 0. 
 
CYLINDRICAL LENSES 
 
 129 
 
 The result is the same 
 in the two cases, li the 
 two cylindrical lenses have 
 different focal lengths, 
 their combined effect will 
 not be equivalent to that 
 of a spherical lens, be- 
 cause their focal lines 
 will not intersect. If 
 the focal length of the 
 first lens is less than that 
 of the second, it is clear 
 that when a ray parallel to 
 the axis of the refracting 
 system has been brought 
 by the first lens to its in- 
 tersection with the first 
 focal line, it will not yet 
 have reached its intersec- 
 tion with the second focal 
 line. 
 
 Let Q (Fig. 37) repre- 
 sent a combination of two 
 cylindrical lenses whose 
 axes are vertical and hori- 
 zontal respectively. The 
 line AB is conjugate to P 
 as regards the lens whose 
 
130 HANDBOOK OF OPTICS 
 
 axis is vertical ; and A'£', at ri^ht angles to AB, 
 is conjugate to _P as regards the lens whose axis 
 is horizontal. We take two rays, PH and PH', 
 so situated that they pass through the axis of the 
 second lens. Hence tliey will be deviated only 
 by the first lens ; they will intersect at U in the 
 first conjugate focal line, and after crossing at this 
 point they will meet the second conjugate focal line 
 at T and T'. Likewise, if we take two rays, PS and 
 PS', which pass through the axis of the first lens, 
 they will be deviated only by the second lens and will 
 intersect at Jl' in the second conjugate focal line, 
 meeting the first conjugate focal line before intersec- 
 tion at Zand /'. All rays which do not pass through 
 the axis of either lens will be deviated by both lenses, 
 and the deviation by one lens will be superposed upon 
 that of the other. The action of the first lens is such 
 as to cause all rays from P to pass through, the verti- 
 cal line AB conjugate to P ; and since the second 
 lens deviates light in the vertical meridian, its effect 
 is to change the position of rays in the line AB, but 
 not to deviate them out of this line ; hence all rays 
 from P must pass through AB. Similarly, the action 
 of the second lens is such as to cause all rays from P 
 to pass through the horizontal line A'B', and since 
 the first lens deviates light only in the horizontal 
 meridian, its effect is to change the position of rays 
 in the line A'B', but not to deviate them out of this 
 
CYLINDRICAL LENSES 131 
 
 line. From this we see that rays proceeding from a 
 point will never be united in a focus by such a lens ; 
 they will, however, all pass through two lines which 
 are at right angles to each other, called focal lines. 
 If the point is so far distant that the rays may be 
 regarded as parallel, the lines AH and A'B' are the 
 principal focal lines. The interval between the prin- 
 cipal focal lines is called the focal interval of Sturm.* 
 
 It is clear that the more nearly equal the two cylin- 
 drical lenses are in power, the shorter will be the 
 focal interval, and the more closely will the image of 
 a point resemble a point. 
 
 Since two equal cylindrical lenses with axes at 
 right angles are equivalent to a spherical lens, it 
 follows that two unequal cylindrical lenses, similarly 
 combined, are equivalent to a sphero-cylindrical lens ; 
 for we may regard the unequal lenses as composed 
 of two equal cylindrical lenses with the addition of 
 another cylindrical lens. 
 
 In order to investigate the image of a point after 
 refraction by an astigmatic surface, we must suppose 
 a screen to be placed at different positions in the 
 path of the rays so as to intercept them. If a 
 screen be placed at AB, the image of the point P, 
 as formed on the screen, will be the vertical line //'. 
 
 *Tlie theory of refraction by asymmetrical surfaces was first 
 demonstrated by Sturm in 1845. See " Comptes Eendus de I'Aoad. 
 des Sci. de Paris," torn, xx., pp. 554, 761, 1238. 
 
132 HANDBOOK OF OPTICS 
 
 Likewise, the image of the point as formed at A'B' 
 is TT'. To the left of AB the image will be ellip- 
 tical, for the rays .will not yet have intersected 
 either focal line, but they will be nearer the vertical 
 than the horizontal intersection. To the right of 
 A'B' the image will be elliptical, for the rays have 
 passed both vertical and horizontal intersections, but 
 are nearer to the horizontal than to the vertical 
 intersection. Between AB and A'B' the image will 
 also be elliptical, except in one position, in which 
 the distance of the rays from the axis is the same 
 in the vertical as in the horizontal meridian ; it will 
 then be a circle. This is called the circle of least 
 confusion. To the left of this circle the ellipse has, 
 in our case, the long axis vertical, and to the right 
 the long axis is horizontal. 
 
 The image of a line will vary according to its 
 position in relation to the focal lines. If the line 
 be a vertical one passing through P, then at AB 
 every point of the line will have as its image a 
 vertical line such as //'. The image of the vertical 
 line will be a lengthened and intensified line. At 
 A'B', however, every point of the vertical line will 
 have as its image a horizontal line such as TT', and 
 the image of the vertical line will be an aggregation 
 of horizontal lines ; it will therefore be a broad and 
 indistinct line. At any other point tlie image will con- 
 sist of a superposition of confusion ellipses or circles. 
 
CYLINDRICAL LENSES 133 
 
 If the line be horizontal, we shall have a broad 
 blurred image at AB and a clear intensified image 
 at A'B'. If the line be neither vertical nor hori- 
 zontal, its image will be blurred in all positions. 
 
 From what has been said we see that a vertical 
 line appears distinct at the focus of the cylindrical 
 lens whose axis is vertical, and a horizontal line ap- 
 pears distinct at the focus of the lens whose axis is 
 horizontal; or, since the meridians of refraction are 
 at right angles to the axes, a vertical line appears 
 distinct at the focus of the horizontal meridian, and 
 a horizontal line appears distinct at the focus of the 
 vertical meridian.* 
 
 The formulae which we have deduced for spherical 
 lenses are also applicable to cylindrical lenses. We 
 have only to bear in mind that the action of the 
 latter is confined entirely to the meridian at right 
 angles to the axis of the lens. What has been said 
 of the effect of spherical lenses upon the size of reti- 
 nal images applies, therefore, to cylindrical lenses, 
 with the understanding that this effect is confined 
 to the refracting meridian, no effect being produced 
 by the cylindrical lens in the meridian of its axis. 
 
 Astigmatism of the eye is a curvature defect, 
 
 * The student who finds it difficult to comprehend refraction 
 through the double cylindrical or sphero-cylindrical lens should 
 construct for himself, or procure from an instrument maker, thread 
 models, which illustrate very clearly astigmatic refraction. 
 
134 HANDBOOK OF OPTICS 
 
 while hyperopia and myopia are, in a large majority 
 of cases, axial defects. The effect of hyperopia or 
 myopic astigmatism upon retinal images will, con- 
 sequently, not be analogous to that of axial hyper- 
 opia or myopia. If the eye is hyperopic in one 
 meridian and emmetropic in the meridian at right 
 angles to this, the defect in curvature in the hyper- 
 opic meridian is the same as if a concave cylindrical 
 lens were placed in contact with a normal cornea. 
 The effect of such a lens, since it would be within 
 the anterior focus of the eye, would be to enlarge 
 images in the refracting meridian of the lens. In 
 other words, since the curvature of the eye is less 
 in the hyperopic than in the emmetropic meridian, 
 the anterior focal distance is greater in the faulty 
 than in the normal meridian ; and consequently the 
 image, of an object will be too large in the former 
 meridian, for the size of the image is proportional 
 to the anterior focal distance. Similarly, in myopic 
 astigmatism, the image is too small in the myopic 
 meridian. But we must remember that the retina 
 is not in the proper position to receive an accurately 
 focused image in the faulty meridian ; and, to dem- 
 onstrate the effect of this faulty position of the 
 retina, let 0, Fig. 38, be the optical centre of a lens 
 which may, for the present purpose, represent the 
 eye ; then, if A be conjugate to P, the image oi PQ 
 will be AB. If, now, the screen or retina remain at 
 
CYLINDRICAL LENSES 135 
 
 A, while the lens is increased in power so that P' is 
 conjugate to P, the true image, P' Q', will be smaller 
 than AB, but the indistinct image, as intercepted -by 
 the screen, will be AC, which is larger than AB.* 
 
 This is why a round object such as the full moon 
 appears greater in the meridian of myopic refraction 
 than in the emmetropic meridian, though the accurate 
 image is less in the myopic meridian. In like man- 
 
 Q. 
 
 ^^:^^ 
 
 
 
 c 
 
 
 ^^==::^^ A 
 
 Q'^ 
 
 
 B 
 
 _p 
 
 ^^>^^ 
 
 ^V 
 
 
 A P, 
 
 
 Us^^ 
 
 
 ^ 
 
 ^^ 1 
 
 Q 
 
 Fig. 38. 
 
 
 
 ^. 
 
 ner, if the screen remain at A while the conjugate of 
 P is changed from A to Pj', it is apparent that by 
 the change both the focused and blurred images will 
 be rendered larger than the normal, but the blurred 
 image will be less enlarged than the other. 
 
 We can now understand the influence upon retinal 
 images of cylindrical lenses used as spectacles. A 
 proper convex cylindrical lens worn as a spectacle 
 
 * The change in position of the optical centre, being too slight 
 to affect the result, is neglected in the figure. 
 
136 HANDBOOK OF OPTICS 
 
 brings the image of an object to an accurate focus on 
 the retina ; but it also enlarges the image in the 
 meridian at right angles to the axis of the lens. For, 
 as we h^ve seen in Fig. 38, if the lens be worn at the 
 anterior focus of the eye, the new image will be of 
 the same size as Pj' Q^', since the effect of the lens is 
 to bring the image forward without changing its size. 
 If the lens be worn without the anterior focus of the 
 eye, then the new image will be larger than Pj' Qj' ; 
 in either case it will be larger than the blurred image 
 AC, which the eye receives without the lens, and 
 larger than AB', the normal image. A proper con- 
 cave cylindrical lens throws the image back upon the 
 retina ; it also minifies the image in the refracting 
 meridian of the lens. If the lens be worn at the an- 
 terior focus of the eye, P' Q' will represent the size of 
 the new image ; and if the lens be worn without this 
 focus, the image will be smaller than F'Q' ; in either 
 case it will be smaller than the blurred image AC, 
 which the eye receives without the lens, and smaller 
 than AB, the normal image. Hence we see that 
 cylindrical lenses worn as spectacles do not, under 
 any circumstances, produce normal retinal images ; 
 all objects are magnified in the refracting meridian 
 b}'^ a convex lens, and minified in this meridian by a 
 concave lens. If the lens could be worn in contact 
 with the cornea, the seat of defective curvature, nor- 
 mal images would result. The same position of the 
 
CYLINDRICAL LENSES I37 
 
 spherical lens would be necessary to produce images 
 of normal size in curvature hyperopia and myopia ; 
 but, as shown in the preceding chapter, in ordinary 
 axial hyperopia and myopia, the lens must be placed 
 at the anterior focus of the eye in order that images 
 of normal size be produced. 
 
CHAPTER IX 
 
 THE TWISTING PROPERTY OF CYLINDRICAL LENSES 
 
 As a consequence of the effect of cylindrical lenses 
 upon retinal images, it follows that if one hold such 
 a lens in front of the eye, and through the lens look 
 at a distant rectangular object as a picture frame or 
 test-type card, there will be observed a distortion of 
 the object, which will vary with every variation in 
 the position of the lens. If the axis of the lens be 
 parallel to one of the sides of the object, the rectan- 
 gular form of the object will be retained, but the 
 ratio of the sides will be altered. The side which is 
 parallel to the axis will not be changed, while that 
 which is perpendicular to the axis will be increased 
 or diminished. If now the lens be rotated in its own 
 plane, the distortion will no longer be confined to 
 the apparent size of the object ; it will also affect the 
 direction of the lines forming the sides, so that the 
 rectangular object will assume the form of an oblique 
 parallelogram. Use is made of this phenomenon to 
 determine the position of the axis of a cylindrical 
 lens. Looking through the lens at a distant straight 
 line, we observe the position of the lens in which 
 
 138 
 
CYLINDRICAL LENSES 
 
 139 
 
 there is no apparent deviation of the line ; the axis 
 of the lens must be either parallel or perpendicular to 
 this line. 
 
 We have seen that a cylindrical lens has in its 
 refracting meridian the same effect that a similar 
 spherical lens would have. The formula by which 
 the magnifying or minifying power is obtained has 
 been given in Chapter VII. (page 112). 
 
 Fig. 39. 
 
 Let OA (Fig. 39) be a line parallel to the axis of 
 a cylindrical lens, and let BA be perpendicular to 
 the axis. If we look through the lens at the line 
 OA, the image of this line on the retina will be the 
 same as without the lens. If we look at the line 
 AB, and if the lens be convex, then AB will be 
 magnified in its own direction, and will appear as 
 the line AB'. If we now look at an oblique line 
 OB, its direction will be changed, for B will appear 
 at B' and U will appear at U', and so on for every 
 
140 HANDBOOK OF OPTICS 
 
 other point of the line. The line OB will conse- 
 quently assume the position OB' Hence any line 
 not parallel or perpendicular to the axis of the lens 
 undergoes an angular deviation when viewed through 
 a cylindrical lens. If a is the angle which the line 
 makes with the axis, and x the angle which the line 
 appears to make with the axis, we have 
 
 AB . AB' 
 
 tana = — ; tan.= — ; 
 
 AB' 
 
 from which tan x = —-—tan a. 
 
 AB 
 
 AB' 
 
 — — being the magnifying power of the lens may 
 
 AB 
 
 be called m. Then tan x = m tan a. When the lens 
 
 is concave m is less than unity, and the line OB' 
 appears as OB. 
 
 Taking a convex lens, we look through it at a 
 pencil held at arm's length from the eye. When 
 the pencil is parallel to the axis of the lens there is 
 no apparent deviation, but as we turn the pencil 
 through an angle a it appears to make the greater 
 angle x with the axis of the lens. At first x in- 
 creases more rapidly than a, and x— a, which is the 
 angle made by the apparent position with the real 
 position, becomes rapidly greater. As the pencil is 
 turned farther, the angle x — a changes more slowly, 
 then comes to a standstill ; and finally a increases 
 
CYLINDRICAL LENSES 141 
 
 more rapidly than x, and a; — a diminishes, so that when 
 the pencil has been turned through 90 degrees a has 
 overtaken x and there is no deviation. If for the 
 pencil we substitute a distant straight line and turn 
 the lens while the line remains stationary, we shall 
 have the same effect. As we turn the lens to the 
 right, the line appears deviated to the left, since the 
 apparent position always makes a greater angle with 
 the axis than the real line. After reaching a maxi- 
 mum deviation the apparent line now travels back- 
 ward, coinciding with the real position when the 
 lens has been turned through 90 degrees. If we 
 take a concave lens, m being less than unity, the 
 angle which the apparent position of the line makes 
 with the axis will be less than that which the real 
 position makes; consequently, when we look at a 
 distant line and turn the lens to the right, the ap- 
 parent position also moves to the right, and after 
 reaching a maximum deviation it travels backward, 
 coinciding with the real position when the lens has 
 been turned through 90 degrees. 
 
 This to-and-fro deviation, familiar to every one who 
 uses the oculists' trial lenses, vanishes under cer- 
 tain conditions when we experiment with the convex 
 lens. If we take a convex cylindrical lens of four 
 dioptres and, holding it about \ metre in front of 
 the eye, look at a vertical line across the room, we 
 shall have, upon turning the lens, the to-and-fro 
 
142 HANDBOOK OF OPTICS 
 
 motion of the line ; but as we move the lens farther 
 from the eye, the line becomes so indistinct that 
 we cannot determine its movement ; continuing to 
 increase the distance between the eye and lens, we 
 now notice that as we turn the lens the behavior 
 of the observed line has entirely changed. It no 
 longer moves to and fro, but as the axis of the lens 
 is turned through 90 degrees the apparent position 
 of the line moves through twice this angle, or 180 
 degrees. This phenomenon can be most easily ob- 
 served by holding a pencil or similar object at arm's 
 length and viewing it through a cylindrical lens of 
 ten or twelve dioptres held about ^ metre in front 
 of the eye. 
 
 To explain this we shall compare the action of the 
 cylindrical lens with that of the spherical lens of the 
 same radius of curvature and refractive index. It 
 has been shown that if a convex spherical lens be 
 held before the eye at a sufficient distance, an in- 
 verted image of an object will be formed in the air in 
 front of the eye, and a second image will be formed 
 by the eye. 
 
 In Fig. 40 let (1) represent a rectangular object 
 such as a picture frame ; then if it be viewed through 
 a spherical lens held beyond its focal length from the 
 eye, (2) will represent the object as it will appear 
 to the observer. If we replace the spherical lens 
 by a similar cylindrical lens with axis vertical, (3) 
 
CYLINDRICAL LENSES 
 
 143 
 
 will represent the object as it will appear to the ob- 
 server. The cylindrical lens has the same effect as 
 the spherical one in deviating the rays in the merid- 
 ian at right angles to the axis of the lens, that is, 
 rays from the right of the object are made to cross 
 over and intersect on the left, and vice versa. The 
 object therefore appears reversed in this direction ; 
 but it is not reversed in the meridian parallel to the 
 axis of the lens; in this meridian the rays are un- 
 aifected by the lens. Similarly, (4) represents the 
 
 (1) 
 
 □ (.2) 
 
 \ 
 
 (3) ..^ 
 
 (4) 
 
 Fig. 40. 
 
 object as it would appear when viewed through a 
 cylindrical lens with axis horizontal. 
 
 The image as formed with the cylindrical lens 
 will be much blurred, because the rays will, in one 
 meridian, enter the eye diverging from the aerial 
 image ; in the meridian at right angles to this 
 they have been unaffected by the lens, and con- 
 sequently diverge from the more distant object. 
 They cannot, therefore, be accurately focused on 
 the retina. 
 
 To this reversal of the object in one meridian 
 is due the apparent rotation of a line through 
 
144 
 
 HANDBOOK OF OPTICS 
 
 180 degrees, while the lens is turned through 90 
 degrees.* 
 
 In Fig. 41 let OB represent a line making the 
 angle a with the line OA, which is parallel to the 
 axis of the lens. From what we have just shown, it 
 
 Fig. 41. 
 
 is clear that if we view the figure AOB through a 
 convex cylindrical lens so placed as to form the aerial 
 image, then AB will appear in the position AB', and 
 OB will appear at OB' . Hence, if we suppose the 
 original position of the axis to be parallel to OB, in 
 this position there will be no apparent deviation of 
 
 * Dr. Carl KoUer, the discoverer of tlie anaesthetic property of 
 cocaine, lias demonstrated this phenomenon by means of analytical 
 geometry. See Graefe's "Archives," 1886. 
 
CYLINDRICAL LENSES 145 
 
 the line OB, but when the axis of the lens is turned 
 into the direction OA, the apparent position of OB is 
 changed from OB to OB'. The angle AOB', which 
 the apparent position of the line makes with the axis 
 of the lens, is as before x. From the figure it is seen 
 
 that tan x = — — - tan a ; where — — - represents the 
 AB AB 
 
 magnifying power of the lens at right angles to its 
 axis. Hence, tan x = m tan a. This is the same 
 equation as that derived for the to-and-fro deviation, 
 but the angle which the apparent position OB' makes 
 with the real position OB is now equal to the sum of 
 X and a, whereas in the former case it was equal to 
 their difference. Since m becomes negative when the 
 reversed aerial image is formed, x is also negative, 
 and adding a negative value is equivalent to sub- 
 tracting a positive value. When a is zero, x is also 
 zero ; as a increases, the negative value of x increases ; 
 and when a is equal to 90 degrees, x is equal to — 90 
 degrees. That is, if we turn the axis through 90 de- 
 grees from.OD to OA, in so doing the line OD will 
 be apparently rotated through 180 degrees and will 
 appear in the position OE. 
 
 It was shown in Chapter VII. that the reversed 
 image of a distant object is formed when the distance 
 of the lens from the anterior focus of the eye exceeds 
 the focal distance of the lens, but that as the object 
 approaches the lens the distance between the eye and 
 
146 HANDBOOK OF OPTICS 
 
 lens must be increased in order that the negative 
 image be formed. Hence, if a pencil or similar 
 object be held so that we have the continuous devia- 
 tion through 180 degrees, and if the pencil be then 
 moved nearer the lens, a point will be reached at 
 which this phenomenon will disappear, and upon 
 further approximation of the pencil to the lens, the 
 to-and-fro deviation will be seen. Since spectacle 
 lenses are never worn beyond their focal distance 
 from the eye, it is clear that, so far as this continuous 
 deviation is concerned, nothing analogous occurs in 
 the use of cylindrical spectacles. 
 
 As was shown in the preceding chapter, retinal 
 images in astigmatic eyes are not normal in their 
 proportions ; we have seen that the effect of astig- 
 matism upon images is similar to that of a cylindri- 
 cal lens placed in contact with a normal cornea. 
 A cylindrical lens so placed would have a magnify- 
 ing or minifying effect on images in the refracting 
 meridian of the lens ; and, since upon this property 
 depends the apparent deviation of lines, it is clear 
 that in astigmatic eyes all lines not parallel or per- 
 pendicular to the axis of the astigmatism are twisted 
 out of their proper relations. A rectangle whose 
 sides do not correspond in direction with the merid- 
 ians of greatest and least refraction, appears as an 
 oblique parallelogram. This distortion is, however, 
 slight, for the dioptric power of the eye is great in 
 
CYLINDRICAL LENSES 147 
 
 comparison with the amount of astigmatism. The 
 defect is not appreciable to the person whose eyes 
 are astigmatic, even if the astigmatism is of high 
 degree ; but when the astigmatism is corrected by 
 a suitable lens, complaint is frequently made of 
 annoying distortion of lines. This annoyance is 
 fortunately transito.ry. Since a cylindrical lens, as 
 worn before the eye, cannot reduce the retinal image 
 to its proper proportions, it is evident that it cannot 
 correct the distortion of lines ; furthermore, by 
 recalling what was said on this subject in the pre- 
 ceding chapter, it will be seen that the eflfect of the 
 lens is to increase the distortion in hyperopic astig- 
 matism ; in myopic astigmatism the distortion pro- 
 duced by the lens is in the opposite direction to that 
 existing in the blurred image without the lens. In 
 either case it is easy to see why annoyance should 
 arise when glasses are first worn.* 
 
 * Aside from this actual distortion of the retinal image, there 
 frequently occurs a distortion due to mental influence. For the 
 explanation of this peculiar phenomenon the following articles may 
 be consulted : 
 
 "The Effect of a Cyl. .Lens with Vert. Axis Placed before One 
 Eye," by O. F. Wadsworth, Trans. American Ophth. Soc, 1875; 
 "Binoc. Metamorphopsia," Archives of Ophthalmology, 1889, and 
 " New Tests for Binoc. Vision," Trans. American Ophth. Soc, 
 1890, by J. A. Llppinoott ; " Stereo. Illusions Evoked by Prismatic 
 and Cyl. Spec. Glasses," by John Green, Trans. American Ophth. 
 Soc, 1889. 
 
CHAPTER X 
 
 THE SPHERO-CYLINDRICAL EQUIVALENCE OF 
 BI-CYLINDRICAL LENSES 
 
 It was shown in Chapter VIII. that two cylin- 
 drical lenses whose axes are at right angles are 
 equivalent to a spherical lens if the two lenses have 
 the same dioptric power, and to a sphero-cylindrical 
 lens if the two lenses differ in power. It remains 
 now to find the equivalent of any two cylindrical 
 lenses whose axes are not at right angles. Formerly 
 it was not uncommon for oculists to prescribe 
 glasses consisting of a combination of two cylindri- 
 cal lenses obliquely inclined to each other. This 
 was done notwithstanding that attention had already 
 been called to the fact that such a combination is 
 equivalent to a sphero-cylindrical lens. Sir G. G. 
 Stokes, a celebrated English physicist, first demon- 
 strated this problem, at least for the special case 
 in which two lenses of equal but opposite curvature 
 are combined. He devised this combination as a 
 test for astigmatism. The two lenses are placed 
 the one over the other so that by rotating one lens 
 any angle between the axes can be obtained. This 
 
 148 
 
BI-CYLINDRICAL LENSES I49 
 
 combination is called the Stokes lens. To use it 
 as a test for astigmatism it should be placed before 
 the eye to be examined, and the angle between the 
 axes varied until the position of best vision is found, 
 when further improvement can usually be obtained 
 by adding a spherical lens. From a table con- 
 structed by calculation the amount of astigmatism 
 can be deduced.* 
 
 Bonders in his book on refraction refers to Stokes' 
 demonstration, and also presents a solution appli- 
 cable for any two cylindrical lenses ; but while his 
 conclusion is correct, the demonstration is defective ; 
 for, as will be subsequently shown, his assumption is 
 not generally true. In 1886 Dr. Jackson, of Phila- 
 delphia, read before the American Ophthalmological 
 Society a complete demonstration of the problem of 
 cylindrical refraction, and at the same meeting Dr. 
 Gustavus Hay, of Boston, offered a somewhat dif- 
 ferent solution, j- In 1888 Prentice J published a 
 solution, similar in principle to those of Jackson and 
 Hay, and in 1893 Dr. Weiland, § of Philadelphia, 
 published a solution based upon that of Donders, but 
 
 * The " Stokes lens" is chiefly of historical interest ; it is rarely 
 if ever used as a test for astigmatism at the present day, for other 
 more convenient tests have supplanted it. 
 
 t Trans. American Ophth. Soc, 1886. 
 
 } Dioptric Fnrmuloe, Cylindrical Lenses, 1888. 
 
 % Archives of Ophthalmology, Vol. XXII., No. 4, and Vol. 
 XXIII., No. 1. 
 
150 handbook; op optics 
 
 this, containing the same error as that of Bonders, 
 is not a general solution. There is also given in 
 Heath's " Geometrical Optics " * a solution by means' 
 of analytical geometry, which leaves nothing to he 
 desired except that this method is unsuitable for the 
 use of students who are not familiar with the higher 
 mathematics. Thus we see that this question has 
 not lacked investigation. With our more accurate 
 methods of examination it is of less practical im- 
 portance than formerly, for it is seldom necessary in 
 testing the vision of an eye to resort to two cylin- 
 drical lenses obliquely inclined; moreover, if, in any 
 case, it should be found convenient to place two 
 lenses in this manner, the sphero-cylindrical equiva- 
 lent can be found without formulae or calculation. 
 Nevertheless this subject must always be of interest 
 to the ocialist who wishes to have a scientific knowl- 
 edge of optics, and therefore, before giving the prac- 
 tical method of determining the equivalent in any 
 case, we shall demonstrate that any two cylindrical 
 lenses acting in combination are equivalent to two 
 other cylindrical lenses whose axes are at right 
 angles to each other, and consequently to a sphero- 
 cylindrical lens. Our apology for adding still 
 another solution to the list of those already pub- 
 lished is that in all these solutions the process of 
 
 * "Geometrical Optics," Heath, 2d ed., p. 186. 
 
BI-CYLINDRICAL LENSES 151 
 
 eliminating the special point at which the ray meets 
 the lens is unnecessarily tedious. 
 
 In studying refraction by two cylindrical lenses at 
 right angles to each other, we learned that a ray of 
 light in it& progress from a point, P, on a spherical 
 lens. Fig. 36, to its intersection with the axis at the 
 focus, -F, undergoes the same change in position as 
 if it were first moved from P to in the plane of 
 the lens, and then moved along the axis from to F. 
 Furthermore we learned that the change of position 
 from P to 0, in the plane of the lens, is identical 
 with the resultant of the two motions from P to M 
 and from M to 0, which would be produced by two 
 equal cylindrical lenses at right angles to each other. 
 In the same way we may study the result of the 
 deviation of a ray of light by two cylindrical lenses 
 obliquely inclined. In Fig. 42 let J.0 and JBO rep- 
 resent the axes of the two lenses; then a ray per- 
 pendicular to the plane of the lens, and meeting the 
 lens at P, would, if acted upon by the first lens only, 
 be so deviated in the meridian, PM, as to intersect 
 the principal focal line of the lens at the distance 
 F from the lens. The motion of the ray from A to 
 F (P is not represented in the figure) is equivalent 
 to the resultant of the two motions from P to M 
 and from M to F. Similarly this ray, if acted upon 
 by the second lens only, would be deviated in the 
 meridian PN, and would intersect the focal line of 
 
152 HANDBOOK OF OPTICS 
 
 this lens at a distance F^ from the lens. Now each 
 of these displacements, JPM and PiV^, in the plane of 
 the combined lenses, is equivalent to two displace- 
 ments at right angles ; * thus the displacement from 
 P to iJf is the resultant of the displacements from 
 
 Pig. 42. 
 
 P to U and from E to M, and the displacement 
 from P to iV is the resultant of the displacements 
 from P to 5" and from Hto N. If the two lenses act 
 in combination, the deviation produced by one lens 
 must be superposed upon that produced by the 
 other. Each lens produces a certain deviation in 
 
 * Since we neglect the thickness of the lenses, they both lie in 
 the same plane. 
 
BI-CYLINDRICAL LENSES I53 
 
 the direction PK or L 0, and also a deviation in the 
 direction PK' or Q 0, at right angles to L 0. If we 
 add the deviation which the first lens produces in 
 the direction LO to that which the second lens pro- 
 duces in this direction, the result is the deviation in 
 this direction which the two lenses would produce 
 acting in combination; and this same deviation might 
 evidently be produced by a certain piism whose edge 
 is perpendicular to LO, or by a certain cylindrical 
 lens whose axis is represented by QO. In the same 
 way the deviation produced by the two lenses in the 
 direction QO might also be produced by a cylindrical 
 lens whose axis is represented by L 0. 
 
 From this we see that any two cylindrical lenses 
 have upon a ray of light the same deviating effect 
 as two other lenses whose axes are at right angles. 
 This is true for any point on the lens and for any 
 position oi LO and Q ; but we do not mean that 
 the same two lenses at right angles are equivalent 
 to the obliquely inclined lenses for different posi- 
 tions of the point P. The problem that we wish 
 to prove is that for a particular position of the lines 
 L and QO the same two cylindrical lenses are equiv- 
 alent to the obliquely inclined lenses irrespective of 
 the position of the point P at which the ray meets 
 the lens. 
 
 If the focal length of the two lenses were equal, 
 we should find the combined action of the two lenses 
 
154 HANDBOOK OF OPTICS 
 
 in the direction PKhj adding the displacement PE, 
 produced by the first lens, to PH, produced by the 
 second lens ; but when the strength of the lenses is 
 not the same, these distances do not measure the dis- 
 placements which the two lenses produce in the same 
 time.* 
 
 Hence we must find an expression from which we 
 can reckon the deviating power of the two lenses 
 acting through the same distance. Since all rays 
 parallel to the axis of the refracting system are re- 
 fracted to the focus or focal line of the lens, it is 
 evident that the displacement which any ray under- 
 goes in. the plane of the lens is proportional to the 
 distance from the axis at which the ray meets the 
 lens ; and so long as the focal length remains un- 
 changed, this displacement measures the deviating 
 power of a lens at any point on its surface. It is 
 also evident that the time required to produce a 
 certain displacement varies inversely as the focal 
 length of the lens, or directly as the dioptric power. 
 Hence the deviating power of a lens at any point 
 
 on its surface is measured by the expression — , in 
 
 which D represents the distance from the axis at 
 which the ray meets the lens and F the focal 
 length. 
 
 Thus the deviating power of the first lens in the 
 
 * Compare Chap. I., p. 22. 
 
BI-CYLINDEICAL LENSES 155 
 
 PM 
 
 meridian PM is expressed by — — , P being any 
 
 P 
 
 point on the lens; and the deviating power of the 
 
 PN 
 
 second lens for the same point is expressed by —- ; 
 
 or if we replace the focal lengths by the dioptric 
 values, and represent these by and C^, respec- 
 tively, we have PM • C as the measure of the devi- 
 ating power of the first lens, and PN ■ (7j as the 
 measure of this power for the second lens. 
 
 Let the angle AOB, which is included between 
 the axes of the two cylindrical lenses, be represented 
 by the letter a ; and let us assume that in a certain 
 position of LO and QO the two obliquely inclined 
 lenses may be replaced by two other lenses at right 
 angles. The unknown angle LOA, which the axis 
 LO must make with OA, is denoted by x, and the 
 angle LOB, which is equal to a — x, is denoted by y. 
 It is readily seen from the figure that PME, MPB', 
 and OBK are each equal to x, and that PNH, S'PN 
 and OSK are each equal to y. 
 
 The dioptric value of the assumed lens whose 
 axis is OQ is denoted by Cj, and that of the lens 
 whose axis is LO is denoted by Cg. If the lens Cg 
 is equivalent to the combined action of the two 
 lenses O and Cj in the direction LO, we have the 
 equation 
 
 PK- 0^=PE-Q + PH. 0^, (1) 
 
156 HANDBOOK OF OPTICS 
 
 for PK- C^ expresses the deviating power of the 
 lens O2 in its refracting meridian PK, and PE • C 
 and PH • O^ express the deviating power of the 
 first and second lens respectively in tlie direction 
 PK; and, since when both lenses are convex the 
 displacements PE and PH are both toward the 
 axis OQ, we must add the deviating powers of 
 the two lenses to find the equivalent lens.* In 
 the direction PK or QO the displacements EM and 
 HN are in opposite directions, and the difference in 
 power of the two lenses in this direction will express 
 the power of the equivalent lens. Hence we have 
 also the equation 
 
 PK ■ C3 = HN-C^- EM- 0. (2) 
 
 But PE = PM-smx, PH=PN-&iny, 
 
 HN = PN • cos ?/, and EM = PM •cos.x. 
 
 Hence equation (1) becomes 
 
 n PM . n , PN . n 
 <^2 = p^- sin a; • C' + -^^- siny • Q^. 
 
 But PM= PR ■ sin x, and PiV= PS • sin ?/ ; 
 also, PR = PK- RK, and PS=PK+ KS. 
 
 * We take as the typical case two convex lenses ; the same 
 demonstration is applicable if one or both lenses be concave, it 
 being only necessary to change the sign of C or of Ci, or of both. 
 
BI-CYLINDRICAL LENSES I57 
 
 Hence we have 
 
 or, Cg = sin^ x- Q + sin^ y • Cj 
 
 sin^ X 
 
 C + ^.sin^y.C,. (3) 
 
 FK PK 
 
 If this is true for all positions of P, it must be 
 true when P lies on the line LO. It is readily seen 
 that when P is moved over to K\ BK and KS 
 both become zero, and our equation reduces to the 
 form 
 
 C2 = sin^ X- O + sin^ V '^\\ (4) 
 
 and since O^ is a constant quantity for all positions 
 of P, if it is an equivalent lens for G and (7j in its 
 refracting meridian, then the algebraic sum of the 
 last two terms of the second member of equation (3) 
 must be equal to zero.* Thus we have as the con- 
 dition of equivalence for all points on the lens that 
 the expression 
 
 BK .9 /> , KS ■ n /Y 
 ; sm'' X- (J + — — sm-* y • Cj 
 
 PK PK 
 
 * Bonders and Weiland in their demonstrations have assumed 
 that a lens C is in the meridian PK equivalent to another lens 
 whose dioptric power is expressed by C • sin" x. This we readily 
 see is true only for points on the axis LO. 
 
158 HANDBOOK OF OPTICS 
 
 should be equal to zero ; or, 
 
 but we see from the figure that RK= OK- cot x, 
 and KS = OK ■ cot y. Substitutmg and replacing 
 
 the cotangent of these angles by — -, we have 
 
 sin 
 
 C • sin X • cos a; = Cj sin y ■ cos t/ ; 
 
 or, (7 • sin 2 a; = {7j sin 2 y ; 
 
 or, C • sin 2a; = 0^ sin 2(a — x). 
 
 By reduction we obtain the equation 
 
 , o C'l sin 2 a 
 
 tan 2 a; = — — L. 
 
 (7 + Ci cos 2 a 
 
 and from this we know the value which must be 
 assigned to x. 
 
 In the same way we find from equation (2), 
 
 rt PN n PM n 
 
 ^ = YK' '^' ^ ~ ~PK' *^°^ "^ ' ' 
 
 but PN= PS' . cos y, and PS' = PK + K'S', 
 also PM= PR' ■ cos a;, and PR' = ^'i?' - P^'. 
 
BI-CYLINDRICAL LENSES 159 
 
 Hence 
 n PK'+K'S' 2 n K'E'-PK' . ^ 
 
 ^ = — ^K' — """"^y ■ ^1 — -pK> — ' ' 
 
 or, Cg = cos^ 2/ ■ ^1 + ^^^ * ■ ^ 
 
 , K'S' 2 n ^'^' 2 /y 
 
 "•" tri y ' ^~ :pk^ ' 
 
 'when P lies on the axis OQ, K'S' and K'B' 
 both become zero, and 
 
 Og = cos^ y ■ Cj + cos^ a; • C. (5) 
 
 Hence 
 
 -^^ cos2 ?/ • (7t - -p^ cos2 X-O = 0, 
 K'S' cos^x-C 
 
 from which 
 
 K'JR' Gos" yC^' 
 
 or, since K'S' = OjK"' • tan y, 
 
 and ^'^' = 0^'tan2:, 
 
 sin «/ cos X _ cos^ x- G _ 
 cos «/ sin a; eos^ y • 6\ ' 
 
 or, Csin 2 a; = Cj sin 2 «/ ; 
 
 from which, as above, 
 
 , n G-, sin 2 a 
 
 tan 2 a; = — — i- 
 
 O' + Cj cos 2 a 
 
160 HANDBOOK OF OPTICS 
 
 Thus we find that the angle L OA is the same in 
 order that 0^ should be equivalent to and C^ in 
 the direction OQ as when O^ is equivalent to these 
 lenses in the direction LO, and therefore it is proved 
 that by giving a suitable value to the angle LOA, 
 we may substitute for the obliquely inclined lenses 
 two other lenses at right angles, or, the equivalent 
 of the latter, a sphero-C}dindrical combination. To 
 find the dioptric value of 0^ and Cg we have only to 
 substitute the values of x and y which we now know 
 in equations (4) and (5). If we add these two 
 equations, we see that 
 
 ^2 + C^ = G -\- (7j, 
 
 since sin^ x + cos^ x and sin^ y + cos^ y 
 
 are each equal to unity. 
 
 In practice it is not necessary to resort to this cal- 
 culation, as was stated in the first part of the present 
 chapter. The equivalent may be found in the fol- 
 lowing manner : Place the two lenses in a trial 
 frame with their axes in proper position ; and, hold- 
 ing the frame about i metre in front of the eye, look 
 through the lenses at a test-type card or other rec- 
 tangular object distant five metres or more. Rotate 
 the frame in the plane of the lenses until the posi- 
 tion is found in which there is no angular deviation 
 of the vertical and horizontal edges of the card; 
 
BI-CYLINDRICAL LENSES 161 
 
 then move the frame slightly from right to left or 
 vice versa until there is no break in the vertical line 
 as seen through the lens and above it. Notice where 
 this unbroken line cuts the trial frame, and thus read 
 off on the frame the angular marking, which gives 
 the position of the axis of one of the equivalent 
 lenses; the axis of the other is at right angles to 
 this. Having found the position of the axes, we 
 next neutralize the meridian of least refraction by 
 means of a spherical lens ; adding now a suitable 
 cylindrical lens, with axis in the meridian already 
 neutralized, we neutralize the meridian of greatest 
 refraction. The sphero-cylindrical combination of 
 equal but opposite power to that required for neu- 
 tralization represents the equivalent of the two ob- 
 liquely inclined lenses. The accuracy of this method 
 is limited only by the intervals between lenses in the 
 trial case; it is therefore sufficiently accurate for 
 practical purposes. The nearest equivalent which 
 exists in the trial case can be found in a few 
 moments. 
 
 In this way also can the lenticular astigmatism of 
 the eye be found.* Suppose-, for instance, that the 
 entire astigmatism of the eye, as found by subjective 
 or objective test, is three dioptres, the meridian of 
 
 * It was for the purpose of showing how to find the lenticular 
 astigmatism of the eye that Bonders gave his solution of the bi- 
 cylindrical problem. 
 
162 HANDBOOK OF OPTICS 
 
 least curvature being 60 degrees from the horizontal 
 line. This can be represented by a convex lens of 
 three dioptres, with its axis at 60 degrees. If the 
 corneal astigmatism be found by the ophtlialmometer 
 to be two dioptres, with the meridian of least, cur- 
 vature at 130 degrees, this will be represented by 
 a convex lens of two dioptres with axis at 130 de- 
 grees. The lenticular astigmatism is evidently equal 
 to the entire astigmatism less that of the cornea. 
 To find this we place in the trial frame a convex 
 cylindrical lens of three dioptres with axis at 60 
 degrees, which represents the entire astigmatism. 
 If we neutralize the corneal astigmatism, we have 
 remaining the lenticular astigmatism. The corneal 
 portion is neutralized by a concave lens of two 
 dioptres with axis at 130 degrees. Placing this lens 
 in the trial frame, we have a combination of two 
 cylindrical lenses obliquely inclined, and the astig- 
 matism of tills combination represents the lenticular 
 astigmatism of the eye. By neutralizing the combi- 
 nation, we derive the lens equivalent of the lentic- 
 ular astigmatism and the angle of its axis. 
 
CHAPTER XI 
 
 OBLIQUE REFRACTION THROUGH LENSES 
 
 In our study of refraction we have considered 
 only pencils of light whose central ray or axis is 
 perpendicular to the refracting surface. Such a 
 pencil is called direct. We have also learned that 
 a spherical surface has greater refractive power for 
 rays which meet it at a distance from the axis than 
 for those which pass near the axis. We have been 
 obliged in order to escape spherical aberration to 
 limit our consideration to those rays which do not 
 deviate far from the axis. We are justified in this, 
 since only small pencils can enter the eye. 
 
 We shall now investigate the refraction of small 
 pencils, the axis of which is not perpendicular to the 
 refracting surface. Such a pencil is called oblique. 
 Refraction by oblique pencils takes place when we 
 look through a tilted lens, as is frequently done in 
 the use of spectacles. The complete analysis of this 
 subject is difficult and requires a knowledge of the 
 higher mathematics, but we can study oblique 
 refraction in an elementary manner, and this will 
 enable us to understand the problems which present 
 
 163 
 
164 HANDBOOK OF OPTICS 
 
 themselves in ophthalmology. A complete solution, 
 so far as relates to tilted spectacle lenses, has been 
 worked out by Dr. John Green of St. Louis ; * and 
 a general solution of the problem of oblique spher- 
 ical refraction is given in Heath's " Geometrical 
 Optics." f From the formulae derived by these 
 investigations, the exact value of a lens when tilted 
 at any angle can be found. We shall suppose the 
 lens to be tilted only in the meridian of vertical 
 
 Fig. 43. 
 
 refraction. Let A (Fig. 43) represent the princi- 
 pal point of a refracting surface. The direct pencil, 
 whose axis is PA and whose peripheral ray is PT, 
 will after refraction meet the axis at Q. Let 
 R'PR" represent a pencil meeting the surface 
 obliquely, PR being the axis of this pencil. We 
 have learned from our previous studies that it is 
 sufficient to determine the deviation of light in two 
 meridians at right angles to each, other. We shall 
 
 * Trans. American Ophth. Soc, 1890. 
 t Heath's "Geom. Optics," 2d ed., p. 179. 
 
OBLIQUE REFRACTION THROUGH LENSES 165 
 
 therefore investigate the refraction of the oblique 
 pencil first in the vertical meridian or plane of tlie 
 paper, as represented by R'R", and then in the 
 horizontal meridian, as represented by ^*S". We 
 may consider the rays PR', PR, and PR" as rays 
 of the direct pencil which are so far from the axis 
 that spherical aberration cannot be neglected. 
 These rays will after refraction meet the axis at Q-^^, 
 Q^, and ^3, respectively. The refracted rays R'Qi 
 and RQ^ will meet at q; RQ^ and R" Q^ will meet 
 at q^. If the pencil be small, q and q^ will be so 
 near to each other that we may regard them as 
 identical. Then the point of intersection q will 
 be the focus of the pencil in the vertical meridian, 
 and Rq will be the focal distance in this meridian. 
 If now we take the rays PS and PS' in the hori- 
 zontal meridian, it is evident that after refraction 
 they will meet the axis at Q^ ; for their vertical 
 distance from A is the same as that of the ray PR, 
 the axis of the pencil. Hence Q^ is the horizontal 
 focus, and RQ^ is the horizontal focal distance. 
 From this we see that oblique spherical refraction 
 is astigmatic. Since Rq is the focal distance in 
 any vertical meridian, all rays of the pencil must 
 intersect a straight line passing through q and par- 
 allel to SS'.* Likewise all rays must have their 
 
 * S8', being a small arc, does not materially differ from a 
 straight line. 
 
166 HANDBOOK OF OPTICS 
 
 horizontal intersections on the line Q-^ Q^ ; and since 
 the pencil is small, the error will be inappreciable 
 if we replace Q1Q3 by a line through ^^parallel to 
 R'R" . These lines drawn through q and Q^ are 
 the focal lines of the pencil, and qQ^ is the focal 
 interval. It is clear that the focal interval increases 
 as the distance of the oblique pencil from the axis 
 PA increases. Ks, AQ is the focal distance for the 
 direct pencil, it is seen that by the change from 
 direct to oblique refraction the focal distance is 
 
 Fig. 44. 
 
 shortened in both vertical and horizontal meridians, 
 but more in the former than in the latter; for AQ,^ 
 and Rq are both less than AQ, and Rq is less than 
 AQ^. In the question before us we have oblique 
 refraction at two surfaces. We shall consider only 
 the case in which the axis of the pencil passes 
 through the optical centre of the lens, so that after 
 the two refractions its direction is parallel to that 
 before refraction ; and we shall disregard the lateral 
 displacement due to the thickness of the lens. 
 
 In Fig. 44 let R'PR" represent the vertical 
 section of a pencil meeting the lens obliquely at 
 
OBLIQUE REFRACTION THROUGH LENSES 167 
 
 R'B" ; then since the axis of the pencil, PR, passes 
 through the optical centre of the lens, its direction 
 after emerging from the lens will be parallel to PR. 
 Let Q be conjugate to. P in the vertical meridian. 
 We have seen the refractive power of the first sur- 
 face is increased by tilting the lens; consequently 
 a shorter incident pencil R'PR" will cause the 
 refracted pencil to assume the convergence R'qR" 
 than if the pencil were direct. Likewise if we 
 suppose a pencil T'QT" to proceed from Q so that 
 after refraction it assumes the divergence R'qR", 
 appearing to proceed from q, the pencil T'QT" will' 
 be shorter than if it were direct. Since the path of 
 light is reversible, the same will be true when 
 T'QT" is an emergent pencil; in other words both 
 the incident and emergent pencils are shortened by 
 tilting the lens, and the lens is increased in refrac- 
 tive power. The same reasoning applies .also to 
 the refraction in the horizontal meridian, but in 
 this meridian the increase of power is less at each 
 surface than in the vertical meridian. Hence a 
 spherical lens tilted in its vertical refracting me- 
 ridian is equivalent to a sphero-cylindrical lens. 
 Reference to Fig. 43 will show that as the tilting 
 is increased, the amount of astigmatism increases 
 very rapidly. 
 
 If a cylindrical lens whose axis is horizontal be 
 tilted in its vertical refracting meridian, the increase 
 
168 HANDBOOK OF OPTICS 
 
 in power will be the same as that of a similar spher- 
 ical lens in the vertical meridian. If the axis of the 
 lens be vertical and it be tilted in this meridian, the 
 increase in power will be the same as that which a 
 spherical lens undergoes in the horizontal meridian 
 when tilted in the vertical meridian. 
 
 We should learn two practical points from this 
 study : first, the necessity of giving the proper 
 inclination to spectacles, especially when strong 
 lenses are used ; and, secondly, the uncertainty of 
 effect of very weak cylinders in combination with 
 strong spherical lenses. A spherical lens of four 
 dioptres acquires more than one-quarter of a dioptre 
 of astigmatism by tilting it through an angle of 15 
 degrees. Hence a convex cylindrical lens of .25 D., 
 axis vertical, combined with a convex spherical lens 
 of 4 D., would be completely nullified by a slight 
 amount of tilting ; on the other hand, if the axis 
 of the lens be horizontal, its action will be practi- 
 cally doubled by the same tilting. 
 
CHAPTER XII 
 
 THE EFFECT OF PRISMATIC GLASSES UPON 
 EETINAL IMAGES 
 
 In Chapter I. we considered refraction of rays 
 through prisms. We must now investigate in an 
 elementary way the more difficult subject of refrac- 
 tion of pencils of light such as enter the eye. The 
 study of the effect of prisms upon stereoscopic vision 
 belongs to physiology, and is discussed in treatises 
 on physiological optics. We shall confine our atten- 
 tion to the influence of prisms upon the size and form 
 of retinal images. 
 
 For convenience we repeat the following : 
 
 1. A principal plane of a prism is a plane perpen- 
 dicular to the edge of the prism, and therefore to 
 each face of the prism.* 
 
 2. By the law of refraction, the incident and re- 
 fracted rays and the normal to the surface aU lie in 
 the same plane. 
 
 3. From the equation sin i = n ■ sin r, it follows 
 that the greater the angle of incidence, the greater is 
 
 * The principal plane of a prism bears no analogy to the princi- 
 pal plane of a spherical refracting surface. 
 
 169 
 
170 
 
 HANDBOOK OF OPTICS 
 
 the deviation of the ray ; and the greater the angle 
 of incidence, the greater is the increase in deviation 
 for a fixed increase in the angle of incidence, the rate 
 of change in deviation increasing very rapidly as the 
 angle approaches 90 degrees. 
 
 From (1) and (2) we see that a ray which enters 
 the prism in a principal plane must lie in this plane 
 after emergence. 
 
 A ray which enters in any other plane passes out 
 of the prism in a plane parallel to that in which it 
 
 Fig. 45 
 
 entered, the amount of separation between the planes 
 depending upon the thickness of the prism ; and the 
 deviation which this ray undergoes in the principal 
 plane is greater than if the ray were in this plane.* 
 
 The effect which these facts have upon images will 
 appear subsequently ; we must first examine the re- 
 lation between the length of the pencil and the size 
 of the image. Helmholtz' formula applies here as in 
 all cases, but this relation is indicated clearly in Fig. 
 45. Let AP represent the linear dimension of an 
 
 * For the demonstration of this see Appendix II. 
 
PRISMATIC GLASSES 171 
 
 object, then '^^S* represents its image on the retina. 
 The distance from P to the cornea is the length of 
 the pencil before refraction, RPO is the angle of 
 divergence of the pencil, and AOP or QOS is the 
 visual angle which the object subtends. 
 
 Thus we see that as the incident pencil becomes 
 shorter the image on the retina increases. If, by any 
 means, rays of light from an object AP are rendered 
 more divergent, so that the pencils appear to proceed 
 from A'P', the image will be enlarged from QS to QS'.* 
 
 The effect of prisms upon the length of pencils 
 varies greatly with position of the prism as regards 
 the light which passes into it. If the prism be of 
 small refracting angle, all rays which pass through it 
 near the position of minimum deviation will undergo 
 practically the same amount of deviation, f Small 
 pencils passing through such a prism in this position 
 undergo no change in length, for if the rays all have 
 the same amount of deviation their relative diver- 
 gence will be unaltered in their passage through the 
 prism. But all other pencils will be altered in 
 length ; and to prove this let Fig. 46 represent a 
 principal section of a prism. From draw R OR', 
 representing a pencil in the position of minimum 
 deviation, and also SOS' and TOT', two pencils, the 
 
 * We neglect as inappreciable the slight change in position of the 
 optical centre O, due to change in refractive state of the eye upon 
 approximation of the object. t Chap. I, p. 21. 
 
172 HANDBOOK 01" OPTICS 
 
 former being nearer the apex and the latter nearer 
 the base of the prism than BOM'. As has been 
 stated, the pencil SOB' undergoes no change in 
 length. By the first refraction the pencil SOS' is 
 increased in length, for OS makes a greater angle 
 with the normal to the surface than does OS' ; its 
 deviation is consequently greater and the divergence 
 of the pencil is diminished. By the second refrac- 
 tion the divergence of the pencil is for a similar 
 
 Fig. 46. 
 
 reason increased ; but since the angles of refraction 
 are greater at the first surface, the difference in the 
 amount of deviation undergone by OS and OS' is 
 greater at this than at the second surface ; conse- 
 quently the lengthening of the first surface out- 
 balances the shortening at the second surface. In 
 the pencil TOT' the angles of refraction are greater 
 at the second surface, and the shortening at this sur- 
 face outbalances the lengthening at the first surface. 
 Furthermore it will be observed that pencils proceed- 
 
PRISMATIC GLASSES 173 
 
 ing from a point will not, after passing through a 
 prism, appear to come from a point ; but if tlie axis 
 of the pencil lie in a principal plane of the prism, as 
 we suppose in Fig. 46, rays proceeding from will 
 appear to come from two focal lines parallel and per- 
 pendicular, respectively, to the edge of the prism. 
 This follows from the analogy to astigmatic refrac- 
 tion at cylindrical surfaces ; for the rays of the 
 pencil will not be deviated in the direction of the 
 edge of the prism, the alteration in length taking 
 place in the meridian at right angles to this edge. " 
 Since we deal only with small pencils, we may for 
 our present purpose ignore the astigmatic effect and 
 regard the pencil SOS' as proceeding from a point 0" 
 after refraction by the prism. An object seen by 
 pencils such as SOS' appears minified in the direc- 
 tion of the principal plane, since the incident pencils 
 •as received by the observer's eye are lengthened ; on 
 the other hand, an object seen by pencils such as 
 TOT' appears magnified in this direction, since the 
 incident pencils are shortened. 
 
 From this we see that as the apex of a prism is 
 turned toward the object, the image is magnified in 
 the principal plane of the prism ; and as the base of 
 the prism is turned toward the object, the image is 
 minified in this plane.* 
 
 * Upon this principle is constructed a toy, — the "laughing 
 camera," — which is sold on the streets and in the shops. 
 
174 
 
 HANDBOOK OF OPTICS 
 
 Let US now study the effect of a prism placed 
 before the eye with its principal plane horizontal. 
 If the prism have a refracting angle of 15 or 18 de- 
 grees, objects in the field of view will undergo 
 marked distortion. ' This is due not only to the 
 actual distortion of the retinal image, but also, in 
 part, to mental impressions of previous experience. 
 
 (a) 
 
 o 
 
 
 / 
 
 / 
 
 
 
 
 / 
 
 B 
 
 \ 
 
 (b) 
 
 \ 
 
 \ 
 
 
 
 D 
 
 H 
 Fig. 
 
 47. 
 
 C 
 
 When prisms are worn before both eyes there is still 
 further confusion arising from altered convergence 
 and perspective.* 
 
 In Fig. 47 (a) represents a square object so placed 
 that 0, its middle point, is seen by rays passing 
 through the prism with minimum deviation; then 
 if the base of the prism is placed to the right, (6) is 
 a representation of the object as it appears on the 
 
 * Articles bearing upon this subject have been published by a 
 number of writers. Reference may be made to the following : 
 Helmholtz' "Physiological Optics"; "Stereoscopic lUuisions 
 Evoked by Prismatic and Cylindrical Spectacle Glasses," John 
 Green, Trans. American Ophth. Soc, 1889. 
 
PRISMATIC GLASSES 175 
 
 retina. The portion of the figure near is un- 
 affected by the prism, but the portion toward the 
 base of the prism is minified, and that toward the 
 apex is magnified in the horizontal meridian. The 
 point A is displaced toward 0, and any other point 
 of the line AE is displaced in this direction to a 
 greater extent than is -4, because a ray not in a 
 principal plane is more deflected than a correspond- 
 ing ray lying in this plane. As this increase in 
 deviation continues at an increasing rate with the 
 increase of the distance of the point from A, the 
 line AE appears curved. For the same vqSS>'i'^^ON 
 and BE appear curved. 
 
 Neglecting, as we do, the thickness of the prism, 
 there is no displacement at right angles AE, and 
 therefore EE is unchanged in direction in the retinal 
 image. Nevertheless EE does seem to be changed 
 in such manner as to assume the form represented 
 in (c) — a change which is due to illusion. The 
 line AE, as seen through the prism, appears at an 
 increased distance while its image is unchanged ; 
 consequently the line seems to be increased in length. 
 For a similar reason BE appears shortened. This 
 increasing apparent distance, as the attention is 
 directed farther to the left, also causes this part of 
 the figure to assume a convex cylindrical form ; 
 while that portion toward the extreme right appears 
 concave. This effect can be obtained by looking 
 
176 HANDBOOK OF OPTICS 
 
 through a strong prism at the opposite wall of a 
 room, turning the prism so as to get the extreme 
 magnifying and minifying power. Since the lower 
 half of the figure is a reproduction of the upper half, 
 it needs no explanation. 
 
 Similar distortions and illusions must be produced 
 by weak prisms, though in a less marked degree ; 
 it is not surprising therefore that unpleasant sen- 
 sations arise when persons attempt to wear com- 
 paratively strong prismatic glasses as spectacles. In 
 many cases, however, this disturbance passes away 
 afte^^e glasses have been worn for a short time. 
 
CHAPTER XIII 
 THE REFLEXION OF LIGHT 
 
 The law of reflexion — namely, the angles of inci- 
 dence and reflexion lie in the same plane and are 
 equal — was, as we have stated, known to the 
 ancients. 
 
 As we have found the law of refraction to follow 
 as a necessary consequence of the wave theory of 
 light, being due to the varying velocity of light in 
 different media, so also the law of reflexion cor- 
 roborates this theory. In reflexion the light does 
 not pass out of the first medium ; its direction, 
 however, is reversed. Hence, if we make n equal 
 to minus 1 in any equation pertaining to refraction, 
 we ought to get the corresponding equation for 
 reflexion. By making this substitution we arrive 
 at identical formulae with those obtained from in- 
 dependent geometrical construction. 
 
 The phenomena of reflexion, so far as they con- 
 cern the ophthalmologist, are exceedingly simple. 
 
 The illumination of the interior of the eye is ac- 
 complished by means of a mirror, which reflects light 
 from a flame, while the observer is so situated as to 
 
 177 
 
178 
 
 HANDBOOK OF OPTICS 
 
 be in the path of the light returning by reflexion 
 and refraction. 
 
 If a plane mirror is used, light is thrown into the 
 eye by pencils which appear to come from behind 
 the mirror and from a distance equal to that of the 
 
 V' 
 
 ^ 
 
 Fig. 48. 
 
 lamp from the mirror. This is illustrated in Fig. 48. 
 It can easily be proved that the angles A and Al are 
 equal. 
 
 If a concave mirror is used, the light appears to 
 come from a point in front of the mirror. This we 
 deduce from equation (a), page 30. If we make n 
 equal to minus 1 in this equation, the result is an 
 equation expressing the relation between the conju- 
 
THE KEFLEXION OE LIGHT 179 
 
 gate foci after reflexion at a spherical surface. Mak- 
 ing this substitution, equation (a) becomes 
 
 1_ 1_ 2 1 _2 , 1 
 
 f is the distance of the flame from the mirror, /' is 
 the distance of its conjugate from the mirror, and r 
 is the radius of curvature. Since the mirror is con- 
 cave, r is negative. Hence, / being positive, /' is 
 
 2 1 
 
 negative when - is greater than - and positive when 
 
 »• / 
 
 - is less than -. 
 '^ f 
 
 In other words, if / is greater than -, the two 
 
 conjugate foci lie on the same side of the mirror ; 
 if / is less than -, both / and /' are positive, and 
 
 consequently lie on opposite sides of the mirror. 
 
 The radius of curvature of the concave mirrors 
 used in ophthalmoscopy does not exceed J metre ; 
 and as the distance of the flame from the mirror is 
 greater than half this radius, the point from which 
 light proceeds is in front of the mirror. The 
 . illumination of the eye by the concave mirror is 
 therefore more intense than that by the plane 
 mirror. 
 
 Figure 49 illustrates reflexion by the concave 
 mirror ; light from A is focused at A' . li A rep- 
 
180 HANDBOOK OF OPTICS 
 
 resent a gas flame, an image of this flame will be 
 formed at A! . 
 
 Continuing the study of reflexion at a spherical 
 surface, we see that if we make / equal to infinity, 
 
 /' is equal to -, that is, the principal focal distance 
 of a spherical mirror is equal to one-half of the 
 
 radius of curvature.* If the mirror is concave, - 
 
 2 
 
 is negative, and the principal focus is half way be- 
 
 FiG. 49. 
 
 tween the centre of curvature and the surface of the 
 mirror. If the mirror is convex, - is positive, and 
 
 the principal focus lies behind the mirror. Clearly 
 such a focus must be virtual, while the negative 
 focus of the concave mirror is real. Hence if we 
 use equation (a), real foci in reflexion are negative 
 while virtual ones are positive. 
 
 *It -will be observed that in reflexion the two principal foci 
 coincide. 
 
THE REFLEXION OF LIGHT Igl 
 
 It must be borne in mind that the equation ap- 
 plies only to small pencils near the, axis of the sur- 
 face, for spherical aberration occurs in reflexion as 
 in refraction. 
 
 The same relation exists between the size of object 
 and image as in refraction, viz. : 
 
 _u 
 
 If u be positive, that is, if the object be without 
 the principal focus, as in Fig. 49, then the image 
 will be positive or negative according as -F is posi- 
 tive or negative. When F is negative, as in the 
 concave mirror, the image is real and inverted; 
 when F is positive, as in the convex mirror, the 
 image is virtual and erect. 
 
 In the concave mirror u is negative when the ob- 
 ject lies nearer the mirror than the principal focus ; 
 in the convex mirror u cannot be negative, since 
 the principal focus is virtual and behind the mirror. 
 Hence in reflexion at concave mirrors the image is 
 real and inverted when the object lies without the 
 principal focus ; virtual and erect when the object 
 lies within this focus. In reflexion at convex 
 mirrors the image is always virtual and erect. 
 
 Since the size of the image is equal to o • —, the 
 
 u 
 
 virtual image formed by the concave mirror is larger 
 
182 HANDBOOK OF OPTICS 
 
 than the object, for u, lying between F and the 
 mirror, must be less than F. 
 
 The real image formed by the concave mirror is 
 less than the object when u is greater than F, and 
 vice versa. 
 
 The erect virtual image formed by the convex 
 mirror is always smaller than the object, since u is 
 necessarily greater than F. 
 
 Since all refracting surfaces act also as reflecting 
 surfaces, the three refracting surfaces of the eye, 
 namely, the cornea, the anterior, and the posterior 
 surface of the lens, must furnish also three reflect- 
 ing surfaces. Consequently, when an object is held 
 before the eye, there must be formed by reflexion 
 three images of the object. The first, formed at 
 the convex surface of the cornea, is virtual and 
 erect; the second, formed at the anterior surface of 
 the lens, is also virtual and erect; while the third, 
 formed at the concave posterior surface of the lens, 
 is real and inverted. These images, as seen with 
 the aid of a lighted candle in a darkened room, 
 are of great interest to . ophthalmologists. The 
 formation of all three images is conclusive evidence 
 of the presence of the crystalline lens in the eye. 
 Furthermore, by the change in relative size of 
 tliese images, the increase in curvature of the lens 
 during the act of accommodation can be demon- 
 strated. 
 
THE EEITLEXION OF LIGHT 183 
 
 Since the size of the image varies with — , and F 
 
 r ^ 
 
 is equal to — , we have a means of measuring the 
 
 curvature of the refracting surfaces of the eye, 
 provided we can measure the size of the reflected 
 images. A brief description of the way in which 
 this can be done will be given in the next chapter. 
 
CHAPTER XIV 
 
 THE OPTICAL PEINCIPLES OF OPHTHALMOMETEY 
 AND OF OPHTHALMOSCOPY 
 
 To the genius of Helmholtz we owe the invention 
 of the ophthalmometer, an instrument of great pre- 
 cision for measuring the curvature of the refracting 
 surfaces of the eye. In the construction of this 
 instrument Helmholtz employed a device already in 
 use by astronomers for the measurement of the stars, 
 namely, the production of double images of a single 
 object. 
 
 This is possible by means of several contrivances. 
 The simplest is the double prism, such as the Mad- 
 dox prism found in the oculists' trial case. Helm- 
 holtz' device consisted of two plates of glass of 
 equal thickness inclined at an angle, as is shown in 
 Fig. 50. A pencil of light from o meets the plate 
 D, and is refracted as in the figure. The rays, after 
 emergence, are parallel to their direction before 
 entering the plate, but they undergo a lateral dis- 
 placement due to the thickness of the glass, so that 
 they all appear to come from A. Likewise, that 
 part of the pencil which passes through the plate 
 
 184 
 
OPHTHALMOMETRY AND OPHTHALMOSCOPY 185 
 
 E appears to come from B. Hence, if o represent 
 a small object, there will appear after refraction 
 through the plates two similar objects at A and B, 
 respectively. 
 
 Fig. 50. 
 
 If a convex lens be placed at L, a real image of 
 A will be formed at A\ and an image of B will 
 be formed at B' . A second convex lens, M, whose 
 principal focal plane is A'B', will render rays from 
 the image A'B' parallel ; and the 
 image will be focused on the retina /' ^^^ 
 
 without accommodation. 
 
 If the circle whose centre is (Fig. 
 51) be viewed through a double prism, 
 or through two inclined plates ; and, 
 if the double images are separated to 
 such an extent that the two circles A 
 and B appear to touch at 0, it is clear 
 that the amount of separation of the 
 images will be equal to the diame- 
 ter of the circle 0. It is also clear that if the 
 angle between the two plates can be varied, then, by 
 changing this .angle to the proper degree, the two 
 
 y 
 
 Fig. 51. 
 
186 HANDBOOK OF OPTICS 
 
 images can be made to touch as in the figure. 
 Knowing the angle between the plates, the thick- 
 ness and refractive index of the glass, the amount 
 of displacement from to A and from to B can 
 be calculated. 
 
 If the object at be a reflected image as seen in 
 the cornea, we can obtain its size from the data 
 above mentioned. The size of the object, the size 
 of the image, and the distance of the object from 
 the cornea being known, we deduce the curvature of 
 the cornea from the equation, 
 
 _ u __2u 
 
 i F r ' ^ 
 
 The greater the curvature of the surface, the 
 smaller is the image ; consequently, if the two im- 
 ages touch in the meridian of least curvature of an 
 astigmatic cornea, they will be separated by an 
 interval in the meridian of greatest curvature ; 
 while, if they touch in the meridian of greatest cur- 
 vature, they will overlap in that of least curvature. 
 
 The construction of the modern ophthalmometer 
 of Javal and Schiotz is somewhat different from that 
 of Helmholtz ; but the essential optical principles 
 are the same in both instruments. 
 
 By the aid of ingenious mechanical devices, ob- 
 servations of the corneal curvature have become a 
 matter of the greatest ease. In the instrument of 
 
OPHTHALMOMETRY AND OPHTHALMOSCOPY 187 
 
 Javal and Schiotz the glass plates are replaced by a 
 Wollaston prism, which, like the plates, produces 
 two images of a single object. 
 
 Certain crystalline substances possess the peculiar 
 property of double refraction. Iceland spar is a 
 familiar example of a double refracting substance. 
 Part of the light entering this material undergoes 
 refraction in the ordinary way, while a part possesses 
 the property of having different velocities, and hence 
 different refractive indices in different directions. 
 
 This is due to the fact that the constitution of the 
 substance is such as to offer unequal resistance to 
 the passage of light in different directions or axes. 
 The first portion of light which follows the ordinary 
 law of refraction is called the ordinary ray ; the 
 second portion, whose index varies for different 
 meridians, is called the extra-ordinary ray. 
 
 If we take a piece of double refracting substance, 
 as Iceland spar or quartz, and through it look at an 
 object placed in such position as regards incident 
 light that the difference in index between the ordi- 
 nary and extra-ordinary rays causes a separation of 
 these rays, a double image of the object will be 
 formed. WoUaston's prism is based upon this 
 principle.* 
 
 * For a detailed account of tlie phenomenon of double refrac- 
 tion and the construction of WoUaston's prism, consult Preston's 
 " Theory of Light," or other complete treatise on optics. 
 
188 HANDBOOK OF OPTICS 
 
 In the modern ophthalmometer, in which the Wol- 
 laston prism is used, the objective L is composed of 
 two lenses separated by an interval ; and the prism 
 is placed in this interval between the lenses. 
 
 The exact amount of separation which the prism 
 produces in its fixed position is known. The di- 
 ameter of the object from which light is reflected to 
 the cornea can be varied at will. This object con- 
 sists of two sets of white enamelled disks called mires, 
 equally distant from the centre of a connecting bar. 
 By increasing or diminishing the distance between 
 the mires their separation may be made such' that the 
 double images are tangent to one another. From a 
 scale which has been constructed by previous calcula- 
 tion, the curvature of the cornea or its refractive 
 power in dioptres can be read off on the bar separat- 
 ing the mires. 
 
 The ophthalmoscope is a contrivance by which the 
 observer reflects light into an eye, while he is in 
 such position as to receive in his own eye the light 
 which returns by reflexion and refraction from the 
 observed eye. As Helmholtz invented the ophthal- 
 mometer, so to him are we indebted for the gift of 
 the ophthalmoscope. Prior to this invention, in 
 1851, the question of seeing the fundus of the eye 
 had attracted much attention. It was of course 
 known that the eyes of some animals emit a reddish 
 or greenish tint under certain circumstances, and 
 
OPHTHALMOMETRY AND OPHTHALMOSCOPY 189 
 
 many absurd speculations were indulged in for the 
 explanation of this phenomenon. Briicke made a 
 thorough study of this subject, and in 1847 gave the 
 true explanation.* Indeed, he came so near to the 
 invention of the ophthalmoscope as to place in a 
 flame an iron tube, through which he could see the 
 fundus of the eye. It was also known prior to these 
 discoveries that the fundus of the eye would become 
 visible if the eye were immersed in water. The ex- 
 planation of this is similar to that of the glow of a 
 cat's eye. 
 
 The ophthalmoscope in its simplest form is a plane 
 or concave mirror, having in its centre a small circu- 
 lar opening through which the observer receives the 
 light returning from the observed eye. Reference to 
 Fig. 52 shows why the fundus of an eye cannot be 
 seen without a special contrivance. If light be re- 
 flected into the eye from a flame A along the axis 
 AH, and along other axes near AU, a small image 
 of the flame will be formed on the retina, and at the 
 same .time a small portion of the fundus adjacent to 
 the image will be illuminated by irregular reflexion. 
 Light from this portion of the fundus passes out of 
 the eye. If the eye is emmetropic, the pencil from 
 U, having as its axis UA, is after refraction changed 
 to parallel rays ; similarly, pencils from other points 
 
 * Mueller's " ArcMv ftlr Anat. und Phys.," 1845, S. 387; and 
 1847, SS. 225, 479. 
 
190 
 
 HANDBOOK OF OPTICS 
 
 Fig. 52. 
 
 near E will have as axes lines ly- 
 ing very near IIA, and these rays 
 will also, after refraction, be par- 
 allel to their axes ; thus it is evi- 
 dent that the light which returns 
 from the examined eye cannot 
 within a short distance from the 
 eye deviate far from the axis HA. 
 Hence we see that without a spe- 
 cial contrivance the observer's 
 head would necessarily cut off the 
 light which illuminates the eye. 
 
 In the hyperopic eye pencils of 
 light from any point on the fundus 
 are divergent after leaving the eye. 
 If the fundus be at H, then BHQ 
 will represent a pencil from M. If 
 the rays diverge considerably it 
 will be possible for an observer 
 to place his head in position to 
 receive some of the rays, and yet 
 not obstruct those from the illu- 
 minating source. It is on this 
 account that a cat's eye glows in 
 the dark when the observer is not 
 far from the path of the rays which 
 enter the eye. Placing the eye 
 under water has the same effect 
 
OPHTHALMOMETKY AND OPHTHALMOSCOPY 191 
 
 upon pencils as hyperopia, for the refractive index 
 of water is nearly the same as that of the aqueous, 
 and, the external surface of the water being plane, 
 we have in the immersed eye a high degree curva- 
 ture hyperopia. 
 
 By referring to Fig. 52 we see that when the fun- 
 dus is conjugate to the position from which the illumi- 
 nation proceeds, only a small portion of the fundus 
 is illuminated, and that the more remote the fundus 
 is from this conjugate position, the greater is the 
 portion of fundus illuminated. This is the prin- 
 ciple upon which is based the method of examina- 
 tion known as skiascopy. If the examination be 
 conducted with a plane mirror so placed that light 
 enters the eye in pencils diverging from a distance 
 of one metre in front of the eye, and if the far point 
 of the eye be also distant one metre, that is, if the eye 
 have one dioptre of myopia, a very small part of the 
 fundus will be illuminated. Hence, if the mirror be 
 slightly tilted, the area of illumination will at once 
 be thrown out of the line of vision of the observer. 
 If the eye be hyperopic, then as the mirror is tilted 
 the area of illumination will move in the same direc- 
 tion, but it will not pass out of view so rapidly, and 
 we can observe the motion of the reflex and its at- 
 tendant shadow as they move across the pupil. 
 When the mirror is so tilted that the light which 
 enters the eye appears to come from L, then a 
 
192 HANDBOOK OF OPTICS 
 
 straight line, drawn through the optical centre of 
 the eye, connecting L, S\ E', N', and M', replaces 
 the axis AM; hence, no matter what may be the 
 refractive condition of the eye, the area of illumina- 
 tion always moves in the direction of the tilting of 
 the mirror. But in myopia such that the fundus is 
 at M, beyond the conjugate N, there will be formed 
 to the left of A an aerial image of the illuminated 
 area, and this will evidently move in the opposite 
 direction to the tilting of the mirror. If the ob- 
 server is farther from the eye than this image, he 
 must therefore see the reflex and shadow move in 
 the opposite direction to the tilting. 
 
 By observing the movement of the shadow in dif- 
 ferent meridians of the eye, the test can also be used 
 for the detection of astigmatism. 
 
 If a concave mirror be used so that light enters 
 the eye diverging from a real image in front of the 
 mirror, the movements of the shadow will be oppo- 
 site to those which occur with the use of the plane 
 mirror ; the reason for this is apparent. 
 
 With such accuracy can the motion of the shadow 
 be observed that this method, in skilful hands, sur- 
 passes all other objective means of examining the 
 refractive condition of the eye. 
 
 Since light leaves the emmetropic eye in parallel 
 rays and the hyperopic eye in divergent pencils, 
 these rays may be brought to a focus on the retina 
 
OPHTHALMOMETRY AND OPHTHALMOSCOPY 193 
 
 of the observer, thus forming an image of the illumi- 
 nated part of the fundus under examination, so that 
 in both hyperopia and emmetropia the details of the 
 fundus can be seen, provided the observer is suffi- 
 ciently near the examined eye to receive light from 
 an appreciable area of the fundus. Moreover, since 
 in hyperopia a larger part of the fundus is illumi- 
 nated than in emmetropia, the details can be seen at 
 a greater distance in hyperopic eyes. 
 
 The rays from hyperopic and emmetropic eyes, 
 being either divergent or parallel, will never meet 
 in a real aerial image ; hence the image, as seen by 
 the examiner, will always be erect. The examina- 
 tion of the erect image is called the direct method of 
 ophthalmoscopy. 
 
 Light emerging from a myopic eye is convergent, 
 and if the eye of an emmetropic observer be nearer 
 the eye than the far point, a concave lens must be 
 used to see clearly the details of the fundus. The 
 image as thus seen will be erect; but at the far 
 point of the eye, which we know is conjugate to the 
 retina, an inverted aerial image will be formed, and 
 from this, divergent pencils will enter an observer's 
 eye situated beyond the image. A concave lens will 
 be no longer required ; on the other hand, exercise 
 of the accommodation or a convex lens varying with 
 the distance of the eye from the image is necessary. 
 The examination of the inverted image is called the 
 
194 HANDBOOK OF OPTICS 
 
 indirect method of ophthalmoscopy. As thus de- 
 scribed, this method would be practicable only in 
 highly myopic eyes ; . but we may produce the aerial 
 image in all cases by holding a strong convex lens 
 in front of the eye to be examined. The stronger 
 the lens the larger will be the field of view and 
 the smaller the image ; hence the strength of the 
 converging lens may be varied to suit the purpose of 
 the examiner. 
 
 Finally let us investigate the apparent size of the 
 optic disk — the most conspicuous object revealed 
 by the ophthalmoscope — as affected by the various 
 refractive conditions of the eye. We shall first con- 
 sider the examination by the direct method. When 
 the eye under examination is emmetropic, no change 
 in apparent size is produced by varying the distance 
 between the observed and observer's eyes ; for we 
 see from Fig. 60 (in which A!B' may represent the 
 disk, the lens M may represent the observed, and ffiV 
 the observer's eye respectively) that the visual angle 
 HON under which the disk is seen does not vary 
 with the distance between the two eyes.* The field 
 of view becomes smaller as this distance increases, for 
 more rays pass outside of the eye s but no change in 
 size is produced. If the observed eye be hyperopic, 
 rays after leaving it will be divergent, and this same 
 divergence might be caused by placing a suitable 
 
 * See also Fig. 45, p. 170. 
 
OPHTHALMOMETRY AND OPHTHALMOSCOPY 195 
 
 concave lens in contact with an emmetropic eye. A 
 lens in this position is without the anterior focus of 
 the observer's eye. We know that a concave lens 
 placed without the anterior focus of an optical sys- 
 tem diminishes the size of images formed by it, and 
 that the minifying effect increases as the distance of 
 the lens from the focus. Hence in hyperopia the 
 disk appears smaller than in emmetropia, and its 
 apparent size diminishes as the distance between 
 the eyes increases. In myopia the pencils are con- 
 vergent after leaving the eye, just as if a convex lens 
 were placed before the eye ; thus the disk in this 
 case appears larger than in emmetropia, and the 
 apparent size increases with the distance between 
 the eyes. 
 
 In astigmatism the disk appears smaller than 
 normal in the meridian of hyperopic and larger than 
 normal in the meridian of myopic refraction. Hence 
 supposing the disk to be circular in form, it appears 
 oval in astigmatism with the long axis in the merid- 
 ian of greatest refraction. 
 
 In the indirect method of examination the image 
 as seen by the observer varies in size with the aerial 
 image formed by the convergent lens ; hence we 
 must investigate the size of this image as affected by 
 the refractive condition of the examined eye. In 
 hyperopia the concave lens, which we suppose to be 
 placed in contact with the cornea of a normal eye, 
 
196 HANDBOOK OF OPTICS 
 
 has a magnifying or minifying effect upon the aerial 
 image according as the distance of the lens from the 
 eye is less or greater than the focal length of the 
 lens. Therefore in the indirect examination the disk 
 appears larger than normal when the distance of the 
 lens from the examined eye is less than the focal 
 length of the lens; when this distance is equal to 
 the focal length of the lens, the disk appears nor- 
 mal in size, and when the distance of the lens from 
 the eye is greater than the focal length of the former, 
 the disk appears smaller than normal. 
 
 In myopia we have the opposite conditions, that is, 
 the disk appears diminished in size when the distance 
 between the eye and lens is less than the focal length 
 of the latter ; and, increasing with the distance of 
 the lens, the disk appears larger than normal when 
 the distance of the lens from the eye is greater 
 than the focal length of the lens. 
 
 Thus also in astigmatism the disk undergoes the 
 opposite distortion to that which occurs in the direct 
 examination, provided that the distance of the lens 
 from the eye is less than the focal length of the lens, 
 and when the lens is farther from the eye than this 
 length -the distortion is the same in both methods. 
 
 This distortion, first described by Knapp, was 
 formei'ly used as a test for astigmatism. By placing 
 a suitable lens before the eye the distortion can be 
 made to disappear, and this lens represents the 
 
OPHTHALMOMETRY AND OPHTHALMOSCOPY 197 
 
 amount of astigmatism present. While this test 
 has given way to other more delicate ones, the 
 phenomenon is, in high degrees of ametropia, so 
 striking that it must attract the attention of every 
 student of ophthalmoscopy. 
 
APPENDIX I 
 
 For the convenience of those who may not be familiar 
 with trigonometrical formulae, we append the following 
 synopsis : 
 
 In the figure, ABCDE represents a quadrant, or 90 
 degrees of the circumference of a circle, of which AB, 
 
 BC, OD, and DE are equal 
 arcs. The angles AOB, 
 BOG, GOD, and DOE are 
 also equal. 
 
 If BF be drawn perpen- 
 
 dicular to OA, then — is 
 ' OB 
 
 called the sine of the angle 
 
 GH 
 AOB; similarly, —— is the 
 
 Fig. 53. 
 
 sine of GOH. 
 
 BF 
 
 OF 
 OB 
 OF 
 
 is called 
 
 the cosine of AOB, y— is the tangent, and — - the cotan- 
 OF Bh 
 
 gent of this angle. 
 
 The angle AOG is twice the angle AOB, but it is 
 readily seen that the perpendicular GH is less than 
 twice the perpendicular BF\ hence the sine of twice an 
 angle is less than twice the sine of the angle. Similarly 
 
 198 
 
APPENDIX I 199 
 
 the sine of AOD is less tlian three times the sine of 
 AOB; moreover it is evident that the increase in the sine 
 caused by adding the angle COD is less than that caused 
 by adding the equal angle BOO. As the angle approaches 
 90 degrees, an increase in this angle will produce much 
 less increase in the length of the perpendicular DM than 
 the same increase of a small angle. The opposite to this 
 is true of the cosine of an angle ; as the angle increases 
 the cosine diminishes, and at an increasing rate as the 
 angle approaches 90 degrees. 
 
 The sine of an angle increases, as we see, from zero to 
 unity as the angle increases from zero to 90 degrees, and 
 the cosine diminishes from unity to zero as the angle in- 
 creases from zero to 90 degrees. The value of the sine, 
 cosine, or tangent of any angle can be found from tables 
 which have been constructed by calculation. 
 
 All the trigonometrical formulae used in the preceding 
 pages are easily deduced from the foregoing. They are 
 as follows : 
 
 , sin a , cos a 
 
 tan a = ; cot a = — — ; 
 
 cos a sm a 
 
 sin(a±&)=sina'COS&±cosa sin 6; sin2a=2 sina -cosa; 
 sin (180 — a) = sin a ; sin^ a + cos^ a = 1. 
 
 Also in any triangle ABC, 5HLd = :?^ BC being the 
 •' ^ sin 5 AG 
 
 side opposite to the angle A, and AC being opposite to B. 
 
APPENDIX II 
 
 The following demonstration is based upon that given 
 in Heath's " Geometrical Optics " : * 
 
 In the figure, the normal, MN, to the refracting surface 
 lies in the plane of the paper represented by ABGD, and 
 
 A 
 
 
 M 
 / 
 
 / 
 
 ^_-. 
 
 / 
 
 / 
 
 / 
 
 
 F 
 
 E 
 
 \ 
 
 
 
 ■ 
 1 
 
 1 
 
 />/ 
 
 / 
 
 / 
 f 
 1 
 
 
 
 u 
 
 
 1 
 
 / 
 
 N 
 
 r\ 
 
 Q 
 
 Fig. 54. 
 
 the plane EFOH is perpendicular to the plane of the 
 paper. For the sake of clearness all lines which lie in 
 
 * " Geometrical Optics," Heath, 2d ed., p. 21. 
 200 
 
APPENDIX n 201 
 
 the plane ABCD- are drawn as continuous, and all not in 
 this plane are drawn as interrupted lines. 
 
 Let MO represent an incident ray which lies in front 
 of the plane ABCD, then OS, the refracted ray, will lie 
 behind this plane. If the first medium be air, whose 
 index is 1, and if the index of the second medium be n, 
 then we shall have the equation 
 
 sin MOB = n . sin SON. 
 
 Since the incident and refracted rays must lie in the 
 plane MENS, the deviation produced by the refraction is 
 in this plane, but we have learned that a deviation in any 
 plane may be resolved into two deviations at right angles 
 to each other ; and therefore a deviation in the plane 
 MBNS may be considered as the resultant of two deviar 
 tions lying respectively in the plane ABCD and in PQRS 
 at right angles to ABCD. To find these resultant de- 
 viations, lay off the distances BO and OS such that 
 OS = n- BO. From B, which lies in front of the plane 
 ABCD, draw BM perpendicular to MN, and BP perpen- 
 dicular to ABCD; from S, which lies behind the plane 
 ABCD, draw in like manner SN and SQ. Then as RMO 
 is a right angle, we have MB = BO ■ sin MOB, and from 
 the triangle SON we have SN= OS • sin SON, or, since 
 OS = 71- BO, SN=BO-n- sin SON; but sin MOB = 
 n • sin SON, hence MR = SN. MB is also parallel to 
 SN, since both lie in the same plane and are perpen- 
 dicular to MN. We see also that PB is parallel to SQ 
 and PM to QN; thus the triangles PMB and NQS are 
 equal, and PB = QS and PM= QN. 
 
202 HANDBOOK OF OPTICS 
 
 Let the angle FOR, wliich the incident ray makes 
 
 with the plane ABGD, be denoted by I'l, and let the 
 
 angle QOS, which the refracted ray makes with this 
 
 plane, be denoted by r^; then PB, = RO • sin i-^, and 
 
 QS =0S ■ sin n, or, since PE = QS and OS = n- BO, 
 
 we have 
 
 sm I'l = w • sm Vi. (1) 
 
 From this we see that there is the same relation be- 
 tween % and ri as between the angles of incidence and 
 refraction, MOB and SON. 
 
 If the angle POM be denoted by 12, and QON by 
 rs, we shall have PM = PO ■ sin (2 = BO • cos ii • sin i^, 
 and Q]Sf= QO ■ sin r-j = OS ■ cos rj sin rj = ?i • BO • cos r^ 
 sin rj. 
 
 From this, since PM= QN, we have 
 
 cos I'l sin 12 = M • cos rx sin rj, 
 
 • . cos r, . ,„. 
 
 or sm Zg = « * • sm r^. (2) 
 
 cos Ij 
 
 Thus we see that Jg and r^ are connected by the law 
 of refraction, the index of the first medium being 1 and 
 that of the second being n • '. From equations (1) 
 
 cos I'l 
 
 and (2) we can find the deviation of the ray BOS in the 
 planes PBQS and MPNQ, or ABCD. 
 
 When n is greater than unity, as in glass, it follows 
 from (1) that «, is greater than r-^, and since the greater 
 
 the angle the smaller is the cosine, ^ must be greater 
 
 cos «i 
 
 than unity ; hence n • is greater than n. In other 
 
 cos ii 
 
APPENDIX II 203 
 
 words, the deviation of the ray in the direction of the 
 plane ABGD is the same as would be produced in a 
 ray, PO, upon entering a medium of greater index than 
 that of the medium which we are considering. As we 
 know that the deviation increases with the index, it fol- 
 lows that the ray BO undergoes a greater deviation in 
 the plane ABGD than would a ray in the line PO, which 
 is the projection of BO on this plane. 
 
 We can now apply these deductions to refraction 
 through a prism. The plane ABGD represents a prin- 
 cipal plane of the prism; the ray BO, not in this plane, 
 will, at the first face of the prism, undergo deviation in 
 the plane MBN8, and this deviation is equivalent to a 
 certain deviation in the principal plane superposed upon 
 a deviation in the plane PBQS, which is perpendicular 
 to the principal plane. The plane PBQS, being perpen- 
 dicular to the principal plane, cuts the two faces of the 
 prism in two parallel lines. From equation (1) we have 
 seen that the angles POB and QOS, whose difference 
 expresses the deviation in the plane PBQS, are con- 
 nected by the law of refraction; and, since this plane 
 cuts the faces of the prism in parallel lines, it is clear 
 that the deviation in this plane at the first face must be 
 neutralized by that at the second face. Thus we see 
 that in any position of the ray no deviation is produced 
 in the direction of the edge of the prism. * 
 
 Let us now investigate the deviation by the prism in 
 
 * We have already assumed this to be true for the cylindrical 
 lens, which we may consider as composed of an infinite number 
 of prisms whose edges are all parallel to the axis of the lens. 
 
204 HANDBOOK OF OPTICS 
 
 its principal plane. The edge of the prism -would be 
 represented by a line perpendicular to ABCD; such a 
 line would be parallel to PR, and consequently R and P 
 are equally distant from the edge. At the first face the 
 deviation in the principal plane is greater for the ray 
 RO than for a ray, PO, equally distant from the edge of 
 the prism. We have seen that the deviation in a prin- 
 cipal plane which the ray RO undergoes at the first face 
 is the same as would be produced in the ray PO upon 
 
 entering a medium whose index is n • !. We have 
 
 cos I'l 
 
 seen also that in its passage through the prism there is 
 no deviation of the ray in the plane PRQS; hence, if 
 we trace the ray backward, we shall have at the second 
 face the same angles, FOR and QOS, that we have at 
 the first face. At the second face also, then, the oblique 
 ray is deviated in the principal plane to the same extent 
 as the corresponding ray in this plane would be deviated 
 
 by a prism whose index is n • -. Since the total 
 
 cos I'l 
 
 deviation produced by a prism increases as the index 
 increases, it is proved that the deviation in the principal 
 plane is greater for rays not in this plane than for those 
 which lie in it. 
 
 When POR (i,) increases, the index of the hypo- 
 thetical prism increases at a continually increasing rate. 
 
INDEX 
 
 Aberration, chromatic, 29. 
 spherical, 28, 181. 
 of crystalline lens, 64. 
 Accommodation, 95. 
 
 effect of, on retinal images, 118. 
 Airy, Sir George, 12i. 
 Ametropia, 94. 
 
 Angles of incidence and refrac- 
 tion, 7. 
 Aphakic eye, curvature hyperopia 
 of, 120. 
 dioptric power of, 100. 
 effect on images of correcting 
 
 leiis in, 120. 
 focal distances of, 100, 120 
 with axial elongation, 101. 
 Aqueous humor, 62. 
 
 refractive index of, 83. 
 Astigmatism, 124, 133. 
 effect of, on retinal images, 134. 
 lenticular, 161. 
 
 produced by oblique spherical 
 refraction, 165. 
 the action of a prism, 172. 
 Axis of cylindrical lens, 125. 
 of optical system, 65. 
 refracting surface, 27. 
 
 Bruecke, 189. 
 
 Cardinal planes and points, 37, 65. 
 
 of the eye, 86. 
 
 iu combination with a lens, 90. 
 Cartesian oval, 29. 
 Cataract, vision after extraction 
 of, 121. 
 
 Catoptrics, 1. 
 
 Caustic, 28. 
 
 Centrad, 22. 
 
 Centre, optical, 43. 
 
 Circle of least confusion, 132. 
 
 Colors of the spectrum, 17. 
 
 Conjugate focal planes, 36. 
 
 Conjugate focal distances and foci. 
 
 See Focal distances. 
 Constants, optical, of the eye, 67. 
 Continuous fraction, 69. 
 Convergent, 70. 
 Cornea, 62. 
 curvature of, asymmetrical, 124, 
 method of determining, 186. 
 normal, 83. 
 Corpuscular theory, 8. 
 Critical angle, 12. 
 Crystalline lens, change in curva- 
 ture of, in accommodation, 
 95. 
 curvature of, 83. 
 effect of extraction of, in axial 
 myopia, 101. 
 in curvature myopia, 101. 
 in emmetropia, 100. 
 on retinal images, 120. 
 effect of variation in position of, 
 
 104. 
 equivalent refractive index of, 
 
 64. 
 normal position of, 83. 
 oblique position of, in astigma- 
 tism, 124. 
 structure of, 62. 
 thickness of, 83. 
 
 205 
 
206 
 
 INDEX 
 
 Cylindrical lens, 125. 
 action of, 125. 
 axis of, 125. 
 combination of, with spherical 
 
 lens, 125. 
 lenses, combination of, at right 
 angles, 126. 
 at oblique angles, 148. 
 effect of, on retinal images, 
 
 135, 146. 
 twisting property of, 138. 
 
 Dennett, 22. 
 
 Descartes, 8. 
 
 Deviation, increase of, with in- 
 crease of angle of incidence, 
 14. 
 
 Dioptre, 61. 
 
 Dioptrics, 1. 
 
 Direct method of ophthalmoscopy, 
 193. 
 
 Direct pencil, 163. 
 
 Dispersion, 17. 
 
 Donders, 88, 149, 157, 161. 
 
 Double refraction, 187. 
 
 Electro-magnetic theory of waves, 
 
 4. 
 Ellipsoidal surfaces, 40. 
 Emission theory, 3. 
 Emmetropia, 94. 
 Errors of refraction, 94. 
 Ether, 3. 
 
 Eye, adaptation of, to varying dis- 
 tances, 95. 
 
 aphakic, dioptric power of, 100. 
 focal distances of, 100, 120. 
 
 artificial, 89. 
 
 as an optical system, 62. 
 
 cardinal points of, 86. 
 
 conjugate foci and focal dis- 
 tances of, 98. 
 
 far point of, 97. 
 
 illumination of, 177. 
 
 lengthening of, in myopia, 98. 
 
 nodal points of, 85. 
 
 Eye, normal, dioptric power of, 
 100. 
 focal distances of, 85. 
 optical constants of, 67, 83. 
 principal points of, 85. 
 reduced, 87. 
 schematic, 86, 88. 
 shortening of, in hyperopia, 98. 
 
 Fizeau and Foucault, 3. 
 
 Focal distance of cylindrical lens, 
 
 126. 
 Focal distances and foci, conju- 
 gate, of single spherical re- 
 fracting surface, 30. 
 of a lens, 45. 
 
 of a system of surfaces, 65. 
 in reflexion, 179. 
 principal, of single spherical re- 
 fracting surface, 32. 
 of a lens, 48, 54. 
 of a system of surfaces, 78. 
 of the eye, 85. 
 
 of the aphakic eye, 100, 121. 
 of the eye in accommodation, 
 95, 118. 
 relation between principal and 
 conjugate, 32, 57, 102. 
 Focal interval of Sturm, 131. 
 Focal length of spectacle lenses, 
 
 ,54. 
 Focal line of cylindrical lens, 126. 
 lines of bi-cylindrical lens, 131. 
 Focal planes. See Planes. 
 Foci. See Focal distances. 
 Focus, 27. 
 conjugate, 30. 
 principal, 32. 
 real, 33. 
 virtual, 33. 
 Fresnel, 3. 
 
 Galileo, 2. 
 Gauss, 64, 65. 
 Green, John, 164. 
 
INDEX 
 
 207 
 
 Hay, Gustavus, 149. 
 Heath, 150, 164. 
 Helmholtz, 62, 64, 79, 184, 188. 
 Helmholtz' formula, 39. 
 Huyghens, 3. 
 Hyperopia, 94. 
 
 effect of correcting lens on im- 
 ages in, 118. 
 
 use of convex lenses in, 96. 
 
 Image, formed by astigmatic re- 
 fraction, 131. 
 by refraction at one spherical 
 
 surface, 36, 37. 
 by refraction through a lens, 
 
 52, 58. 
 by a system of spherical re- 
 fracting surfaces, 79. 
 Helmholtz' formula for deter- 
 mining size of, 39. 
 real, 58. 
 reflected, 181. 
 
 relation between length of pencil 
 and size of, 170. 
 size of, and of object. See 
 Object, 
 retinal, effect on, of cylindrical 
 lens, 135. 
 of prism, 169, 173, 174. 
 of spherical lens, 109, 117, 118, 
 122. 
 virtual, 59. 
 Incident ray, 7. 
 
 Index of refraction. See Refrac- 
 tive index. 
 Indirect method of ophthalmo- 
 scopy, 193. 
 
 Jackson, Edward, 22, 149. 
 Javal and Schiotz, 186. 
 
 Kepler, 2, 8. 
 Knapp, 196. 
 Roller, 144. 
 
 Law, Snell's, of refraction, 8. 
 
 of reflexion, 177. 
 Lens, 40. 
 
 crystalline. See Crystalline. 
 Lenses, action of concave and con- 
 vex, 56. 
 bi-cylindrical, 127, 148. 
 classification of, 40. 
 collective, 57. 
 combination of, 60. 
 cylindrical. See Cylindrical, 
 deviating power of, 154. 
 dispersive, 57. 
 focal length of, 48, 54. 
 nodal points of, 41. 
 oblique refraction through, 163. 
 periscopic, 41, 59. 
 piano-curved, 41, 59. 
 position of, when used as spec- 
 tacles, 97, 99, 104, 118. 
 principal planes and points of, 
 
 49. 
 refractive power of, 59. 
 spherical, effect of, on images, 
 109, 117, 118, 122. 
 in hyperopia, 96. 
 in myopia, 97. 
 in presbyopia, 105. 
 sphero-cylindrical, 131, 148. 
 systems of numbering, 59, 61. 
 thickness of, 40. 
 thin, 54. 
 toric, 125. 
 Lens, Stokes, 149. 
 Lenticular astigmatism, 161. 
 Light, method of transmission of , 3. 
 reflected and scattered, 6. 
 velocity of, 3. 
 in media of different densities, 
 8,17. 
 Lippershey, Hans, 2. 
 Listing, 65, 87. 
 
 Maddox, 22. 
 prism, 184. 
 Meniscus, 40. 
 
208 
 
 INDEX 
 
 Meridian, principal, 124. 
 Metre-angle, 22. 
 Minimum deviation, 20. 
 Moebius, 64. 
 Myopia, 94, 95. 
 concave lenses in, 97. 
 
 effect of, on images, 119. 
 extraction of crystalline, ior cure 
 of, 101, 120. 
 
 Nagel, 22. 
 
 Near vision, effect of changing 
 position of lens in, 104, 122. 
 in emmetropia, 94. 
 in hyperopia, 95, 105, 119. 
 in myopia, 119. 
 in presbyopia, 105, 122. 
 Newcomh, 3. 
 Newton, Sir Isaac", 2. 
 Nodal point of single spherical re- 
 fracting surface, 37. 
 points, of lens, 41, 54. 
 of system of refracting sur- 
 faces, 81. 
 of the eye, 85. 
 
 Object, relation between size of, 
 and Image, after one spheri- 
 cal refraction, 37. 
 after refraction by a lens, 52. 
 after refraction by a system of 
 
 surfaces, 79. 
 in reflexion, 181. 
 Oblique pencil, 163. 
 Oblique refraction, cylindrical, 167. 
 
 spherical, 163. 
 Ophthalmometer, 184. 
 Ophthalmometry, optical prin- 
 ciples of, 184. 
 Ophthalmoscope, 188. 
 Ophthalmoscopy, direct, 193. 
 indirect, 194. 
 optical principles of, 189. 
 Optical centre, 43. 
 Optic disk, apparent size and 
 shape of, 194. 
 
 Optics, definition and subdivision 
 of, 1. 
 
 Paraboloid surfaces, 40. 
 Parallel rays, 31. 
 Pencil of light, 6. 
 direct, 163. 
 oblique, 163. 
 Planes and points, cardinal. See 
 Cardinal, 
 conjugate. See Conjugate, 
 principal. See Principal. 
 Prentice, 22, 149. 
 
 Presbyopes, near vision of, 105,119. 
 Principal focal distances and foci. 
 
 See focal distances. 
 Principal meridians, 124. 
 Principal plane and point, of single 
 spherical refracting surface, 
 36. 
 Principal plane of prism, 14, 169. 
 Principal planes and points of a 
 lens, 49. 
 of a system of surfaces, 72. 
 of the eye, 85. 
 
 of the eye in combination with 
 a lens, 93. 
 Prism, 14. 
 
 Wollaston's, 187. 
 Prism-dioptre, 22. 
 Prisms, astigmatic effect of, 172. 
 deviation produced by, 19, 21. 
 effect of, on pencils of light and 
 
 images, 169. 
 refracting angle of, 18. 
 refraction of pencils by, 169. 
 of rays by, 14. 
 
 of rays not in principal plane, 
 170. 
 combination of, 22. 
 numbering of, 21. 
 use of, in ophthalmology, 17. 
 
 Ray, 6. 
 extraordinary and ordinary, 187. 
 incident and refracted, 7. 
 
IKDEX 
 
 209 
 
 Keduced eye, 87. 
 Reflected images, 181. 
 
 formation of, at surfaces of 
 cornea and crystalline lens, 
 182. 
 Keflexion at a plane surface, 178. 
 at a spherical surface, 178. 
 law of, 177. 
 total internal, 12. 
 Refractive index, 7. 
 absolute, 13. 
 of spectacle glass, 21. 
 Refraction by one spherical sur- 
 face, 26. 
 by cylindrical surfaces, 125. 
 by spherical lenses, 41. 
 by system of spherical surfaces, 
 
 65. 
 by toric surfaces, 125. 
 double, 187. 
 law of, 8. 
 oblique, 163. 
 Retinal images, as affected by 
 accommodation, 118. 
 by cylindrical lenses, 135. 
 by prisms, 169. 
 by spherical lenses, 109. 
 In aphakia, 120. 
 in hyperopia, 118. 
 in myopia, 119. 
 in presbyopia, 119. 
 Reversibility of path of light, 12. 
 Romer, 2. 
 
 Schematic eye, 86, 88. 
 Skiascopy, 191. 
 Snell, 2, 8. 
 Spectacles, 2, 54. 
 Spectrum, colors of, 17. 
 Sphero-cylindrical equivalence of 
 
 bi-cylindrical lenses, 131, 
 
 148. 
 equivalent, practical method of 
 
 finding, 160. 
 Stokes, Sir G. G., 148. 
 Stokes lens, 149. 
 Sturm, focal interval of, 131. 
 
 Toric surface, 124. 
 
 Units used in numbering lenses, 
 59, 61. 
 prisms, 21. 
 
 Vision, theory of ancients, regard- 
 ing, 2. 
 Visual angle, 171. 
 Vitreous, 64. 
 refractive index of, 83. 
 
 Wave theory, 3. 
 Weiland 149, 157. 
 Wollaston's prism, 187. 
 
 Young, Dr. Thomas, 3, 124. 
 
DEFECTIVE EYESIGHT: 
 
 The Principles of Its Relief by Glasses. 
 
 BY 
 
 D. B. ST. JOHN ROOSA, M.D., LL.D., 
 
 Professor Emeritus of Diseases of the Eye and Ear; Post-Graduate Medical 
 
 School and Hospital ; Surgeon to the Manhattan Eye and Ear Hospital^ 
 
 etc. t etc. ; author of ^^A Clinical Manual of Diseases of the Eye" ; 
 
 "Ophthalmic and Otic Memoranda " ; "A Practical Treatise 
 
 on the Diseases of the Ear" ; "The Old Hospital and Other 
 
 Papers "; "A Vest-Pocket Medical Lexicon." 
 
 Cloth. i6mo. $1.00, net. 
 
 Dr. St. John Roosa has revised and very carefully enlarged the first edi- 
 tion of this vifork, which was published under the title, "The Determination 
 of the Necessity for Wearing Glasses," so as to make it a complete manual 
 for the student and practitioner. The treatise takes up all conditions re- 
 quiring the use of glasses, and indicates in the most careful manner the 
 indications and rules for describing them. It contains six chapters, and 
 is illustrated, so that a perfect understanding of the text is made easy. It is 
 well known that the author is aconservative in regard to the value of glasses, 
 believing that there is a limitation to their use, and that they ought not to 
 be prescribed unless of positive value. The revolution which has been pro- 
 duced in modern ophthalmic practice by the invention of a practical and 
 exact instrument for measuring the radius of the cornea, is fully dwelt upon. 
 No pains have been spared to make the manual a. complete guide to the 
 practitioner who wishes to understand and practise the rules for the pre- 
 scription of lenses for the improvement of impaired sight. The book may 
 also be interesting to educated men in all departments of life, who desire to 
 be informed as to advances that have been made in this interesting subject, 
 one which concerns such a large proportion of the human race. 
 
 THE MACMILLAN COMPANY, 
 
 66 FIFTH AVENUE, NEW YORK. 
 
 I