A FIfiST 800J 
 
 OF THE L : 
 
 C. Welborne Piper 
 
A FIRST BOOK O 
 THE LENS. 
 
 AN ELEMENTARY TREATISE 
 ON THE ACTION AND USE OF THE 
 PHOTOGRAPHIC LENS. 
 
 BY 
 
 C. WELBORNE PIPER. 
 
 ILIFFE & SONS LIMITED 
 
 20, Tudor Street, London, E.C.4 
 
PRINTED BY 
 HAZEI.L, WATSON, AND VINEY, T.D., 
 LONDON AND AYLESBURY. 
 
CONTENTS 
 
 CHAPTER I. 
 
 PAGE 
 
 LIGHT AND OPTICS 5 
 
 CHAPTER II. 
 
 ACTION OP LENSES . . . . . . .21 
 
 CHAPTER III. 
 
 FOCAL POINTS AND DISTANCES ..... 37 
 CHAPTER TV. 
 
 COMBINING LENSES 50 
 
 CHAPTER V. 
 
 ABERRATION . . Gl 
 
 CHAPTER VI. 
 
 SCALE, INTENSITY AND ILLUMINATION . . . .92 
 
 CHAPTER VII. 
 
 DEPTH . . . . . . . . .110 
 
4 Contents. 
 
 CHAPTER VIII. 
 
 PACK 
 
 FOCUSSING-SCALES • .127 
 
 CHAPTER IX. 
 
 MEASURING LENSES 133 
 
 CHAPTER X. 
 
 TYPES OF LENSES 143 
 
 APPENDIX. 
 
 TABLES, ETC ^3 
 
 INDEX . . 167 
 
A FIRST BOOK OF THE LENS. 
 
 CHAPTER I. 
 
 LIGHT AND OPTICS. 
 
 1. Light and Optics. — -The science of optics is de- 
 rived from the study of the natural composition and 
 propagation of light; visible light being a manifestation 
 of certain forms of energy. A knowledge of optics 
 enables us, with the aid of certain appliances, to so control 
 the composition and transmission of light as to produce 
 certain desired effects. 
 
 In photography we employ a lens to produce inside a 
 camera an image of such objects as come within a certain 
 outside range of view; and the purpose of this book is 
 to explain the manner in which such an image is produced, 
 and the rules governing its production under varying 
 conditions. Our main subject of study is, therefore, the 
 photographic lens, but its action cannot be well under- 
 stood or explained without a preliminary knowledge of 
 the theories of light and optics. 
 
 All that we know of light is derived from the study of 
 its effects, and by a logical process, supplemented as far 
 as possible by experimental tests, a theory with regard 
 to the constitution and propagation of light has been 
 evolved, upon which is based the science of optics. This 
 theory of light is of such a nature that we can look at 
 
6 
 
 A First Book of the Lens. 
 
 it from two points of view, and evolve from it two optical 
 theories, perfectly compatible with one another in all 
 respects, and dependent on each other ; one is known as 
 physical optics, and the other as geometrical optics. Both 
 lead to the same result ; but while the geometrical theory 
 is in most cases more easily understood and applied, an 
 appeal to the physical theory will usually more quickly 
 elucidate a knotty point. As the result of the existence 
 of two optical theories, there are always two methods of 
 expressing an optical fact, and one mode of expression 
 being sometimes more convenient than the other, it is 
 advisable to have at least a distinct appreciation of the 
 difference between the two theories. 
 
 2. The Theory of Light. — Under the accepted wave 
 theory, light is nothing but a travelling undulation in 
 an imperceptible hypothetical medium called ether, which 
 is supposed to pervade all space throughout the universe. 
 Such a travelling undulation may be illustrated by dropping 
 a stone into a pond of still water, and observing the 
 circular series of ripples that apparently move outward 
 from the point of impact in all directions. This apparent 
 outward movement is an illusion : it is not the water 
 but the undulation only that travels. All the move- 
 ment that really takes place is a circular over-and-over 
 motion in a vertical plane of each particle of water in 
 succession ; as the particles rise they form ripples, as they 
 fall troughs are produced between the ripples. But, though 
 the water does not move outwards, an impulse is imparted 
 from one particle of water to the next which eventually 
 causes the undulations to extend over the whole surface of 
 the pond. This impulse may be considered to travel in 
 straight lines radiating outwards in all directions from 
 the centre of the ripples, while the circular ripple that at 
 any particular period is farthest from the centre denotes 
 the limit to which the impulse has at that moment reached. 
 The primary cause of the ripples is of course the impact 
 of the stone upon the water, and the point of impact is 
 the starting point of the impulse which causes the ripples. 
 
 Applying this analogy to the case of light, we may look 
 upon a source of light as the source of an impulse, or 
 energy, that imparts an undulatory wave or ripple-like 
 
Light and Optics. 
 
 7 
 
 motion to the particles of ether.* These wave movements 
 do not, however, take place on a plane surface, but in 
 the interior of a mass ; and therefore tbe movement which 
 denotes the limits to which the impulse at any particular 
 moment has reached, does not take the form of a circular 
 ripple on a plane surface, but forms the surface of a 
 sphere, the centre of which is the source of energy, and 
 the radius of which is equal to the distance travelled by 
 the impulse. As the light travels farther and farther 
 into space there is produced a continually advancing and 
 expanding spherical wavefront due to the continually pro- 
 gressing impulse, which travels at a certain velocity in 
 straight lines direct from the source of light. 
 
 In optics we are concerned mainly with the effect of 
 placing obstructions of various kinds in the way of the 
 travelling energy and wavefront. A lens is nothing but 
 a temporary obstruction placed in the way of the light, 
 and the effect of such an obstruction is that the direction 
 of the impulse is changed, and the form of the wavefront 
 altered, just as the circular form of the water ripple is 
 altered if it meets an obstruction in the pond. This is 
 the point where the two theories of optics start. We 
 may study only the effect of the obstruction on the wave- 
 front, which is physical optics ; or Ave may confine ourselves 
 to the study of the effect of the obstruction on the direction 
 of the impulse, which is geometrical optics. In either 
 case we shall arrive at the same result, for under all 
 conditions the direction of impulse is normal to the wave- 
 front, and so the two change together. 
 
 3. Physical Optics. — Assume that we have a source 
 of light in the shape of a point as at P (figs. 1 and 2), 
 and that at L we have a lens. Under the physical theory, 
 from P there is a progressive expanding series of con- 
 centric wavefronts, a, «, a, etc., the advance of which 
 (disregarding all light that does not actually meet the 
 lens) is temporarily hindered by the lens, which, though 
 transparent, is of denser substance than the air. The 
 thickness of the lens being variable, the obstruction offered 
 
 * These ether ripples are not exactly comparable with the water 
 ripples, and the analogy must not be pushed too far, 
 
8 
 
 A First Book of the Lent 
 
 by it also varies ; hence the curvature of the wavefront 
 is altered. If the greatest obstruction is offered by the 
 centre of the lens, as in fig. 1, the convex wavefront is 
 flattened during its passage through the glass and becomes 
 concave as it leaves it, as shown at b, b, b, etc., and when 
 free of the lens it contracts as it progresses towards the 
 point Q, on which point the altered directions of impulse 
 now converge. This point is a focus. If there is no 
 
 further obstruction in the way after having contracted 
 to a focus the continued impulse causes the wavefront to 
 again expand. The focus is then practically a new source 
 of light, and the new expanding wavefront is convex as 
 shown at c, c, c, etc. 
 
 If the greatest obstruction to the passage of the light 
 is offered by the margins of the lens, as in fig. 2, the centre 
 of the wavefront progresses more rapidly than the margins, 
 and the whole becomes more convex, and, when free from 
 the lens, expands more rapidly, and as if it originated from 
 a point Q between the lens and the original source of 
 light 7*. The point Q is then a focus, differing from Q 
 
Light and Optics. 
 
 9 
 
 in fig. I in that it is an unreal or " virtual " focus instead 
 of a real one. In fig. 1 the light really contracts on 
 to Q, but in fig. 2 it only appears to expand from the 
 focus. This virtual focus is also in effect a new source 
 of light. Fig. 1 illustrates the effect of the type of lens 
 known as positive, fig. 2 that of the* one known as a 
 negative lens. 
 
 4. Geometrical Optics. — We will now consider the 
 action of a similar pair of lenses from the point of view 
 of geometrical optics. In figs. 3 and 4 again let P 
 represent the source of light, but, instead of showing the 
 
 expanding wavefronts, let the lines Pa represent a few 
 of the directions of impulse of the energy emanating from 
 P • as before ignoring all light that does not actually 
 meet the lens. Altering the nomenclature to suit the 
 accepted conventions of geometrical optics, we call these 
 lines of direction of impulse or energy " light rays." 
 
 All rays that are obstructed by the lens, excepting only 
 the one that is exactly normal to the lens surface, are bent 
 at the points of contact a, a, etc., and after passing through 
 the substance of the lens they are again bent at the second 
 surface at the points b, b, etc., again excepting the one 
 ray that meets that surface normally. After the second 
 
10 
 
 A First Book of the Lens. 
 
 bending they are either actually convergent on (as in fig. 3) 
 or apparently divergent from (fig. 4) the focus Q. After 
 intersecting each other at the focus Q in fig. 3, the rays 
 again diverge, in the absence of an obstruction, as shown 
 at c, c, c, etc. 
 
 This is a very Convenient but conventional theory ; light 
 rays do not actually exist while wavefronts are a real con- 
 ception. To explain why the conventional light rays are 
 bent at the surfaces of the lens we must look upon them 
 as normals to wavefronts, and resort to the true physical 
 theory; but, in most cases, we can more easily trace and 
 explain the action of lenses with the aid of the geometrical 
 theory. It must, in future, be understood that reference 
 to light rays implies that we are looking at whatever 
 matter is under consideration from the geometrical point 
 of view, while mention of wavefronts implies that we are 
 taking up a physical standpoint. 
 
 It is easy to remember the distinction between the two 
 theories if we clearly understand that : 
 
 (a) In physical optics we consider only the form of the 
 travelling wavefront. 
 
 (b) In geometrical optics we consider only the direction 
 of movement of particular points in the travelling 
 wavefront. 
 
 (c) Any point in a wavefront moves in a direction 
 normal to the wavefront. 
 
 5. Spherical Aberration. — If from any cause a focus, 
 such as that shown in figs. 1 to 4, is either not formed 
 perfectly, or is misplaced, aberration is considered to exist. 
 If in either of the illustrated cases the altered wavefront 
 is not perfectly spherical, the various altered directions 
 of impulse, which are always normal to the curved surface, 
 cannot either converge upon or diverge from one point, and 
 a true or sharp focus cannot be formed. This particular 
 aberration, due to non- sphericity of the wavefront, is known 
 as spherical aberration. A lens so designed and constructed 
 as to ensure perfect sphericity of the altered wavefront, 
 is called aplanatic, while if spherical aberration exists the 
 lens is non-aplanatic* 
 
 * Special attention should be given to these definitions, as various 
 
Light and Optics. 
 
 I J 
 
 6. Colour. — -We cannot thoroughly understand the 
 theory of light without studying that of colour, for 
 colour is an effect of light, which indicates by its variations 
 the existence of varieties of light. 
 
 Going back to our simile of the ripples on the pond, it is 
 evident that the size of the ripples depends on the size of 
 the stone, and the force with which it strikes the water, 
 for a small pebble will raise only small ripples while a large 
 rock will produce big waves. Also, a succession of stones 
 thrown into the water will produce a succession of systems 
 of ripples. Something similar to this takes place at a 
 source of light. The ether undulations vary in size with 
 the force of the impulse that starts them, and the prim- 
 ary causes of the impulse being various, and continually 
 at work, innumerable series of undulations of different sizes 
 are continuously travelling from the source. Ordinary 
 daylight, as we appreciate it, is composed of a number of 
 undulations varying in size — both as regards wavelength, 
 or the length of an undulation measured from crest to 
 crest of each ripple, and as regards amplitude, or height 
 measured from the top of the crest to the bottom of the 
 trough. The effect produced by the undulations depends on 
 their wavelength and amplitude, colour varying with the 
 former and intensity with the latter. 
 
 A wavefront of white light is therefore not formed by 
 a simple undulation, but by a number of undulations, 
 and when it meets the surface of a medium such as glass 
 the resistance offered varies with undulations of different 
 Avavelength. Glass offering the greatest resistance to the 
 passage of undulations of the shortest wavelength the 
 different sets of undulations become separated, each 
 forming its own wavefront ; those which meet with the 
 greatest resistance forming wavefronts of the greatest 
 curvature, or shortest radius. 
 
 writers attach various meanings to the terms " spherical aberration " 
 and " aplanatic." This note applies also in the case of other defini- 
 tions of technical terms. No two authors seem to agree throughout 
 in matters of nomenclature, and students are often confused by the 
 various applications of the same word. The knowledge that such 
 differences of expression exist, should, however, be an efficient 
 safeguard against any misapprehension. 
 
12 
 
 A First Book of tJie Lens. 
 
 7. Chromatic Aberration. — Suppose P (fig. 5) to be a 
 source of compound light emitting two series of undulations 
 of different wavelength or colour, say red and violet. The 
 colour of the light will then be a compound of red and 
 violet. Let a, a, a represent the expanding wavefront of 
 compound light progressing from P to the lens L. During 
 the passage through the lens the violet light, which is 
 composed of undulations of very short wavelength, is more 
 retarded than the red, and it therefore forms a wavefront 
 of its own (as indicated by the dotted curves b, b, b), which, 
 being of greater curvature than that of the red light (in- 
 dicated at c, c, c), contracts to a focus at a nearer point, at Q x 
 instead of Q. Thus there are two foci formed, one of violet 
 and one of red light. 
 
 
 <Cj i | Nl i | i 
 
 
 
 
 p^-l; - / • i 
 
 
 Q 
 
 
 
 
 L 
 
 
 If we draw rays normal to the different wavefronts, 
 as shown by straight lines on the diagram, we find that 
 what we may call a single compound ray from P splits up 
 at the point of contact with the lens into two rays, one of 
 which we may call the violet ray, and the other the red, 
 each of which goes to its own focal point. To understand 
 this it is necessary to remember the conventional nature 
 of a ray of light. 
 
 This chromatic aberration can be corrected by combining 
 different materials in one lens to so control the direction 
 of the separated rays of various colours that they may 
 re-combine at the focal point, and so produce a focus of 
 the same colour as the original light. A lens so adjusted 
 is styled achromatic for such colours as are thus recom- 
 bined, while it is non-achromatic, or possesses chromatic 
 aberration, in the case of any colours which come to 
 
Light and Optics. 
 
 13 
 
 different foci. If the light is very complex the lens is 
 generally achromatic for certain colours only, and non- 
 achromatic for the rest. If achromatic for more than 
 two colours it is sometimes styled apochromatic. 
 
 It should be understood that in considering light and 
 colour we deal only with a comparatively small number 
 of ether waves. The longest visible ethereal wave is only 
 about twice the length of the shortest, and light and colour 
 only covers an octave or so in a mighty scale of graduated 
 ethereal waves of unknown limits. Some of the invisible 
 waves beyond the visible light octave are of importance to 
 the photographer. The visible vibrations give the colour 
 effects of red, orange, yellow, green, blue, and violet, with 
 intermediate shades ; and, in addition to such colour effects, 
 these same vibrations give effects of heat and chemical 
 action, the former being most intense at the red end of the 
 scale, and the latter at the violet end. Beyond the visible 
 red we have infra-red vibrations which produce more 
 powerful heat effects, and beyond the visible violet we have 
 ultra-violet vibrations of great chemical or actinic energy, 
 and it is by this ultra-violet invisible " light " that the 
 photographic image is mainly produced. In an achromatic 
 photographic lens the visible vibrations by which we focus, 
 and the actinic vibrations by which the image is produced 
 must come to the same focus, but in a lens that is to be 
 used solely for visual purposes, the latter vibrations need 
 not be considered, and the lens need only be achromatic 
 for the brighter visible light ; hence a lens may be perfectly 
 achroinatised for use in a lantern, but non- achromatic for 
 use in a camera. Thus, there are varying degrees of 
 achromatism. 
 
 8. Light Pencils. — Referring again to figs. 1 to 5, 
 it is easily seen that only a small portion of the total 
 amount of light emanating from the source P is actually 
 transmitted by the lens. No light outside the dotted lines 
 in figs. 1 and 2, and the corresponding rays in figs. 3 and 
 4, is utilised. The limited quantity of light that falls 
 on the lens forms a pencil of light incident to the lens, 
 and this pencil has the form of a cone with a convex 
 spherical base or wavefront. On the opposite side of the 
 lens we have an emergent pencil also forming a cone, the 
 
14 
 
 A First Book of the Lens. 
 
 apex of which is the focus. The incident and emergent 
 pencils together form a double pencil of light, which in 
 the case of a positive lens consists of two cones base to 
 base. The incident pencil is divergent, while the emergent 
 pencil is convergent. The diameter of the pencil is (in the 
 diagrams) limited by the diameter of the lens; and, the 
 lens being circular, a transverse section of the pencil is 
 also circular, if the source of light is directly in front of 
 the lens, but elliptical if the pencil passes obliquely through 
 the lens. 
 
 If the source of light is removed to an extreme distance 
 from the lens, the length of the incident pencil becomes so 
 much greater than its diameter that the rays composing 
 it become practically parallel, and its wavefront practically 
 plane. In such a case it is sufficient to consider the pencil 
 to be truly parallel with a plane wavefront, or to be 
 cylindrical instead of conical in form. 
 
 When light passes through a series of lenses, a pencil 
 emergent from the first lens is incident to the second, and 
 so on. Such intermediate pencils may have either concave, 
 convex, or plane wavefronts — i.e. they may be convergent, 
 divergent, or parallel. If the series of lenses is used as a 
 photographic combination, the incident pencil to the first 
 lens must be either parallel or divergent, while the emergent 
 pencil from the last lens must be convergent. 
 
 9. Aperture. — In the preceding diagrams the greatest 
 diameter of the incident pencil is limited by the diameter 
 of the lens, but, actually, it is limited either by the brass 
 mounting of the lens, or by a metal diaphragm or stop placed 
 near the lens. However limited, the greatest diameter 
 of the incident pencil is the diameter of the <! effective " 
 aperture of the lens. It is important to remember 
 that the effective aperture of a lens does not necessarily 
 correspond with the actual aperture of the diaphragm, and, 
 also, that it is liable, under certain conditions, to vary in 
 diameter with variations in the distance of the object. 
 This is especially the case when a diaphragm is placed a 
 little distance from the lens. The effective aperture for 
 a parallel pencil is then less than it is for a divergent 
 pencil from a near object, and the nearer the object the 
 greater becomes the aperture, until its expansion is limited 
 
Light and Optics. 
 
 15 
 
 by some other mechanical obstruction. This inconstancy of 
 aperture is one of the small matters that are commonly 
 neglected by photographers. 
 
 The aperture for a parallel pencil is always taken as 
 the standard or nominal aperture of the lens, and variations 
 are universally ignored, with the result that some apparently 
 sound theoretical principles in connection with the use of 
 lenses break down to a certain extent in practice. The 
 matter of aperture will be frequently alluded to again. 
 The term nominal aperture is frequently applied to the 
 actual diaphragm aperture to distinguish it from the effective 
 aperture. The term is not, however, used in that sense 
 in this book. 
 
 10. Refraction. — So long as a pencil of light is travel- 
 ling in one medium only, its wavefront remains unaltered 
 and its rays continue to travel in straight lines. When, 
 however, it meets the surface of a second medium of 
 either greater or less density, or resisting power, the form 
 of the wavefront is altered, while the rays are bent or 
 deviated from their original course, as has been shown in 
 figs. 1 to 4. This deviation, or alteration in form of 
 wavefront, is called refraction, and a surface at which 
 it takes place is a refractive surface. The degree to 
 which a refracted ray of a certain wavelength is deviated 
 from its original course, depends, first upon the angle at 
 which the incident ray meets the surface, and second 
 upon the relative density, or resisting power, or relative 
 refractive power of the two media ; every two media 
 capable of transmitting light having a certain constant 
 refractive power for light of a certain wavelength or 
 colour. 
 
 Relative refractive power is represented by the ratio 
 of the relative velocities of light in the two media, and 
 this ratio is styled the relative refractive index. The 
 velocity of light of any colour is constant in a vacuum, 
 and different media are classified according to the refrac- 
 tive indices from a vacuum. The ratios are then styled 
 absolute refractive indices, but they do not differ greatly 
 *from the relative refractive indices from air into the 
 several media. 
 
 The greater the refractive index of any medium, the 
 
L6 
 
 A First Book of the Lens. 
 
 greater the deviation produced in a ray entering that 
 medium. Thus the absolute refractive indices of flint 
 and crown glass being respectively 1 "57 and 1'52, any 
 particular ray falling on a surface of flint glass suffers 
 greater deviation than a similar ray meeting a surface of 
 crown glass at the same angle. 
 
 The angle of incidence of a ray on a refractive surface 
 is measured, not from the surface, but from a normal to 
 the surface. In fig, 6 two surfaces (cibc) are shown, one 
 curved and the other plane, the description applying to 
 either indiscriminately. The surface may be considered 
 to be one separating air and glass. Consider db to be 
 a ray of light of a certain fixed colour, or wavelength, 
 passing through air and striking the surface of the denser 
 
 medium in the point b. Instead of continuing straight 
 on to e, it suffers deviation from its original course to 
 the extent of the angle ebf, which is the angle of deviation. 
 Through b draw gh normal to the surface — that is, at 
 aight angles to the plane surface, and radial to the curved 
 surface, the centre of which is at h. The direction of the 
 light being from d to b, the ray db is incident to the 
 surface abc, its degree of incidence being measured by 
 the angle dbg, which is known as the angle of incidence. 
 The angle Jbh, made by the refracted portion of the ray 
 with the normal bh, is the angle of refraction, and the 
 sine of this angle is equal to the sine of the angle of 
 incidence divided by the relative refractive index from 
 air into glass ; or, to put the relationship of the angles 
 in another form, the ratio of the sine of the angle of 
 incidence to the sine of the angle of refraction equals 
 
Light and Optics. 
 
 17 
 
 the relative refractive index. This is called the law of 
 sines. 
 
 The course of a ray of light is always reversible, and we 
 may assume that in fig. 6 the light travels from / to b in 
 the glass and is then refracted to d. The angle of incidence 
 is then the angle fbh, and the angle of refraction is gbd, 
 while the deviation is the same as before : that is, it is equal 
 to ebf. The ratio of the sines of the angles of incidence and 
 refraction is now the converse of the former ratio — i.e. if, 
 when the light passes from air into glass the ratio (or the 
 relative refractive index) equals then when the light 
 passes from glass into air the ratio or index is l/p. Sup- 
 posing light to pass through two refractive surfaces, such 
 as those of a lens, it first passes from air into glass and 
 then from glass into air, so that the ratio of the sines 
 at one surface is the reciprocal of that at the other. 
 
 If the surfaces are parallel the angle of incidence at the 
 second surface is equal to the angle of refraction at the first, 
 and the angle of refraction at the second surface is equal 
 to the angle of incidence at the first, therefore the first 
 incident and last emergent rays are parallel. The surfaces 
 may be considered to be parallel when their normals are 
 parallel, and this condition may be fulfilled with any form 
 of lens as well as with plane surfaces. 
 
 When a ray, after undergoing refraction and deviation at 
 the several surfaces of a lens, finally emerges in a direction 
 parallel to its original course we may consider that the lens 
 has produced refraction without deviation, or that the ray 
 is displaced but not deviated from its original direction. 
 Sometimes, however, with a certain arrangement of lenses 
 the ray may ultimately be free even from displacement, 
 and pursue a course in continuation of its original one. 
 When a pencil of light falls upon a lens, one ray (if the 
 whole surface of the lens is utilised) must pass free from 
 deviation, and that ray is called an axial ray, but it is not 
 necessarily the central (or " chief") ray of the pencil. 
 
 In fig. 6 there are some points to be noticed in connection 
 with the deviation of rays. 
 
 (a) When a ray passes from a light medium into a dense 
 one it is refracted towards the normal — i.e. bf is 
 nearer bh than is be. 
 
 ■1 
 
18 
 
 A First Book of the Lens. 
 
 (b) When a ray passes from a dense medium into a light 
 one it is refracted away from the normal— i.e. db 
 is farther from gb than would be a continuation of fb. 
 
 (c) The angle of deviation is equal to the difference 
 between the angles of incidence and refraction, — i.e. 
 ebf=dbg—fbh. 
 
 (d) The greater the angle of incidence the greater are 
 the angles of deviation and refraction. 
 
 11. Dispersion. — As the refractive power of a particular 
 medium varies with rays of different colours or wave 
 lengths, compound rays are separated at the point of in- 
 cidence on that medium. Let a ray db (fig. 7) of combined 
 red and violet light fall on a glass surface abc at b, and, 
 as before, let gbh be normal to that surface. As the violet 
 
 light suffers greater deviation than the red, the former 
 takes the direction bv, while the latter takes that of br. 
 The degree to which the rays are separated depends on the 
 difference between the refractive powers of the glass for 
 red and violet rays. This separation is known as dispersion, 
 and the degree of dispersion is represented by the angle 
 rbv which equals the difference between the deviations of 
 the two rays. The dispersive power of a medium for two 
 rays of particular colours thus varies with its refractive 
 indices for those two colours. The dispersive power of any 
 one medium varies with different pairs of rays, and, to ex- 
 press it, it is necessary to specify the particular colours and 
 to take some one particular colour as a standard of reference. 
 
 12. Reflection. — When light falls on a surface at an 
 angle with the normal, part only of the light suffers 
 
Light and 0])tics. 
 
 19 
 
 refraction ; a certain quantity is reflected in an opposite 
 direction. Thus in fig. 8 a ray db, incident on the 
 surface abc, of either a dense or rare medium is partly 
 refracted in the direction of bf, or bk, and partly reflected 
 in the direction bm, the angle of reflection rjbm being 
 (under the law of reflection) equal to the angle of incidence 
 dbg, both angles being measured from the normal gh. 
 There is therefore a certain loss of intensity in the re- 
 fracted ray, and if there are a number of refractive 
 surfaces in succession, there may be considerable total 
 diminution of light owing to the loss at each surface. 
 
 The greater the angle of incidence, the greater is that 
 of reflection, and the greater is the amount of light 
 reflected and lost. To obviate excessive loss of intensity 
 
 of light, it is advisable to avoid an excessive number of 
 reflecting surfaces, and also to avoid very oblique angles of 
 incidence. Oblique light must be employed, but by careful 
 selection of curvatures for the refracting surfaces very 
 oblique incidence may be to a certain extent obviated. 
 
 When a refracting surface is a separating surface 
 between two different kinds of glass, reflection may be 
 diminished by cementing the two surfaces together with 
 some medium, the refractive index of which is approxi- 
 mately the same as the glass. This renders the lens more 
 nearly homogeneous in structure : that is to say, it brings 
 the two different glasses into optical contact without any 
 separating film of air, and, as the less the difference in 
 density of the media bounding a refractive surface the 
 less is the loss of light by reflection, surfaces between 
 
20 
 
 A First Book of the L&iis. 
 
 glass and glass are of less consequence than those between 
 glass and air. 
 
 If we assume the surface abc to be that of a rare 
 medium, and the angle of incidence to be so great that 
 the still greater angle of refraction would come to 90°, 
 refraction does not take place, and the whole of the light 
 is reflected. This is known as total reflection, and the 
 angle of incidence at which total reflection takes place is 
 called the " critical " angle of the dense medium. Reflection 
 and refraction together combine to produce false images, in 
 addition to the true one due to refraction only, and these 
 false images are the cause of the t-ffect known as Flare. 
 
CHAPTER II. 
 
 ACTION OF LENSES. 
 
 13. Lenses. — A lens may be defined as an isolated portion 
 of a medium bounded on two opposite sides by refractive 
 surfaces of regular geometrical form having a common 
 normal. These surfaces are usually either plane or 
 spherical ; non-spherical, or conic curvatures possessing no 
 advantages for photographic purposes. In fig. 9 a series 
 of lenses of various forms are shown in section. Each 
 lens has either two spherical, or spherical and plane 
 surfaces ; and, as all centres of curvature are on the 
 line PQ, which is also normal to every plane surface, 
 that line is a common normal to all the surfaces, and to 
 each lens. 
 
 The common normal to any lens is called the principal 
 axis of the lens, and a ray of light coinciding with the 
 principal axis passes through the lens without suffering 
 any refraction, its angle of incidence being nil at each 
 surface. When several lenses are employed in combination, 
 their principal axes are made to coincide so that there 
 is one common normal for all the lenses ; this adjustment 
 (a very delicate one to carry out) is called centring the 
 lenses. 
 
 14. Positive and Negative Lenses. — The lenses shown 
 in fig. 9. represent two classes, positive or convergent 
 lenses (A, £, and C), and negative or divergent lenses 
 (D, E, and F). The former are thicker in the centre than 
 at the edges and the latter thickest at the edges. This is 
 an essential distinction in simple single lenses, though it 
 is not so with compound ones. 
 
 Figs. 1 to 4 (Sees. 3 and 4) showed the effect of 
 21 
 
22 
 
 A First Book of the Lens. 
 
 typical positive and negative lenses on a pencil of light 
 coming from a certain near source ; we now consider the 
 case of the incident pencil being parallel, or having a 
 plane wavefront, as is the case when the source of light 
 is at an infinite distance from the lens. Under these 
 conditions — assuming the lens to be aplanatic (Sec. 5) — with 
 a positive lens (Li fig. 10) the emergent pencil is always 
 composed of converging rays and has a concave wavefront ; 
 contracting on to a real focus, With a negative lens Z 2 
 the emergent pencil is always divergent, and has a convex 
 wavefront expanding from a virtual focus Q. In fig. 10 
 rays are shown in full, and wavefronts in broken lines, and 
 the rays are continued to the virtual focus by dotted lines. 
 
 From these figures it is evident that with a positive lens 
 all non-axial rays suffer deviation towards the axis of the 
 light pencil, but with a negative lens they are deviated 
 from the axis. This is the characteristic distinction 
 between the two types of lenses, for, while positive lenses 
 with very near objects may fail to produce a real focus 
 they always produce positive deviation (towards the axis) 
 in non-axial rays ; and negative lenses, which do not always 
 produce virtual foci, always produce negative deviation 
 away from the axis in non-axial rays. These conditions are 
 universal, whether the light pencils are " direct " (axial ray 
 coinciding with principal axis) or " oblique," and, as a result, 
 with positive lenses incident parallel rays are rendered 
 convergent, convergent incident rays more convergent, and 
 
Action of Lenses. 
 
 2:5 
 
 divergent rays either less divergent, parallel, or convergent. 
 With negative lenses parallel rays become divergent, 
 divergent rays more divergent, and convergent rays either 
 less convergent, parallel, or divergent. The emergent rays 
 come to a real focus if convergent, lead from a virtual 
 focus if divergent, and come to no focus, or only to an 
 infinitely distant one, if parallel. The formation of a focus 
 of either a real or virtual type thus depends with either 
 class of lenses on the nature of the incident pencil. 
 
 15. Forms of Lenses. — In fig. 9 six typical forms of 
 
 simple single glass lenses are shown. These are distin- 
 guished by the following names descriptive of their forms, 
 surfaces, or properties. 
 
 Positive or Convergent Lenses. — 
 
 A. A double convex lens ; " symmetrical " if the sur- 
 faces are of the same curvatures, " crossed ' if of 
 different curvatures. 
 
 B. A plano-convex or convexo-plane lens, according 
 as we consider the light to be first incident on 
 the plane or on the convex surface. 
 
 C. A positive or convergent meniscus lens ; or 
 
24 
 
 A First Book of the Lens. 
 
 convexo-concave or concavo-convex positive lens, 
 also called a periscopic lens. 
 Negative or Divergent Lenses. — 
 
 D. A double concave lens, " symmetrical " or " crossed " 
 according as the surfaces are of the same or 
 different curvatures. 
 
 E. A plano-concave or concavo-plane lens. 
 
 F. A negative or divergent meniscus lens ; or concavo- 
 convex or convexo-concave negative lens. 
 
 The above are, properly speaking, single lenses, but the 
 term single is also applied to a combination of two or more 
 lenses cemented together, or very slightly separated, to 
 distinguish such a close combination from a double or triple 
 combination, of widely separated lenses. Thus a " single " 
 lens may be a single simple lens or a single combination of 
 several simple lenses, and a doublet or triplet may consist 
 of two or three separated simple lenses, or of two or three 
 separated single combinations. 
 
 A doublet consisting of two separated simple lenses, or 
 single combinations, is called a symmetrical doublet if the 
 two lenses or combinations are similar in all respects. If 
 they vary it is non-symmetrical. If one single lens or 
 combination is positive and the other negative, the doublet 
 may be further distinguished as periscopic.* In fig. 11 
 some typical forms of single, double, and triple lenses are 
 illustrated. 
 
 1 G. Lens Surfaces. — Simple lenses such as those shown 
 in fig. 9, have two refractive surfaces, and it is sometimes 
 convenient to look upon compound lenses as combinations 
 of surfaces rather than of lenses. 
 
 The refractive surfaces of lenses may be classed as 
 positive or negative according as non-axial rays are deviated 
 to or from the principal axis. Speaking generally, a 
 refractive surface between two media is positive when it 
 is convex to the rarer medium, but negative when concave. 
 
 * Sec footnote to Sec. 5. Also see Sec. 95 in Chap. 10. The 
 term "periscopic" being commonly applied to a single simple 
 positive meniscus lens with one negative and one positive surface,' 
 is here applied to a doublet formed of one negative and one 
 positive lens or single combination. A distinctive title for such 
 doublets is desirable, but a better one might be devised. 
 
>3impie. 
 
 Combinations 
 
 tfymmetricaL. -Nonsymmetrical 
 
 Triplets . 
 
 Fiji 11 
 
 scop i c do ulslef. 
 
26 
 
 A First Book of the Lens. 
 
 Therefore, it follows that a double convex lens has two 
 positive refractive surfaces, while a meniscus has one 
 positive and one negative surface. By combining lenses 
 together we can produce a positive combination possessing 
 three or more surfaces. For example, lens A in fig. 12, 
 has two positive surfaces. If we combine it with the 
 negative lens F, as shown at a, the combination as a whole 
 may be still positive, but it then has four refractive surfaces, 
 three positive and one negative. If the adjacent curvatures 
 of the lenses are of the same radius we may cement them 
 together as at b, and we then have a positive lens with 
 three refractive surfaces, if the component lenses are of 
 different densities. If F is of denser material than A, the 
 contact surface is negative, but if A is denser than F, the 
 
 A F AF 
 
 contact surface is positive, and we have three "positive 
 surfaces instead of two positive and one negative. The 
 addition of a third lens cemented to A or F, and of different 
 material, adds a fourth surface. If the third lens is 
 separate we have five surfaces, and so a positive lens of 
 any number of surfaces can be built up. We may then 
 look upon the result, either as a combination of so many 
 lenses, positive or negative, or as a combination of so 
 many refractive positive or negative surfaces. To under- 
 stand the purpose and effect of combining lenses it is advis- 
 able to adopt both points of view, as will be apparent later. 
 
 It should be observed that A and F must be of different 
 densities to produce three surfaces when cemented; if of 
 the same density the cemented contact surface is not a 
 refractive surface (Sec. 10), and the lens as a whole has 
 
Action of Lenses. 
 
 27 
 
 then only two surfaces. If the lenses are separated as 
 at a, whatever the respective densities, there must he 
 four surfaces. 
 
 17. Formation of Images. — In order that an object 
 may be photographed it must be luminous ; that is, it 
 must either be a source of light or must reflect light 
 emanating from some other source. The surface of the 
 object may therefore be considered to consist of an infinite 
 number of infinitely small luminous points, each of which 
 emits light in all possible directions. By the employment 
 of suitable optical appliances we select certain rays proceed- 
 ing from each luminous point in the direction of the 
 camera, and so control their course as to produce upon 
 a screen, or sensitive surface, an image which may also 
 be considered to consist of an infinite number of infinitely 
 small points, each corresponding with one of the infinitely 
 small luminous points in the object. For this purpose we 
 employ either a pinhole, which only permits the passage 
 of the few rays that are naturally travelling in the right 
 direction, or a positive lens, which collects a number of 
 rays travelling approximately in the right direction from 
 the object point, and refracts them in such a way as to 
 eventually bring them all to the proper image point. 
 
 To illustrate the formation of an image we need only 
 consider two or three luminous points in the object and 
 their corresponding image points, assuming that if these 
 two or three image points are perfectly formed and correctly 
 situated then the rest of the image lying between these 
 points is also perfect and correct. The screen on which 
 the image is received is in general a plane surface, and 
 we assume that the object is also plane and parallel with 
 the screen, both being intersected normally by a line which 
 also passes normally through, or forms the principal axis 
 of, the pinhole or lens. It is desirable to take the inter- 
 section of the principal axis with the object plane as one 
 of the selected object points. 
 
 18. Pinhole Image. — In fig. 13 A and B represent two 
 infinitely small luminous points in the object. At C we 
 place a screen containing a small pinhole (h) and a second 
 screen at J) to receive the image. Of all the light rays 
 emanating from A only a few can pass through the pinhole, 
 
L!8 
 
 A First Book of the Lens. 
 
 the others being stopped by the screen G. The few rays 
 that do pass the hole will, however, continue straight on 
 until intercepted by the screen D, on which they form a 
 point of light at a, the size of the point varying with 
 the size of the pinhole. Similarly, a few rays from B 
 will also pass through the pinhole and form a second point 
 of light at b. As it is practically impossible to employ 
 an infinitely small pinhole the points of light a and b are 
 produced by straight rays forming narrow, slightly diverging 
 pencils of light ; a and b are therefore discs rather than 
 points, and form a somewhat ill-defined inverted image of 
 AB. The definition is further affected by the optical 
 phenomenon called diffraction, but this matter need not 
 be now considered.* 
 
 As the light rays suffer no deviation at the pinhole the 
 image, as a whole, geometrically corresponds with the object, 
 the latter being plane and parallel to the image screen. 
 
 19. Size of Pinhole Image. — Assuming that the pin- 
 hole is capable of producing a well-defined image, it is 
 apparent that the linear dimensions of the image will vary 
 directly with the distance of the screen from the pinhole. 
 And as AhB and ahb (fig 13) are similar triangles, therefore 
 ab : AB : : ha : Ah, or the ratio of the image to the object, 
 
 * Diffraction is o£ considerable importance in the formation of 
 pinhole images, but has little practical effect on the lens image, 
 though it does play a part in its formation. It is an intricate 
 subject and can well be neglected in an elementary book on lenses 
 only. To understand diffraction, it is necessary to study minutely 
 the structure of a wavefront of light, a matter which was only 
 dealt with in very general terms in Chapter 1. 
 
Action of Lenses. 
 
 ■20 
 
 is equal to the ratio of the distance of the image from the 
 pinhole to that of the object from the same point. If we 
 let v represent the distance of the image, u that of the 
 object, and R the ratio of image to object, then 
 
 20. Lens Image. — If for the pinhole we substitute a 
 positive lens (Z), as shown in fig. 14, a similarly inverted 
 image is again produced ; but, as a much greater number 
 of rays from the object points A and B pass through the 
 lens than through the pinhole, the image is brighter ; and, 
 as all the transmitted rays converge on infinitely small 
 image points a and b, the image is perfectly sharp. (We 
 are of course assuming that the lens is aplanatic, and 
 
 
 
 
 
 
 
 b- 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 i 
 
 
 
 
 
 perfectly adapted to the purposes of photography ; and 
 this assumption must be maintained until we can consider 
 aberration and its effects.) A lens thus produces a better 
 denned image than a pinhole; and, as it also transmits 
 more light, less exposure is necessary to produce a develop- 
 able image. 
 
 21. Curvature op Field. —Under the conditions as- 
 sumed in Sec. 17, the lens must be capable of producing 
 a plane image of a plane object ; that is to say, all image 
 points must lie in the plane of the screen. If this condition 
 is not fulfilled the aberration known as curvature of the 
 field exists. It must be particularly noted that both the 
 object and image are assumed to be plane. A lens that 
 has a curved image field for a plane object may produce a 
 plane image from a curved object, but is not therefore to 
 
30 
 
 A First Book of the Lens. 
 
 be looked upon as a flat field lens. This is the third of 
 the three primary aberrations to which lenses are liable, 
 and which should all three be absent in the perfectly 
 adjusted or " corrected " photographic lens. In studying 
 the theoretical general action of lenses it is necessary to 
 assume in every case that the lens is aplanatic (Sec. 5), 
 achromatic (Sec. 7), and that it possesses flatness of field. 
 
 22. Thin and Thick Lenses. — Fig. 14 may be con- 
 sidered to represent the manner in which the image is 
 formed with a lens of infinite thinness, if we disregard the 
 fact that such a lens is impracticable. Waiving this 
 point, we may say that with a thin lens the image is pro- 
 duced by the agency of comparatively wide double pencils 
 
 
 
 
 
 
 
 
 A 
 B 
 
 
 
 a. 
 
 ^ ~>^L^^^^ ^'l ^^^^ 
 
 
 
 
 
 of light, the axial rays only of which are straight. As the 
 object and image points are situated on these axial rays, 
 which all intersect in the point N, the image geometrically 
 corresponds with the pinhole image, and is equally accurate, 
 while its size is governed by the same rule (Sec. 19). In 
 fact a pinhole in the position of the point iV" will produce 
 a similar image, differing only in quality of definition. 
 
 All practicable lenses are, however, of appreciable 
 thickness, and a single thick lens produces an image by 
 the agency of double pencils of light with which the only 
 straight ray is that of the one direct pencil — i.e. the ray 
 coinciding with the principal axis of the lens (see Aa, 
 fig. 15). In oblique double pencils, such as Bb, there can 
 be no straight ray, but there is one axial ray (Bcdb), 
 which pursues parallel courses before and after refraction 
 (Sec. 10), and on these axial rays all image points are 
 
Action of Lenses. 
 
 31 
 
 situated. If we produce the rays Be and bd to meet the 
 principal axis in the points N 2 and i\ r i, it will be apparent 
 that the image is formed on precisely the same geometrical 
 principles that govern its formation by the thin lens, the 
 only essential difference in the geometrical construction of 
 tigs. 14 and 15 being the existence of the space N 2 N X in 
 the latter figure. 
 
 In fig. 16 this matter is more clearly illustrated, only 
 axial rays being shown. BN 2 being parallel to N x b, the 
 two diagrams are similar, excepting for the space N 2 N\. 
 The object and image being parallel, ANB and aNb, or 
 AN 2 B and aA^b, are similar triangles, and the scale of the 
 
 image is governed by the same principles as with a pinhole. 
 That is to say, B = v/u if v and u are measured as in- 
 dicated in fig. 16. 
 
 23. The Nodes op a Lens.— The points ]V, JV h and N 2 , 
 illustrated in tigs. 14 to 16, are called the nodes of the 
 lenses, the thin lens having one node and the thick lens 
 two. The separation of the two nodes {NS) is the nodal 
 space of the thick lens ; and if in fig. 16 we assume that 
 the distance AN is, equal to AN 2 , and Na to i^a, then, 
 the objects being the same, the images must be of equal 
 size; but the distance Aa with the thick lens is greater 
 than the distance A a with the thin lens by NS, or the 
 amount of the nodal space N 2 N X . With a simple single 
 lens this nodal space is very approximately equal to 
 
32 
 
 A First Book of the Lens. 
 
 one-third the thickness of the lens, and is therefore, a 
 negligible quantity if the lens is thin, but its amount 
 varies with compound lenses of various types. 
 
 In fig. 15 the nodes were found by producing the axial 
 rays Be and bd to meet the principal axis. In the 
 ideally perfect lens the nodes are fixed points, and the 
 axial rays of all oblique pencils would pass through 
 the nodes if produced in the same way. The nodes are 
 virtual points in the figure,' but with some types of lenses 
 one of them may be a real point ; that is, axial rays may 
 actually pass through it without any prolongation. In 
 the example illustrated all incident axial rays will converge 
 upon the point N 2 , which is called the node of admission. 
 All emergent axial rays will diverge from N x , which is the 
 node of emission. But if we reverse the lens, or look 
 upon AB as an image of ab, these titles are reversed, and 
 JVi becomes the node of admission, while N 2 is that of 
 emission. 
 
 These nodal points, or nodes, are sometimes termed 
 principal points, or Gauss points. They can be shown 
 to exist in every possible form of lens, simple or compound, 
 that is capable of producing a perfect image, but their 
 positions, relative to the lens and to each other, vary with 
 different types of lenses. 
 
 24. The Theory of the Equivalent Lens. — If we know 
 the positions and separation of the nodes of a lens we can, 
 as indicated in figs. 15 and 16, virtually trace the course 
 of all oblique axial rays without paying any regard to the 
 actual deviations produced in those rays by the various 
 surfaces of which the lens is composed. Thus in fig. 15 
 the actual course of the refracted portion of the ray from 
 c to d is of no consequence if the points N x and N 2 are 
 fixed, and we may similarly disregard the actual course 
 of an axial ray through a system of lenses of the most 
 complicated nature if only the nodes are known. There- 
 fore, Ave may, when considering the manner in which the 
 image is formed, disregard all peculiarities of construction, 
 and with a lens of the most complicated form proceed just 
 as we should with a lens of the simplest possible character. 
 Thus we may look upon the nodes of a complex doublet 
 as being those of a simple lens " equivalent " to the 
 
Action of Lenses 
 
 33 
 
 doublet m all respects. As, however, no simple single lens 
 can practically produce a perfect image, we must adopt a 
 conventional method of representing such an equivalent 
 lens if we want to trace the course of non-axial rays 
 and this conventional representation is dependent on the 
 Gauss theory. 
 
 25. The Gauss Theory.— Assume that in fig. 17 we 
 have a perfectly adjusted doublet formed of the two thin 
 lenses G and D, the nodes of the doublet being at N x and 
 JV 2 Let BN 2 N L 6 represent the axial ray of an oblique pencil 
 of light forming the image b of an object point B, as in the 
 other diagrams. Through N x and N 2 draw the perpen- 
 dicular lines shown, and assume them to represent sections 
 of two parallel planes which are intersected normally by 
 
 the principal axis Aa. These are called nodal planes and 
 are represented by the letters NP X and NP 2 in the diagram 
 Now any non-axial ray (Be) starting from B and forming 
 part of the pencil Bb, will be deviated by the lenses C and D 
 at the points c and d, and will eventually arrive at b 
 Produce Be and bd to meet the nodal planes in g and h 
 If then the lens is perfectly adjusted these two points 
 will be opposite one another, and be equidistant from the 
 principal axis,— that is, gh will be parallel to Aa. The 
 course of any non-axial ray, even if exterior to the lenses 
 can be traced with the aid of the nodal planes if we know 
 the positions of the object and image points, just as well 
 as axial rays can be virtually traced through the nodal 
 
 fu^'j ?" he rule being that aU incident rays intersect 
 the nodal plane of admission in a point normally opposite 
 the point in the nodal plane of emission from which 
 
 A 
 
 3 
 
34 
 
 A First Book of the Lens. 
 
 emergent rays virtually start. The planes are considered to 
 be unlimited in extent. 
 
 These two parallel planes, together with the nodal points 
 and principal axis, thus form a conventional representation 
 of a perfectly adjusted lens, and if the equivalent lens 
 to a system of lenses be indicated in this manner, we can 
 disregard all constructional complications. If, however, 
 the system of lenses is not perfectly adjusted, or free from 
 aberration, this Gauss theory does not quite hold good; 
 but that is a matter for after consideration. It may be 
 noted that the effective aperture of the lens, or the diameter 
 of the incident pencil, is measured on the nodal plane of 
 admission. 
 
 We have here only considered certain facts, the truth 
 of which can be proved under the Gauss theory. The 
 proofs of the existence of nodal points and nodal planes 
 which act in the way described are beyond our scope. 
 It must be pointed out that the Gauss theory is a modern 
 conception, and that in the older text-books no mention 
 will be found of either nodes or nodal planes. Under 
 the old theory, instead of two nodes, there was one optical, 
 or focal, centre, which under the new theory is a point 
 of very minor importance. 
 
 26. The Minor Points of a Lens.— Along the principal 
 axis of a lens there are (excluding the nodes) certain minor 
 points which require definition. 
 
 (a). The poles of a lens are the two points in which 
 
Action of Lenses 
 
 35 
 
 the principal axis intersects the outer surfaces of the lens. 
 The term pole is a convenient one, due to Professor Sylvanus 
 Thompson (See P 2 and P x in fig. 18). 
 
 (b). The rotary centre is a point on the principal axis 
 about which the lens may be rotated to a certain extent 
 without producing any movement of the image. This 
 point can be found by tentative experiment with a lens. 
 On the diagram it is situated at the point where a straight 
 line joining B and b intersects the principal axis (see r). 
 The point is variable, but is always either between the 
 nodes or at one of them. When the lens is focussed on 
 an infinite _ distance the rotary centre coincides with the 
 node of emission, but as the object is brought nearer the 
 
 point moves towards the node of admission. The rotary 
 centre is of little consequence excepting for the purpose 
 of finding the nodes. A pinhole substituted for the lens 
 in the position of the rotary centre will produce an image 
 exactly similar to that formed by the lens, object and image 
 being at a fixed distance apart. 
 
 (c) . Crossing-Point. All oblique axial rays intersect one 
 another, and also the principal axis, in one fixed point, 
 the position of which varies according to the type of lens 
 (see c in figs. 18 and 19). The crossing-point is always 
 either at one of the nodes or between the two, but it does 
 not necessarily coincide with the rotary centre (see fig. 18). 
 
 (d) . Optical Centre. Just as all incident and emergent 
 oblique axial rays meet at the nodes of admission and 
 emission, so the refracted portions of all oblique axes meet, 
 
36 
 
 A First Book of the Lens. 
 
 or will meet if produced, at one point. In the case of 
 simple single lenses there is only one such point, as shown 
 at o in figs. 18 and 19, and with such lenses that point 
 is called the optical centre. It will be noticed that in the 
 case of the meniscus lens the optical centre is virtual, out- 
 side the lens, and beyond the crossing-point ; in the other 
 cases it is real and coincides with the crossing-point. With 
 doublets we can also find a point which in certain respects 
 corresponds to the optical centre of a single lens, but such 
 a point is not usually recognised as an optical centre. 
 
 (e). The focal centre is a term applied in pre-Gauss days 
 somewhat indiscriminately. It was intended to apply to 
 the point from which all distances from the lens should 
 be measured, but, as with single lenses the optical centre 
 was taken, and with doublets the rotary centre (two quite 
 distinct points), there is a good deal of confusion surround- 
 ing the term. Under the Gauss theory all measurements 
 are made from the nodes (see fig. 16), which are the only 
 points to which the term focal centre can now be applied. 
 
 It should be clearly understood that the rotary centre is 
 the only variable point in the lens, while the nodes and 
 optical centre are fixed points. The crossing-point is also 
 fixed, but not independent of the others. It agrees with 
 the optical centre if the latter is real, and with one of the 
 nodes if the optical centre is virtual ; see fig. 19, in which 
 the positions of the nodes are indicated by the nodal planes. 
 The crossing-point is the most important of the minor 
 points. The various points of a lens are frequently con- 
 fused. The term " optical centre " is often applied to a 
 node or to the rotary centre. Still more often is it applied 
 to the point described in this book as the " crossing-point," 
 though the two points exist separately in a meniscus lens. 
 The optical centre is sometimes incorrectly defined and 
 described, but correctly illustrated. 
 
CHAPTER III. 
 
 FOCAL POINTS AND DISTANCES. 
 
 27. The Principal Foci op a Lens. — When the object 
 is at an infinite distance, so that the incident pencil is 
 parallel, the focus of the emergent direct pencil is called 
 a principal focus of the lens ; and, as the lens may have 
 either side turned towards the object, there are two principal 
 foci„ both situated on the principal axis and equidistant 
 from the nodes of the lens. Thus, with the object on 
 the side A of the positive lens L (fig. 20) the principal 
 focus is on the principal axis at F h but with the object 
 on the side B the principal focus is at F 2 , the distances 
 F^¥ 2 and F X N X being equal. 
 
 With a negative lens (fig. 21), the principal focus is on 
 the same side as the object, and is virtual, and, the lens 
 being again reversible, there are two principal foci (F x and 
 F 2 ), and F x Ni is again equal to F 2 N 2 . 
 
 28. Focal Length and Focal Power.— The distance 
 from the node of emission of a lens to its principal focus 
 is cadled the focal length of the lens. Thus in figs. 20 and 
 21 the distance F^ or F 2 N~ 2 is the focal length of either 
 the negative or positive lens. 
 
 • If we compare the effect of lenses on parallel pencils 
 of equal diameter, or compare lenses of equal aperture (see 
 Sec. 9) it will be noticed that the deviation produced in 
 the outermost rays of the pencil is equal if the focal 
 lengths are equal, while if the focal lengths differ the 
 greatest deviation produces the shortest focal length. Thus, 
 if ini figs. 20 and 21 the incident parallel direct pencils 
 are of the same diameter, and the angle F^b in fig. 20 
 
 37 
 
38 
 
 A First Book of the Lens. 
 
 is equal to Fica in fig. 21, then the focal lengths FxN~x 
 are equal ; but if Ficb is greater than F x ca the focal length 
 of the positive lens must be less than that of the negative 
 lens. Short focal length thus denotes that the lens has 
 the power of producing great deviation, or is of great 
 "focal power," and with very narrow pencils the focal 
 power of a lens may be considered to vary inversely 
 with its focal length. If, therefore, one lens has twice 
 the focal length of another, it has only half the focal 
 
 power; or if we represent the focal lengths of two 
 lenses by fx and f 2 , their focal powers are in the ratio of 
 I /fx : l// 2 . The greater the focal power the greater is 
 the curvature of the emergent wavefront, and with the 
 same two lenses the curvatures will be in the ratio of 
 1/fx to l/f 2 ; thus the focal power may be taken to re- 
 present the curvature imprinted by the lens on the plane 
 wavefront of the incident parallel pencil. 
 
 Focal length is a simple dimension that can be expressed 
 in inches or centimetres (see Appendix, Table A). Focal 
 
Focal Points and Distances. 
 
 39 
 
 power is a relative term, and conveys nothing definite 
 unless we fix upon some particular unit of power. This 
 unit is taken to be the focal power of a lens, the focal 
 length of which equals one metre, and this unit of power 
 is called a dioptrie. Thus a lens of half a metre focal 
 length has a focal power of two dioptries, and so on. This 
 method of expressing the power of lenses is in common 
 use among oculists, while photographers always distinguish 
 lenses by their focal lengths, using the expression " focal 
 power " in a relative sense only (see Appendix, Table B). . 
 
 29. Conjugate Foci. — When the distance of the object 
 is less than infinity, any object point and its corresponding 
 image point are called conjugate foci. In practical photo- 
 graphy we are concerned solely with one class of conjugate 
 foci. We always use a positive lens, and the distance 
 of the object is always greater than the focal length, and, 
 under these conditions, the object focus is always on the 
 opposite side of the lens and never nearer to it than 
 the principal focus. To clearly understand conjugate foci 
 it is, however, necessary to consider other cases, and to 
 study the various classes of real and virtual foci produced 
 under varying conditions. 
 
 In each of the following figs. (22 to 27) the points F 1 
 and F 2 are the two principal foci of the lens L, the nodal 
 planes of which are shown in full lines, while its character 
 (negative or positive) is indicated by dotted curvatures. 
 In all cases we assume that the light travels from left 
 to right. 
 
 With a Positive Lens. — If the object point is at an infinite 
 distance the image is at F±. 
 
 Iff the object is at P, farther from the lens than F 2 
 (see fig. 22), the image is at Q, and is real. P and Q 
 are conjugate foci. 
 
 With the object at F 2l the image is at an infinite 
 distance on the other side of the lens. 
 
 If the object P is between F 2 and the lens (fig. 23), the 
 image is virtual and is at Q. P and Q are conjugate foci, 
 Q being farther from the lens than P. In the particular 
 case of P being half-way between F 2 and the lens, or 
 at tiie semi-focus point, the conjugate focus Q is at F 2 . 
 
 In the above cases the incident light is either parallel 
 
40 
 
 A First Book of the Zens. 
 
 or divergent ; it may, however, be convergent on to a point 
 P behind the lens, as in fig. 24, in which case the con- 
 jugate focus is at Q, between F\ and the lens. In this 
 case the object is virtual and the image real. If con- 
 vergent on the principal focus F^ Q is at the semi-focus, 
 half the focal length from F\. 
 
 With a Negative Lens. — The object being at an infinite 
 distance a virtual focus is at F\. 
 
 t 
 
 If the object is at P (fig. 25) the conjugate focus Q is 
 between the lens and the principal focus F\. 
 
 If the object is at the principal focus F\, the virtual 
 image Q is half-way between the principal focus and the 
 lens, or at the semi-focus. 
 
 If the incident pencil is convergent on to a point P 
 beyond the lens, but between it and its principal focus 
 F 2 , a real image Q is formed at a greater distance 
 from the lens (fig. 26). P then represents a virtual 
 object. 
 
Focal Points and Distances. 
 
 41 
 
 If P is half-way between the lens and F 2 , or at the 
 semi-focus, the real image is at F 2 . 
 
 If P is at F 2 the image is at an infinite distance. 
 
 If P is at a finite distance beyond F 2 the image is 
 virtual, and is at Q (fig. 27) on the incident side of the 
 lens, and farther from it than F h In this case P and Q 
 are both virtual points. 
 
 With either class of lens any movement of the object 
 
 point is accompanied by a movement in the same direction 
 of the conjugate image point. 
 
 30. Conjugate Focal Distances. — The distance of the 
 object measured from the node of admission, and that of 
 the conjugate image focus measured from the node of 
 emission are conjugate focal distances, which are always 
 connected by a certain formula. 
 
 Jjet u represent the distance of the object, v that of 
 the image, and / the focal length of the lens ; then, with 
 a positive lens, so long as the conditions illustrated in 
 
42 
 
 A First Book oj the Lens. 
 
 fig. 22, or the ordinary conditions of photography, are 
 fulfilled, 
 
 U'-l (2). 
 
 v u f 
 
 Knowing the values of two of the terms u, v, and /, 
 the other can be easily found, either from the above funda- 
 mental equation, or from one of the following variations : 
 
 (3) 
 
 -(4) 
 
 v-f 
 
 ft 
 « -f 
 
 ■■ (5). 
 
 The conditions under which the above formulae, as they 
 stand, hold good, must be carefully noted ; under any other 
 conditions the following convention with regard to the 
 signs must be observed. 
 
 31. Popular Sign Convention. — Under the above- 
 mentioned conditions all the focal points to which the 
 distances v, u, and / relate are real, and the distances 
 are positive in formula 2. Under other (non-photographic) 
 conditions, one or more of the focal points might be virtual, 
 in which case the corresponding distances would be nega- 
 tive. Briefly stated, we adopt the convention that the 
 distance of a real focus is positive, while that of a virtual 
 focus is negative, and from this it follows that 
 
 / is positive with a positive lens and negative with 
 
 a negative lens. 
 v is positive so long as a real image is formed, and 
 
 negative when the image is virtual. 
 u is positive so long as the incident pencil of light 
 is divergent from a real source, and negative 
 when the incident pencil is convergent on to a 
 virtual source. 
 Applying the formulae to the various cases illustrated 
 in figs. 22 to 27, under this popular convention, in 
 fig. 22, representing the conditions under which a photo- 
 graph is taken, all distances are positive. 
 
 In 23 v is negative, while u and / are positive. 
 
Focal Points and Distances. 
 
 43 
 
 In 24 u is negative, and v and / positive. 
 
 In 25 v and /are both negative, u being positive. 
 
 In 26 u and/ are negative, and v positive. 
 
 In 27 u, v, and / are all three negative. 
 
 Any problem relating to conjugate foci can be worked 
 out by the formulae given if the signs are changed in the 
 case of any distance relating to a virtual focus. If in any 
 case the result is a negative quantity, we know that the 
 distance found relates to a virtual point, and, therefore, 
 that the conditions are non-photographic. From formula 5 
 it is manifest that if u is less than /, the resultant 
 value of v is negative, and the image is virtual. To 
 give a real image, u must be not only positive but 
 greater in value than /. Further, it may be noted that 
 when these conditions are fulfilled, v is also greater 
 than / 
 
 32. The Standard Sign Convention. — As misconceptions 
 frequently arise from confusing sign conventions, it is 
 advisable to explain the convention that is generally 
 adopted in standard text-books upon general optics. 
 
 All distances being measured from the lens to the focal 
 point, any distance measured in the same direction as that 
 of the incident light is negative, while any distance 
 measured in an opposite direction to that of the incident 
 light is positive. 
 
 This is a very complete definition, but it is somewhat 
 confusing to the beginner. It may perhaps be summed 
 up as implying that the length of a convergent pencil 
 of light is negative, while that of a divergent one is 
 positive ; but it is instructive to look at the matter from 
 a physical point of view. 
 
 If we consider that the curvature of the wavefront of 
 an incident or emergent pencil varies inversely with the 
 distances of the object or image, then, if the distances are 
 in the ratio of u to v, the curvatures are in that of l/u 
 to I /v. 
 
 "When an incident pencil with a wavefront of a certain 
 curvature falls upon a lens of a certain focal power (see 
 Sec. 28), the curvature of the wavefront of the emergent 
 pencil is equal to the sum of the curvature of the incident 
 wavefront and the focal power, which also represents a 
 
44 
 
 A First Booh of the Lens. 
 
 curvature : thus ; 1/v = 1/u + 1/f When, however, the focal 
 power represents a curvature of an opposite nature to 
 that of the incident pencil, one neutralises the other to a 
 certain extent, and a universally applicable formula can 
 be found if we adopt the convention that a convex wave- 
 front is positive, while a concave one is negative. This 
 renders the focal power of a positive lens negative, while 
 that of a negative lens is positive. Tf then we have a 
 divergent pencil incident on a negative lens, both the 
 incident and emergent pencils have positive curvatures, 
 thus all the terms are positive and the formula mentioned 
 above applies as it stands, though it is generally transposed 
 into 1/v — 1/u = 1/f. 
 
 With a positive lens giving a real focus the focal power is 
 
 i e 4 
 
 
 * — a — *— 
 
 *~OC-->< 
 
 / • 
 
 
 
 negative, while the emergent wavefront is concave, hence 
 we have : 
 
 111 111 
 
 = — . ; or, - + - = - 
 
 -V U -J V U J 
 
 which is the same formula as that given under the popular 
 convention. It is the conventions only, not the formulae, 
 that differ, and while under the standard convention the 
 fundamental formula is the one relating only to negative 
 lenses, under the popular convention it is that relating 
 to positive lenses working under ordinary photographic 
 conditions. 
 
 In studying general optics the adoption of the standard 
 convention is imperative, but the popular convention is 
 preferred in this book as it is more easily understood, and 
 is applicable to all the problems that photographers meet 
 with in connection with lenses. 
 
Focal Points and Distances. 
 
 45 
 
 33. Extra-Focal Distances. — As both v and u must 
 be greater than / when a real image is produced with a 
 positive lens, a pair of conjugate fociP and Q (fig. 28) must 
 be outside the principal foci F x and F 2 . The distances 
 u and v of these conjugate foci, measured from the 
 nodes of the lens, are called focal distances ; but we may- 
 measure the distances of P and Q from the principal foci F x 
 and F 2 , and distinguish them as the extra-focal distances d 
 and x. This method of measuring has advantages inasmuch 
 as the formula connecting the extra-focal distances is 
 simpler than that given for focal distances. If in formula 
 2 we substitute d+f for u, and f+x for v, we have 
 
 dx = p (6). 
 
 The fundamental formulae for conjugate focal and extra- 
 focal distances are of constant use in connection with lenses, 
 and will be frequently referred to. The following formula 
 expresses the relationship between focal and extra-focal 
 distances and scale of image : 
 
 V f X 
 
 u - 3 " / = R (7)> 
 
 34. Conjugate Symmetric Foci. From the formula 
 1/u + 1/v — l/f it is apparent that when u equals v they 
 each equal 2/; that is to say, when the object is at a 
 distance of two focal lengths from a positive lens, the image 
 is at an equal distance on the other side. The image is 
 then exactly the same size as the object (see Sees. 19 and 22), 
 and thus the term symmetric foci (due to Professor Sylvanus 
 Thompson) is very suitable. It is also useful, as these 
 particular conjugate foci have often to be specially referred 
 to. There are no symmetric foci to a negative lens. 
 
 35. Conjugate Principal Foci. — If two lenses are used 
 in combination the principal foci of the individual com- 
 ponent lenses are conjugate foci, though the corresponding 
 conjugate focal distances are not necessarily equal to the 
 focal lengths. It is apparent that if the object is at the 
 principal focus of the first lens the pencil emergent from 
 that lens and incident on the second is a parallel pencil, 
 hence the image is at the principal focus of the second lens. 
 
 36. The Cardinal Points op a Lens. — We have con- 
 sidered several particular points situated on the principal 
 
46 
 
 A First Book of the Lens. 
 
 axis of a positive lens, but the two nodes and the two 
 principal foci are so essentially important that they may 
 be looked upon as cardinal points. The principal, foci are 
 always equidistant from their respective nodes but the 
 relative positions of the nodes are liable to variations. 
 Fig. 29 shows four different possible relative positions for 
 the cardinal points. In each case the focal lengths N\Fi 
 and AT 2 F 2 are equal, but in case A the nodes are separated 
 so that the distance from F 2 to F x equals /+ NS +f. In 
 case B the nodes are coincident, and FiF 2 = 2/. In case 0 
 the nodes are crossed, so that F X F 2 =f+f—NS. In case 
 
 (A)" 
 (c) 
 
 Ti Ni 
 
 Ni Ti 
 ^_ 
 
 _) 
 
 \ J.— , f 
 
 -3L— 1 
 
 5 e ' 
 
 H Ji 
 
 k_— y 
 
 1— -J-— J 
 
 
 
 
 k f 
 
 
 —i 1 
 
 IV. 
 
 Tx Ft 
 1 1 
 
 Hi 
 
 
 t ^_J___> 
 
 \ 
 
 
 - -* 
 
 D the nodes are crossed, but outside the principal foci ; F^F 2 
 still being equal to f+f— A T S. 
 
 Separated nodes are most common, and fig. 29A may 
 be taken to illustrate the cardinal points of a thick simple 
 single lens, or of a doublet of two thick lenses, or of the 
 majority of compound single lenses. Coincident nodes are 
 rare, and fig. 29B may illustrate the points of a hypothetical 
 infinitely thin single lens, or of a doublet of two lenses of 
 medium thickness. Crossed nodes, as in fig. 29C, exist with 
 some forms of compound single lenses, and with doublets 
 of thin lenses, but fig. 29D illustrates a very rare case, 
 only met with in microscope lenses. 
 
 Though coincident nodes are unusual, it often happens 
 that separated or crossed nodes are so close together that 
 
Focal Points and Distances. 
 
 47 
 
 for all practical purposes they may be looked upon as 
 coincident,. Sometimes, however, this is so far from being 
 the case, that the nodal space becomes a very important 
 factor in calculating distances. With single simple lenses 
 the nodes must be separated, and the nodal space is very 
 approximately equal to one-third the thickness of the lens. 
 The nodes may be crossed with compound single lenses. 
 
 With crossed nodes the nodal space has to be deducted 
 from the sum of two focal distances to find the actual 
 distance from object to image, and the space may be looked 
 upon as negative. With separated nodes the space may 
 be considered positive. 
 
 37. The Positions of the Nodes. — The positions of the 
 
 nodes of a single lens, relative to the lens, vary in different 
 types of lenses, and in fig. 30 a series of typical single 
 positive and negative lenses are shown. The light is 
 assumed to be passing in the direction of the arrow, and 
 the nodal planes of admission and emission are marked 
 N% and N\ above and below the lenses. When the 
 curvatures are unequal the nodes are nearer the side of 
 greatest curvature, one node being on the curved surface 
 with a piano lens, and outside the lens, or " free," with 
 a meniscus. When a node is on the surface, it coincides 
 with both the crossing-point and the optical centre. 
 When outside the lens it coincides with the crossing- 
 point only (see Sec. 26). 
 
 With combinations of lenses there is a very much greater 
 
48 
 
 A First Book oj the Lens. 
 
 variety of possible positions for the nodes. They may 
 both be not only outside the lens, but a very long way 
 outside. The telephoto lens is specially designed with a 
 view to throwing the nodes so far outside and in front 
 of the lens as to produce the effect of a lens of very long 
 focal length, though the lens itself is comparatively close 
 to the plate. On the other hand, with the type of com 
 bination known as a single landscape lens, the nodes are 
 between the lens and the plate, and a greater extension 
 of camera is therefore required than would be necessary 
 if the nodes were in front of the lens, or within it, as 
 is the case with the ordinary doublet. 
 
 With separated photographic combinations of the doublet 
 form, if the lenses are both positive or both negative, 
 
 
 i 
 
 HI 
 1111' 
 
 
 
 
 
 the nodes (which may be separated, crossed, or coincident), 
 are between them, but nearest to the lens of greatest focal 
 power. If one lens is positive and one negative, the nodes 
 (usually separated) are free, or outside the combination 
 altogether, being beyond the positive lens if the combina- 
 tion as a whole is positive, and beyond the negative lens 
 if the combination is negative. 
 
 With cemented or close combinations the positions of 
 the nodes are somewhat similar to those which they would 
 occupy in the case of a simple lens of the same outward 
 form, excepting that the nodes may be crossed or possibly 
 coincident ; and, in the case of a meniscus, may both be 
 free, instead of one being free, and one inside the lens. 
 With the single meniscus combination of an achromatic 
 landscape, or rectilinear, both nodes are usually free and 
 
Focal Points and Distances. 
 
 49 
 
 beyond the convex surface, as shown in fig. 31, in which 
 an achromatic meniscus is compared with a single meniscus. 
 
 Mr. T. R. Dallmeyer, in the first Traill Taylor Memorial 
 Lecture, stated that in the stigmatic of 6'4 ins. focal 
 length, the nodes "of the front combination are slightly 
 crossed, near together and about six-tenths of an inch 
 in front ; in the back combination they are not crossed 
 near together, and as much as one and a quarter inches 
 behind the lens." These are exceptional examples of single 
 combinations, but many variations are possible. 
 
 38. Equivalent Lenses. — The theoretical equivalent lens 
 to a combination has already been defined (Sec 24), but 
 we may describe two separate lenses, capable of producing 
 similar effects under all similar conditions, as equivalent 
 to one another. With two such lenses the cardinal points 
 are exactly the same; that is to say, the nodal spaces 
 are equal and of the same sign, and the focal lengths are 
 equal. Equal focal length alone does not render the 
 lenses truly equivalent ; the nodal spaces must also be 
 equal if the lenses are to produce exactly similar effects 
 at all distances. On the other hand, equivalency does not 
 necessarily imply that the positions of the cardinal points 
 relative to the lens are the same in both cases; in this 
 respect the lenses may differ widely, and they may have 
 no outward resemblance to one another, even though they 
 are strictly equivalent. It need hardly be added that 
 to be truly equivalent, lenses should be equally free from 
 aberration. 
 
 Strictly equivalent lenses are necessary in a stereoscopic 
 camera, as lenses of equal focal length but of unequal 
 nodal spaces produce similar images only whon the object 
 is at an infinite distance. With nearer objects the images 
 vary in size. 
 
 4 
 
CHAPTER IV. 
 
 COMBINING LENSES. 
 
 39. Effect of Combining Lenses. — When lenses are 
 combined to form a system the focal length of the whole 
 combination depends upon the focal powers of the component 
 lenses, and also upon whether they are in nodal contact 
 or nodally separated. The positions of the cardinal points 
 of the combination depend on the same conditions, and 
 also on the nodal spaces of the component lenses. Lenses 
 are combined mainly for the purpose of producing a system 
 free from aberration, but sometimes we combine them for 
 the express purpose of altering focal length, and the effect 
 produced upon focal length and the cardinal points is the 
 matter of present consideration. First, however, we must 
 consider the meaning of nodal contact between two lenses, 
 which, is a matter very liable to be misunderstood, and 
 demands particular explanation. 
 
 40. Nodal Contact and Separation. — A condition of 
 perfect nodal contact is secured between two lenses when 
 the node of emission of the front lens (the one nearest the 
 incident light) coincides with the node of admission of 
 the back lens (the one nearest the image). When the 
 lenses are not thus in contact the distance between the two 
 nodes mentioned represents the separation of the lenses. 
 In some cases the node of emission of the front lens may 
 be behind the node of admission of the back lens, which 
 case is distinguished as crossed, or negative separation. It 
 must be clearly understood that nodal contact does not 
 necessarily involve the touching of the lenses, it is, in fact, 
 an unusual condition even with cemented lenses, and there 
 
 50 
 
 / 
 
52 
 
 A First Book of the Lens. 
 
 are many forms of lenses that cannot possibly be placed 
 in nodal contact. 
 
 Figs. 32 to 35 show various combinations of two lenses, 
 each marked to correspond with typical lenses shown in 
 fig. 30. The nodal planes of the front lenses are marked 
 above the axis, those of the back lenses below the axis. 
 The direction of the light is shown by the arrow. 
 
 Fig. 32 practically includes all the possible forms of 
 cemented double combinations of positive and negative 
 lenses with which the condition of nodal contact can be 
 secured. It is impossible to cement two positive or two 
 negative lenses in nodal contact. 
 
 Fig. 33 gives a few examples of lenses that touch 
 and are in contact, but cannot be cemented. Pairs 
 of positive or of negative lenses are possible in this 
 case. 
 
 Fig. 34 shows some lenses in contact without touching ; 
 a condition that cannot well be fulfilled with a pair of 
 negative lenses. 
 
 Fig. 35 gives a few only out of a great variety of possible 
 cemented combinations in which the condition of contact 
 is not secured at all. Many of these are to be found in 
 photographic lenses. CI and HI are examples of crossed 
 or negative separation. In the rest the separation is 
 positive. 
 
 It will be noticed from these figures that nodal contact 
 cannot be secured when each node is within its respective 
 lens. One or both nodes must be free to a certain extent 
 dependent on the form of the lens. The same conditions 
 hold good when one of the two lenses is a doublet ; one 
 node must be sufficiently free to reach the other. In the 
 case of a doublet a free node must be outside, not between 
 the lenses composing the doublet. No form of simple single 
 lens can be placed in contact with an ordinary doublet con- 
 sisting of two positive combinations, though nodal contact 
 can be readily secured if the doublet is of the periscopic 
 form, as it then has a free node. 
 
 41. Focal Length of Combination. — If to a positive 
 lens of focal length fi we add another positive lens of 
 focal length f 2 , and nodal contact is secured, then the 
 focal power of the combination is equal to the sum of* 
 
Combining Lenses. 
 
 53 
 
 the focal powers of the component lenses. Let F equal 
 the focal length of the combination. Then 1/F equals its 
 focal power and, 
 
 or, 
 
 F = Mh (9). 
 
 /l + J-z 
 
 If the lenses are not in contact, but nodally separated 
 by a positive distance equal to a, then 
 
 Ilia /1AN 
 
 75 =-7 +y --rj- (10) 
 
 * h J-i J1J2 
 
 or, 
 
 F = , =6/2 (11). 
 
 fi + A - a 
 
 By adding a positive lens to a positive we increase focal 
 power, or diminish focal length, and the less the separation 
 the shorter is the focal length. 
 
 By adding a negative lens to a positive we reduce focal 
 power or increase focal length, and the less the separation 
 the greater is the focal length. 
 
 In either case the nearer the second lens is to the first 
 the greater is its effect upon focal length. 
 
 The formulae given apply to the case of negative lenses 
 if a minus sign is prefixed to the term representing the 
 focal length or power of the negative lens. If the nodal 
 separation is crossed or negative the sign prefixed to a 
 is changed. 
 
 When the resultant value of F is positive the combina- 
 tion is positive, or has a real focus. If the result is 
 negative the combination has a virtual focus, or is negative. 
 If the result is infinite, there is no focus ; that is to say, 
 one lens has neutralised the other. To preserve a real 
 focus in a combination of negative and positive lenses, 
 the negative lens must be of less power or greater focal 
 length than the other, if they are in nodal contact. If 
 separated the negative lens may be of greater focal power, 
 provided the separation is greater than the difference 
 between the focal lengths. Thus, suppose we combine a 
 
54 
 
 A First Book of the Lens. 
 
 positive lens of 6 ins. focal length with a negative lens 
 of 3 ins. focal length. Placing them in contact we have 
 a negative combination of 6 ins. focal length. With 
 a separation of 3 ins. the focal length is infinite, or 
 the combination is neutral. With a separation of 
 4 ins. we have a positive combination of 18 ins. focal 
 length. 
 
 It should be noted that the formulae only apply to 
 combinations of distinct lenses. They do not apply if 
 one lens is placed inside the other, as it may be when one 
 of the lenses is itself a separated combination. 
 
 42. Nodes of a Combination. — When two lenses are 
 in nodal contact the other two nodes form the nodes of 
 
 the combination, as shown in figs. 32, 33, and 34, where 
 the nodes of the combination are marked by black dots. 
 If nodal contact is not secured the nodes take different 
 positions. 
 
 As an example we will take a combination of two 
 positive lenses, forming a doublet as shown in fig. 36. 
 The position of the nodes of this combination can be 
 calculated if we know those of the single lenses. In the 
 following formulae, d 2 is the distance of the node of 
 admission of the combination from the node of admission 
 of the front single lens (which may be either a simple 
 lens or compound as shown), and d-i is the distance of the 
 node of emission of the combination from the node of 
 emission of the back lens. The other letters are the same 
 as in formulae 8-11, f x being the focal length of the front 
 
Combining Lenses. 
 
 55 
 
 and/ 2 °f the back lens, while a is the separation of the 
 two lenses. 
 
 a>f\ 
 
 d, = 
 
 fx +A 
 
 fx + /, 
 
 oF 
 _ aF , 
 
 (12) 
 (13). 
 
 If negative lenses are involved, or the separation a is 
 crossed, the signs must be varied as before explained. If 
 the result is positive, d 2 is measured towards the image 
 and di towards the object, as in fig. 36. If either result 
 is negative, the distance is measured in the opposite 
 
 n___ 
 
 a 2. * 
 
 |*_L 
 
 I I 
 
 -c«. ^--.BF-- 
 
 k F 
 
 direction. Thus, in the case of a telephoto lens, illustrated 
 by the periscopic doublet fig. 37, the back lens is negative 
 and the front positive; a is greater than f\—f 2 , hence 
 d 2 becomes negative and d\ positive, both are therefore 
 measured in the same direction, towards the object. 
 
 To find the separation of the nodes, or the nodal space 
 of the doublet, we must know the nodal spaces of the 
 component lenses. Let ns^ and ns 2 be the nodal spaces 
 of the front and back lenses respectively, and that of 
 the combination (see figs. 36 and 37). Then 
 a 2 
 
 NS = ns, + ns„ — - (14). 
 
 A + A - a 
 
 If the result is negative the nodes are crossed. 
 
 In some cases the values of ns\ and ns 2 are so small 
 
56 
 
 A First Book of the Lens. 
 
 that they may be neglected. This is especially the case 
 with the telephoto lens with which the value of JVS may 
 be many feet; but when IPS is very small, ns x and ns 2 
 are of importance. If they are neglected in the case of 
 an ordinary rectilinear combination, a negative value for 
 j¥S is always obtained, whereas with the majority of such 
 combinations it is positive. 
 
 43. Back-Focus op Combination. — This term is used 
 to denote the distance between the image and the node 
 of emission of the nearest single lens of the combination 
 (see BF in figs. 36 and 37). It is not a correct term, 
 as a focus is a point and not a dimension, but it is a 
 useful expression. The back-focus is a very important 
 dimension with the telephoto lens, though not of much 
 consequence with other forms of lenses. As a rule it 
 indicates, more or less accurately, the extension of camera 
 required. With a rectilinear or single landscape lens, the 
 extension measured to the lens flange, or face of camera 
 front, is usually less than an inch greater than the back- 
 focus, but with telephoto lenses the extension is from 2 
 to 6 ins. greater. The back-focus is less than the focal 
 length with all combinations, excepting such as have their 
 nodes behind the back lens. The ordinary type of single 
 landscape lens is an example. Using the same symbols as 
 in formula 11 we can find the back-focus, or BF, from the 
 general formula : 
 
 This is equal to the difference between formulae 11 and 
 13, or to F—d x in figs. 36 and 37. 
 
 In the case of a telephoto lens, with which this formula 
 is most often required, a negative sign has of course to 
 be prefixed to f 2 . 
 
 44. Combinations op more than Two Lenses. — The 
 formulae given, and the principles laid down, with regard 
 to combinations of two lenses apply equally well to com- 
 binations of three or more lenses, if we take the lenses in 
 pairs, and remember that the formulae only apply when 
 combining distinct lenses, or lens systems. Thus, if we 
 have three lenses in series A, B, and C, we can first find 
 
Combining Lenses. 
 
 57 
 
 the focal length and nodes of a doublet consisting of A and 
 B, and then find similar particulars for a combination of 
 G with the equivalent lens to A and B. Or we can reverse 
 the procedure ; first combine B with G, and then the result 
 with A. We cannot, however, combine the centre lens 
 B with the equivalent lens to A and G. 
 
 With a series of four lenses, A, B, G, and D, we can 
 proceed in two ways. We may combine A with B, and 
 also G with B, and then combine the results ; or we can 
 combine A with B, the result with G, and this result with 
 B. It may be useful to give formulae for the direct com- 
 bination of three lenses A, B, and G, separated by intervals 
 
 equal to a and b, and of focal lengths J h / 2 , and f 3 , as 
 shown in fig. 38. 
 
 F = 
 
 AAA 
 
 AAA 
 
 *i = 
 
 «) + (/, - *) (A + A - «) 
 
 bf 2 + a(f 2 + /, - V) 
 
 A(fi - + (/, - ») (/i + /, - a) 
 d = qf* + KA + A 2 - a) 
 
 (16) 
 (17) 
 (18). 
 
 A 2 (A - *) + (/, - ft)(/i +/ 2 - «) 
 
 45. Supplementary Lenses. — It is sometimes desirable 
 to add a supplementary lens to an existing combination 
 for the purpose of increasing or diminishing its focal length. 
 The formulae previously given are sufficient to meet all 
 requirements, but, as sometimes their application is im- 
 perfectly understood, it is as well to illustrate some actual 
 cases. Mistakes most usually occurring in connection with 
 
58 
 
 A First Book of the Lens. 
 
 the matter of separation, attention is more particularly- 
 directed to that point. We may assume that the original 
 combination is adjusted so as to produce an image free 
 from the effects of aberration, and as it is essential that 
 this adjustment shall not be appreciably disturbed by the 
 addition of the extra lens, the supplementary lens must 
 be thin with practically only one node and of weak focal 
 power. We may add also that it is generally safer to use 
 a negative lens for the purpose of increasing focal length 
 than to add a positive lens to reduce focal length, and 
 that some consideration should be given to the form of 
 the supplementary lens. 
 
 Adding Supplementary Lens to Single Landscape Lens. — 
 Suppose we want to add a supplementary lens B to a single 
 
 landscape lens A, as shown in fig. 39. Landscape lenses 
 are of the meniscus form, and present their concave surface 
 to the object, hence their nodes are usually behind, as 
 shown at N 2 and N x . 
 
 If the supplementary lens (which should be of meniscus or 
 piano form) is placed at B x (near the stop of the landscape 
 lens) it is in front of A ; the distance of separation is 
 measured to JV 2 , is positive, and equals a x . 
 
 If B is at B 2 it is behind A, but its node of admission 
 is in front of JV X (the node of emission of A), hence the 
 separation is negative and equals a 2 . In many cases 
 the amount of separation may be practically negligible, 
 but the example is a good illustration of negative separation 
 (see Sec. 40). If B is at B 3 it is behind A, and its 
 node is also behind JV X , hence the separation a s is positive. 
 
 Adding Supplementary Lens outside Doublet. — Fig. 40 
 
 A 
 
 ?W9- I 
 
 >w- — 
 
Combining Lenses. 
 
 59 
 
 represents a doublet composed of two positive single 
 combinations A and B. The nodes of the doublet are 
 marked N 2 and JV h and those of the component lenses n 2 
 and n\. The supplementary lens C (preferably of piano 
 form) may be placed either in front of the doublet, at C h or 
 behind it, at C 2 ; and in either case the separation 6 1} or 
 b 2 , is positive. A minimum amount of separation exists 
 when C is close to the most powerful component lens 
 
 of the doublet, therefore in this position we secure the 
 nearest possible approach to nodal contact with the 
 doublet. If A and B are of equal focal length the minimum 
 separation of G is approximately half the extreme length 
 of the doublet. 
 
 With a periscopic doublet, such as that shown in 
 fig. 41, nodal contact can be secured by placing C in 
 position C 2 , at the node of admission of the doublet ; at 
 C\ the separation equals e h and is positive ; at 0 3 the 
 
60 
 
 A First Book of the Lens. 
 
 separation equals e 3 and is negative ; at C± the separation 
 equals e 4 , and is positive. 
 
 Adding Supplementary Lens inside Doublet. — If the sup- 
 plementary lens C is placed between the positive lenses 
 A and B in fig. 40, the ordinary formulae no longer 
 apply, and to arrive exactly at the focal length of the 
 combination we must have all the data necessary to 
 determine the focal length of a triplet (see Sec. 44) ; 
 but the accurate result thus arrived at will not differ 
 to an appreciable extent from that obtained when the 
 lens C is placed outside and close to one of the component 
 lenses A, or B, if they are of approximately the same 
 focal length. Thus, when the supplementary lens is 
 inside the doublet it is generally sufficient to apply the usual 
 formulae, taking the separation as equal to approximately 
 half the extreme length of the doublet. A double convex 
 or concave is the best form of supplementary lens to 
 use inside a doublet. 
 
 If the doublet is of the periscopic form, as in fig. 41, 
 we must work out the whole combination as a triplet, for 
 slight variations in the position of the lens C greatly 
 affect the result. 
 
 Use of Magnifiers. — We do not always add a supplemen- 
 tary lens for the express purpose of altering focal length, 
 though that is a necessary consequence. With cameras 
 that have no focussing arrangement, and with which the 
 focussing-screen is fixed at the principal focus of the lens, 
 any near object can be brought into focus on the screen 
 by adding in front of the lens a supplementary lens the 
 focal length of which is equal to the distance of the object. 
 The addition of this lens brings the near object into focus, 
 but does not materially alter the size of the image, hence 
 the term magnifier, which is commonly applied to 
 supplementary lenses when used in this way, is hardly a 
 happy one. The resultant focal length of the whole 
 combination can of course be found by the formulae given, 
 but it is of little consequence, for, in the absence of any 
 focussing arrangement the only conjugate foci that we can 
 make use of are the principal foci of the two lenses, which 
 form the conjugate principal foci of the combination 
 (see Sec. 35). 
 
CHAPTER V. 
 
 ABERRATION. 
 
 46. The Primary Aberrations. — As before explained, 
 the ideal photographic image of a plane object may be 
 considered to be formed of an infinite number of infinitely 
 small points, each being the image of a corresponding small 
 point in the object, all such image points being situated in 
 a plane parallel to that of the object, and both planes 
 being intersected normally by the principal axis of tbe 
 lens. The essential requirements of the ideal photographic 
 image therefore are, 
 
 1. Perfect formation of the image points. 
 
 2. Perfect complaneity of all image points. 
 
 3. Correct relative situation of each image point. 
 The presence of aberration is denoted by non-compliance 
 
 with one or more of these requirements.* They are, 
 however, all fulfilled with any simple positive lens, provided 
 that the light is monochromatic, or of one colour only, 
 that the pencils of light are infinitely narrow, and that 
 oblique pencils are only inclined at infinitely small angles 
 to the principal axis. In photography these conditions 
 cannot be fulfilled : the light employed is compound ; the 
 production of sufficient exposure effect within a reasonable 
 time necessitates the employment of wide pencils, or of 
 large apertures ; and pencils of a considerable degree of 
 
 * If these requirements are fulfilled the lens may be described 
 as " collinear " ; a term which implies that it is also "aplanatic," 
 "achromatic," " stigmatic," or " anastigmatic," and " orthoscopic," 
 or " rectilinear." From these terms the distinctive titles applied 
 to particular lenses are in many cases derived, though a truly 
 " collinear " lens has not yet been manufactured. 
 
 61 
 
62 
 
 A First Book of the Lens. 
 
 obliquity are necessary to the production of an image 
 of any size. 
 
 The employment of wide pencils renders apparent the 
 effects of spherical aberration ; the light being compound, 
 chromatic aberration appears ; and the employment of 
 pencils of considerable obliquity accentuates the effects of 
 curvature of the field. Either of these three primary 
 aberrations causes the production of blurred image points, 
 and a consequent loss of definition. 
 
 47. Secondary Aberrations. — These may be due to 
 several causes. 
 
 1st. The primary aberrations become complicated under 
 varying conditions, and produce distinctive effects, which 
 are sometimes classed as separate aberrations, though they 
 would be more properly described as secondary effects of 
 the primary aberrations. For example, spherical aberra- 
 tion in an oblique pencil is complicated by two effects 
 known as " coma " and " astigmatism " ; and the latter 
 causes complicated effects of curvature of the field. 
 
 2nd. The attempt to correct one form of aberration 
 may introduce another secondary form. Thus, in modify- 
 ing certain forms of aberration by the use of a stop or 
 diaphragm, diaphragm distortion may be introduced. 
 
 3rd. Correction may be either incomplete, or carried 
 too far ; and in either case the remaining aberration may 
 be classed as secondary. 
 
 48. Principles of Correction. — There are three general 
 principles of correction. — 1st, The elimination of aberrated 
 rays ; 2nd, Diminution of effect of blur ; 3rd, Compensation 
 of the aberration. 
 
 The first principle is carried out by adjusting a stop or 
 diaphragm to cut out all rays excepting such as come to foci 
 at (or very near) the proper points. This generally re- 
 quires careful adjustment of the position of the diaphragm 
 relative to the lens, and is a matter for the optician to 
 settle. 
 
 The second principle, diminution of effect, is applied by 
 reducing the diameter of the aperture of the diaphragm. 
 This reduces the effective aperture of the lens, the diameter 
 of the light pencil, and the size of the blur at the focus. 
 This is a matter under the control of the photographer, 
 
Aberration. 
 
 63 
 
 and a very small aperture will practically obviate, though 
 it may not actually correct, the effects of any form of 
 aberration excepting such as its use may introduce. 
 
 The third principle, compensation, is the most important, 
 as it is carried out independently of the aperture and does 
 not interfere with rapidity. Aberration is compensated by 
 combining lenses or refractive surfaces that individually 
 produce opposite forms of the same aberration, and that 
 in combination neutralise each other's effects. If the 
 aberration varies to any considerable extent with the form 
 of the lens, two positive lenses of opposite forms may com- 
 
 A 
 
 
 
 cx 
 
 ou 
 
 
 
 , r~- — 
 
 
 
 
 
 
 pensate each other • but if it varies more especially with 
 the power of the lens, lenses of opposite (positive and 
 negative) poAvers are combined. 
 
 49. Positive and Negative Aberration. — If we com- 
 pare the aberration of a positive lens with the similar 
 aberration of a negative lens acting under the same 
 conditions, it will be found that while the effects produced 
 at the foci are similar, the emergent pencils are affected in 
 exactly opposite manners ; this is due to the fact that while 
 in the one case the focus and emergent pencil are on the 
 same side of the lens, in the other they are on opposite 
 sides. 
 
 Suppose the aberration with a positive lens A (fig. 42) 
 
64 
 
 A First Book of the Lens. 
 
 is such that a certain pair of rays aa come to too near a 
 focus (to q instead of Q), or are too convergent ; then with 
 a negative lens B the same pair of rays have a virtual 
 focus which is also too near the lens, which renders them 
 too divergent ; hence the ultimate effects are opposite, and 
 if the two lenses are combined, the excessive convergency 
 given by the one may be neutralised by the excessive 
 divergency given by the other. Similar conditions prevail 
 with positive and negative surfaces. 
 
 To perfectly compensate aberration the positive aberration 
 of the one lens must be exactly neutralised by the negative 
 aberration of the other, but, at the same time, there must 
 be an excess of positive focal power to preserve a real 
 focus. It is therefore necessary to very carefully adjust 
 tbe conditions that affect the extent of the aberration. 
 
 It should be noted that negative aberration can only 
 be observed on the focussing-screen with the aid of an 
 over-compensated positive combination. Over correction 
 introduces negative aberration, just as under correction 
 leaves a residue of positive aberration. 
 
 50. General Principles of Compensation. — If the two 
 lenses are separate the required adjustment can be made 
 by varying the materials and curvatures of each, and the 
 degree of separation between them, and additional control 
 is obtained by multiplying the number of lenses or sur- 
 faces. If, however, the two lenses are cemented in contact 
 so that the combination only possesses three surfaces, 
 the power of correction is somewhat limited. !For example, 
 suppose A and B (fig. 42) to be cemented ; and the light 
 to be monochromatic. A must be of greater focal power 
 than B to preserve a real focus, and the two lenses must 
 be of different material, otherwise the contact surface 
 will be neutral and we shall have a simple positive lens of 
 two surfaces only and possessing positive aberration. If, 
 however, B is of greater refractive power than A, the 
 contact surface is negative and gives negative aberration, 
 the amount of which can be adjusted by suitable selection 
 of materials to just counteract the positive aberration of 
 the outer surfaces, provided the latter is not excessive. 
 Similarly, with compound light, if B is of greater dis- 
 persive power than A, the contact surface giving negative 
 
Ahermtion* 
 
 ffi 
 
 dispersion can be adjusted to counteract the positive 
 dispersion of the outer surfaces.. 
 
 If the positive and negative lenses are separate, and 
 perfectly adjusted to one another, a reduction of the 
 amount of separation produces- too much negative effect, 
 and introduces over-corrected aberration of a negative 
 character. If, on the other hand, the separation is in- 
 creased,, the effect of the negative lens is diminished, and 
 under correction results. 
 
 When lenses are separated it is not in all cases absolutely 
 necessary that they should be of opposite power, thus in 
 a telescopic eye-piece two separated positive lenses may 
 counteract each other's spherical or chromatic aberration ; 
 but in photographic lenses these two aberrations are cor- 
 rected by the employment of compensating negative lenses, 
 the result being only slightly affected by the separation 
 of the positive combinations forming a doublet. Positive 
 compensating lenses are in general only employed for 
 correcting particular secondary effects of aberration, such 
 as coma and distortion. 
 
 Compensation is the most important principle of cor- 
 rection, for the more use we can make of it the less are 
 we dependent on the diaphragm, and the greater is the 
 rapidity and general usefulness of the lens. In the 
 ordinary landscape lens so much depends on the diaphragm 
 that a small aperture has to be used under almost all 
 circumstances. With the ordinary rectilinear doublet a 
 larger relative aperture can be used ; but much still depends 
 on the diaphragm, and while such a lens is faster than a 
 single lens of the same focal power it is much slower than 
 the " anastigmat," in which the diaphragm is responsible 
 in only a slight degree for the correction of aberration. 
 
 Portrait lenses are the quickest in use, but large apertures 
 are employed with these, not because the lenses are better 
 compensated, but because the presence of certain aberra- 
 tions is of less consequence in the particular case of por- 
 traiture. If employed for other purposes the aperture 
 has to be reduced considerably, and other forms of lenses 
 will serve fairly well for portraiture if their apertures are 
 greatly increased. It is simply a matter of compromise, and 
 the science of optics is, unfortunately, not yet sufficiently 
 
 5 
 
66 
 
 A First Book of the Lens. 
 
 far advanced to render compromises unnecessary. A lens 
 that will perfectly fulfil every possible condition has not 
 yet been designed. If perfect for copying a near object it 
 is likely to be deficient if used on an object at a great 
 distance, and vice versa. If it produces a perfect image on 
 a small plate, it almost certainly fails to do so on a large 
 plate. And if it covers a large plate satisfactorily, it may 
 be deficient if used for a very small image. The use of a 
 small aperture and the sacrifice of rapidity is the only 
 method of making a lens produce good definition in cases for 
 which it is not expressly designed ; and, conversely, great 
 
 extra rapidity can only be gained by sacrificing definition 
 to a certain extent. 
 
 51. Curvature of the Field. — In explaining the different 
 forms of aberration, it is desirable for the sake of clearness 
 to assume that one form only is present at one time. This 
 can seldom be the case with any of the aberrations, as each 
 of them is usually complicated by the presence of others ; 
 but, assuming simple curvature of the field (see Sec. 21) 
 to be alone present, then the foci of the direct and 
 various oblique pencils will lie, not in one plane, but on 
 a curved surface, which may be considered to be approxi- 
 mately spherical in curvature. With a simple positive lens 
 the image surface is concave to the lens, but over correction 
 
Aberration. 
 
 67 
 
 may render it convex, or introduce negative curvature of 
 the field. 
 
 To produce a plane image of an infinitely distant plane 
 object the oblique pencils should be of greater focal length 
 than a direct pencil, compare Nq x with NQ in fig. 43 ; but 
 with an uncorrected positive lens their focal length is too 
 short, as shown at JVq 2 , and at qi a blur is produced upon 
 the screen, as shown by broken lines. The oblique pencils 
 heing too convergent, can be corrected by a negative 
 compensating lens, which, if compensation is alone relied 
 upon (a somewhat unusual condition), must be a fairly 
 powerful lens and be separated to a certain extent from 
 the positive lens. Such a combination will give a flat 
 field only for an object at a particular distance. The 
 correction is likely to be insufficient for a more distant 
 object and excessive for a near one.* 
 
 If a lens possesses spherical aberration the curvature of 
 the field is affected very considerably by the position of 
 the diaphragm, but in the complete absence of spherical 
 aberration the diaphragm has no effect upon curvature, 
 though, as with all other aberration, a small aperture will 
 produce better definition. The effect of the diaphragm 
 is of great importance, but it must be considered later. 
 
 In an imcorrected lens the actual curvature of the field 
 varies with the curvatures, materials, thicknesses, and 
 separations of the component lenses; and also with the 
 position of the diaphragm and distance of object. If, 
 however, we consider only a simple single lens without a 
 diaphragm, we may say that the radius of curvature of 
 the field, when the incident light is parallel, is usually 
 between one quarter and one half the focal length of the 
 lens, while in exceptional cases it may be greater than half 
 the focal length ; for example, in the case of a spherical lens, 
 the radius of the field must be equal to the focal length 
 (see fig. 44). 
 
 It may be said that a perfectly plane field is hardly to 
 be attained. At the best we must expect the field to have 
 either a slight curvature of positive or negative nature, 
 
 * This applies more or less accurately iu the case of all the 
 aberrations. Each can, however, be approximately corrected for 
 any distance, but not all simultaneously. 
 
68 
 
 A First Booh of the Lens. 
 
 or a reflex curvature that does not depart very far from 
 the plane. 
 
 52. Spherical Aberration in Direct Pencils. — Spherical 
 aberration has been defined as want of sphericity in the 
 wavefront of an emergent pencil. A longitudinal central 
 section of a direct pencil then intersects the wavefront in 
 a curve of a symmetrical, but non-circular form. 
 
 Suppose in figs. 45 and 46 ac to be the trace of 
 a spherical wavefront of a perfect direct pencil, and die 
 
 31 
 
 — i — i 
 
 
 -■ 
 
 i ' 
 
 
 i 
 
 Bl 
 
 1 
 
 L 
 
 
 1 
 i 
 
 
 
 to be that of the non-spherical real front. All rays 
 normal to ac would come to one focus at Q, but normals 
 to dbe will come to different foci q u g 2 , etc., and Q will 
 be the focus only for infinitely close central rays ; marginal 
 rays coming to either a nearer focus, as in fig. 45, or to a 
 more distant one as in fig. 46. The first figure illustrates 
 the positive spherical aberration characteristic of un- 
 corrected positive lenses; the marginal rays are too 
 convergent, a defect that can be compensated by a negative 
 lens. Over compensation will then render marginal rays 
 
A berratiou. 
 
 69 
 
 insufficiently convergent, and produce the negative form 
 of aberration shown in fig. 46. 
 
 It will be noticed that the foci of corresponding pairs 
 of rays are displaced longitudinally along the axis of the 
 pencil. This particular form of aberration may therefore 
 be distinguished as longitudinal spherical aberration, and 
 as all foci lie between Q and q 2 , the distance Qq 2 may be 
 taken as a measure of the aberration. 
 
 In fig. 47 is shown on a larger scale the distribution 
 of the light between the extreme foci Q and q 2 . 
 Negative aberration being precisely the opposite of positive 
 aberration, these foci may be considered to belong to a 
 
 II 
 ii 
 
 
 II 
 
 §4 
 
 ii \ 
 
 II ^ < — m 1 
 
 / 
 
 / 
 
 - n 
 
 : n 
 
 !! 
 
 pencil emergent either from a lens Ly with positive 
 aberration, or from L 2 with negative aberration. The 
 intersections of consecutive rays cause the illumination to 
 be more intense along the axis and the outside of the pencil, 
 and there is the greatest accumulation of light in the 
 smallest compass, and therefore the brightest focus, at a 
 point q, very near Q. This may be considered to be the 
 best attainable focus, but it is surrounded by a considerable 
 amount of blur of less intensity. If we move the focussing- 
 screen in the direction of the arrow the blur diminishes, 
 while the central spot increases and becomes less bright, 
 until at qi * the blur has disappeared and we have a ring 
 of light surrounding a central bright spot, or disc of light. 
 
 This point does not agree with in figs, 45 and 46, 
 
70 
 
 A First Book of the Lens. 
 
 Still farther towards q- 2 the centre spot disappears and 
 beyond q 2 we have the appearance of a ring of light 
 surrounding a comparatively dark centre. The points 
 of best focus q, and of mean focus q h are easily distinguished 
 on the screen, and if the mean focus is between the best 
 focus and the lens the aberration is positive, while if the 
 mean focus is farther from the lens than the best focus the 
 aberration is negative. This affords a good means of testing 
 aberration, and so also does the appearance of the dark 
 centre beyond q 2 . 
 
 Referring again to figs. 45 and 46, it will be evident 
 that by placing a diaphragm at B the outer rays of the 
 pencils are eliminated and the longitudinal aberration is 
 reduced in extent to Qq x . The size of the mean focus and 
 of the blur around the best focus are reduced also, and 
 there is an alteration in the distribution of the light shown 
 in fig. 47 ; the points q x and q both moving towards Q. 
 Therefore, if we take the best focus q as the principal 
 focus of the lens, the focal length of the lens apparently 
 increases with the small aperture if the aberration is positive, 
 and diminishes if the aberration is negative. With any 
 lens in which the diaphragm is mainly relied on for the 
 purpose of correcting spherical aberration, and especially 
 with single combinations, the best focus is thus liable 
 to shift with variations in the size of the aperture ; hence, 
 whenever possible, it is desirable to focus with the working 
 stop that is to be used for exposure. 
 
 The amount of spherical aberration possessed by a single 
 lens depends to a great extent on the form of the lens, a 
 minimum amount being present in the case of a crossed 
 lens in which the surface nearest the principal focus has 
 a curvature of one-sixth that of the other surface. 
 
 With all forms of lenses the aberration in direct pencils 
 is greatest when the side of greatest curvature is nearest 
 to the image. Hence with a positive meniscus there is 
 less spherical aberration when the concave side is towards 
 the image; but this position is disadvantageous as re- 
 gards oblique pencils, and the opposite position, concave 
 towards the object, has perforce to be adopted in spite of 
 the large amount of spherical aberration that is then 
 introduced. 
 
Aberration. 
 
 71 
 
 Form has also to be considered in the compensating lens, 
 and as form is restricted if the lenses are cemented, greater 
 control is obtained by keeping them separate. If too far 
 apart the aberration is under corrected, and if too near it 
 is over corrected. With cemented lenses the contact surface 
 must be negative, and one such surface alone will not com- 
 pensate more than a moderate amount of spherical aberra- 
 tion. If then the combination as a whole is of the meniscus 
 form, and is vised with the concave side to the object, the 
 one contact surface will fail to completely compensate the 
 aberration. If, however, a small aperture is used to cut 
 out the more violently aberrated outer rays the small 
 central pencil may be fairly well compensated. In teles- 
 copic objectives more complete compensation is possible 
 because the meniscus form of lens can be reversed. The 
 slight amount of positive aberration possessed by a crossed 
 lens can be corrected by one contact surface. 
 
 53. Spherical, Aberration in Oblique Pencils. — With 
 an oblique pencil longitudinal sections taken through the 
 central ray are dissimilar, varying according to the inclina- 
 tion of the section. To take the simplest possible case, 
 assume that the lens is infinitely thin so that all oblique axial 
 rays are straight, and also central to their respective 
 pencils. The lens being circular, a pencil passing obliquely 
 through it is elliptical in transverse section, and a longi- 
 tudinal section of the pencil taken through the shortest 
 diameter of the ellipse will lie in one plane with the principal 
 axis of the lens. Such a section is designated as a section 
 in primary plane. Thus, when we show on a diagram a 
 section of both a direct and an oblique pencil, the latter 
 must be shown in primary plane, which is the plane of 
 the paper on which the diagram is drawn. 
 
 A section of the oblique pencil made in a plane at right 
 angles to the primary plane and taken through the longest 
 diameter of the ellipse is called a section in secondary 
 plane. In the simple case we have assumed this section 
 would pass through the axis of the oblique pencil, but 
 if the axial ray was not the central ray, the section might 
 be some way from the axis. Between these two sections 
 we might take any number of others at varying degrees 
 of inclination from the primary plane, but, as sections in. 
 
72 
 
 A First Book of the Lens. 
 
 primary and secondary plane differ the most it is sufficient 
 to consider them only. 
 
 Assuming an oblique emergent pencil to be free from 
 spherical aberration, its wavefront is spherical, and sections 
 of the wavefront in primary and secondary planes are both 
 circular and concentric to the same centre or focus. If 
 the wavefront is non-spherical varying conditions may exist. 
 1st. The sections may be non-circular, but each of sym- 
 metrical curvature, in which case there is longitudinal 
 spherical aberration as already described. 2nd. The primary 
 section of the wavefront may be not only non-circular but 
 also unsym metrical in form, in which case we have the effect 
 of coma. A longitudinal section in secondary plane is 
 always symmetrical. 3rd. Both sections may be circular, 
 or approximately so, but be concentric to different centres 
 or foci, in which case the effect of astigmatism is produced. 
 
 If the aberration is longitudinal only, the effect is similar 
 to that in direct pencils; but if a diaphragm is used to 
 correct it, the curvature of the field is considerably affected 
 by varying the position of the diaphragm. Suppose, for 
 example, we have a simple lens L, producing either positive 
 longitudinal spherical aberration as shown in primary 
 plane in fig. 48, or negative similar aberration as in 
 fig. 49. If the diaphragm is placed at the crossing-point 
 C, its effect is similar to that of a diaphragm in the case 
 of a direct pencil. The focal length of the oblique pencil 
 is slightly altered by variations in the size of the aperture, 
 and therefore the curvature of field is very slightly 
 affected; but if the stop be placed at A, away from the 
 crossing- point, so as to eliminate all rays in primary plane 
 excepting such as come to foci at q, a very considerable 
 effect is produced upon curvature. With positive aberra - 
 tion, as in fig. 48, the emergent pencil is shortened and 
 curvature of the field increased ; and with negative aberra- 
 tion, as in fig. 49, the field is flattened (assuming that 
 curvature already exists) or even rendered convex, the 
 focus q being thrown farther from the lens. Thus the 
 existence of negative aberration assists in the attainment 
 of a flat field so far as the primary plane of the oblique 
 pencil is concerned. The farther the stop is from the 
 crossing -point the greater is its effect. 
 
A berraticm. 
 
 IS 
 
 Incidentally it may be noted that the foci q, q, are 
 thrown off the axial ray, hence the image is geometrically 
 inaccurate, or distorted. 
 
 Variations in the position of the stop similarly affect 
 the focus of the oblique pencil in secondary plane, but 
 in a somewhat different manner and to a different extent. 
 There is a different secondary plane with its own focus 
 for every position of the stop, and the farther the stop 
 is from the lens the farther is the secondary plane from 
 
 1 
 
 
 
 i ^ 
 
 L 
 
 A 
 
 1 
 
 / 
 
 a Fi|M-9. 
 
 the axial ray of the oblique pencil. The focal length is 
 shortened if the aberration is positive, and lengthened if 
 negative ; but the effect of the stop on the focal length 
 in secondary plane is only one-third the amount of its 
 effect in primary plane.* 
 
 * It may be observed that the effects of a diaphragm may be 
 produced, though no actual diaphragm is present* Some portion of 
 the lens mount may cut off parts of oblique pencils and act as a 
 partial diaphragm ; or, one lens of a doublet may act as a diaphragm 
 to the other. An alteration in the length of the mount, or in the 
 separation of the lenses, then has a similar effect to that produced 
 by a similar alteration jn the distance between a single lens and 
 its stop. 
 
74 
 
 A First Book of the Lens. 
 
 It should be noted that if the stop is at C, in figs. 48 
 and 49, the secondary plane passes through the straight 
 oblique axial ray ; but if the stop is at A , the secondary 
 plane passes through the central ray of the narrow pencil 
 transmitted by the stop, and is not straight but bent at 
 the lens. 
 
 As the effect of the stop differs in primary and secondary 
 plane, the two planes may have different foci, or the 
 pencil may be astigmatic. This illustrates the fact that 
 astigmatism is only a particular phase of spherical aberra- 
 tion, and also that it may be produced by a misplacement 
 of the stop under certain conditions. In a simple lens, 
 however, it exists independently of the stop which may 
 be employed to cure it. 
 
 54. Coma. — In fig. 50 assume we have a lens L, the 
 principal axis of which is represented by PQ. The axis 
 of an oblique pencil is represented by pgr, and the section 
 of this pencil is shown in primary plane. In the absence 
 of spherical aberration, the emergent plane pencil has 
 a circular wavefront ac, and comes to a focus at q. If, 
 however, coma exists, the section of the wavefront is 
 both non-circular and non-symmetrical, being, with the 
 form of lens shown, more sharply curved below the axis 
 than above it, as shown at dbe, with the result that 
 corresponding opposite pairs of rays come to foci q\, q%, 
 etc., above the axis, and this produces a blur on one side 
 of the true focus-point q in a direction away from the 
 principal axis PQ. Assuming a screen to be placed at 
 q, then, instead of an image point, we shall have a one- 
 sided blur ; while, if we assume curvature of the field to 
 exist, and the screen to be at Q, the additional blur, due 
 to the curvature, will not be circular, as it would be in 
 the absence of coma, but of an irregular pear shape ; this 
 appearance being a test for coma. 
 
 If this figure is compared with figs. 48 and 49, it will be 
 noticed that while those two show longitudinal displacement 
 of the foci of corresponding pairs of rays, fig. 50 shows 
 lateral displacement of similar foci. Coma may therefore 
 be described as due to lateral spherical aberration. 
 
 Fig. 50 illustrates outward coma, but if the lens is 
 reversed, as in fig. 51, the foci q\, q%, etc., are laterally 
 
Aberration. 
 
 7:> 
 
 displaced towards the principal axis, and produce the re- 
 versed effect of inward coma. If two such lenses are com- 
 bined in reversed positions, as in the ordinary doublet, the 
 inward coma of one is neutralised by the outward coma of 
 the other, and lateral aberration is compensated, this being 
 an example of positive compensation. Further, as opposite 
 forms of the aberration are given by simply reversing the 
 
 curvatures of the lens, it is possible, as pointed out by Mr. 
 Dennis Taylor, to select an intermediate form of lens that 
 shall be free from coma. This will be a crossed lens of 
 approximately the same form as that which possesses a 
 minimum amount of longitudinal spherical aberration 
 (see Sec. 52). 
 
 With single lenses, or combinations of the meniscus 
 form, it is usual to rely on the diaphragm to eliminate the 
 majority of the rays that produce the coma. Thus if 
 
76 
 
 A First Book of the Lens. 
 
 we place a~ diaphragm at B (tigs. 50 and 51) only a 
 portion of the large pencil is allowed to pass, and this 
 comes to a focus at q s . In connection with this effect of 
 the diaphragm, there are several matters to notice. 
 
 First, the focus q s is farther from the lens than the 
 other foci q, q h and q 2 , and by carefully adjusting the 
 position of the diaphragm we can arrange matters so that 
 q 3 comes exactly on the screen passing through Q, and so 
 produces a flat field image. The farther the diaphragm is 
 from the lens the farther is q s , hence, as in the diagram, 
 q, is beyond the screen and the field is over corrected, B is 
 too far from the lens. If the diaphragm were on the 
 other side of the lens at A, curvature would be increased, 
 not diminished (see foci q 4 ). 
 
 Second, it will be noticed that q 5 is below the axis in 
 fig. 50 and above it in 51, being on the opposite side to 
 the coma in each case. This lateral displacement of the 
 foci off the axis renders the image geometrically incorrect, or 
 distorts it in a direction opposite to that of the coma. It 
 is thus apparent that the effect of the stop with coma is 
 very similar to its effect with longitudinal aberration. If 
 positive longitudinal aberration exists simultaneously with 
 coma, the effect of the stop with the one aberration may be 
 counteracted by its effect with the other, and the lens may 
 be considered to be free from diaphragm corrections so 
 far as they affect curvature. This is taken advantage of 
 in the Cooke lens, in which compensation is alone relied 
 upon to cure curvature. 
 
 55. Astigmatism. — The curvatures of the wavefront in 
 primary and secondary plane may both be circular, but 
 they may not have a common centre, in which case each 
 plane has a different focus. This produces the effect known 
 as astigmatism, which is better described as astigmatic 
 spherical aberration. As before, we need consider only 
 the extreme sections in primary and secondary plane. 
 
 The existence of astigmatism is shown by the blurring of 
 the image in two particular opposite directions which vary 
 with the position of the screen. If the screen is at the 
 focus of the primary plane the image of a point is blurred 
 in secondary plane and becomes a line. Conversely if the 
 screen is at the secondary focus, the image is blurred in 
 
Aberration. 
 
 77 
 
 primary plane. At intermediate positions,, between the 
 primary and secondary foci, the image is out of focus in 
 both planes, and at one point it will be blurred equally 
 in both directions, so that the image of a point becomes 
 approximately a disc ; this, being the' nearest approach 
 to a regular point that can be attained, may be called the 
 mean focus.* Thus there are three astigmatic foci to the 
 oblique pencil, the primary, mean, and secondary foci, and 
 while the best image of a point is to be obtained at the 
 mean focus-, the best image of a line lying in primary plane 
 
 is to be obtained at the secondary focus, and that of a line 
 in secondary plane at the primary focus. 
 
 Suppose that in the object there are a series of lines all 
 radiating from a point on the principal axis of the lens, as 
 in fig. 52. As a line may be considered to consist of an 
 infinite number of points, the image of one of these lines 
 will be formed by an infinite series of oblique pencils, a few 
 of which are shown in fig. 53, all in primary plane. As 
 the object line itself is in primary plane a blur in primary 
 
 * Some writers state erroneously that the mean astigmatic image 
 of a point is a cross. If a lens has a circular aperture the primary 
 and secondary foci are very attenuated ellipses (practically lines) at 
 right angles to one another, and it is not possible for a cross to be 
 formed between these two foci. 
 
78 
 
 A First Book of the Lens. 
 
 plane of each image point will be of no consequence ; it will 
 simply overlap the other points and .slightly lengthen the 
 line ; hence, if the light pencils are astigmatic, the best 
 image of a radial line will be produced at their secondary 
 foci, at which any blur is in a primary direction. If, 
 however, in addition to the radial lines, we have a circle, 
 as shown in fig. 52, the image of that circle, being 
 blurred radially, will be quite out of focus. If now the 
 screen is moved to the primary focus of the oblique pencils, 
 any point in the circle will be blurred in secondary plane, 
 or at right angles, or tangential, to the radii ; hence the 
 
 circle will be in focus while the radii, being blurred tan- 
 gentially, will be out of focus. We cannot in any way 
 bring both the circle and the radii into focus together, 
 which fact is a proof of the existence of astigmatism. An 
 object such as that shown in fig. 52 is a very convenient 
 test object for this aberration. On account of these radial 
 and tangential effects, the primary focus is often called the 
 tangent focus, and the secondary focus the radial focus. 
 They are also sometimes respectively distinguished as the 
 meridional and sagittal foci. 
 
 The separation of the astigmatic foci increases with 
 the obliquity of the pencils «(as do all other aberrations to 
 which oblique pencils are subject), and as there are three 
 
Aberration. 
 
 70 
 
 foci to every pencil there are also three focal fields — the 
 primary, mean, and secondary ; two of which, at least, 
 must be curved fields. With a simple positive lens 
 all three fields are curved, see P, M, and S in fig. 54A, the 
 primary field being of greater curvature or shorter radius 
 than the others. The three fields coincide only at the 
 direct focus Q, at which there is no astigmatism. This is 
 an illustration of positive astigmatism, and the amount 
 of the astigmatism may be represented by the ratio of the 
 curvatures of the primary and secondary fields. 
 
 It is apparent that to cure astigmatism, or to render 
 the lens anastigmatic, all three fields must be brought 
 together ; and to cure curvature the one field produced 
 must be flat. If we attempt to flatten the fields by any 
 means not affecting the material or the curvatures of 
 the lens (see Sec. 51), the primary field is affected just 
 three times as much as the secondary field ; hence, if the 
 primary field has originally three times the curvature 
 of the secondary field, the correction of curvature involves 
 that of astigmatism, the two fields coinciding at the moment 
 when both are flat. This cannot, however, be accomplished 
 with ordinary single lenses, for with such the primary 
 field has only about twice the curvature of the secondary 
 field, and therefore the two come together before either of 
 them is flat, and they produce one curved non-astigmatic 
 field, as shown in fig. 54B. If the flattening process 
 is carried further the primary field passes the secondary 
 field and the astigmatism is over corrected and becomes 
 negative, as shown in 54C to F. The extent to which this 
 over correction may be carried is considerable. In 54C 
 slight negative astigmatism has been introduced, but the 
 average curvature is less than in 54B. In 54D the 
 primary field is flat, but negative astigmatism has in- 
 creased. In 54E, the mean field is flat, while the primary 
 and secondary field are of opposite curvatures: In 54F 
 the secondary field is approximately flat, while the mean 
 and primary fields have negative curvatures. Figs. 
 52D, E, F, may be taken to represent the average state 
 of correction of " flat field " astigmatic lenses of the 
 ordinary types, E being perhaps the most useful degree 
 of correction. Fig. 52C may represent a " round field " 
 
80 
 
 A First Book of the Lens. 
 
 liens, the curvature of the anastigmatic field in 52B> 
 being too great to render such a type of lens of much 
 utility. 
 
 To render all three fields flat the materials and surfaces 
 of the lens must be so adjusted as to increase the initial 
 astigmatism to the required ratio of 3 — 1. This can be 
 accomplished by combining a negative and positive lens 
 so that the contact surface is positive ; but this condition 
 is directly opposite to that essential to the correction 
 of longitudinal spherical aberration, which requires the 
 eontact surface to be negative. Thus, with two cemented 
 lenses, the preliminary increase of astigmatism necessary 
 to its correction simultaneously with curvature of field 
 involves an increase of longitudinal aberration, while the 
 compensation of the latter reduces astigmatism to such 
 an extent as to render it impossible to simultaneously 
 correct it and secure a flat field. 
 
 The difficulty can be got over by employing extra 
 surfaces. Thus by combining four lenses, so arranged that 
 the contact surface in one pair is negative and in the 
 other pair positive, the required condition for longitudinal 
 correction can then be fulfilled in the first pair, and that 
 for astigmatism and curvature in the second ; while any 
 residue of aberration of a positive nature in one combination 
 cati be neutralised by an excess of negative aberration in 
 the other. This method is employed in certain of the Zeiss 
 lenses. Such a doublet is non- separable, as the removal 
 of either combination disturbs the correction of the whole. 
 
 The arrangement may be varied by employing one lens 
 in place of the two interior ones. That is, by combining 
 three lenses with two contact surfaces, one negative and 
 one positive. A corrected single combination can be formed 
 in this way, and two such combinations may form a 
 separable doublet, such as the Goerz double anastigmat. 
 
 The system may be further varied by combining two 
 separated lenses so that of the adjacent surfaces one may 
 be positive and the other negative. We then practically 
 have a lens of air enclosed between two of glass. This 
 method is also employed in the Goerz lenses. Further ; 
 two such combinations may be combined to form a doublet, 
 and if the two interior negative lenses are coalesced into 
 
A berration 
 
 81 
 
 one, we have a triplet very similar to the Cooke lens. The 
 one central negative lens then compensates the aberrations 
 of the two other positive lenses. The Cooke lens is specially 
 distinguished by the employment of single lenses free from 
 diaphragm corrections (see Sec. 54), and of only one com- 
 pensating lens to simultaneously correct the three primary 
 aberrations. 
 
 Thei ■e are many possible modifications, but the systems 
 mentioned sufficiently illustrate the methods employed, 
 as a general rule, in lenses that are corrected for astig- 
 matic and longitudinal aberration, and for curvature ; but 
 there are a few exceptional methods. As, for example, 
 we may ignore the compensation of longitudinal spherical 
 aberration and rely upon the stop to eliminate it as far 
 as possible. This renders the lens slow and Ross' Concentric 
 is a type of such a slow anastigmatic fiat-field lens. It may 
 be mentioned that this lens owes its name to the fact that 
 in each of its two combinations the outer lens surfaces 
 are approximately concentric, so that the lenses are of 
 approximately equal thickness in all parts. This is a 
 condition that is very closely observed in all combinations 
 that produce primary and secondary fields with curvatures 
 in the ratio of 3 to 1. 
 
 56. Explanatory Notes.— It should be understood that 
 the terms coma and astigmatism are only descriptive of 
 secondary effects produced by spherical aberration in 
 oblique pencils. Similar effects might be due to other 
 quite different causes. For example, if there is any defect 
 in the curvatures of the lens surfaces, either coma or 
 astigmatism will appear in direct pencils as well as in 
 oblique ones. The appearance of these effects only in 
 oblique pencils indicates that spherical aberration exists. 
 Some photographic writers apply the term spherical aber- 
 ration only to the longitudinal displacement of the foci 
 of pencils of different diameter, and apparently look upon 
 astigmatism as a separate aberration altogether, and this 
 erroneous distinction has led to a good deal of confusion. It 
 is worthy of note that Glazebrook in his " Physical Optics " 
 avoids the use of any distinctive terms, describing all the 
 effects of spherical aberration under the simple heading 
 of aberration. If spherical aberration is considered to be 
 
 G 
 
82 
 
 A First Book of the Lens. 
 
 divided into three varieties ; longitudinal, lateral, and astig- 
 matic spherical aberration —respectively distinguished by 
 symmetrical, non-symmetrical, or linear blurs— the sub- 
 ject is much simplified. 
 
 57. Chromatic Aberration. — This has been defined as 
 the separation of a wavefront of compound light into separate 
 wavefronts of light of different colours, or, as the dispersion 
 of a ray of compound light into rays of different colours, 
 which converge on different foci. 
 
 Chromatic aberration is cured by the employment of 
 compensating lenses. With a positive lens violet rays are 
 rendered too strongly convergent as compared with, say, the 
 yellow; but with a negative lens they are too strongly 
 divergent, hence one lens will compensate the other. 
 
 The peculiar difficulty of chromatic correction is the 
 simultaneous correction of more than two colours, and as 
 the photographic plate is sensitive to a considerable range 
 of colours, correction for two only is generally insufficient. 
 If the dispersive powers of various materials for different 
 pairs of rays were always in the same proportion, the cor- 
 rection of any two rays would involve that of the others. 
 This not being the case, it is generally necessary to employ 
 additional lenses or surfaces to eliminate all such secondary 
 or residual aberration as may be of importance. 
 
 With modern varieties of glass dispersive power and 
 refractive power do not necessarily vary together, and it 
 is possible for a contact surface between two lenses to 
 produce positive refraction and negative dispersion ; that 
 is to say, the material of one lens may be of higher 
 refractive power but of lower dispersive power than the 
 other. Thus chromatic aberration, astigmatism, and 
 curvature may be simultaneously corrected. Before the 
 invention of Jena glass, however, dispersive and refractive 
 power always varied together ; hence it was only possible 
 to simultaneously correct longitudinal spherical aberration 
 and chromatic aberration. The chromatic correction being 
 of great importance, astigmatic flat field lenses could not 
 be produced. The general principles of their construction 
 were understood, but the necessary material was not available. 
 
 A pair of lenses arranged to correct spherical and chro- 
 matic aberration is called a "normal" or "old" achromat, 
 
Aberration. 
 
 83 
 
 A pair arranged for the corrections of astigmatism, 
 curvature, and chromatic aberration as an " abnormal " or 
 "new" achromat. Thus, the Concentric is formed of two 
 abnormal achromats. The ordinary " rectilinear " of two 
 normal achromats. The Zeiss, referred to in Sec. 55, of 
 one normal and one abnormal achromat. 
 
 The diaphragm has no curative effect upon chromatic 
 aberration, excepting so far as the aperture can be reduced 
 to diminish the diameter of the blur. The presence of 
 chromatic aberration is indicated by the colouration of the 
 margin of the blur around the focus, but the best test is 
 to expose a plate on a series of numbered cards at various 
 distances from the lens, focussing carefully on one particular 
 number. If that number is the best defined in the developed 
 image the lens is achromatic, but if a nearer number is 
 better defined, positive aberration exists, or if a farther 
 number, then there is negative aberration. 
 
 58. Diaphragm Distortion. — It has been incidentally 
 explained (Sees. 53 and 54) that the use of a diaphragm 
 to flatten the field of oblique pencils possessing either 
 longitudinal spherical aberration or coma throws the image 
 of a point off the oblique axial ray and renders the image 
 geometrically incorrect. The distortion thus produced 
 may be either inward (all oblique pencils coming to foci 
 too near the principal axis) or outward (the foci being too 
 far from the principal axis). The former variety is produced 
 with single lenses of the usual landscape type with the 
 diaphragm on the object side of the lens; the latter is 
 produced if the lens is reversed so that the diaphragm is 
 between the lens and image. 
 
 The amount of the distortion increases with the obliquity 
 of the pencils. Thus, if the object happens to be a series 
 of concentric equidistant circles, - as shown by dotted lines 
 in fig. 55, inward distortion will produce a contracted 
 image, as shown by full lines in A, and outward distortion 
 an expanded image, as shown in B. If the distortion is 
 very appreciable, a square object, such as shown in fig. 56A 
 will not only be contracted, but its diagonals act will suffer 
 greater contraction than its diameters bb, so that the sides 
 will become curved as in B. This is known as barrel distor- 
 tion. Similarly if the square suffers expansion from outward 
 
84 
 
 A First Book of the Lens. 
 
 distortion it will take the form shown at G, which is known 
 as cushion distortion. All lines that pass through the 
 principal axis of the lens remain straight while all others 
 become curved, on which account diaphragm distortion is 
 often termed curvilinear distortion, and a lens free from 
 distortion is called rectilinear.* It appears, however, to be 
 possible for a certain amount of distortion to be present 
 without the production of any marked curvilinear effects. 
 The best test for diaphragm distortion is therefore the 
 comparison of proportions. For example, if we take an 
 equally divided object, and the lens throws an equally 
 
 divided image on a screen parallel to the object, diaphragm 
 distortion must be absent. 
 
 Speaking generally, distortion is diminished by placing 
 the diaphragm at or near the crossing-point of the lens. 
 This condition cannot be fulfilled with a single meniscus 
 lens, as the crossing-point is on the convex side of the lens 
 and the diaphragm must be on the concave side to diminish 
 curvature of the field (figs. 50 and 51). In a doublet with 
 the crossing-point between the lenses the diaphragm can 
 be placed in the required position, but with a doublet of 
 the periscopic form it cannot. In the telephotographic 
 lens, which is essentially a combination of a positive and 
 
 * It may also be correctly described as " Orthoscopic." The par- 
 ticular lens (now out of date) known by this title was, however, thus 
 christened in error, for it eventually proved to be non-rectilinear. 
 
A berratipn. 
 
 85 
 
 a negative system, distortion may be obviated by making 
 eacli lens itself rectilinear. Practically each lens has a 
 diaphragm at its own crossing-point, hence the whole is 
 free from distortion. 
 
 It should be noted that combinations of lenses may 
 show distortion even though no actual diaphragm is present. 
 This may be due to the lens mount limiting the aperture 
 and serving as a diaphragm, or to each lens serving as a 
 diaphragm to the other (see footnote to Sec. 53), but in the 
 absence of anything capable of serving as a diaphragm, any 
 apparent distortion is really an effect of coma, which, from 
 its nature, not only blurs but also distorts the image. In 
 
 A JB C 
 
 « -/-'J Ju 
 
 such a case, outward coma is accompanied by inward dis- 
 tortion and vice versa. This effect of coma must be carefully 
 distinguished from diaphragm distortion. 
 
 If a lens is perfectly free from spherical aberration in 
 oblique pencils no distortion is produced by any position 
 of the diaphragm, and the less the part played by the 
 diaphragm in correcting the aberrations the less is the 
 distortion. If spherical aberration is well compensated, 
 as it is in a high class single lens, we may expect less 
 distortion than with a cbeap achromatic single lens, even 
 though the diaphragm is equally far removed from the 
 crossing-point. 
 
 It should be noted that in subjects possessing no straight 
 lines, diaphragm curvilinear distortion may have no dis- 
 advantage ; on the contrary, inward or barrel distortion 
 
86 
 
 A First Book of the Lens. 
 
 has the advantage of simulating the proportions of the 
 object as they appear to the eye in visual perspective. 
 The image is not in plane perspective, but in a sort of 
 pseudo-spherical perspective, and is less likely to convey 
 a false impression of distance and comparative size. On 
 the other hand, the presence of distortion renders a photo- 
 graph quite valueless as a record from which dimensions 
 and distances are to be measured. 
 
 59. Effect of Aberration on the Action of Lenses. — ■ 
 When studying the theoretical action of a lens in forming 
 the image, it was assumed that the lens was perfectly free 
 from all forms of aberration. If spherical aberration is 
 present the cardinal points and image field of a lens are 
 rendered more or less variable. The nodes and foci are 
 liable to shift for pencils of different widths, or of different 
 obliquity, or for objects at varying distances. The focal 
 length may vary with different-sized apertures, and the 
 form of the image field may vary with the aperture and 
 distance of objects. 
 
 With chromatic aberration the nodes, foci, focal length, 
 and field will all vary for light pencils of different wave- 
 lengths, or for objects of different colours. 
 
 Chromatic aberration should be practically absent in a 
 lens of moderately good quality, but a slight amount of 
 spherical aberration is likely to remain, and also there is 
 very probably a certain amount of variable curvature of 
 the field for objects at varying distances. 
 
 The results of calculations with regard to foci, etc., 
 based on the assumption that the lens acts with theoretical 
 accuracy are likely to be slightly upset by these residual 
 aberrations. For example, it is not safe to rely solely on 
 calculated conjugate distances when adjusting an enlarging 
 camera ; a visual test is necessary. 
 
 60. Flare. — This is not an aberration of the true image, 
 but an effect due to the production of sundry false images 
 by the agency of reflection. 
 
 Flare may be due solely to reflections from the lens 
 surfaces, or partly to reflections from the lens mount ; in' 
 the former case the effect may be distinguished as optical 
 flare, in the latter as mechanical flare. Mechanical flare 
 leads to a local or general degradation of the image 
 
A berration. 
 
 87 
 
 by reflected diffused light ; optical flare produces similar 
 effects by the agency of numerous false images, which, 
 being non-coincident with the true image, confuse it to a 
 certain extent. There is no perfect cure for flare, as no 
 lens can be absolutely free from it. Mechanical flare can 
 be diminished by careful design and construction ; and by 
 adjustment of surfaces, separations, and diaphragm, false 
 images of optical flare may be thrown into positions where 
 their effect is least detrimental. 
 
 The false images due to optical flare may be divided into 
 two classes — the secondary inverted images, formed by 
 single lenses, and tertiary erect images, formed by doublet 
 combinations. 
 
 In the case of a single meniscus lens, concave side to 
 
 the object, an oblique pencil Pq (fig. 57) produces three 
 false images of the object p, in addition to the one true 
 image q. Two of these images (q t and q 2 ) are in front of 
 the lens, and one (q 3 ) behind it, all three being inverted 
 in the same way as the true image. The image q 2 is caused 
 by reflection alone from the front surface of the lens, which 
 acts as a concave mirror. The image by reflection from 
 the back surface of the lens, which acts also as a concave 
 mirror; in this case all rays that reach q v twice suffer 
 refraction at the front surface. The third image (q 3 ) is 
 formed by rays that suffer refraction at the first surface, 
 reflection at ' the second, a second reflection at the first 
 (which serves as a convex mirror) and then refraction 
 at the second surface. 
 
 All three false images can be seen upon screens placed 
 at the respective foci if the light from p is sufficiently 
 
88 
 
 A First Book of the Lens. 
 
 intense, but if a. diaphragm is placed at A in front of the 
 lens the pencils forming q } and q 2 are likely to be ob- 
 structed, and the back of the diaphragm being illuminated 
 by them, may produce a certain amount of mechanical 
 flare. The pencil forming q 3 is not thus obstructed 
 by the diaphragm, but it should be noted that if the 
 aperture is large q s will lie within the pencil forming q, 
 while a small aperture will so reduce the width of the 
 pencil as to leave q% outside it, and therefore more distinct. 
 All three false foci are on the same side of the principal 
 axis as the true image (q). 
 
 If qz is close to the plate on which the true image is 
 received, an out-of-focus false image is apparent on the 
 plate between the true image and the principal axis ; but 
 if q 3 is close to the lens, this out-of-focus image is so 
 expanded as to produce simply a general effect of fog, and 
 the detrimental effect of the false focus is minimised. The 
 position of q% depends on the surfaces and thickness of the 
 lens, and the distance of the diaphragm A from the lens. 
 If the distance of the diaphragm is increased, q s moves 
 towards the lens ; if A is brought close to the lens, q 3 
 comes nearer the plate. If we have a number of oblique 
 pencils, each one has a false focus similar to q%, and the 
 greater the obliquity of the pencils the farther ai-e the 
 false foci from the principal axis ; but in no case does the 
 false focus move very far from the axis, hence it follows 
 that if these foci are near the plate they produce an 
 accumulated effect in the centre of it, with the result that 
 an over-exposed blurred patch, commonly called a flare-spot, 
 appears. If the false foci are near the lens the flare-spot 
 is so expanded as to produce uniform slight fog over the 
 whole image. By shifting the diaphragm the flare-spot 
 may be thus distributed and the defect practically, though 
 not actually, cured. 
 
 With a rectilinear doublet, (see fig. 58) similar reflected 
 secondary images are produced by each of the lenses, but 
 the reflections for which the front lens is alone responsible 
 are of little importance. Its front surface being convex 
 light reflected from it is dispersed. From its back surface 
 a false front focus (q 5 ), is produced ; but all this light is 
 lost. The back false focus corresponding to q 3 in fig. 64 
 
Aberration 
 
 89 
 
 is also of no account, as it is not likely to be anywhere 
 near the plate. As regards the back lens the conditions 
 are similar to those in fig. 57. There is a back false focus 
 (g 3 ) which may be productive of flare-spot, and also front 
 false foci corresponding to q x and q 2 . The nearer of these 
 two foci (q{), is not likely to produce any effect, but the 
 reflected pencil forming the farther one (72), will be reflected 
 from the back surface of the front lens, producing a false 
 tertiary focus between the back lens and the plate, and 
 on the opposite side of the principal axis to the true focus 
 q. This tertiary focus in the only one of consequence ; 
 it is shown at g 4 . Any image formed by such a focus is 
 erect, not inverted. 
 
 The effect of this tertiary focus depends again on its 
 
 distance from the plate, and the distance is affected very 
 considerably by the separation of the lenses. At a certain 
 distance apart the tertiary erect image is sharply defined 
 on the screen ; a very slight alteration in either direction 
 will, however, throw it right out of focus and render it 
 negligible. 
 
 Tertiary flare is not in any way responsible for flare- 
 spot, which, as already described, is purely an effect of 
 secondary flare. Flare-spot is a very much out-of-focus 
 small scale inverted image of the whole view situated in 
 the centre of the true image, and the spot will resolve 
 itself into a fairly well defined image if the camera is 
 racked in sufficiently. Tertiary flare, on the other hand, 
 is a false image on much the same scale as the true image, 
 
9(1 
 
 A First Book of the Lens. 
 
 but reversed in every particular. If we focus a gas flame 
 near one side of the plate a corresponding erect image 
 of the gas flame will appear on the opposite side of the 
 plate. This is tertiary flare, and it is detrimental if botli 
 images are visible at the same time, but of little moment 
 if we have to throw the true image out of focus to render 
 the tertiary image apparent. If we rack in quite close 
 to the lens secondary flare will appear in the form of a 
 small inverted image of the flame between the true image 
 and the centre of the plate. If there is no out-of-focus 
 trace of this image when the true image is in focus there 
 is not likely to be any flare spot. In this way we can 
 readily test whether flare of an objectionable nature is present. 
 If we photograph a landscape with a very brilliant sky, 
 flare spot will appear in the centre of the plate if secondary 
 flare is present in a disadvantageous form, and the fore- 
 ground may be covered and degraded by a reversed, out-of- 
 focus image of the sky if tertiary flare is a serious defect ; 
 this can, however, only be the case with doublet lenses, 
 while flare-spot may appear with any form of lens. 
 
 The various secondary and tertiary foci that have been 
 described as of little moment may contribute in some 
 degree to mechanical flare, and to the production of general 
 fog. Probably the whole of the interior metal work of 
 the lens is illuminated directly or indirectly during exposure, 
 and a good deal of the light reflected or dispersed from 
 the metal work may reach the plate. Further, the plate 
 itself, if brightly illuminated in any part, reflects light, and 
 may serve as a source of light that may be reflected from 
 the mount or the lenses on to another portion of the plate. 
 So far as the mount is concerned, careful design and dead- 
 black will minimise reflections. It is obvious that by 
 combining a number of lenses the number of reflective 
 surfaces is increased, and there is therefore greater likeli- 
 hood of flare appearing in some form. By cementing the 
 lenses together the reflective surfaces are reduced in number 
 (see Sec. 12) and so also are the false foci. For example, 
 suppose we have a combination of three lenses. If the 
 three are separated there are six refractive and reflective 
 surfaces ; if cemented, there are four refractive and only 
 two reflective surfaces. There is then a gain both in 
 
A berration. 
 
 91 
 
 rapidity (less light being lost) and freedom from flare. 
 But such gain may be counter- balanced by inefficient 
 correction of the refractive aberrations, owing to the 
 reduced number of refractive surfaces; and inefficient 
 correction may require a reduction of aperture and a loss 
 of rapidity greatly in excess of the gain due to minimised 
 reflections. It may be noticed that in modern lenses there 
 is a tendency to use separated rather than cemented lenses, 
 and as a result there is a considerable number of reflective 
 surfaces. The absence of obvious flare with such lenses 
 must therefore be put down to the careful adjustment 
 of separations, etc. 
 
 Many other false foci beyond those that have been 
 mentioned are produced in a doublet lens, but they are 
 generally so faint as to be of no moment. 
 
CHAPTER VI. 
 
 Scale, Intensity and Illumination. 
 
 61. Scale of Image — -It has already been explained that 
 the ratio of the linear dimensions of the image to those of 
 the object is the ratio of the focal distance of the image to 
 that of the object ; or that, as stated in formula 1 (Sec. 19), 
 
 R= t 
 it 
 
 This general formula covers all cases, but there are special 
 cases to consider. 
 
 If we compare the value of R with different lenses 
 focussed on the same object at a fixed distance u, it is 
 apparent that in all cases R must vary with v, excepting 
 only when the object is at a truly infinite distance so that 
 the image is infinitely small and R = o. The size of the 
 image is then unaffected by the value of v, which under 
 these conditions is equal to the focal length of the lens. 
 In other words, all lenses produce images of the same 
 size from infinitely distant objects. This is a fact that 
 becomes partially apparent in astronomical work, but only 
 in the case of the farther fixed stars.* In photography 
 
 * No optical system could produce a visible image of an object at 
 a truly infinite distance. The image being infinitely small, or a 
 mathematical point, would be invisible, and could not be magnified. 
 The distance of the known fixed stars is not truly infinite. They 
 are near enough to produce visible images ; but distant enough to 
 prevent the size of the image from being appreciably affected by the 
 focal length of the object glass of a telescope. This is due to the 
 virtual parallelism of the various light pencils forming the image. 
 The production of a visible image shows that the pencils are not 
 truly parallel to one another, but we cannot construct a lens of 
 sufficient focal length to enable us to take advantage of their want 
 of parallelism, and so increase the size of the image. 
 
 92 
 
Scale, Intensity and Illumination. 
 
 93 
 
 we never deal with truly infinite distances, but, if we 
 focus on such a distance by placing the plate at the 
 principal focus of the lens, all objects beyond a certain 
 near finite distance are also in focus ; and if we assign a 
 definite finite value to u then li varies strictly with v, 
 which under the conditions equals /. Thus, when different 
 lenses are working at their focal lengths, the size of the 
 image of an object at a finite distance varies strictly with 
 the value of /, so that an 8-in. lens will produce an image 
 twice the size of that formed by a 4-in. lens. 
 
 If, then, we bring the object so near that we have to 
 rack out the camera and make v greater than j\ the size 
 no longer varies with J] but simply with v. Thus, if the 
 object is at a focal distance of 32 ins. an 8-in. lens produces 
 an image, not twice, but 2^ times the size of one formed 
 by a 4-in. lens. The mistake is frequently made of 
 assuming that in all cases the size of the image is pro- 
 portional to the focal length of the lens. 
 
 We have here only considered cases in which u remains 
 constant with different lenses. If we keep to one lens and 
 vary the distance of the object, it is manifest that if an 
 alteration in u involves an alteration in v to secure focus, 
 then R varies with the value of v/u as in the formula ; but 
 if the object is at such a distance from the lens as to render 
 an alteration of v unnecessary, then R varies inversely with 
 u. For example, if with a 6-in. lens we reduce the 
 distance of the object from 80 ft. to 40 ft. vefocussing 
 on the nearer distance is unnecessary, and the size of the 
 image is simply doubled ; but, if we reduced the distance 
 from 10 to 5 ft. we should have to refocus and alter v, 
 and as the result the larger image would be 2^ times 
 the size of the smaller. 
 
 62. Enlargement and Reduction.— From the funda- 
 mental formula R — v/u we can find one term, if the 
 other two are known, but in the case of enlarging or 
 reducing on a particular scale we know only R, and 
 must find both the others for a lens of a particular focal 
 length /, in order that we may adjust object, lens, and 
 image at proper distances. From formula 7 in Sec. 33 
 we can find the extra-focal distances of image and object 
 in terms of R and /, 
 
94 
 
 A First Book of the Lens. 
 
 The extra-focal distance of the image equals 
 
 v~Rf (19). 
 
 The extra-focal distance of the object equals 
 
 d . = I (20). 
 
 By adding f to the values of x and cl we can express the 
 focal distances v and u in terms of It and f ; thus 
 
 v = (1 + R)f (21) 
 
 « = (1 + ±)f (22). 
 
 In these formulae we may consider the extra-focal dis- 
 tances x and cl, and the focal distances v and u, to be 
 represented in terms of the focal length / as a unit, which 
 is a very convenient method of describing them. To give 
 a few examples. If we wish to produce an enlarged image 
 on a scale of 3 to 1 diameters, from the formulae we can 
 see at once that the extra-focal distance of the image is 
 three times the focal length, and that of the object one- 
 third the focal length, one multiplier being the reciprocal 
 of the other ; or that the focal distance of the image equals 
 four focal lengths, and that of the object one and a third 
 focal lengths. If then we are using a 6-in. lens the 
 distance v equals 24 ins. and the distance u equals 8 ins. 
 
 Or to take the example given earlier in this section. If 
 the focal distance of the object is 32 ins. and the focal 
 length of the lens is 8 ins., u equals 4/, hence d = 3f and 
 x = if. Therefore 1/R = 3 or H =? \. 
 
 If the focal length is 4 ins., then u = 8f d — If and 
 A' = \. Hence the 8-in. lens gives an image 2^ times as 
 large as the 4-in. lens. 
 
 To take an example of reduction on a particular scale. 
 Assume R to equal \ and f to be 8 ins. The distance 
 v must then equal (l+A J )/=l|x8 = 10 ins. The 
 distance u = (1 + l/R) /= (1 + 4) 8 = 40 ins. Or, we 
 might find u by dividing v by R, thus 10 -4- | — 40 ins. 
 
 In practice it is a very difficult matter to tentatively 
 adjust the apparatus so as to give an image on an exact 
 scale. The formulas enable us to place the object, lens, and 
 screen in approximately the correct positions, but almost 
 always a slight extra adjustment is necessary to secure 
 
tScale, Intensity and Illumination. 
 
 9o 
 
 sharp focus. The focal length may not be exactly known, 
 the nodes of the lens are probably unmarked, and there 
 are likely to be mechanical difficulties in the way of 
 measuring to the focussing-screen, so that an uncertain 
 amount of error frequently appears as regards scale. 
 
 In the Appendix, Table C gives the focal distances of 
 object and image for various values of R from -}. up to 25, 
 for a lens of 1-in. focal length. From these it is easy 
 to arrive at the corresponding distances for a lens of any 
 focal length, as we need only multiply the figures given 
 by the focal length in inches. 
 
 63. Relative Intensity of Illumination. — This is a 
 matter that directly affects exposure. The intensity of 
 
 the illumination received from a constant source of light 
 varies inversely with the square of the distance of the 
 light. Suppose, for example, we have an infinitely small 
 source of light a (fig. 59) from which light radiates in all 
 directions. Placing a diaphragm b with a circular aperture 
 near the light we can isolate one divergent pencil which 
 will form a disc of light of a certain size on a screen 
 placed at c, or a larger disc on a more distant screen 
 at d. The areas of these discs vary with the squares of 
 their diameters, and their diameters vary with the dis- 
 tance of the screen from the source of light «, hence the 
 areas of the discs vary with the squares of the distances ac 
 and ad. The total amount of light passed by the diaphragm 
 is constant, but being distributed over a larger area on the 
 more distant screen, the intensity of its effect is less than 
 on the nearer screen, the intensity at corresponding 
 
96 
 
 A First Book of the Lens. 
 
 portions of the discs being inversely proportional to the 
 area of the disc, or to the distance of the screen, or as 
 (l/acf : (I /ad) 2 . 
 
 This "law of inverse squares" governs all problems 
 relating to the intensity of the illumination produced by 
 a lens, but it is important to remember that the intensity 
 is only relative. We do not measure the actual intensity 
 at either screen, only the comparative intensity at the two 
 distances. Further, intensity is distinct from quantity. 
 The two screens receive equal quantities of light, but the 
 illumination is of different intensity. 
 
 Relative quantity may be represented by intensity 
 multiplied by area, thus the quantity of light passed by the 
 diaphragm is proportional to the intensity of the light at 
 the aperture multiplied by the area of the aperture ; that 
 is, to the square of the diameter of the aperture divided 
 by the square of its distance from the light. If we vary 
 both diameter and distance so as to preserve a constant 
 ratio between them, the quantity of light passed remains 
 constant. In the case of images projected by a lens under 
 varying conditions, if we estimate the relative quantities 
 of light passed by the lens, and divide those quantities by 
 the relative image areas, we arrive at the relative intensities 
 of illumination at the various images, and from these the 
 relative exposures required to produce a developable image. 
 Not the actual exposures, as they require, in addition, 
 measurement of the power of the light at its source, and of 
 the sensitiveness or speed of the plate. 
 
 64. Relative Intensity and Exposure. — The exposure 
 required to produce a developable image in a camera 
 varies inversely with the relative intensity of the light 
 received by the plate, and this intensity can be found in 
 the manner just described. 
 
 The amount of light transmitted by a lens from a certain 
 luminous object point varies inversely with the square of 
 the focal distance of the point, and directly with the area 
 of the lens effective aperture, or, the aperture being 
 circular, with the square of its diameter. Thus if u 
 represents the focal distance of the point, and k is the 
 diameter of the aperture, the light transmitted varies with 
 the value of k 2 /u 2 . The total quantity transmitted by the 
 
Scale, Intensity and Illumination, 
 
 97 
 
 lens from the whole of the object may be considered to be 
 governed by the same rule if we look upon the object as 
 consisting of a number of points. The size of the image 
 varies, under the laws of conjugate foci, with the relative 
 distances of image and object from the lens (see formula 
 1), or with the ratio of the focal distances v and u; and its 
 area varies with the value of v 2 /u 2 . Dividing this relative 
 area into the relative amount of light transmitted by the 
 lens, we have at the image a relative intensity of illumina- 
 tion equal to k 2 /v 2 . If then we represent the diameter of 
 the aperture in the form of a fraction of the focal distance 
 v or as v/a* then k 2 /v 2 = I /a 2 , and this is the usual and 
 most convenient method of representing the relative 
 working intensity of the illumination of the image. 
 
 Relative exposure being inversely proportional to relative 
 intensity, exposure is always proportional to the value of a 2 . 
 
 As k = v/a, it follows that a = v/k, varying directly with 
 the value of v and inversely with that of k. The distance 
 v depends upon the focal length of the lens and the scale 
 of the image, and it is important to understand clearly the 
 effect upon intensity and exposure of any alteration in one 
 of the three factors — size of aperture, scale of image, and 
 focal length. Assume that we know the correct intensity 
 and exposure for a certain object with a lens of certain 
 power and aperture, when photographing the object on a 
 certain scale. Under all conditions the intensity is pro- 
 portionate to the value of I /a 2 and exposure to that of a 2 , 
 but it is not always convenient to look at the matter solely 
 from that point of view. 
 
 Assume first that we vary the size of the aperture only • 
 such an alteration affects a in inverse proportion — that is, 
 if we halve the diameter of the stop we double a, and there- 
 fore increase the value of a 2 four times, hence the intensity 
 is reduced to a quarter, and the exposure increased four 
 times — therefore exposure varies inversely with the square 
 of the diameter of the aperture. 
 
 Second, suppose we alter the scale of the image, using 
 the same lens and aperture. An alteration in scale may 
 be brought about indirectly by altering the distance 
 
 * This fraction is sometimes called the working " angular aper- 
 ture," but I prefer the term ,; fractional diameter " used later on. 
 
 7 
 
98 
 
 A First Book of the Lens. 
 
 between lens and object and refocussing. This alters the 
 value of v, and as a varies directly with v, 1/a 2 varies with 
 1/v 2 , and therefore intensity is proportional to 1/v 2 , and 
 exposure to v 2 . Thus, if by bringing the object nearer to 
 the lens so as produce a larger image, the value of v is 
 increased from 7 to 10 ins. the relative exposures are in the 
 ratio of 49 : 100, or approximately 1 : 2. 
 
 We may, however, directly alter the scale, or the value of 
 v/v or R, and then a is still proportional to v, and v equals 
 (R + 1) F. As F is constant v varies with ^ + 1 and 
 the intensity varies with the value of l-±-(R + l) 2 , and the 
 exposure with (R + l) 2 . In the example given above, 
 assuming the focal length of the lens to be 4 ins., an 
 extension of 7 ins. would produce an image on a reduced 
 scale of |, and an extension of 10 ins. represents an 
 enlargement of 1| diameters. The exposures are then in 
 the ratio of (f + l) 2 : (1| + l) 2 or 49 : 100 as before. 
 
 Third, we may vary the focal length alone, retaining an 
 aperture of the same diameter and producing an image 
 of the same size. Here again a varies with v, while v 
 equals (R + 1) F ; and R being constant, v varies with 
 F. Therefore intensity varies with the value of 1/J* 2 and 
 exposure with F 2 . For example, if we know the exposure 
 required to produce a full size copy with a 6-in. lens and 
 | in. aperture, and want to know the exposure required 
 to make a full size copy with an 8-in. lens and | in. 
 aperture, the exposures are in the ratio of 36 : 64 or 9 : 16. 
 
 These rules with regard to intensity and exposure for 
 different sized images produced by one lens, or equal 
 sized images produced by different lenses, apply especially 
 to cases in which the object is so near the lens that the 
 distance v and u have to be adjusted strictly in accordance 
 with the laws of conjugate foci to secure sharp focus. 
 When the object is anywhere beyond a certain distance 
 from the lens, v equals f, and is not subject to variation. 
 The size of the image then varies inversely with the distance 
 of the object, but intensity and exposure for a certain 
 lens remain constant with a given aperture for any 
 distance beyond the minimum, while with different lenses 
 of equal aperture exposure varies with the value of 
 f 2 , whether equal sized images are produced or not. 
 
Scale, Intensity and Illumination. 
 
 99 
 
 Apparent variations from these rules, in the case of vary- 
 ing distances, are due to the effect of the intervening 
 atmosphere, which, being itself a luminous body, adds to 
 the apparent luminosity of the object and renders shorter 
 exposure desirable in the case of very distant objects. 
 
 It must be understood that however we vary scale 
 or focal length, exposure is uniform if we preserve a 
 constant value for a. That is, if we vary the size of 
 the aperture so as to preserve a constant ratio between 
 its diameter and the distance v. 
 
 Table D in the Appendix gives varying relative exposures 
 for different degrees of enlargement or reduction with 
 any one particular lens and aperture. 
 
 It should be observed that as with near objects an altera- 
 tion in scale necessitates alterations in the distances of object 
 and image, and such alterations with some lenses cause the 
 size of the effective aperture of the lens to vary, or to be 
 inconstant, the value of a may not always directly vary 
 with that of v, and error may thus be introduced, especially 
 when dealing with enlarged images. This error is probably 
 in most cases a negligible quantity and may not exist at all 
 with a doublet. The reversing of a lens with an inconstant 
 aperture will also cause intensity to vary to a very small extent. 
 
 It will be noticed that so long as we consider relative 
 exposures only, though so much depends on the intensity or 
 the value of a 2 , a knowledge of that value is not of much 
 practical importance. If we vary the aperture alone we 
 want to know simply the comparative diameters or areas 
 of the apertures ; if we vary scale we want the comparative 
 values of either v or r + 1 ; if we vary focal length we 
 simply want to know /. The value of l/a 2 , which may be 
 distinguished as the working intensity of the lens, is not 
 actually required, excepting for the purpose of estimating 
 the absolute exposure which serves as the basis of com- 
 parison, and to estimate which we must know not only 
 the working intensity of the lens, but also the speed of 
 the plate and power of the light. For the purpose of 
 calculating absolute exposure, and for that of comparing 
 the rapidity of different lenses, intensity must be known, 
 but it is customary to record only the nominal in- 
 tensity of the lens with each aperture. This nominal 
 
100 
 
 A First Book of the Lens. 
 
 intensity is the working intensity or value of 1/a 2 in the 
 particular case when v = f or when the lens is focussed 
 on a distant object ; but, usually, we record simply the 
 square root of the intensity, or the value of 1/a which is 
 the ratio of the diameter of the aperture to the focal length, 
 and may therefore be styled the nominal ratio aperture. 
 
 65. Marking Apertures. — In describing the manner in 
 which we arrive at the fraction 1/a 2 , representing the 
 working intensity of the lens for a particular aperture, 
 the actual diameter of the aperture was represented in 
 the form of a fraction of the focal distance of the image, 
 or as v/a. As v is a variable quantity, a (which may be 
 styled the ratio number) also varies, but in the particular 
 case when v is equal to f the diameter of the aperture is 
 equal to f/a. This may be styled the fractional diameter 
 of the apei-ture, to distinguish it from the absolute diameter 
 expressed in the form of a dimension. If we divide the 
 fractional diameter f/a by / we have 1/a, which is the 
 nominal ratio aperture, as described in the previous section. 
 Squaring the nominal ratio aperture we have 1/a 2 , which 
 is the nominal relative intensity with aperture f/a. The 
 reciprocal of the intensity, or a 2 , which is the square of 
 the ratio number, then represents the relative exposure. 
 Thus, the fraction f/a gives directly, or indirectly, the 
 nominal diameter, ratio, and intensity of the aperture, 
 and also the exposure. These are practically all the 
 particulars necessary, for if we know the nominal intensity 
 or exposure for any aperture, the working intensity or 
 exposure for a different value of v is readily found by 
 the rules previously given. Therefore, as a rule, all 
 apertures are marked either with their fractional diameters 
 or nominal ratio apertures. Under this system of marking, 
 an aperture that is equal in diameter to the 'focal length is 
 the unit aperture, possesses a unit intensity, and requires 
 a unit exposure ; as indicated in the following table, which 
 shows a series of apertures decreasing in diameter in 
 geometrical progression. 
 
 Fractional diameter //, //., 
 
 Ratio aperture ... 1 -\ 
 
 Relative intensity ... 1 \ 
 
 Relative exposure ... 1 4 
 
 f/\ f/s //l6 
 111 
 T ¥ TT 
 
 1 1 1 
 
 TIT 
 
 16 64 256 
 
Scale, Intensity and Illumination. 101 
 
 Instead of marking each aperture with its fractional 
 diameter, or its ratio aperture, we may mark it, either with 
 its relative intensity or its relative exposure. Both 
 systems of marking are in use, but the advantage of 
 marking the intensity is very problematical, while that 
 of marking the exposure is more apparent than real. If 
 either system is adopted it is evident an aperture of f/1 is 
 an inconvenient unit, as the numbers representing intensity 
 and exposure soon expand into formidable values. Taking, 
 for example, apertures f/8 and //16 : with //l as the unit 
 the intensities are as 1/64 : 1/256, and the exposures as 
 64 to 256. If we take f/8 as unity, then the intensities 
 are as 1:1/4 and the exposures as 1 : 4. Further, the 
 fractions representing intensity can be eliminated if we 
 take the smallest instead of the largest stop as the unit ; 
 thus, if //16 is the unit the intensities are as 4 : 1. In all 
 the recognised systems, a small aperture is the unit if 
 intensities are marked; and a large one if the apertures 
 are classified by exposure. 
 
 There are three systems of classification according to 
 exposure ; one known as the Uniform System (U.S.) of 
 the Royal Photographic Society, in which an aperture 
 of //4 is taken as requiring a unit exposure. Another 
 is the Dallmeyer System, in which /3 - 16 gives a unit 
 exposure. The third is the Continental or International 
 System, also known as the C.I. system, in which f/10 
 requires a unit exposure. The two former are now 
 practically obsolete, but the third is still in use. Then 
 there are two Zeiss systems, in which apertures are classified 
 according to intensity,// 100 having a unit intensity in the 
 "old" system, and f/50 in the "new" system. These 
 different systems with their varying units may cause a good 
 deal of confusion. For example, a stop marked simply 4 may 
 be under the U.S. system equal to f/8, or under Dallmeyer's 
 Systemy/6'3; under the Continental System //20, under Zeiss' 
 new system //25, and under the old system //50. It is neces- 
 sary to know what system is adopted before interpreting 
 the number. If the apertures are marked under the U.S. 
 system, the Royal Photographic Society recommend that both 
 the relative exposure number and the fractional diameter 
 should be given. This is equally desirable with any system. 
 
102 
 
 A Fi?'st Book of the Lens. 
 
 All these systems suffer through the adoption of fixed 
 unit apertures. If the actual apertures do not form part 
 of a series including the unit, or multiples of it, the 
 advantages of the system are nullified by the awkwardness 
 of the stop numbers. 
 
 All series of apertures should be arranged so that each 
 aperture is half the area of the next larger one, and therefore 
 requires double the exposure ; but, unfortunately, this rule 
 is not always observed. If the apertures are of unusual 
 sizes, it may not be apparent that they are in proper 
 progression, and then it is advantageous to mai-k them 
 with the relative exposures as well as their fractional 
 diameters, taking the largest, or most generally useful 
 aperture as the unit. A similar mode of marking is 
 desirable if the apertures are not in a proper order, and 
 the largest aperture is very often not in regular series 
 with the rest. The various methods of marking apertures 
 are given in the Appendix, Tables E to H, and rules for 
 finding numbers in Appendix I. 
 
 It can be easily understood that in all cases the effective 
 aperture (Sec. 9), not simply the diaphragm aperture, should 
 be recorded. The effective aperture varies in very ap- 
 proximately the same ratio as the diaphragm aperture, 
 and hence the latter may be sufficient if we only want 
 to consider aperture from a relative point of view ; but at 
 times the actual effective aperture is absolutely necessary, 
 and if the diaphragm aperture is substituted for it errors 
 must arise. Unfortunately, lens makers seem to have no 
 uniform system, some marking effective aperture and others 
 only the diaphragm aperture. The actual aperture should 
 be tested by the owner of a lens in the way that will be 
 described in the chapter on measuring lenses. 
 
 It may also be added that apertures are often marked in 
 a very approximate manner. Thus we frequently hear of 
 aperture /ll, but possibly such an aperture is never used. 
 It is usually/11-3, but may be /10-9, or /1M8. 
 
 66. Uniformity op Illumination. — Strictly speaking, 
 the intensity of illumination is greatest in the centre of 
 the image, and falls off more or less from the centre to the 
 edges of the plate. This lack of uniformity is quite apart 
 from the question of flare, and is due to several causes. 
 
Scale, Intensity and Illumination. 
 
 103 
 
 First, to the fact that oblique light produces less effect 
 upon the film than direct light, owing to there being a 
 greater loss by reflection from the film and from the lens 
 surfaces. Second, oblique pencils, passing through a lens 
 or circular aperture, are smaller in sectional area than 
 direct pencils ; they are of necessity narrower in primary 
 plane and therefore contain less light. Any of the diagrams 
 showing oblique pencils will illustrate this fact. Third, 
 oblique pencils are liable to be further reduced in area 
 owing to obstruction offered by the lens mount. The first 
 and second causes only will operate in the case of a single 
 
 lens mounted in an aperture in a plane screen ; the third, 
 if the lens is mounted in the usual way in a tube. 
 
 The falling off in uniformity of illumination owing to the 
 first two causes is gradual and continuous, but very slight 
 and practically negligible, excepting when pencils of very 
 great obliquity are in question. A mechanical limit is, 
 however, set to the employment of very oblique pencils by 
 the third cause of unequal illumination ; that is, by the 
 construction and mounting of the lens which produces 
 very marked effects within certain fixed limits. 
 
 67. Angles and Circles op Illumination. — In fig. 60 
 we show a doublet, and the pencils pq represent the most 
 oblique pencils that can pass through the lens without 
 
 P 
 
104 
 
 A First Book of the Lens. 
 
 being in any way obstructed by the lens mount. These 
 pencils are narrower than the direct pencil PQ, and there- 
 fore in the image there is a slight falling off in illumination 
 from Q to q owing to the second cause of inequality, and also 
 a slight falling off in effect due to the first. Disregarding 
 these two matters, however, the illumination from q to q 
 is as complete and regular as the construction of the lens 
 will permit, and the distance qq represents the diameter 
 of a circle of complete illumination which is subtended by 
 an angle of complete illumination qNq. 
 
 Light of greater obliquity can pass the lens, but the 
 third cause of inequality then operates. An extreme limit 
 is set to the illumination at the points q 2 , which can only 
 be reached by single theox-etical rays, and thus q 2 q% is 
 the diameter of a circle, and q^Nq 2 is an angle, of extreme 
 illumination. Between these angles of complete and 
 extreme illumination, there is one, q\Nq x , bounded by 
 pencils the sectional area of which is reduced one half by 
 the obstruction of the lens mount. This is the angle of 
 semi-illumination, and it subtends a circle of semi-illumina- 
 tion, the diameter of which is q\q x . It should be noted 
 that all three angles must be bisected by the principal axis 
 of the lens. 
 
 The falling off in illumination from q to q\ is at first 
 slight, but it increases very rapidly. At qi there is a 
 marked reduction of light, and beyond q 1 the loss is so 
 rapid and so great that little effect is produced on the 
 plate. Practically, all light effect ceases long before q 2 is 
 reached. To avoid apparent inequality of illumination in 
 the image, no part of it should be outside the circle of 
 complete illumination ; but, as the diameter of this circle 
 can be increased by reducing the aperture, it is possible to 
 secure satisfactory illumination over a fairly large plate 
 by sacrificing rapidity. From fig. 60 it is evident that 
 while the extreme angle must be constant with any 
 aperture, the use of an infinitely small aperture would 
 make all three angles coincide ; hence a reduction of 
 aperture increases the angles of complete and semi-illumi- 
 nation. With a single landscape lens with diaphragm in 
 front the angle of semi-illumination is practically constant 
 for any aperture, and a reduction of aperture increases the 
 
Scale, Intensity and Illumination. 
 
 105 
 
 angle of complete illumination and reduces that of extreme 
 illumination. 
 
 As the aperture cannot be reduced below a certain size 
 without inconveniently reducing the intensity, we cannot 
 with any particular lens increase the circle of complete 
 illumination beyond certain limits considerably short of 
 those of the largest circle possible with a minute aperture. 
 Thus, in fig. 60, with the smallest practicable aperture the 
 angle of complete illumination must fall short of the angle 
 q<iNq%. The extreme limiting angle can, however, be 
 increased by bringing the lenses closer together, and this 
 increases all the angles and converts a "narrow angle" 
 lens into a relatively "wide angle" one. With a single 
 landscape lens the angles can be similarly widened by 
 bringing the diaphragm closer to the lens. 
 
 68. Angle and Circle op Definition. — Just as a lens 
 has an angle and circle of complete illumination dependent 
 on its mechanical construction, so it has an angle and 
 circle of mean, or approximately uniform, definition, 
 dependent on the extent to which oblique pencils are 
 corrected for aberration, more particularly for curva- 
 ture and astigmatism. The angle of definition must 
 be bisected by the principal axis similarly to the other 
 angles. 
 
 Speaking generally, the size of the circle of definition 
 increases as the aperture is diminished, but much, of course, 
 depends on the nature of the uncorrected aberration and 
 the degree to which it is effected by the aperture. A 
 lens free from astigmatism and curvature will include a 
 wider angle of definition than an astigmatic or curved field 
 lens of the same aperture, and will include an equal angle 
 at a larger aperture. In fact, lenses might well be 
 classified in order of merit according to their angles of 
 perfect definition at particular apertures. The best lens 
 including the widest angle. 
 
 It may be noted that the shortening of the lens mount 
 to increase the angle of illumination will either introduce 
 aberration or render its correction more difficult, and the 
 angle of definition cannot be very large if aberration is 
 present. Obviously, uniform defining power is as important 
 as uniformity of illumination, but it is not so easily secured, 
 
106 
 
 A First Booh of the Lens. 
 
 and at wide angles it cannot be expected except in lenses 
 of the very highest class. 
 
 69. View and Covering Angles. — The view angle may- 
 be defined as the angle subtended by the diagonal of the 
 plate. This angle is only bisected by the principal axis 
 when the plate is centrally opposite the lens. The covering 
 angle may be defined as the angle subtending the diameter 
 of the smallest circle that will include the plate within its 
 limits; this angle always being bisected by the principal 
 axis. When the plate is centred opposite the lens the 
 view and covering angles coincide, but when the plate is 
 not central the covering angle is necessarily greater than 
 the view angle. 
 
 To secure even illumination and definition over the 
 whole surface of the plate the covering angle must not 
 exceed either the angle of complete illumination, or that 
 of definition. 
 
 It should be noted that while the angles of illumination 
 and definition limit the capabilities of the' lens, the terms 
 view and covering angles are only descriptive of the 
 conditions under which the lens is used. 
 
 From Table J in the Appendix, we can find the angle 
 covering a circle of definite diameter, and we can thus 
 determine either the covering angle or the view angle 
 included on a plate centrally opposite the lens. If the 
 plate is not centred the table will give the covering angle, 
 but will over estimate the view angle. Table K gives the 
 diagonals of plates of various sizes. 
 
 70. Wide and Narrow Angle Lenses. — While the terms 
 wide and narrow may be applied in a relative sense to any 
 of the angles that have been described, the particular term, 
 " wide angle lens " is generally used to convey the fact that 
 the lens is constructed in such a manner as to permit 
 the passage of very oblique light pencils j it thus applies 
 only to illumination. We may assume that if the diameter 
 of the circle of illumination is greater than the focal 
 distance of the lens, or the angle is greater than about 53°, 
 the lens has a wide angle ; if the diameter is less than 
 two thirds of the focal distance, or the angle is under 36°, 
 the lens is narrow angled ; and anything between the two 
 is a mid angle lens. Or we may put it this way : if the 
 
Scale, Intensity and Illumination. 
 
 107 
 
 focal distance of the lens is less than the diameter of 
 the circle the angle is wide ; if from one to one and a 
 half times the diameter, the angle is a mid angle ; if 
 beyond one and a half times the diameter, the angle is 
 narrow. 
 
 Other angles may also be conveniently distinguished as 
 wide, medium, or narrow, when similar proportions are 
 observed between focal distance and diameter of circle sub- 
 tended by the angle. 
 
 The angle of definition being independent of that of 
 illumination, while the two may be equal in a very narrow 
 angle lens, the angle of definition is always the smaller in a 
 
 
 ^ — " 
 
 — ?k 
 
 c I 
 
 7S- I 
 
 ol ; ; 
 
 -TV 1 
 
 fei • ; 
 
 
 
 
 
 £?■_ • 
 i 
 
 wide angle lens of similar aperture ; hence a small aperture 
 is necessary if we wish to take advantage of the wide angle, 
 or, in other words, if we wish to employ a large covering 
 angle. 
 
 A wide angle lens may be of any focal length, but it 
 is evident that if we wish to include a wide view angle on 
 a small plate, we must use a lens of comparatively short 
 focal length. For example, assume that we have a wide 
 angle 8 -in lens (L in fig. 61). A quarter plate ab will 
 subtend a narrow view angle, aNb ; a whole plate cd 
 a medium view angle cNd ; and a 12x10, gh, a wide 
 view angle gNh. The lens is of short focal length for the 
 12x10 plate, but of long focal length for the quarter plate. 
 
108 
 
 A First Book of the Lens. 
 
 To produce a wide angle view on the quarter plate we 
 should require a wide angle lens of less than 4 in. in 
 focal length. 
 
 Though the 8-in. lens will not include a wide view 
 angle on a quarter plate, the position of the plate may 
 necessarily produce a wide covering angle. If the rising 
 front is used so that the quarter plate is at bh, the view 
 angle bNh is still narrow, but the covering angle is 
 increased to gNh. The altered relative position of the 
 plate thus requires the insertion of a smaller diaphragm 
 to produce satisfactory illumination and definition with the 
 larger covering angle. Under these conditions we must 
 have a wide angle lens, even though we are only making a 
 narrow angle view, and as such a lens passes a' great deal 
 more light than is actually required, the image may suffer 
 a certain amount of degradation through this surplus light 
 being reflected on to the plate from the sides of the camera. 
 To prevent this effect it is advisable to intercept the 
 surplus light by means of a screen either in front of or 
 behind the lens. 
 
 71. Covering Power.— The catalogue statement that a 
 lens covers a certain angle or size of plate with a given aper- 
 ture is somewhat indefinite. Sometimes it evidently is only 
 true as regards illumination (not always complete illumi- 
 nation), and quite incorrect in respect to definition. 
 The modern anastigmats will generally be found to cover 
 with complete illumination and good definition the whole 
 surface of the plate they are listed to cover ; but this 
 is by no means the case with the cheaper and less 
 perfectly corrected lenses, and purchasers should be careful 
 to test the point. It should, however, be understood that 
 many cheap lenses produce excellent results with fair 
 rapidity if used only for narrow covering angles ; in fact 
 we may say that any lens will produce good results 
 within a certain limited angle, but with a poor lens the 
 angle is very small. To render this quite clear, a cheap 
 rectilinear may within an angle of 30° produce an image 
 equal^ in all respects to that formed by an expensive 
 anastigmat of the same rapidity; but increase the angle 
 to 50°, and while the rectilinear has to be severely stopped 
 clown, the anastigmat is just as efficient as before, and 
 
Scale, Intensity and Illumination. 109 
 
 it will probably remain so if the angle is increased to 
 70°. Emphasis is laid upon tbis matter, because many 
 seem to think that a cheap lens must be inferior to an 
 expensive one under all conditions, and colour is given 
 to this belief by the fact that the cheap lens will seldom 
 perform well at the angle that the catalogue states it 
 will " cover." 
 
 Some catalogues give the " covering angle " in degrees ; 
 some give the diameter of the circle subtended by the angle ; 
 and others only the size of the plate that can be included 
 in that circle. Occasionally, however, they give what they 
 call the angular aperture in degrees ; meaning, probably, the 
 angle of illumination. This is a misuse of a term which is 
 often used in quite a different sense ; see footnote to Sec. 64. 
 
CHAPTER VII. 
 
 DEPTH. 
 
 72. Depth. — Under the laws of conjugate foci, when a 
 point P (fig. 62) is at a certain distance u from the lens 
 L, an absolutely sharp image of that point can only be 
 obtained at the point Q, at a distance v from the lens; 
 and, conversely, if the focussing-screen is fixed at Q, P is 
 the only point that will be represented in absolutely sharp 
 focus. 
 
 It is, however, manifest that a focus such as Q may 
 be blurred to a certain slight extent, without that fact 
 becoming noticeable. Therefore, either the screen or the 
 object may suffer a certain amount of displacement without 
 the production of appreciable blur. For example, the 
 screen may be anywhere between, say, 5-1 and q 2 without 
 throwing the image of P obviously out of focus, and P may 
 be anywhere between, say, P x and P 2 and still be represented 
 with sufficient sharpness on a screen at Q. 
 
 The points P h P 2 limit depth of field, while q x , q 2 
 limit depth of definition, and it is necessary to clearly 
 understand the distinction between depth of field and 
 definition. 
 
 Depth of field gives the limits of all axially situated 
 object points that can be simultaneously represented in 
 approximate sharp focus, when P is sharply defined on a 
 screen at Q. 
 
 Depth of definition limits the permissible position of the 
 screen, if it is desired to keep P in approximate focus. 
 
 When photographing a solid object we consider depth of 
 field, but if the object is plane we mainly consider depth 
 of definition, which must obviously affect focussing, the 
 use of the swing back, and the scale of the image. 
 
 110 
 
Depth. 
 
 Ill 
 
 Depth of field is always considered relatively to the 
 position of P ; depth of definition relatively to Q, the 
 conjugate focus of P. 
 
 The points qi, q 2 are not conjugates to, nor have they 
 any definite relationship with Pi and P 2 , but still, depth 
 of field and definition are in a way connected. Assume 
 Qi and Q 2 to be the conjugate foci to Pi and P 2 , so that P 2 
 will be sharp with the screen at Q 2 , and Pi be in focus 
 with the screen at Q v If then P] and P 2 limit the 
 depth of field when we focus on P with the screen 
 at Q, it follows that Q must limit the front depth of 
 definition for Q 2 and the back depth for Q\. The points 
 Qi and Q 2 , though near, are quite distinct from the points 
 
 *k Lit 
 
 qi and q 2) which limit the front and back depth of definition 
 for Q. 
 
 The distinction between the terms " depth of field " and 
 "depth of definition " is not always strictly observed. The 
 latter term (and also " depth of focus ") is frequently 
 applied to matters that solely relate to depth of field, 
 which is also sometimes called " depth of distance." 
 
 Depth is a matter of importance in either direct or 
 oblique pencils, but only with the former is it possible 
 to arrive at practically useful formulae, unless we assume 
 the lens to be so perfect as to produce under all conditions 
 a plane image from a plane object ; in which case the 
 formulae applying to direct pencils cover oblique ones also, 
 if all measurements are taken on, or parallel to, the 
 principal axis. If curvature of the field exists, depth in 
 oblique pencils can only be considered in a generally vague 
 
112 
 
 A First Book of the Zens. 
 
 manner, and as it concerns the whole image or object, or 
 what we may call the focal volume of the image or object. 
 Thus there are four matters to consider — " depth of field," 
 " depth of definition," " definition volume," and " field 
 volume." 
 
 73, The Circle of Confusion. — The amount of any 
 kind of depth is dependent on the amount of out-of -focus 
 blur that may be deemed to be permissible. This varies, 
 under certain conditions dependent on the subject, size, 
 and purpose of the photograph, but, as a general rule, we 
 may allow T ^ in. as a maximum. This means that 
 the image of an infinitely small object point (or of an 
 infinitely thin line), is considered to be in adequately sharp 
 focus so long as its breadth does not exceed in., 
 or that sufficient sharpness is secured so long as the 
 image of any larger point is not widened to a greater 
 extent than T ^ in. That dimension may then be described 
 as the maximum diameter of the permissible circle or disc 
 of confusion that will adequately represent an infinitely 
 small point. 
 
 It should be remembered that if the image is subsequently 
 enlarged any out-of-focus effect is also enlarged, and 
 while an amount of confusion of in. may be of no 
 
 consequence in a direct print, an enlargement of four 
 diameters will show a blur of -Jv in. which may be 
 very detrimental. If an effect of extreme sharp focus is 
 desired, T ^ in. blur is too much in a small print, and 
 it may be reduced to in. with advantage. It may 
 further be noted that a certain amount of blur is inevitable 
 when objects at widely varying distances are included in 
 the one image, hence, in view of future enlargement, it is 
 important to ensure perfect focus on the most important 
 part of the subject. If this precaution is not taken it may 
 be found that in the enlarged image sharp definition 
 exists only in the less important features of the view. 
 
 For the sake of explaining matters connected with 
 depth we need only consider the effect of allowing a circle 
 of confusion of T ^ in. but it must be understood that this 
 factor is subject to modification. 
 
 74. Measurement of Depth. — The positions of the 
 points limiting depth may be denoted in several ways. 
 
Depth. 
 
 113 
 
 Consider, for example, the matter of depth of field as 
 illustrated in fig. 62, in which the total depth of field 
 for the point P (or the distance u) is denoted by the 
 distance P X P 2 . This may be divided into near depth 
 of field PP 2 , or far depth of field P X P (m 2 and mf). Or 
 we may fix P x and P 2 by taking their focal distances 
 from the lens, u 2 being the near focal depth and 
 Ui the far focal depth of field. Or, again, we may 
 take the extra-focal distance from F 2 , the front principal 
 focus of the lens ; P 2 F 2 (d 2 ) is then the near, and P X F 2 
 (cli) the far extra-focal depth. Sometimes one mode of 
 expression is more convenient than the other ; but, 
 generally, the focal depths may be considered as of first 
 importance for depth of field, while in the case of depth of 
 definition we need consider only the total depth q t q 2 , or 
 the front depth q x Q and the back depth Qq 2 . 
 
 It will be obvious that when ,we know u and its near 
 or far focal depth of field, the difference between the two 
 dimensions gives the near or far depth, while by deducting 
 the focal-length of the lens from the focal depth we get 
 the extra-focal depth. 
 
 75. Axial Depth of Field.— The factors that have 
 to be considered are four : the distance of the point P 
 on which we sharply focus, which may be expressed either 
 by the focal distance u, or the extra -focal distance d; 
 the focal length of the lens, expressed by /; the diameter 
 of the effective aperture, which may be expressed by its 
 nominal fractional diameter of f/a; and the diameter 
 of the disc of permissible confusion, which we assume 
 to equal T ^ in. (It is important to note that the 
 effective, and not the diaphragm aperture must be 
 taken.) With any particular lens and aperture the 
 last three factors are constants, and in the formula; they 
 appear invariably in one form, which may be termed the 
 depth constant of the lens for that particular aperture. 
 This depth constant is represented by H, and it is always 
 equal to 10U/ 2 /a, or to the focal length multiplied into 
 the diameter of the aperture and divided by the diameter 
 of the circle of confusion. Knowing the value of // 
 (see Appendix, Table L) for a particular aperture, depth 
 of field can be calculated by the aid of very simple 
 
 8 
 
114 
 
 A First Book of the Lens. 
 
 formulae that can be easily remembered. In exact 
 formulae we must employ both the focal distance u and 
 the extra-focal distance d, but when comparatively long 
 distances are in question the difference between u and d 
 may well be disregarded. 
 
 It should be noted that H varies directly with the 
 diameter of the aperture, and with the focal length, 
 while it varies inversely with the diameter of the circle 
 of confusion. 
 
 All the formulae relating to depth of field for a point 
 P at a focal distance u, or at an extra-focal distance d, 
 may be collected in a tabulated form as follows : 
 
 Depth op Field. 
 
 Neae Depths. Fae Depths. 
 
 Focal depth 
 
 ull 
 
 (23) 
 
 «i 
 
 uH 
 
 (24). 
 
 ** = M + d - 
 
 H-d "'• 
 
 Extra-focal depth 
 
 j d(H-f) 
 H + d 
 
 (25) 
 
 *i 
 
 d{H + f) 
 H-d 
 
 (26). 
 
 Depth 
 
 du 
 
 ???., = ... 
 
 (27) 
 
 
 du 
 
 (28). 
 
 2 H + d 
 
 H-d "' 
 
 The distances u%, U\ and u, are in harmonic progression, 
 and therefore — 
 
 i +■ - = - (29). 
 
 The depths m 2 and m 1; are connected by the formula 
 
 (30). 
 
 m., w, — u K 
 
 The application of these formulae is very simple. Suppose, 
 for example, we have a lens of 6 ins. focal length and 
 aperture/ 10. The depth constant = 100/ 2 /« = 360 ins. or 
 30 ft. If now we focus on a distance of 10 ft. the near 
 focal depth of field equals 360 x 1 20 -4- (360 + 114) = 7 ft. 
 7 ins. about, while the far focal depth of field = 360 x 
 120^(360 - 114) = 14 ft. 7| ins. about. From these 
 results it is evident, without applying formulae 27 and 28, 
 that the near depth of field is 2 ft. 5 ins. while the far 
 depth is 4 ft. 7^ ins. 
 
Depth. 
 
 115 
 
 The application of formulae 25 to 28 is equally simple, 
 but 23 and 24 are, in ordinary cases, the only ones required, 
 and they may be easily remembered. We may, however, 
 disregard the difference between it and d, if these dimen- 
 sions are large compared with /, and then we have the 
 still simpler formulae : 
 
 nH (31); and «, = ... (32). 
 
 - 1L -\- u 1 // 
 
 In words. The near focal depth of field is approximately 
 equal to the product of the distance into the constant, 
 divided by their sum ; and the far focal depth, the product 
 of the distance into the constant, divided by their difference. 
 Extra-focal distances may be substituted for focal distances 
 if desired in these approximate formulas, 
 
 Formula 29 is useful for an altogether different purpose. 
 Assume that we want to find the intermediate distance 
 on which we should focus between two known distances 
 Ui, and w 2 , both of which are to be equally well defined. 
 From the formula it is evident that 
 
 1 (33) ; 
 
 and assuming the subject to extend from 10 to 40 ft. 
 from the lens, the nearest and farthest objects are 
 equally well defined if we focus on a distance equal to 
 2 x 40 x 10-^(40 + 10) = 16 ft. 
 
 The confusion of the near and distant points will then 
 be equal to 1/100 in. if 
 
 „ = ioo/= U -^=p- or 100/= "> 7 (31). 
 
 If / is equal to 6 ins., a must be 11 '6, or the aperture must 
 not exceed //ll '6. 
 
 The last problem is somewhat unusual, and formula? 
 33 and 34 are seldom required. 
 
 In regard to the focal depths of field given in formulae 
 .23 and 24 there are two special cases to consider. If u 
 is infinite, only near depth of field exists, and the near 
 focal depth then is equal to H. 
 
 In the previous example II =* 30 ft. hence when we focus 
 on infinity the nearest object in approximate focus must 
 
116 
 
 A First Book of the Lens. 
 
 be at a focal distance of 30 ft. from the lens ; or all 
 objects beyond 30 ft. must be in focus. 
 
 The second case is this : from formula 24 it is evident that 
 if d = H then the far depth of field is infinite ; therefore, if 
 we focus on a focal distance u equal to H + f, all objects 
 beyond that distance are in focus. If now we substitute 
 H +f for ii and H for d in formula 23, we find that the 
 near focal depth equals \ (H + f), just half the distance 
 u. All objects beyond a distance of \ (H + f) are there- 
 fore in focus, and the depth of field ranges from \ (H + f) 
 up to infinity. This is the maximum amount of depth 
 of field possible, and H -\-f may be styled the distance of 
 maximum depth of field. If we measure this distance 
 extra-focally it is equal to H, and is sometimes called the 
 hyperfocal distance. The depth constant and hyperfocal 
 distance are quite distinct, though of the same value. 
 (Table K serves as a table of hyperfocal distance.) 
 
 It is sometimes found that depths calculated for a near 
 distance do not appear to be strictly accurate when 
 practically tested, even though the exact formulae are 
 used. This apparent error may be due to several causes. 
 The focal length and the effective aperture must both be 
 exactly known to find II, and very often neither is known 
 with sufficient accuracy. Further, the diameter of the 
 aperture may be correct only for a very distant object, and 
 not for a near one. This is a likely source of error with a 
 single lens. Then again it is not always easy to deter- 
 mine the exact distance of the point in sharp focus. All 
 these causes may contribute to produce a fairly consider- 
 able error when focussing on an object only a few feet from 
 the lens, and it is not advisable to trust too much to 
 calculated depth in such cases. Yery commonly the 
 apparent error is due to the fact that the object lies in a 
 portion of the field volume to which the calculated axial 
 depths do not apply. This will be again referred to. 
 
 It should be observed that the formulae given are true 
 only for axial depth of field in the case of a lens that 
 is capable of producing a perfect focus, and that they 
 represent the maximum possible amount of depth. Any 
 form of aberration that deteriorates the focus reduces 
 depth ; hence the better the correction of the lens the 
 
Depth. 
 
 117 
 
 greater is the depth. The exact effect of aberration on depth 
 cannot be allowed for in formula?, hence depth must be a 
 somewhat uncertain quantity in a poor lens. It is a common 
 error to suppose that depth is increased by the presence of 
 aberration ; the exact contrary is the case. The mistake is 
 due to the fact that if the lens will not produce perfect 
 definition under any conditions, we naturally make 
 allowance for a circle of confusion that is much larger than 
 yig- in. No formula? are applicable in such a case, and 
 the matter is purely one of personal judgment. The lack of 
 a critically sharp standard of definition aids to produce an 
 appearance of great depth that is very deceptive in a 
 small image. Enlargement will prove that this pseudo- 
 depth is only universal bad definition. A little considera- 
 tion will show that if a lens cannot represent a point 
 without a blur of T ^ in. there is absolutely no depth at 
 all, unless we allow a permissible disc of confusion larger 
 than Yjyjy in. 
 
 76. Consecutive Depths of Field. — A question that often 
 arises in regard to depth of field is this. If the nearest 
 and farthest points in focus are P 2 and Pi when focussing 
 sharply on P, will P remain in focus if we focus sharply 
 on either Pi or P 2 1 Or, in other words, does P limit the 
 near depth of field for Pi and the far depth of field for P 2 ? 
 As a matter of fact, what has already been said with regard 
 to the connection between depth of field and definition 
 shows that these conditions are not fulfilled ; but practically 
 they are very nearly so, and it is possible to select a 
 continuous series of distances in which each point shall 
 very approximately represent the near depth of field of the 
 next point on one side, and the far depth of field of the 
 next point on the other, the error in depth never exceeding 
 one focal length. For this purpose it is advisable to 
 measure all distances extra-focally, and the required con- 
 ditions are fulfilled, so long as any three distances are in 
 the order H-^(n — 1), H-r-n, H+-{n + 1) ; or by the con- 
 tinuous series. 
 
 Infinity, 77", lift, II/3, 77/4, ff/5, etc. ^ 
 or - 1(35). 
 
 Infinity, 100/ 2 /a, 100/ 2 /2tf, 100/ 2 /3a, 100/ 2 /4a, 100/ 2 /5«., etc.J 
 
 The truth of this can be tested readily. Assume 
 
118 A First Book of the Lens. 
 
 i 
 
 we focus on an extra-focal distance equal to H/u. 
 Substituting Hjn, for d in formulae 25 and 26 we find 
 that the near extra-focal depth is (H-f)-r-(n+ 1), and the 
 far extra-focal depth is (H+f) + {n — 1). If for the former 
 we substitute H -f- (n + 1), the result' is too great by an 
 amount equal to only f ~(n + l). If for the far depth we 
 employ H -4- (n — 1) the result is too small by /"-=- (n — 1). 
 Both errors are on the safe side and from their smallness 
 are of no possible consequence. 
 
 As an example suppose we have a 6-in. lens working at 
 /10. H then equals 360 ins. or 30 ft. Assume we focus 
 on a distance d equal to II /S or 10 ft. The near extra- 
 focal depth by the accurate formula is equal to (ff —/) ■— 
 ( n -j- 1) = (360 — 6) -5- (3 + 1) = 88^ ins. By the approxi- 
 mate formula // -r- (n+ 1) it equals 360/4 = 90 ins. There 
 is then an error on the safe side of 1| ins. Taking the 
 far depth by the accurate formula it equals (Z/+/)-r- 
 ( n — \) = (360 + 6) -7- (3 — 1) = 183 ins. By the approxi- 
 mate formula it equals H -4- (n — 1 ) = 360/2 = 180 ins. 
 The error is here 3 ins. on the safe side. The maxi- 
 mum possible error in near depth exists when d is infinite 
 and n = o, the error then being /. When focussing on 
 a distance H/2 the far depth is in error by one focal 
 length. The error would be greater than this if focussing 
 on a distance between H/2 and II, but such a distance is 
 not included in the series, and, anyway, the error is insig- 
 nificant compared with the distance of far depth, which 
 is infinite when focussing on H. 
 
 A series of distances of consecutive depth may very 
 conveniently be selected for dividing a focussing scale. 
 (Consecutive depths for an aperture of f8 and various focal 
 lengths are given in Table M, see Appendix.) 
 
 77. Variations in Depth of Field.— All lenses give 
 the same amount of depth of field for a given distance, so 
 long as their depth constants are the same ; but the greater 
 the constant the less the depth, and vice versa. Assuming 
 that in all cases we allow a circle of confusion of T ^ in., 
 then the value of H, which equals f ' 2 /ax 100, varies with 
 that of f 2 /a. If we reduce the aperture alone, keeping 
 / constant, a is increased, H diminished, and depth increased. 
 If we reduce the focal length, keeping//^, or the diameter 
 
Depth. 
 
 119 
 
 of the aperture, constant, // diminishes with f, and depth 
 increases. If we preserve the ratio aperture, or ]/a, 
 constant, and reduce /, H diminishes with f 2 , and depth 
 increases rapidly. 
 
 To preserve constant depth a must vary with f 2 ; that 
 is, if we double /, a must be multiplied by 4, which process 
 halves the diameter of the aperture. Here we have 
 assumed that the circle of confusion is of constant diameter. 
 Some have, however, advocated the varying of this factor 
 in proportion to the focal length of the lens, so that the 
 amount of confusion may vary more or less with the scale 
 of the image. This gives a constant value to the quotient 
 of the focal length divided by the diameter of the circle of 
 confusion, and leaves H to vary only with the diameter 
 of the aperture. Thus the depth remains constant for 
 any particular distance with an aperture of, say, 1 in., for 
 a lens of any focal length. To gain any idea of the actual 
 amount of depth we must, however, fix a certain amount 
 of confusion for a lens of particular focal length, and 
 calculate the depths with apertures of various diameters 
 for a certain scale of distances. 
 
 It can be easily understood that this system is incon- 
 venient if obvious confusion is undesirable in large scale 
 images, and extremely acute definition is not wanted in 
 small ones. If we permit in. confusion with a 16-in. 
 lens we can only allow in. with a 4-in. lens, and this, 
 perhaps, excessively fine definition is attained at the 
 expense of depth that might otherwise be taken advantage 
 of. While it may be desirable in pictorial work that 
 definition should vary somewhat with the scale, it is incon- 
 venient to vary it strictly in the ratio of the scale. The 
 circle of confusion and the depth constant should be varied 
 solely in accordance with the necessities of the work, 
 without following any arbitrary rule. 
 
 78. Axial Depth of Definition. — As with depth of 
 field, we must fix upon a maximum permissible amount 
 of confusion, say T ^ in., and we may also conveniently 
 adopt a definition constant to simplify the formulae. 
 
 Referring to fig. 62, front depth of definition is measured 
 by the distance q x Q and back depth by Qq 2 . Under all 
 circumstances front and back depth are equal, therefore 
 
120 
 
 A First Book of the Lens. 
 
 one formula applies to both. The factors affecting axial 
 depth of definition are ; — the focal distance of the true focus 
 from the lens, or the distance v, which is conjugate to u ; the 
 diameter of the circle of confusion, which we assume to 
 be jig- in. ; and the diameter of the lens aperture, which 
 may be expressed by its nominal fractional diameter of f/a. 
 If we divide the diameter of the circle of confusion by the 
 diameter of the aperture we obtain a definition constant h, 
 which has only to be multiplied by v to obtain the depth 
 of definition for any value of v. Thus depth of definition 
 is always equal to 
 
 It may be noted that if we allow the same amount of 
 confusion for either depth of field or depth of definition, 
 then the depth constant h is equal to the focal length of 
 the lens divided by the depth constant H. Thus h = fill 
 = a/100/. 
 
 Assume that we have a 5-in. lens with aperture /10, 
 or ^ in. in diameter. The depth constant // = 250 ins. 
 The definition constant = in. = in. Or, it equals 
 TBiF in. | = -Jg- in. If we focus on a distant object 
 so that v = /, the front or back depth of definition is equal 
 to ^ in. or '1-in. If we focus on a near object so that 
 v = 6 ins., the depth of definition = / ¥ or -12 in. 
 If we are copying full size v = 2/ and the depth is -2 in. 
 If enlarging on a scale of 4 to 1, v = 5/ and the front or 
 back depth of definition is 5 in., while the total depth 
 is 1 in. 
 
 It is manifest that the amount of depth varies directly 
 with the value of h. That is, inversely with the size of 
 the aperture, and directly with the size of the circle of 
 confusion. Thus, if in the above example we substitute 
 aperture/20 for/10 the total depth of definition, when we 
 enlarge four times, is increased to 2 ins. If, on the other 
 hand, we only allow -^hs ir) - confusion, the total depth 
 is reduced to ^ in. 
 
 79. Focussing. — When the depth of definition is great 
 there is obviously a difficulty in focussing accurately on a 
 given object. Take the example of enlarging four times 
 with a 5-in. lens and /10 aperture. If the aperture 
 
Depth. 
 
 121 
 
 cannot be increased the depth of definition is g-in. front 
 or back of the true focus. One hundredth of an inch is, 
 however, a very appreciable dimensiou, and we are not 
 likely to place the focussing-screen as far as | in. out 
 of position ; but if it is only g in. out the blur is only 
 T ^<5- in. and may escape notice, so the screen may be left 
 anywhere within ^ in. of the true position. If, then, 
 we employ a magnifying glass that multiplies twice, the 
 blur of -j-jj-jj in. is magnified to in. and becomes more 
 appreciable, so that we can then place the screen within, 
 say yg- in. of the proper position. A more powerful 
 magnifier will assist still further, but it should be noticed 
 that the use of a ground-glass screen greatly interferes 
 with the use of a focussing-magnifier. A clear glass screen 
 should be used and the magnifier should be focussed care- 
 fully on the front surface of the screen. 
 
 A very near approach to accuracy is obtained by 
 averaging the position of the screen. By racking in the 
 screen until a recognisable amount of blur is produced we 
 can find the limit of front depth of definition for a certain 
 amount of confusion. If then we rack out the screen 
 behind the true focus until we arrive at an equal amount 
 of confusion, the screen is at the limit of back depth, and 
 half-way between these two limits is the position for accurate 
 focus. This is the proper method of focussing under any 
 conditions. 
 
 As depth of definition varies inversely with the size 
 of the aperture, it is easier to focus on near objects with 
 a lai-ge aperture, than with a small one ; but if the lens 
 is not entirely free from longitudinal spherical aberration 
 in direct pencils the after reduction of the aperture for 
 exposure will throw the image out of focus. 
 
 80. Depth op Definition and Scale. — If a very small 
 aperture is used the depth of definition may be so increased 
 as to not only render it difficult to place the screen at the 
 one true focus, but to even render it immaterial, so far as 
 focus is concerned, whether the screen is there or not. 
 Suppose, for example, that we have a 5-in. lens with 
 an aperture of in. ; that is, //200, and that we 
 consider a confusion of hi- to be imperceptible. 
 
 The definition constant is then -f- A- = i, and when 
 
122 
 
 A First Book of the Lens. 
 
 the object is distant, so that the true value of v is f, we 
 have a total depth of definition of 2 ins. The plate may 
 then be anywhere between 4 and 6 ins. from the lens, the 
 latter position giving a larger image than the, former one. 
 Practically we can obtain the same effects with the 5-in. 
 lens as we could with either a 4- or 6-in. lens, so far as 
 scale and view-angle are concerned. It must, however, 
 be remembered that images produced in the extreme 
 positions will not bear much, if any, enlargement, no part 
 being in true focus. The question of depth of field does not 
 come in, as the extremely small aperture renders it practi- 
 cally unlimited. The use of minute apertures to increase 
 depth of definition to an extent sufficient to permit the 
 production of varying size images was apparently first 
 suggested by the Rev. F. 0. Lambert. 
 
 81. Definition Volume. — Assume that we have a lens 
 L (fig. 63) perfectly corrected in all respects and with an 
 absolutely flat field. If q u q 2 limit the axial depth of 
 definition for an object at P, then verticals a\bi and a 2 b 2 , 
 drawn through q± and q 2 , will limit the depths of definition, 
 bib 2 and a^a 2 , in oblique pencils A a and Bb coming from a 
 vertical object APB. If the object is plane and parallel 
 to the focussing-screen then a x bi and a 2 b 2 may be looked 
 upon as the traces of similar parallel planes which form the 
 boundaries of a space called the definition volume. No 
 part of the image will then be obviously out of focus if the 
 focussing-screen is anywhere within this definition volume. 
 Perfect focus is attained only in the position aQb, but 
 approximate focus is secured even if the screen is inclined 
 in the direction a^b 2 , or in that of a 2 b h though the image 
 is then distorted. 
 
Depth. 
 
 123 
 
 We do riot, however, often come across a lens with an 
 ideal definition volume such as that illustrated. If the 
 lens has a curved field, as indicated by the dotted curve 
 passing through Q in fig. 64, the boundaries of the definition 
 volume are curved ; and if the oblique pencils are astigmatic, 
 or in any way less perfectly corrected than direct pencils, 
 they have less depth of definition, and the definition volume 
 is so contracted that a section of it may take the form 
 shown by the thick curved lines. In this case we may have 
 a full amount of depth of definition on the axis which 
 cannot be taken advantage of, for if the plate is at q 2 the 
 
 
 vS. 
 
 \ 
 
 V 
 
 l\ 
 
 M 
 
 H 
 
 [r 
 
 
 
 I 
 
 7-* 
 A 
 
 ;! / 
 fa 
 
 
 points a and b are outside the definition volume and out of 
 focus. To bring them into focus the plate would have to 
 be at q 3 in front of Q, hence the depth of definition is 
 practically limited to q^, while the inclination of the 
 plate may not exceed that shown by the dotted lines. It 
 will be noticed that the angle of definition is affected by the 
 position of the plate. If at q, the angle cNc is at a 
 maximum ; if at q 2 the angle is almost nil, and at Q, dNd 
 is too small to include a and b. The best average definition 
 is obtained at q 3 , but in that position it is evident that the 
 best possible focus is secured about e, where the plate 
 intersects the curved field of mean focus. In the image 
 there is therefore likely to be a circular zone of superior 
 
124 
 
 A Fi?'st Book of the Lens. 
 
 definition concentric to the central axial point, which, in 
 order to compromise matters, has been thrown slightly out 
 of focus. This sort of compromise has often to be effected 
 with inferior lenses, but the reason for it must not be 
 confounded with that for adopting a similar method of 
 excentric focussing with some of the anastigmats, which 
 are so constructed as to produce, under any circumstances, 
 the best definition a little way from the centre. 
 
 It is obvious that as depth of definition is increased by 
 reducing the aperture, the definition volume is also increased. 
 A small aperture is invariably a remedy for deficiency of 
 any kind of depth. 
 
 82. Use of Swingback. — We have referred to the fact 
 
 that while the plate may be inclined from the vertical 
 within the definition volume without destroying focus, such 
 a proceeding produces distortion. Conventionally we look 
 upon all images as distorted when taken upon non-vertical 
 plates, even though they are in perfectly true plane 
 perspective. 
 
 When we tilt the camera to take in a high vertical 
 building, distortion is produced if the plate is not adjusted 
 to a vertical position. The upper part of the building is 
 then farther from the lens than the lower part, and its 
 image is nearer the lens, therefore the true image of the 
 whole lies on a plane inclined away from the lens in an 
 opposite direction to that of the object. This is illustrated 
 in fig. 65 where for convenience the principal axis of 
 
v 
 
 Depth. 
 
 125 
 
 the lens is kept horizontal while the object AB is tilted. 
 Tf the paper is turned so as to make AB vertical, it will 
 be seen that the diagram reproduces the effect of taking 
 a high vertical object with a tilted camera. The point B 
 in the object being nearer the lens than A, b in the image 
 is farther from the lens than a. 
 
 The lens being perfectly corrected, the definition volume 
 of the true image may be represented by the dotted lines 
 passing through q\, q 2 . Suppose now the plate to be in the 
 position cQc, at right angles to the principal axis. The 
 image on the non-vertical plate is distorted, but in fair 
 focus, the plate being inside the definition volume.* To 
 cure the distortion we must incline the plate in the 
 direction dd parallel to AB and therefore vertical, which 
 throws the plate outside the definition volume and out 
 of focus. To remedy distortion and preserve focus we 
 must reduce the aperture to a sufficient extent to increase 
 the depth of definition to q^q^. 
 
 This is an example of the use of the swingback to 
 obviate distortion. If, however, slight distortion is of no 
 consequence, we may leave the plate just within the 
 definition volume of the true image and so secure good 
 focus with a large aperture. A better remedy is to 
 increase the distance between the object and the lens, 
 this alters the slope of the true image, and brings the plate 
 within its definition volume. 
 
 83. Field Volume. — We have here only considered the 
 matter of definition volume as it applies to a plane object. 
 It should be understood that from a solid object a 
 lens projects a solid aerial image ; that is to say, if the 
 object is a cube the true image is a cube. If we 
 focus sharply on a given point the whole will be in 
 approximate focus if the plate is within the definition 
 volume of every separate part of the solid aerial image, 
 and then the whole of the object must of necessity be 
 within the field volume of the plane in sharp focus. The 
 field volume may vary in much the same way as the 
 definition volume, but it may be noted that if the lens 
 has a field with positive curvature for a plane object, 
 
 * This is the case in the diagram. In practice, part of the plate 
 might be outside the definition volume, and, therefore, out-of-focus. 
 
126 
 
 A First Boole of the Lens. 
 
 it has a plane field for an object that is concave to the 
 lens. Hence the near boundary of the field volume is 
 also concave to the lens, and objects lying well off the 
 principal axis may be within the field volume though 
 apparently they are too near the lens to be within the 
 depth of field measured axially. Thus in a group of 
 persons sitting in a semich-cle the outer ones may be 
 in focus though they appear nearer to the lens than 
 the axial point that is the nearest in focus. In other 
 words, there is greater near depth of field at the margins 
 of the view than in the centre. This is not unfrequently 
 the case, though too often any advantage in this respect 
 is counteracted by aberrations in the oblique pencils. The 
 form of the field volume, as of the definition volume, is 
 a variable feature, and it is only possible to obtain an 
 idea of it by experiment. In the ideal lens it should 
 be remembered both volumes are bounded by planes. 
 If bounded by curved surfaces we must be prepared to 
 find that the axial depth formulae give erroneous ideas 
 of the depth that is really available. Thus if the 
 boundaries are concave to the lens, far depth of field 
 and back depth of definition suffer, and if the boundaries 
 are convex to the lens, near depth of field and front 
 depth of definition are practically less than the formulae 
 appear to indicate. 
 
CHAPTER VIII. 
 
 FOCUSSING-SCALES. 
 
 84. Focussing-Scales. — Instead of focussing visually on 
 the ground glass screen, we may measure or estimate the 
 distance of the object and then adjust the screen by pre- 
 determined marks corresponding to particular distances. 
 A series of such marks form a focussing-scale, and there are 
 two matters for consideration in connection with such scales ; 
 first, the manner in which the positions of the scale marks 
 are determined ; and second, the particular distances that 
 they should represent. The simplest way to determine 
 the exact position of the screen for an object at a given 
 distance is to set up an object at that distance, focus 
 upon it, and record the position of the screen; but this 
 is only practically convenient when focussing on an infinite 
 distance, or on one that is easily measured. With inter- 
 mediate distances that are too great to be measured 
 and too small to be infinite, the extension of the camera 
 must be calculated and set off from some convenient 
 datum. 
 
 85. The Infinity Mark. — The most convenient datum 
 point from which the scale can be set out is the infinity 
 mark, which can be found as follows. Focus as carefully and 
 accurately as possible on some object at an infinite distance, 
 and then mark in any convenient fixed spot the exact 
 position of any moving portion of the camera that will 
 serve as an index point, then, whenever we adjust the 
 index to this infinity mark, the focussing-screen is at 
 the principal focus of the lens. It is manifest that if 
 we rack out the camera so that the index point is, say, 
 1 in. beyond the infinity mark, then the focussing-screen 
 
 127 
 
128 
 
 A First Book of the Lens. 
 
 is also 1 in. behind the principal focus. Therefore, any 
 distance measured from the infinity mark corresponds to 
 an extra-focal distance measured from the principal focus. 
 
 To find the infinity mark we may focus on the sun or 
 moon, but these are not generally convenient objects, and 
 we may as well select some terrestrial object at a finite 
 distance sufficiently great to render the error in the 
 position of the screen negligible. Having fixed on a per- 
 missible amount of error, the nearest object on which we 
 may focus is at an extra-focal distance equal to the square 
 of the focal length divided by the error. Assuming we 
 allow a displacement of the screen to the extent of T ^ in. 
 the nearest distance of approximate infinity is lOO/ 2 , 
 or if we allow only ^ in. error the distance is 250/ 2 , 
 and so on. If the infinity mark is to serve as the datum 
 of a scale, in. error is too great ; but it is easy with 
 short focus lenses to employ suitable objects at greater 
 distances which will give a much smaller error. Thus, 
 supposing we have a 6-in. lens and allow in. 
 error we need only focus on an object 100 yds. away ; but 
 it is just as easy to select one 300 or 400 yds. off and 
 so reduce the error to ^ or ^ in., or less. The 
 formula only serves as a guide to the nearest permissible 
 distance, and the farther we go beyond that distance the 
 better. 
 
 (Table N in the Appendix gives distances of appr oximate 
 infinity for lenses of various focal lengths.) 
 
 It will be of course evident that the infinity mark may 
 be found if we can place the focussing-screen at the 
 principal focus of the lens by any method other than 
 that of focussing on a distant object. 
 
 86. Distances Short op Infinity. — If we focus on a near 
 object at an extra-focal distance d, we shall have to rack 
 out the camera beyond the infinity mark for a distance 
 equal to the conjugate extra-focal distance x, and from the 
 formulae relating to extra-focal distances (see Sec. 33) 
 we know that x = f 2 /d. Thus, if with a 6-in. lens we 
 want to mark the scale for an object 6 ft. away from 
 the front principal focus of the lens, we set out from the 
 infinity mark a distance equal to f£ = a in., and if 
 then we adjust the moving index to that mark, an object 
 
Focussing-Scales. 
 
 129 
 
 6 ft. away is in focus. It must be noted that x varies 
 in inverse proportion with d ; hence if d is doubled x is 
 halved, and vice versa. Thus the mark for a distance of 
 12 ft, will be | in. from the infinity mark, and that for 
 a distance of 3 ft. will be 1 in. from the infinity mark. 
 If such a mark proves to be incorrect, either there is an 
 error in the focal length, or aberration in the lens is to 
 blame. A test should be made on a near object at a 
 carefully measured distance. 
 
 87. Selection of Distances. — The simplest possible form 
 of mechanical focussing is adopted in the "fixed focus" 
 camera, in which the lens should be fixed so as to produce 
 a sharp image of an object at the hyperfocal distance of 
 the largest diaphragm, and thus secure a maximum 
 amount of depth of field (see Sec. 75). In such a case 
 there is no focussing arrangement, all objects beyond a 
 certain minimum distance, which can be reduced by 
 diminishing the aperture, being always in focus. With a 
 focussing-camera the hyperfocal distance should always be 
 represented in the scale, and in many cases this distance 
 takes the place of the true infinity mark which is omitted 
 altogether. Practically the true infinity mark is not of 
 much use, as it is seldom desirable to confine the sharpest 
 definition to objects at an extreme distance ; but if the 
 hyperfocal distance takes the place of the infinity mark 
 that fact should be known to the user of the camera. 
 With different cameras there is unfortunately a great deal 
 of uncertainty as to which mark is really shown, and 
 sometimes it is to be feared that the extreme mark is 
 neither one thing nor the other, but represents an inter- 
 mediate distance, in which case it is of no use as a datum 
 for setting out other marks. A distance equal to 100/ is 
 sometimes selected, but this distance is quite meaningless 
 and represents nothing in particular. Assuming that we 
 show the infinity mark and also the hyperfocal mark, for 
 an aperture of f/a, then, as the hyperfocal distance, or H, 
 equals 100/ 2 /a, the two scale marks are separated by a 
 distance equal to f 2 /H = a/100. 
 
 Next the question arises as to shorter distances. These 
 may be selected so as to provide a convenient series of 
 dimensions, such as 5, 10, and 15 ft., but it is best to 
 
 9 
 
130 
 
 A First Book of the Lens. 
 
 adopt one of the more systematic arrangements that 
 follow. 
 
 First, we may select the hyperfocal distances for each 
 diaphragm or stop. Thus if with a 6-in. lens we have 
 stops /8, 11, 16, 22, 32, 44, we might take the distances 
 100/ 2 /8, 100/ 2 /ll, 100/ 2 /16, 100/ 2 /22, 100/ 2 /32, 100/ 2 /44, 
 which equal approximately 37' 6", 27' 3", 18' 9", 13' 7", 9' 4", 
 and 6' 9". Starting from the true infinity mark we should 
 then have a series of divisions at the following distances 
 from the datum, -08, -11, "16, '22, -32, and -44 in. The 
 mark for 27' 3" would have to be omitted as it comes 
 so close to the adjoining ones, and if we also omit the 
 true infinity mark, and start with H for the hyperfocal 
 distance of the largest aperture we shall have a scale 
 as follows : 
 
 /8 /1G /22 f62 fU 
 H 18' 9" 13' 7" 9' 4" 6' 9" 
 or, H 19' 14' 9J' 7' approximately. 
 
 This is a fairly convenient series, and we can with 
 any stop, except /ll, focus at once upon the distance that 
 gives maximum depth, or can tell what stop is required 
 to give maximum depth at one cf the distances. These 
 are advantageous conditions in hand-camera work, but 
 as all stops are not included it is advisable to mark each 
 division with both the distance and the aperture to which 
 it relates, as shown above. It should not be forgotten 
 that the exact distances are extra-focal, and that when 
 we focus on one distance with its own stop the depth of 
 field extends approximately from half that distance up 
 to infinity. 
 
 The second system of dividing the scale has certain 
 advantages over this last method. If we take a series of 
 consecutive depths of field for the largest aperture (see 
 Sec. 76) we can tell the depths of field at near distances, 
 and also the hyperfocal distances for every alternate 
 stop. Assume again that we have a 6-in. lens with a 
 series of stops /8, /ll, /16, /22, /32, /44, /64. Taking 
 the consecutive depths for /8 we have the following 
 series : 
 
Focussing -Scales. 
 
 131 
 
 Infinity, 100/78, 100/-/16, 100/724, 100/732, 100/-/40, 100/748, 
 100/-/56, 100/764 
 
 or, 
 
 Infinity, 37' fi", 18' 9", 12' 6", 9' 4", 7' 6", 6' 3", 5' 4J", 4' 8J" 
 /8 /16 /32 /64. 
 
 This series includes hyperfocal distances for/8, /lfi, /32, 
 and /64, as indicated, and while we can always take 
 advantage of the maximum depth of field for one of those 
 stops, we can also tell the actual depth given by one of 
 them while focussing on one of the marked distances. Thus 
 if we focus by the scale to 7 ft. 6 ins. ; with J8 the depth 
 extends from 6 ft. 3 ins. to 9 ft. 4 ins., being indicated 
 one division away on either side of the focus mark. If we 
 use yi6 the depths are marked two divisions away, thus 
 focussing on 9 ft. 4 ins. with /16 the depth extends from 
 6 ft. 3 ins. to 18 ft. 9 ins. With/32 we take four divisions 
 on either side and find the depth , is from 4 ft. 8 ins. up to 
 infinity. The number of divisions always varies with the 
 value of a in the fraction f/co, which is the fractional 
 diameter of the aperture. The scale divisions for such a 
 series of distances are equally spaced, each space being 
 equal tort/100, or '08 in. in above example. 
 
 It will be noticed that depths are only shown for half 
 the series of stops. We took fS as the basis of the scale 
 and therefore only /16, f32, and /64 can be provided 
 for. We might have taken /ll as the basis, and then 
 depths would have been correct for /22, /44, etc., and 
 not for/8, /16, etc. With a series of stops the diameters, 
 of which were //«, //2a, //3a, //4a, etc., depth would be 
 shown for every aperture, 1, 2, 3, or 4 divisions away 
 from the focus mark ; but stops are arranged in series 
 according to their areas, not their diameters, and so it is 
 impossible to provide for all. Intermediate divisions will, 
 however, approximately represent depths for intermediate 
 stops. It may be noted that /8, is about the largest 
 stop that can well be provided for ; the larger the initial 
 aperture the smaller are the divisions, and the scale 
 becomes confused if the divisions are too minute. It 
 is of course evident that the scale will serve with any 
 aperture for simple focussing purposes, though it may 
 not give correct depths. 
 
132 
 
 A First Book of the Lens. 
 
 With a scale of this kind it is best to record the stop 
 values on the moving portion of the camera, so that they 
 may come opposite the proper divisions when focussing 
 on a given distance. The scale may be arranged, as 
 shown in fig. 66. The portion marked A, which shows 
 distances, is affixed to the base board or non-moving part 
 of the camera; the part marked B to the moving portion. 
 The arrowhead on B is the index point, and the stop 
 values are marked on either side of it at proper distances. 
 
 B 
 
 
 
 A 
 
 1 1 1 1 1 1 1 1 1 1 1 
 
 
 By setting the index point to a certain distance we secure 
 sharp focus on that distance. If then we use one of 
 the particular stops marked we can read off the depth 
 of field at once. As shown, sharp focus is secured on 
 9§ ft., and with stop /8 depth extends from 7| to 12| ft. ; 
 with /1 6 from 6| to 19 ft. ; with /32 from 4| ft. to infinity. 
 This indicates that 9| ft. is the hyperfocal distance for ^32, 
 and we can always obtain the maximum depth with 
 any stop by setting the stop value against the infinity 
 mark. 
 
CHAPTER IX. 
 
 MEASURING LENSES. 
 
 88. Measuring Lenses. — In most calculations concern- 
 ing lenses, focal length and aperture are most important 
 factors, and to apply the results of such calculations it is 
 generally necessary to know the positions of one or more 
 of the cardinal points. Under the heading of measuring 
 lenses we may therefore include the determination of the 
 nodes, principal foci, focal length and aperture. 
 
 89. To Find the Principal Focus. — The most obvious 
 method is that described for ascertaining the infinity mark 
 for a focussing-scale. That is, to focus sharply upon an 
 object at an infinite or approximately infinite distance, 
 and note the position of the screen, which is then at the 
 principal focus. Another method, applicable within the 
 limits of an ordinary room is, however, sometimes useful. 
 For the glass focussing-screen substitute an opaque screen 
 with a small pinhole in the exact centre, and in front of 
 the lens hood fix a diaphragm with an aperture of known 
 dimensions, say \ in. in diameter. If a landscape lens is 
 used the lens diaphragm will serve, but with a doublet an 
 extra front diaphragm with an aperture smaller than that 
 of the lens is essential. Some distance in front of the 
 lens place a white screen, and behind the pinhole place a 
 strong light. A disc of light will then be projected on to 
 the screen. Adjust the distance between pinhole and 
 lens until the projected disc is exactly the same size as 
 the aperture in the front diaphragm. It is then evident 
 that the emergent pencil of light is parallel, and there- 
 fore that the pinhole is at the principal focus. If the 
 lens is well corrected for parallel light, the pinhole is 
 
 133 
 
134 
 
 A First Book of the Lens. 
 
 small, and the light intense, this method of finding the 
 principal focus is fairly accurate and easily carried out. 
 
 When a lens is employed in this manner for the purpose 
 of rendering the emergent light parallel it is styled a 
 collimator, and it is fairly evident that if we have a 
 camera, lens, and pinhole, so adjusted as to project a parallel 
 beam of light, we can use this arrangement to find the 
 principal focus of a second lens. Placing the second lens 
 in the path of the parallel beam a sharp image of the 
 pinhole will be formed at its principal focus. The colli- 
 mator then serves the purpose of an infinitely distant source 
 of light. Any camera and lens can be used as a collimator 
 at a moment's notice, if the true infinity mark is known. 
 
 90. To Find the Nodes. — If we know the principal 
 foci and the focal length of a lens we can of coarse find 
 the nodes by measuring back from the principal foci ; and, 
 conversely, if we know the nodes we can set out the focal 
 length from them to find the principal foci. If, however, 
 the focal length is unknown, the nodes must be inde- 
 pendently found by taking advantage of the fact that the 
 rotary centre coincides with the node of emission when 
 the incident light is parallel. To do this we must be able 
 to tentatively rotate the lens about various points on- its 
 principal axis until we find the rotary centre (fee Sec. 2G), 
 and the operation is not easy without apparatus. The 
 simplest method is to poise the lens on a sharp edge and 
 rotate it horizontally ; but strict accuracy requires the 
 aid of the more elaborate apparatus styled a tourniquet, 
 which is specially designed for the purpose. 
 
 91. To Measure Focal Length. — If we know the 
 principal foci and the nodes it is an easy matter to measure 
 the focal length, which is the distance between them ; but 
 we require the tourniquet to first find the nodes. Or 
 again, if we can find" the principal foci and the symmetric 
 foci (see Sec. 34), the difference between them is the focal 
 length ; but practically it is a very difficult matter to find 
 the symmetric foci without the aid of apparatus such as 
 Professor S. Thompson's focometer. We may guess at the 
 position of the nodes, which in a rectilinear are not far 
 from the diaphragm slot, or we may approximately find 
 the symmetric foci by tentative adjustment of the camera ; 
 
Measuring Lenses. 
 
 135 
 
 but, either way, an uncertain amount of error is introduced 
 which may vitiate future calculations. Methods of calculat- 
 ing focal lengths are therefore more generally useful than 
 any attempt at measuring it without the aid of proper- 
 apparatus. 
 
 92. Calculating Focal Length. — There are numerous 
 methods of calculating focal length without determination 
 of the nodes, and these methods may be divided into two 
 classes : the approximate, in which the fact that a lens 
 has usually two nodes is ignored ; and the theoretically 
 accurate, in which that fact is taken into consideration. 
 With some approximate methods the effect of this funda- 
 mental error is minimised, and, as in all cases we must 
 allow a certain margin of error, it sometimes happens that 
 a method that is not strictly correct in principle may yield 
 as nearly a correct result as another method that, though 
 theoretically correct, in practice favours the perpetration 
 of errors in determining the preliminary data. 
 
 Approximate Methods of Calculating Focal Length. 
 
 (a) Focus an object on a scale of equal size and divide 
 total distance from object to image by four. The result is 
 the focal length plus or minus one quarter the nodal space. 
 The object and image should each be at the symmetric 
 foci, which are very difficult to find; additional error is 
 therefore likely to be introduced by incorrect adjustment. 
 
 (6) The error in method a, due to ignoring the nodal 
 space, may be diminished by copying on a reduced scale. 
 Assume that the ratio of image to object, or the value of 
 E is \fn. Let D equal the total distance of image from 
 object, then the focal length = 
 
 The error due to the nodal space is then equal to plus or 
 minus the product of the nodal space into n-r-(n+ 1) 2 . Say 
 R = I or n = 3, then the error is the nodal space. 
 There is, however, with this method just as much difficulty 
 in copying to an exact scale as in method a. 
 
 (c) The difficulty of copying on an exact scale may be 
 got over by copying on any reduced scale an object of 
 known dimensions ; afterwards finding the value of n by 
 
136 
 
 A First Book of the Lens. 
 
 comparing image with object and then applying the 
 formula. In this case the object must be plane, and 
 parallel with the focussing-screen, otherwise fresh error 
 may be introduced. 
 
 (d) A method due to Mr. W. B. Coventry. Mark 
 extension of camera when an object at an infinite distance 
 is sharply defined, and thus find the infinity mark. Next, 
 focus on a near object and measure extension of camera 
 beyond infinity mark, calling this distance x. Finally 
 measure distance from object to image and call it B 
 Then 
 
 / = lJl)X - X (38) ; 
 
 or repeat operation with an object at a different distance, 
 and thus find two values for D {D and D x ) and two for x 
 (x and x\). . Then 
 
 , x(D - x) - mXD, - x.) 
 
 The error due to nodal space is very small if the value 
 of D is great, say over 10/, and this method is perhaps 
 the best approximate one. It, however, requires nearly 
 as many data as are sufficient to give a theoretically 
 correct result. 
 
 Theoretically Exact Methods. — With these, all dimensions 
 including the nodal space must be avoided, hence the 
 inclusive distance D cannot be employed. 
 
 (e) First find the infinity mark, then focus sharply on a 
 near object and measure the additional extension of the 
 camera beyond the infinity mark. Call this distance x. 
 Next focus on a still nearer object and again measure 
 the extension of the camera beyond the infinity mark, 
 calling this distance y. Let B equal the difference in 
 distance of the two objects. Then 
 
 Note ; if in focussing the lens is moved, B must be ascer- 
 tained by measuring in each case the distance of the object 
 from some particular part of the moving lens or camera- 
 front, and deducting one dimension from the other. If 
 on the other hand the screen is moved when focussing. 
 
Measuring Lenses. 
 
 137 
 
 and the lens remains stationary, we need simply measure 
 the distance between the two positions of the object. 
 
 (f) This method was especially devised by Mr. Dall- 
 meyer for use with his focometer. Focus sharply on a 
 near object, then move the object through a certain 
 distance I farther from the lens, refocus and note the 
 distance c through which the focussing-screen has been 
 moved. Next, move the object through a distance again 
 equal to I farther from the lens, refocus and note the 
 movement of the focussing-screen from its last position, 
 calling the distance a. Then 
 
 = *J 2l x a x c(a + c) 
 c — a 
 
 The lens must remain stationary during the different ad- 
 justments, hence this method is only applicable with a 
 camera in which Ave move the screen to focus. 
 
 (g) This is a very excellent method, also devised by Mr. 
 Dallmeyer. Eeverse the lens in its Uange on the camera, 
 focus accurately on an infinite distance, and measure the 
 distance from the ground glass to some definite point in the 
 lens mount, say the lens hood. Call this distance b. 
 
 Remove the lens, replace it in its ordinary position, 
 and find the infinity mark by focussing on a distant object. 
 The screen is now at the back principal focus of the 
 lens, while the other principal focus is at a distance b in 
 front of the lens hood. 
 
 Next, focus on a near object at a distance B from the 
 lens hood and measure the extension of the camera beyond 
 the infinity mark (the extra-focal distance x). Deducting 
 b from B we have the extra-focal distance of the object, and, 
 therefore, from formula 7 (Sec 33) 
 
 / = yP - b)x (42). 
 
 (h) This method is advocated by Mr. Chapman Jones. 
 Find infinity mark, then rack out and focus sharply on 
 a near object of known size. Find the value of R by 
 comparing size of image with that of object, and measure 
 extension of camera beyond infinity mark to find x, then : — 
 
 X 
 
138 
 
 A First BooTc of the Lens. 
 
 This method is applicable with any form of camera. 
 It is advantageous to use a divided scale as an object, and 
 to measure the image with the same scale. 
 
 (?') This is a modification of the last method. Focus 
 on a certain near object of known size, note the ratio 
 of image to object (R) and mark position of focussing-screen. 
 Hepeat with object at a nearer distance and note new 
 ratio r, and measure the additional extension of the camera, 
 calling the extra distance Z. Then 
 
 f-TTZ-r 
 
 In this method we dispense with the infinity mark. 
 
 (k) Comparative methods. — Focus on an infinitely distant 
 object, observe the size of the image, and then, with a 
 pinhole in the place of the lens, produce an image of equal 
 size from the same object. The distance from the pinhole 
 to the screen is the focal length of the lens. 
 
 Or, if we have a second lens of known focal length, we 
 can compare the size of the image produced by one lens 
 with that produced by the other when each is focussed on 
 the same infinitely distant object. The focal lengths 
 are then proportional to the sizes of the images. 
 
 (I) Mr. Thomas Grubb's method is worked out geo- 
 metrically. 
 
 At each side of the ground glass focussing-screen make 
 a pencil mark, keeping the two marks a certain fixed 
 distance apart, and equi-distant from the centre. Place 
 the camera fiat on a table covered with a sheet of white 
 paper and focus on a very distant object, say a steeple 
 or flag-staff, adjusting the camera so that the image exactly 
 falls on one of the pencil marks. Then, using one side 
 of the camera baseboard as a straight edge, rule a line 
 on the paper to mark the direction of the camera. Next 
 shift camera, altering its direction so as to bring the image 
 over the other pencil mark (it is not necessary to rotate 
 the camera about any paiticular centre) and again rule 
 a pencil line on the paper, using the same ruling edge 
 as before. This is all that need be clone with the camera, 
 which can now be removed. If necessary, produce the two 
 converging lines on the paper until they meet in a point 
 A, as shown in fig. 67. Bisect the angle BAC by the line 
 
Measuring Lenses. 
 
 139 
 
 AD and draw EDG at right angles to AD, making ED 
 equal to DG and the whole EG equal to the separation of 
 the marks on the focussing-screen. Through E and G 
 draw EK and GH parallel to AD and intersecting BA 
 and AC in K and H. Join JTF, cutting ^4Z) in F. The 
 focal length is then equal to AF. 
 
 (m) A modification of the previous method. Grubb's 
 method has the advantage of requiring very simple adjust- 
 ments of the apparatus, and error is not likely to be 
 introduced into the first part of the process if the object 
 is sufficiently distant and the lens is free from distortion. 
 The geometrical work is, however, very liable to error, and 
 
 the length of AF may be better calculated if the angle 
 BA C can be measured, for 
 
 AF = \EG x cot.\ BAC (45). 
 
 The length of EG is known, and the angle BAC can 
 be measured by a protractor, divided by two, and the 
 cotangent of the result found from a table of natural 
 tangents. 
 
 If the camera can be mounted on a pivoted arm that 
 moves horizontally over an accurately divided quadrant 
 circle, the angle BAC can be measured with great accuracy. 
 
 As regards the probable accuracy of the results obtained 
 by any of these theoretically accurate methods, all require 
 great care, while some are more liable to error than others. 
 
140 
 
 A First Book of the Lens. 
 
 Methods that involve the scale of the image are perhaps 
 most liable to error, for it is a difficult matter to find 
 the true ratio with extreme accuracy, and the presence 
 (perhaps unsuspected) of a small amount of diaphragm 
 distortion will upset the calculations. Other methods 
 depend solely on accurate focussing and careful measure- 
 ments, and a focussing eyepiece with clear glass screen 
 is advisable, while an accurately divided measure is in- 
 dispensable. 
 
 Grubb's method requires careful aiming of the camera 
 and accurate draughtsmanship, and it should be noted that 
 if the points K and H do not come exactly opposite one 
 another so that the line joining them is precisely at right 
 angles with AD there is an error somewhere in the draw- 
 ing. It is just the kind of diagram that many draughts- 
 men will fail in, and very great care is necessary. The 
 alternative method m has advantages if it can be carried 
 out conveniently. With any of these methods it is advis- 
 able to make two or three determinations of focal length 
 and average the results. 
 
 93. Measuring Effective Aperture of Lens. — In 
 Sec. 9 it was pointed out that the effective aperture of 
 a lens is represented by the diameter of the widest parallel 
 incident pencil that can pass the lens, and this is not 
 necessarily equal to either the diameter of the lens or of 
 the diaphragm aperture, though it varies with the latter. 
 
 The simplest method of measuring effective aperture is 
 to employ the lens as a collimator as described in Sec. 89. 
 If we place a screen with a pinhole at the principal focus 
 of the lens, and illuminate the pinhole by a powerful light 
 behind, a parallel beam of light is projected by the lens. If 
 this beam is received upon a screen the diameter of the 
 circular disc of light thus produced is the diameter of the 
 effective aperture. To ensure accuracy the screen should 
 be close to the lens. A piece of ground glass may be placed 
 in contact with the lens hood, or a piece of bromide paper 
 may be inserted in the lens cap, and the disc be measured 
 after development. 
 
 It is fairly evident that if a landscape lens with a front 
 diaphragm is used, then the effective aperture is that of 
 the diaphragm, but if a positive lens is in front of a 
 
Measuring Lenses. 
 
 141 
 
 diaphragm, as in a doubtlet or a reversed landscape lens, 
 the effective aperture is greater than that of the diaphragm. 
 If a, negative lens is in front of the diaphragm the effective 
 aperture is less than that of the diaphragm. 
 
 94. Inconstancy op Aperture. — From what has just 
 been said it is clear that the effective aperture of a lens, 
 and therefore its rapidity, may vary according to the 
 direction of the incident light. Probably the only forms of 
 lenses in which rapidity does not so vary are single lenses 
 without a diaphragm, and perfectly corrected positive 
 combinations with a diaphragm placed at the crossing- 
 point. 
 
 In fig. 68 we illustrate a single lens with a diaphragm on 
 the concave side, and therefore away from the crossing- 
 point. The width of a pencil ab which comes to a focus 
 
 
 C 
 
 It- 
 
 t- - --I--J 
 
 |t__^ ~* 
 
 f- a 
 
 at Qi is equal to that of the diaphragm aperture, but the 
 width of a pencil ccl which comes to a focus at Q 2 is much 
 greater ; hence the lens is more rapid, or has greater 
 intensity, if used with the diaphragm behind than it is 
 with the diaphragm in front. 
 
 Qi is the image of a point at an infinite distance, and the 
 pencil cd may be considered to be an emergent pencil 
 forming an infinitely distant image of the point Q 2 . Taking 
 this point of view it appears that by bringing the object 
 from an infinite distance to the point Q. 2 we have increased 
 the effective aperture from ab to cd, and, therefore, if the 
 object had been left at an intermediate distance, the 
 aperture would have been of an intermediate size. The 
 extreme sizes can be easily measured, intermediate ones 
 cannot without intricate calculations. 
 
142 
 
 A First Book of the Lens. 
 
 The formulae for estimating exposure or depth are 
 based on the assumption that ah is the constant aperture 
 for an object at any distance. This not being always 
 correct, the calculations are liable to error with near objects ; 
 exposure and depth both being overestimated, though 
 possibly only to an inappreciable extent in the case of the 
 former. 
 
 Under the Gauss theory we always represent the aperture 
 as situated on the nodes and we measure it on the nodal 
 planes, but make no allowance for variation. It would 
 appear that the effective aperture must be constant if the 
 diaphragm is situated at the crossing-point of a perfectly 
 corrected lens. Inconstancy may be introduced by aberra- 
 tion, and must exist if the diaphragm is situated away from 
 the crossing-point. 
 
CHAPTER X. 
 
 TYPES OF LENSES. 
 
 95. Spectacle Lenses. — The term spectacle lens is applied 
 to a simple single lens that by itself possesses every form 
 of aberration, and with which the stop alone is relied upon 
 as a means of correction. A meniscus lens (periscopic) 
 concave side to the object is generally used, as it lends 
 itself more readily to the production of a flat field, and 
 the aberrations of the oblique pencils are not very violent 
 as compared with direct pencils. On the other hand, 
 longitudinal spherical aberration in direct pencils is 
 considerable and has to be materially reduced by the stop, 
 by which also coma is eliminated and the field flattened. 
 At the best the angle of definition is narrow with any 
 particular aperture. Positive chromatic aberration remains, 
 while negative astigmatism, and inward or barrel distortion 
 are introduced, and though effects of blur can be diminished 
 by a small aperture these three aberrations cannot be 
 corrected. In the case, however, of the positive chromatic 
 aberration, as the visual yellow rays have a more distant 
 focus than the actinic rays that are especially concerned in 
 the formation of the image, the camera may be racked 
 in slightly after focussing, to place the plate in the position 
 of the actinic focus, for which purpose it is generally 
 necessary to rack in for about -J^ the focal length of 
 the lens. 
 
 Another method, due to Mr. Robert H. Bow, C.E., and 
 adopted in the Busch " Vade Mecum '"' casket lens, is to 
 use, when focussing, a positive supplementary lens that 
 will reduce the visual focal length by one fiftieth part. 
 After focussing this supplementary lens is removed, and 
 
 143 
 
144 
 
 A First Book of the Lens. 
 
 the normal focal length is restored, leaving the screen at 
 the actinic focus. 
 
 It should further be noted that when in ortho-chromatic 
 photography a coloured-screen is used to cut out the more 
 actinic rays, and a plate sensitive to yellow and red light is 
 employed, the presence of chromatic aberration may be 
 of little consequence. 
 
 By combining two single spectacle lenses to form a 
 doublet, distortion can be corrected, and also coma. Astig- 
 matism and chromatic aberration remain, but as the second 
 lens may perform a little of the work that otherwise has 
 to be done by the stop alone, a larger aperture can be used 
 and a certain amount of rapidity can be gained. 
 
 Such a doublet is frequently styled a periscopic doublet, 
 being formed of two single periscopic lenses. In this book, 
 however, the term periscopic is applied to doublets in a 
 different sense (see Sec. 15).* 
 
 96. Landscape, or Yiew Lenses. — Generally speaking 
 these titles are applied to single achromatic meniscus 
 lenses, in which a cemented negative compensating lens is 
 employed to secure achromatism and freedom from spherical 
 aberration in small direct pencils. Other corrections 
 depend on the stop, and hence, barrel distortion, astig- 
 matism and secondary residual aberrations remain; the 
 lens is slow and has a narrow angle of definition, and we 
 may interpret the terms " landscape" or " view" to rather 
 imply that the lens is not well adapted for any other 
 purposes. In this case, however, it is not fair to apply 
 the term to all single combinations, for it is possible by 
 careful selection of glasses and curvatures to produce a 
 single combination more or less free from aberration, and 
 differing from a good doublet of similar power only in 
 rapidity and covering angle. 
 
 It has before been pointed out that slight barrel 
 distortion is advantageous rather than otherwise for purely 
 landscape purposes, hence a lens producing this effect is 
 at times very useful, though for many subjects it is not 
 
 * There is a particular lens called the "Periscope," which is a 
 doublet combination of two single periscopic lenses. This was 
 produced by Steinheil in 1865, and seems to have recently again 
 come into use. 
 
Types of Lenses. 
 
 145 
 
 a serviceable lens. The presence of the other aberrations 
 is of course just as detrimental in landscape subjects as 
 in any others. 
 
 If we remove one of the combinations of a rectilinear 
 doublet we can use the remaining combination and stop 
 as a landscape lens, but, as a rule, it is inferior to the 
 purposely made article, for the stop being too close to 
 the lens, the aperture has to be greatly reduced to remedy 
 curvature of the field. Much, however, depends on the 
 importance of the part played by the stop in correcting 
 the doublet, and if the single lenses are well corrected 
 independently of the stop, either of them will make a 
 good single lens. 
 
 Special rectilinear and wide-angled single landscape 
 lenses are made, but these are not necessarily free from 
 astigmatism; on the other hand, an anastigmatic single 
 lens is generally very nearly rectilinear, and will usually 
 include a fairly wide angle. 
 
 97. Rectilinear Lenses. — By combining two single 
 achromatic lenses to form a doublet diaphragm distortion 
 can be corrected, but astigmatism remains with a slight 
 residue of other aberrations. The increase in the number 
 of lenses, however, gives opportunities of more perfectly 
 compensating the aberrations, and hence a larger aperture 
 can be used than with a single lens of the same focal 
 length. Further, the stop may be closer to the lenses 
 and so a wider angle of illumination can be included. 
 In the best form of rectilinear doublet, astigmatism is 
 practically the only form of aberration that is left, the 
 mean astigmatic field being either fiat or only slightly 
 curved. If the lens is of wide-angle construction a small 
 stop is necessary to flatten the field, and astigmatism 
 becomes rather serious towards the margins of the 
 plate. 
 
 It should be noted that, strictly speaking, the term 
 rectilinear simply implies that the lens is free from 
 distortion, and conveys no information with regard to 
 other corrections. In a popular sense, however, the term 
 is limited to doublets of the kind above described, though 
 single landscapes may be, and anastigmats are usually 
 also rectilinear. 
 
 10 
 
146 
 
 A First Book of the Lens. 
 
 A rectilinear (or, in fact, any other lens) may be styled 
 " rapid " if it works well at an aperture not less than 
 about /ll. 
 
 98. Anastigmatic Lenses. — Popularly this term is only 
 given to lenses that are practically free from all the 
 aberrations, but it may be applied to denote the special 
 correction of astigmatism. Thus an anastigmatic lens 
 may possess curvature of the field (see Sec. 55), and Ross' 
 Concenti-ic lens may be described as anastigmatic, it being 
 free from astigmatism and curvature, though longitudinal 
 spherical aberration is not compensated, and has to be 
 reduced by the stop alone, which fact renders the lens 
 slow. 
 
 The modern anastigmats (which are known by a 
 variety of special titles) are distinguished mainly by the 
 fact that spherical aberration is compensated at the same 
 time that astigmatism is corrected, and the stop plays 
 such a small part in the corrections that they can be 
 made very rapid. 
 
 Anastigmatic doublets are of two classes, with separable 
 and non-separable systems. In the former each system 
 is perfect, and therefore the doublet may be divided into 
 two single anastigmatic lenses, each of greater focal 
 length than the doublet. The latter class are only perfect 
 as a whole and cannot be divided. 
 
 99. Portrait Lenses. — A portrait lens need not essenti- 
 ally be different from any other lens, but it is necessary 
 that it should be very rapid for studio work, and that 
 it should be corrected more especially for short distances. 
 Some of the modern anastigmats appear to be equally 
 well corrected for any distance, and to be well adapted 
 for portraiture if rapid enough, but the term portrait 
 lens is generally applied to lenses in which the corrections 
 of oblique rays are to a large extent given up for the 
 sake of extreme rapidity. It is not necessary that they 
 should include a wide angle, in fact a very narrow angle 
 of fine definition is all that is required, and this permits 
 the use of a very large aperture, /3, or even larger. 
 If used for purposes other than portraiture the aperture 
 has to be reduced considerably, and then the portrait lens 
 is likely to prove inferior to the more usual type of 
 
Types of Lenses. 
 
 147 
 
 doublet. The very large aperture and the nearness of 
 the subject of course greatly reduces depth of field in 
 portraiture. (See also Sec. 50.) 
 
 The great rapidity of portrait lenses renders them 
 useful for some very high speed instantaneous work, 
 but only a narrow angle of definition is included, and 
 they are not suited for all subjects. 
 
 We are speaking here of only the ordinary type of 
 rapid portrait lens. The term has also been applied 
 to lenses that are well corrected for an aperture of fQ, 
 and that include moderate angles of definition. 
 
 100. Adjustable Lenses. — All the above types may be 
 described as fixed lenses, but there are others that may 
 be described as adjustable lenses. 
 
 One example of an adjustable lens is Dallmeyer's patent 
 portrait lens, in which by slightly unscrewing one portion 
 of the back combination a certain amount of spherical 
 aberration can be introduced into direct pencils. As before 
 explained, the portrait lens at a large aperture has a very 
 small angle of definition and little depth. Extreme sharp- 
 ness is therefore confined to the centre of the plate and to 
 one plane of the. object, and this effect is sometimes objection- 
 able. The introduction of spherical aberration destroys this 
 extreme local sharpness and renders the definition more 
 uniform and the depth apparently, but not really, greater, 
 (see pseudo-depth, Sec, 75). 
 
 Another adjustable lens is the Dallmeyer-Bergbeim 
 portrait lens. This is a narrow-angled doublet consisting 
 of a front positive and a back negative lens, with a 
 diaphragm in front of the positive lens. Each lens is a 
 simple single uncorrected lens. To a certain extent the 
 negative lens acts as a correcting or compensating lens, but 
 there is a considerable amount of uncorrected spherical and 
 chromatic aberration. This is left intentionally, as the 
 purpose of the lens is to produce a soft image free from 
 either objectionable blur or critical sharpness. The lens 
 is adjustable inasmuch as the separation of the negative 
 and positive lenses can be varied. This gives a considerable 
 range of focal lengths in the one lens. Eor example, the 
 No. 2 lens can be varied to give focal lengths of from 25 to 
 40 ins., and in this respect the Bergheim lens is similar 
 
148 
 
 A First Book of the Lens. 
 
 to the telephoto lens, which is the most important form of 
 adjustable lens. 
 
 101. Telephoto Lens. — This consists essentially of a 
 positive front, and negative back lens, with adjustable 
 separation, differing from the Bergheim lens in that each 
 system is, in itself, a completely corrected combination, 
 usually of the doublet form. This ensures good definition 
 under varying degrees of separation. The range of 
 varying foci is also much greater than with the Bergheim 
 lens. The peculiar advantage of the telephoto is one that 
 is shared by all forms of periscopic doublets in which the 
 front lens is positive and the back negative. The nodes 
 of the lens are so far in advance of the front lens that a 
 comparatively short extension of camera is required. If, 
 for example, we compare the action of a rectilinear of 
 3 ft. focal length with that of a telephoto of the same 
 power, formed of a 6-in. positive and 3-in. negative lens 
 separated by 3| ins., the former requires an extension of 
 the camera to at least 3 ft., the latter to only about 16 ins. 
 Each would produce the same scale image, but the greater 
 convenience of the telephoto lens is obvious. 
 
 Other peculiarities of the telephoto may similarly be 
 illustrated by comparison. With the rectilinear the nodes 
 Avould both be near the diaphragm slot and close together, 
 and the front principal focus of the lens would therefore be 
 about 3 ft. from the diaphragm slot ; further, the lens 
 might be of wide angle construction. With the telephoto 
 lens (which is essentially a narrow angle lens) the node of 
 emission would be 27| ins. in front of the positive lens, 
 and the node of admission 24| ins. beyond the node of 
 emission, while the front principal focus would be 36 + 
 24^ -f 27^ ins. or 7 ft. 4 ins. in front of the positive lens. 
 On account of this great distance of the front principal 
 focus, we can only focus with the telephoto on an object 
 that is apparently a considerable distance away, and the 
 great nodal space of the telephoto greatly increases the 
 distance between object and image. 
 
 Again; with the rectilinear we can only secure sharp 
 focus on a given distance by altering the distance between 
 the lens and the plate. With the telephoto we can focus 
 in the same way, but there is an alternative method, for 
 
Types of Lenses. 
 
 149 
 
 by altering the separation of the lenses we can alter the 
 focal length of the lens to suit the position of the plate. 
 It is recommended to focus first by varying the separation, 
 and to make the final adjustment by the camera focussing 
 arrangement. If we know the focal lengths of the positive 
 and negative lenses, and their separation, the focal length 
 of the whole combination and the positions of its nodes can 
 be determined by the equations given for combining lenses 
 in Sees. 41 to 43. 
 
 The separation is, however, a difficult quantity to deter- 
 mine, and the focal length can be approximately arrived 
 at as follows : — Secure sharp focus, then measure from 
 diaphragm of front positive lens to focussing-screen, and 
 multiply by the focal length of the positive lens divided 
 by that of the negative. This multiplier is a constant 
 for a particular lens ; thus in the example we have con- 
 sidered, it is equal to |- = 2. 
 
 It is frequently convenient to compare the action of a 
 telephoto lens as a whole, with that of its front positive lens 
 used alone, and as the telephoto is commonly used upon 
 objects at a great distance we may compare their actions 
 when working only at their focal lengths. In the example 
 considered we have a 6 -in. positive front lens, and by 
 placing a 3-in. negative lens 3| ins. behind, we have pro- 
 duced a telephoto lens of 36 ins. focal length, which acts 
 precisely as a 36-in. ordinary lens would "do. Therefore 
 it is evident that the telephoto lens will form an image 
 six times the linear dimensions of that produced by the 
 6-in. lens alone, and we may consider that the addition of 
 the negative lens has magnified the image six times. 
 Further, it is evident that this expansion of the image has 
 reduced the intensity of its illumination in inverse propor- 
 tion to its area. Thus, if the intensity with the 6-in. lens is 
 1/a 2 , it is l/(6a) 2 , with the telephoto. The ratio aperture 
 is therefore l/Qa with the telephoto if it is 1/a with the 
 6 in. ; and the fractional diameter of the aperture is f/6a 
 instead of f/a. Dividing the ratio aperture of the 6-in. 
 positive lens by the magnification given by the negative 
 lens, we obtain the ratio aperture of the telephoto lens. 
 Thus the aperture of the 6-in. lens being f&, that of the 
 telephoto is /48 if the magnification is six times. 
 
150 
 
 A First Book of the Lens. 
 
 The degree of magnification can be determined, without 
 knowing the focal length of the telephoto lens, by the 
 following exact rule given by Mr. Dallmeyer. 
 
 Having focussed, measure the distance of the negative 
 back lens from the focussing-screen, divide by the focal 
 length of the back lens, and add 1 to the result. 
 
 102. Casket Lenses. — These generally consist of a set 
 of single lenses of various powers with a mount into which 
 they can be screwed to form either single landscape lenses 
 or doublets of various focal lengths. The diaphragm 
 (generally of the iris variety) forms part of the mount, and 
 there is usually a little peculiarity in the manner in which 
 the various apertures are marked. If they are marked 
 with their fractional diameters it is obvious that these 
 diameters are only correct when employing certain lenses 
 of the set. Suppose a certain aperture to be marked /" 16, 
 this can only apply to a certain combination, which we 
 may assume to be of 8 ins. focal length. If we change the 
 lenses to produce a focal length of say 10 ins., the aperture 
 is no longer f 16 but about / 20. These variations being 
 confusing and misleading it is generally best to mark the 
 actual diameter of the diaphragm aperture, ignoring the 
 effective aperture. Thus if the aperture is | in. the 
 effective aperture is approximately /16 for a lens of 8 ins. 
 focal length and f20 for a 10-in. lens. Another method 
 is to simply put a reference number to the aperture, and 
 provide a table giving the approximate ratio aperture for 
 lenses of various focal lengths. 
 
 Casket lenses differ greatly in their degrees of correction, 
 and can be obtained with spectacle, achromatic, or ana- 
 stigmatic single lenses. It can be readily seen that one 
 particular combination of lenses is likely to be better cor- 
 rected than the others, as the stop cannot be in the best 
 possible position in all cases. 
 
 103. Convertible Lenses. — To a certain degree any 
 separable doublet is a convertible lens, but the term is 
 more generally applied to lenses that are specially designed 
 with a view to conversion. The Dallmeyer Stigmatic is a 
 good example of the type, being a crossed doublet made up 
 of two single combinations of different focal length. We 
 can either use the doublet complete, or we can convert it 
 
Types of Lenses. 
 
 151 
 
 into single lenses of very fine quality, and thus there are 
 three different focal lengths at our service. 
 
 Another example is one of the Cooke lenses. The ordinary 
 Cooke lens is a non-separable triplet, but with the special 
 type, one of the lenses of the triplet can be removed and 
 replaced by a different lens, which alters the focal length 
 of the whole combination without affecting its corrections. 
 
 104. Nomenclature of Lenses. — In many cases the 
 titles given to lenses are self explanatory, but_ special 
 names are sometimes used that do not clearly indicate 
 the character of the lens. 
 
 The type of lenses described as " Rectilinear," in Sec. 97, 
 includes' also the lenses known as " Aplanats" and " Eury- 
 scopes"; the latter being usually of extra rapidity. 
 Busch's " Pantoscope " is another lens of the same class, 
 but made of lenses of very deep curvatures and including a 
 very wide angle. 
 
 The " Anastigmatie " type, described in Sec. 98, includes 
 •not only the lenses known as " Anastigmats," but also the 
 " Collinear," the " Stigmatic," the " Orthostigmat," the 
 " Platystigmat," the " Cooke," the " Planar," the " Unar," 
 and the " Protar," the last being also known as the " Satz- 
 anastigmat," or " Convertible Anastigmat." Among these 
 the "Planar" is first and the "Unar" second as regards 
 rapidity. 
 
 Hoss' " Rapid " and " Universal Symmetrical " lenses 
 belong to the class of rectilinears, and must not be con- 
 fused with their " Universal Symmetrical Anastigmat." 
 Their " Concentric " lens, before referred to, may be looked 
 upon as a transitional type of lens. The " Antiplanet " is 
 similarly a transitional lens between Steinheil's " Aplanat" 
 and "Orthostigmat," being an attempt at producing an 
 anastigmatie lens with the old varieties of glass. 
 
 By " Petzval " lenses we generally mean portrait lenses, 
 made on the principles laid down by Petzval. Other 
 Petzval lenses are practically out of date, but the portrait 
 lens is still the standard type, though it has been modified 
 and slightly improved by Dallmeyer and others. 
 
APPENDIX. 
 
 (A). — Comparative Focal Lengths in Inches 
 and Centimetres. 
 
 INS. 
 
 1 
 
 CMS. 
 
 I NS. 
 
 CMS. 
 
 INS. 
 
 CMS. 
 
 •3937 
 
 1 
 
 9 
 
 
 
 22-86 
 
 19 
 
 
 
 48-26 
 
 1 
 
 2-54 
 
 9-054 
 
 23 
 
 19-685 
 
 50 
 
 575 
 
 
 
 
 20 
 
 OU O 
 
 2 
 
 5-08 
 
 9-842 
 
 25 
 
 21 
 
 53-34 
 
 2-362 
 
 6 
 
 10 
 
 25-4 
 
 21-653 
 
 55 
 
 2-756 
 
 7 
 
 10-236 
 
 26 
 
 22 
 
 55-88 
 
 3 
 
 7-62 
 
 10-63 
 
 27 
 
 22-44 
 
 57 
 
 3-149 
 
 8 
 
 11 
 
 27-94 
 
 23 
 
 58-42 
 
 3-543 
 
 9 
 
 11-022 
 
 28 
 
 23-622 
 
 60 
 
 3-937 
 
 10 
 
 11-416 
 
 29 
 
 24 
 
 6096 
 
 4 
 
 10-16 
 
 11-811 
 
 30 
 
 25 
 
 63-5 
 
 4-33 
 
 11 
 
 12 
 
 30-48 
 
 25-590 
 
 65 
 
 4-724 
 
 12 
 
 12-204 
 
 31 
 
 26 
 
 66-04 
 
 5 
 
 12-7 
 
 12-598 
 
 32 
 
 27 
 
 68-58 
 
 5-118 
 
 13 
 
 13 
 
 33-02 
 
 27-559 
 
 70 
 
 5-512 
 
 14 
 
 13-386 
 
 34 
 
 28 
 
 7112 
 
 5-905 
 
 15 
 
 13-779 
 
 35 
 
 29 
 
 73-66 
 
 6 
 
 15-24 
 
 14 
 
 35-56 
 
 29-527 
 
 75 
 
 6-299 
 
 16 
 
 14-567 
 
 37 
 
 30 
 
 76-2 
 
 6-693 
 
 17 
 
 15 
 
 38-1 
 
 31 
 
 78-74 
 
 7 
 
 17-78 
 
 15-748 
 
 40 
 
 31-496 
 
 80 
 
 7-086 
 
 18 
 
 16 
 
 40-64 
 
 32 
 
 81-28 
 
 7-48 
 
 19 
 
 16-533 
 
 42 
 
 33 
 
 83-82 
 
 7-874 
 
 20 
 
 17 
 
 43-18 
 
 33-464 
 
 85 
 
 8 
 
 20-32 
 
 17-716 
 
 45 
 
 34 
 
 86-36 
 
 8-266 
 
 21 
 
 18 
 
 45-72 
 
 35 
 
 S8-9 
 
 8-661 
 
 22 
 
 18-504 
 
 47 
 
 35-433 
 
 90 
 
 This table can also be used for the purpose of comparing the sizes 
 of English and Continental plates. 
 
 153 
 
154 
 
 Appendix. 
 
 ( B). — Corresponding Focal Powers and 
 Focal Lengths. 
 
 FOCAL POWERS. 
 DIOPTKIES. 
 
 FOCAL LENGTHS. 
 
 CMS. 
 
 INS. 
 
 •25 
 
 400 
 
 15748 
 
 •5 
 
 200 
 
 7874 
 
 •75 
 
 133 "3 
 
 52-49 
 
 1 
 
 100 
 
 39-37 
 
 1-25 
 
 80 
 
 3149 
 
 1-5 
 
 66 - 6 
 
 26-24 
 
 1-75 
 
 57'14 
 
 22-49 
 
 2 
 
 50 
 
 19-68 
 
 2 '25 
 
 44 "4 
 
 17-49 
 
 2-5 
 
 40 
 
 15-74 
 
 2-75 
 
 36*3(5 
 
 1431 
 
 3 
 
 333 
 
 13-12 
 
 3o 
 
 28-57 
 
 11-24 
 
 4 
 
 25 
 
 9-84 
 
 4-5 
 
 22-2 
 
 8-74 
 
 5 
 
 20 
 
 7-87 
 
 
 18-18 
 
 7-15 
 
 6 
 
 16-6 
 
 6-56 
 
 7 
 
 14 28 
 
 5-62 
 
 ' 8 
 
 12-5 
 
 4-92 
 
 9 
 
 11-1 
 
 4-37 
 
 10 
 
 10 
 
 3-93 
 
 11 
 
 9-09 
 
 3-57 
 
 12 
 
 8-3 
 
 3-28 
 
 13 
 
 7-69 
 
 3 02 
 
 14 
 
 7-14 
 
 2-81 
 
 15 
 
 6-b' 
 
 2-62 
 
 20 
 
 5 
 
 1-96 
 
 (C)— Scale Table of Conjugate Focal Distances 
 for a Lens of 1-in. Focal Length. 
 
 To use the following table find the required ratio in top horizontal 
 and left vertical columns. The conjugate distances in inches for a 
 lens of 1-in. focal length are then opposite the ratio numbers, the upper 
 value being that of the shorter conjugate. The table includes 
 ratios of from 1 : 25 to 25 : 1. Eatios between 1 : 10 and 10 : 1 are 
 expressed in multiples ; thus 1 : 4 is found under 3:12, and a scale 
 of equal size by the ratio 10 : 10. 
 
 Example : to enlarge in the ratio of 4 to 3 (or 12 to 9) ; under 9 
 and opposite 12, we have T75 and 2-3 as the lesser and greater 
 conjugates for lens of 1-in, focal length. The ratio of the con- 
 
Appendix. 
 
 155 
 
 jugates being the same for any lens, we have only to multiply 
 them by the focal length of the lens in use to find the actual focal 
 distances. Thus, with a 6-in. lens they will be 7f and 14 ins. 
 
 RATIO 
 
 1 
 
 2 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 in/ 
 10 | 
 
 11 
 LI 
 
 1-2 
 
 6 
 
 1-3 
 4-3 
 
 1-4 
 
 3-5 
 
 1-5 
 
 3 
 
 1- 6 
 
 2- 6 
 
 1- 7 
 
 2- 428 
 
 1- 8 
 
 2- 25 
 
 1- 9 
 
 2- i 
 
 2 
 2 
 
 ii / 
 11 1 
 
 1-09 
 12 
 
 118 
 6-5 
 
 1-27 
 4-6 
 
 1-36 
 3-75 
 
 1-45 
 3-2 
 
 1- 54 
 
 2- 83 
 
 1- 63 
 
 2- 571 
 
 1- 72 
 
 2- 375 
 
 1- 81 
 
 2- 2 
 
 1- 9 
 
 2- 1 
 
 TO f 
 
 12 ( 
 
 1-083 
 L3 
 
 1-16 
 7 
 
 1-25 
 5 
 
 1-3 
 4 
 
 1-416 
 3-4 
 
 
 1- 583 
 
 2- 714 
 
 1-6 
 
 25 
 
 1- 75 
 
 2- 3 
 
 1- 83 
 
 2- 2 
 
 i* f 
 13 | 
 
 1-077 
 14 
 
 1-154 
 7-5 
 
 1-231 
 
 53 
 
 1-308 
 4-25 
 
 1-385 
 3-6 
 
 1-462 
 316 
 
 1- 538 
 
 2- 857 
 
 1- 615 
 
 2- 625 
 
 1- 692 
 
 2- 4 
 
 1- 769 
 
 2- 3 
 
 14/ 
 14 | 
 
 1-071 
 15 
 
 1-143 
 
 8 
 
 1-214 
 5-6 
 
 1-286 
 4-5 
 
 1-357 
 3-8 
 
 1-428 
 33 
 
 
 1- 571 
 
 2- 75 
 
 1- 53 
 
 2- 875 
 
 1- 642 
 
 2- 5 
 
 1- 714 
 
 2- 4 
 
 11 f 
 15 \ 
 
 1-06 
 1G 
 
 1-13 
 
 8-5 
 
 
 1-26 
 4-75 
 
 
 
 1-46 
 3-143 
 
 
 
 if?/ 
 16 ( 
 
 1-062 
 17 
 
 1-125 
 9 
 
 1-187 
 6-3 
 
 
 1-312 
 4-2 
 
 1-375 
 3-6 
 
 1-437 
 3-286 
 
 
 1- 562 
 
 2- 7 
 
 1- 625 
 
 2- 6 
 
 17/ 
 17 ( 
 
 1-058 
 18 
 
 1-117 
 
 9-5 
 
 1-176 
 6-6 
 
 1-235 
 5-25 
 
 1-294 
 4-4 
 
 1-352 
 3-83 
 
 1-41 
 3-428 
 
 1-47 
 3-125 
 
 1- 529 
 
 2- 8 
 
 1- 588 
 
 2- 7 
 
 ia/ 
 18 | 
 
 1-05 
 19 
 
 ii 
 
 10 
 
 
 1-2 
 
 55 
 
 1-27 
 4-6 
 
 
 1-38 
 3-571 
 
 1-4 
 
 3-25 
 
 
 1- 5 
 
 2- 8 
 
 1Q f 
 
 19 | 
 
 1052 
 20 
 
 1-105 
 10-5 
 
 1-157 
 7-3 
 
 1-21 
 5-75 
 
 1-263 
 4-8 
 
 1-316 
 4-16 
 
 1-368 
 3-714 
 
 1-42 
 3-375 
 
 1-474 
 3l 
 
 1- 526 
 
 2- 9 
 
 20 | 
 
 1-05 
 21 
 
 
 1-15 
 7-6 
 
 
 
 
 1-35 
 3-857 
 
 
 1-45 
 
 3-2 
 
 
 n{ 
 
 1-047 
 22 
 
 1-095 
 11-5 
 
 
 1-19 
 
 6-25 
 
 1-238 
 5-2 
 
 
 
 1-38 
 3-625 
 
 
 1-476 
 31 
 
 22 { 
 
 1-045 
 
 23 
 
 
 1136 
 
 8-3 
 
 
 1-227 
 5-4 
 
 
 1-318 
 4143 
 
 
 1-409 
 3-4 
 
 
 23 | 
 
 1-043 
 24 
 
 1-086 
 12-5 
 
 1-130 
 
 8-6 
 
 1-173 
 6-75 
 
 1-217 
 5-6 
 
 1-260 
 4-83 
 
 1-304 
 4-286 
 
 1-347 
 3-875 
 
 1-391 
 3-5 
 
 1-43 
 3-3 
 
 24 { 
 
 , 1-041 
 25 
 
 
 
 
 1 -208 
 5-8 
 
 
 1-291 
 4-428 
 
 
 
 
 25 -j 
 
 1-04 
 26 
 
 1-08 
 13-5 
 
 1-12 
 9-3 
 
 1-16 
 7-25 
 
 
 1-24 
 5-16 
 
 1-28 
 4-571 
 
 1-32 
 4-125 
 
 1-36 
 3-7 
 
 
156 
 
 Appendix. 
 
 (D).— Relative Exposures foe. Different 
 Scales of Image. 
 
 RATIO 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 l.{ 
 
 1-21 
 121 
 
 1-44 
 
 36 
 
 1-69 
 18-7 
 
 1-96 
 1225 
 
 2-25 
 9 
 
 2-56 
 
 7-i 
 
 2-89 
 5-89 
 
 324 
 
 0 Uu 
 
 3-61 
 
 4 
 4 
 
 11 { 
 
 119 
 144 
 
 1-39 
 42-25 
 
 1-62 
 21-7 
 
 1-86 
 14-06 
 
 2-11 
 1024 
 
 2-38 
 8 
 
 2-67 
 6-61 
 
 2-98 
 5'64 
 
 3-3 
 
 3-63 
 
 U { 
 
 1-17 
 169 
 
 1-36 
 49 
 
 1-56 
 25 
 
 1-7 
 16 
 
 2 
 
 11-56 
 
 
 2-5 
 7 - 36 
 
 2-7 
 6'25 
 
 3 06 
 5-4 
 
 3-36 
 
 A ■ QA 
 
 »{ 
 
 1-16 
 
 196 
 
 1-33 
 56-25 
 
 1-51 
 28-4 
 
 1-71 
 18 
 
 ryi 
 
 12-9 
 
 2-13 
 10 
 
 2-36 
 8 - 16 
 
 2-6 
 6 - 8S 
 
 2-86 
 5-97 
 
 3-13 
 
 u{ 
 
 114 
 
 225 
 
 1-3U 
 64 
 
 1-47 
 32 
 
 1-65 
 20-25 
 
 1-84 
 14-4 
 
 2-03 
 111 
 
 
 2-4(5 
 7-56 
 
 2-69 
 6 - 52 
 
 2-93 
 
 O i D 
 
 u{ 
 
 1-13 
 
 256 
 
 1-28 
 72-25 
 
 — 
 
 1-58 
 22-5 
 
 
 
 2-14 
 9-87 
 
 2-35 
 8-26 
 
 
 
 u{ 
 
 112 
 
 289 
 
 1-26 
 81 
 
 1-4 
 
 40 
 
 — 
 
 1-72 
 17-6 
 
 1-89 
 13-4 
 
 206 
 107 
 
 
 2-43 
 7 - 7 
 
 2-64 
 6'76 
 
 »{ 
 
 1119 
 324 
 
 1-24 
 
 90-25 
 
 1-38 
 44-3 
 
 1-52 
 27-5 
 
 1-67 
 19-3 
 
 1-82 
 14-6 
 
 1-98 
 11-7 
 
 2-16 
 9-76 
 
 2-33 
 8 - 34 
 
 2-52 
 
 7-OQ 
 
 
 1-112 
 361 
 
 1-23 
 100 
 
 — 
 
 1-49 
 30-2 
 
 1-63 
 21 -1 
 
 
 1-92 
 127 
 
 2-08 
 10'5 
 
 
 2-41 
 7'84 
 
 » { 
 
 1-106 
 400 
 
 1-22 
 110-25 
 
 1-33 
 53-7 
 
 1-46 
 33 
 
 1-59 
 23 
 
 1-73 
 17-3 
 
 1-87 
 13-7 
 
 201 
 11-3 
 
 2-17 
 9'67 
 
 2-32 
 
 20 | 
 
 1-102 
 441 
 
 — 
 
 1-32 
 58-6 
 
 — 
 
 
 
 1-82 
 14-8 
 
 
 2-10 
 10-3 
 
 
 n { 
 
 1096 
 484 
 
 1-19 
 132-25 
 
 
 1-41 
 
 39 
 
 1-53 
 27 
 
 
 
 1-90 
 13-1 
 
 
 2-17 
 961 
 
 22 | 
 
 1-092 
 
 529 
 
 
 1-29 
 69-3 
 
 
 1-5 
 29-1 
 
 
 1-73 
 17-1 
 
 
 1-97 
 11-8 
 
 
 23 | 
 
 1-087 
 576 
 
 1-17 
 
 13625 
 
 1-27 
 749 
 
 1-37 
 45-5 
 
 1-48 
 31-3 
 
 1-58 
 23-3 
 
 1-70 
 18-3 
 
 1-81 
 15 
 
 1-93 
 12-6 
 
 2-05 
 10-8 
 
 24 | 
 
 1083 
 625 
 
 
 
 
 1-45 
 33-6 
 
 
 1-66 
 19-6 
 
 
 
 
 25 | 
 
 1-081 
 676 
 
 1-J6 
 
 182-25 
 
 1-25 
 87 
 
 1-34 
 52-5 
 
 
 1-53 
 26-6 
 
 1-63 
 20-8 
 
 1-74 
 17 
 
 1-84 
 14-2 
 
 
Appendix. 157 
 
 This table gives relative exposures for the various scales provided 
 for in table 0. In each case the upper figure is the relative ex- 
 posure for reduction, and the lower figure that for enlargement. 
 As an example, suppose we know the required exposure for enlarging 
 on a scale of 1 to 2 (or 5 to 10), and want the exposure for reducing 
 on a scale of 5 to 3 (or 10 to 6) ; from the table the relative 
 exposures are as 9 : 2-56 ; we must therefore divide our known 
 exposure by 9, and multiply the result by 2-56. 
 
 (E, F, G, H).— Systems of Marking Apertures. 
 
 The rapidities of the different apertures included in these tables 
 can be compared by the relative exposure numbers based on a unit 
 exposure for /l as given in the second column. 
 
 Classification by Exposure. 
 
 (E). — Uniform System. 
 
 APERTURE 
 RATIO NO. 
 
 EXPOSURE NOS. 
 
 APERTURE 
 RATIO NO. 
 
 EXPOSURE NOS. 
 
 UNIT fl. 
 
 U.S. 
 UNIT f 4. 
 
 UNIT fl. 
 
 U.S. 
 UNIT /4. 
 
 1 
 
 1 
 
 1/16 
 
 11-312 
 
 128 
 
 8 
 
 1414 
 
 2 
 
 1/8 
 
 16 
 
 256 
 
 16 
 
 2 
 
 4 
 
 1/4 
 
 22-624 
 
 576 
 
 32 
 
 2-828 
 
 8 
 
 1/2 
 
 32 
 
 1024 
 
 64 
 
 4 
 
 16 
 
 1 
 
 45-25 
 
 2304 
 
 128 
 
 5-656 
 
 32 
 
 2 
 
 64 
 
 4096 
 
 256 
 
 8 
 
 64 
 
 4 
 
 90-5 
 
 9216 
 
 512 
 
 This Table includes only the series of apertures recommended by 
 the Royal Photographic Society. 
 
158 
 
 - Appendix. 
 
 (F). — Dallmeyek System. 
 
 APERTURE 
 RATIO NO. 
 
 EXPOSURE NOS. 
 
 APERTURE 
 RATIO NO. 
 
 EXPOSURE NOS. 
 
 UNIT / 1. 
 
 DALLMEYEE 
 UNIT /3-16. 
 
 UNIT /l. 
 
 DALLMEYER 
 UNIT /S-16. 
 
 1 
 
 1 
 
 ■1 
 
 10 
 
 100 
 
 10 
 
 2 236 
 
 5 
 
 •5 
 
 12-25 
 
 150 
 
 15 
 
 2-74 
 
 7-5 
 
 •75 
 
 1414 
 
 200 
 
 20 
 
 3-162 
 
 10 
 
 1 
 
 15-8 L 
 
 250 
 
 25 
 
 3-87 
 
 15 
 
 1-5 
 
 17-32 
 
 300 
 
 30 
 
 4-472 
 
 20 
 
 2 
 
 20 
 
 400 
 
 40 
 
 5 
 
 25 
 
 2-5 
 
 22-36 
 
 500 
 
 50 
 
 5-47 
 
 30 
 
 3 
 
 27-38 
 
 750 
 
 75 
 
 5'9 
 
 35 
 
 3-5 
 
 31-62 
 
 1000 
 
 100 
 
 6-32 
 
 40 
 
 4 
 
 38-73 
 
 1500 
 
 150 
 
 7-U7 
 
 50 
 
 5 
 
 44-72 
 
 2000 
 
 200 
 
 7-42 
 
 55 
 
 5-5 
 
 50 
 
 2500 
 
 250 
 
 8-06 
 
 65 
 
 6-5 
 
 54-77 
 
 3000 
 
 300 
 
 8-66 
 
 75 
 
 7-5 
 
 63-24 
 
 4000 
 
 400 
 
Appendix. 
 
 159 
 
 (G). — International Congress (C.I.) System. 
 
 w 
 N 
 
 APERTURE 
 RATIO 
 NO. 
 
 EXPOSURE NOS. 
 
 SERIES. 
 
 APERTURE 
 RATIO 
 NO. 
 
 
 
 Exposur 
 
 E NOS 
 
 UNIT /l. 
 
 C.I. 
 UNIT f 10. 
 
 UNIT f\. 
 
 C.I. 
 UNIT /10. 
 
 
 
 
 1 
 
 1 
 
 •01 
 
 7, 
 I) 
 
 iX. SO 
 
 'ioV 
 
 t o 
 
 b 
 
 1-985 
 
 3-75 
 
 •037 
 
 C 
 
 OO.'-Kt 
 lii ob 
 
 ouu 
 
 O 
 
 b 
 
 2-74 
 
 7-5 
 
 •075 
 
 ,7 
 (I 
 
 O A ■ i Q 
 
 t)UU 
 
 c 
 O 
 
 a 
 
 3-535 
 
 12-5 
 
 •125 
 
 C 
 
 SO 'iO 
 
 i uu 
 
 ! 
 
 b 
 
 3-87 
 
 15 
 
 •15 
 
 CL 
 
 
 spa 
 
 ov u 
 
 El 
 O 
 
 a 
 
 5 
 
 25 
 
 •25 
 
 
 OA 
 .iU 
 
 yuu 
 
 q 
 y 
 
 b 
 
 5-47 
 
 30 
 
 •3 
 
 b 
 
 oU Jo 
 
 you 
 
 Vl-R 
 v O 
 
 0 
 
 5-59 
 
 31-25 
 
 •312 
 
 c 
 
 n I .ft9 
 Ol OZ 
 
 1UUU 
 
 i n 
 
 (I 
 
 ('■ 
 
 (H2 
 
 375 
 
 ■375 
 
 d 
 
 
 1 _uu 
 
 1 - 
 
 6-612 
 
 43-75 
 
 •437 
 
 e 
 
 0/41 
 
 i jnn 
 1-1 uu 
 
 1 J. 
 11 
 
 a 
 
 7-()7 
 
 50 
 
 •5 
 
 a 
 
 A 1 \ 
 4U 
 
 i ft on 
 louu 
 
 1 n 
 
 10 
 
 b 
 
 7-74 
 
 60 
 
 •6 
 
 b 
 
 43-76 
 
 1920 
 
 19-2 
 
 
 79 
 
 62-5 
 
 •625 
 
 c 
 
 44-72 
 
 2000 
 
 20 
 
 d 
 
 8-66 
 
 75 
 
 •75 
 
 d 
 
 48-98 
 
 2400 
 
 24 
 
 e 
 
 9-35 
 
 87-5 
 
 •875 
 
 e 
 
 52-90 
 
 2800 
 
 28 
 
 a 
 
 10 
 
 100 
 
 1 
 
 a 
 
 56-56 
 
 3200 
 
 32 
 
 b 
 
 10-94 
 
 120 
 
 1-2 
 
 b 
 
 61-92 
 
 3840 
 
 38-4 
 
 c 
 
 11-18 
 
 125 
 
 1*25 
 
 c 
 
 63-24 
 
 4000 
 
 40 
 
 d 
 
 12-25 
 
 150 
 
 1-5 
 
 d 
 
 69'28 
 
 4800 
 
 48 
 
 e 
 
 13-225 
 
 175 
 
 1-75 
 
 
 70-71 
 
 5000 
 
 50 
 
 a 
 
 1414 
 
 200 
 
 2 
 
 e 
 
 74-82 
 
 5600 
 
 56 
 
 b 
 
 15-48 
 
 240 
 
 2-4 
 
 
 80 
 
 6400 
 
 64 
 
 c 
 
 15-81 
 
 250 
 
 2-5 
 
 ■ .! 
 
 87-52 
 
 7680 
 
 76-8 
 
 d 
 
 17-32 
 
 300 
 
 3 
 
 c 
 
 89-44 
 
 8000 
 
 80 
 
 e 
 
 18-7 
 
 350 
 
 3-5 
 
 d 
 
 98 
 
 9600 
 
 96 
 
 a 
 
 20 
 
 400 
 
 4 
 
 
 100 
 
 10,000 
 
 100 
 
 Series of apertures that require exposure to be increased in the 
 geometrical ratio of 2 are marked with corresponding letters. Series 
 a includes the unit aperture of the C.I. system. Series c coincides 
 very closely with the U.S. series given in table E, and is not very 
 different from series b. 
 
160 
 
 Appendix. 
 
 Classification by Intensity. 
 (H).— Zeiss Systems. 
 
 1 SERIES. 
 
 APER - 
 TDHE 
 RATIO 
 NO. 
 
 EXPO- 
 SURE 
 NO. 
 
 INTENSITY NOS. 
 
 I 
 
 § 
 
 S 
 w 
 
 — 
 
 APER- 
 
 RATIO 
 NO. 
 
 SURE 
 NO. 
 
 INTENSITY NOS. 
 
 % . '■ ' 
 
 UNIT 
 /I. 
 
 
 
 OLD 
 UNIT 
 
 /100. 
 
 NEW 
 UNIT 
 
 /50. 
 
 UNIT 
 /I- 
 
 UNIT 
 
 /100. 
 
 UNIT 
 
 /50. 
 
 
 1 
 
 1 
 I 
 
 10,000 
 
 2500 
 
 0 
 
 11-18 
 
 125 
 
 80 
 
 20 
 
 g 
 
 1 JO 
 
 Z 0 
 
 4000 
 
 1000 
 
 f 
 
 12-5 
 
 156 
 
 64 
 
 16 
 
 a 
 
 1 -7(17 
 
 Q.I OK 
 
 3200 
 
 800 
 
 g 
 
 12-65 
 
 160 
 
 62-5 
 
 1 5-625 
 
 c 
 
 1 -07S 
 I it to 
 
 3'9 
 
 2560 
 
 640 
 
 a 
 
 14-14 
 
 200 
 
 50 
 
 12-5 
 
 f 
 
 2-21 
 
 475 
 
 2048 
 
 512 
 
 0 
 
 15-81 
 
 250 
 
 40 
 
 10 
 
 8 
 
 g 
 
 2-236 
 
 5 
 
 2000 
 
 500 
 
 f 
 
 17 68 
 
 312 
 
 32 
 
 a 
 
 25 
 
 6-25 
 
 1600 
 
 400 
 
 g 
 
 17-88 
 
 320 
 
 31-25 
 
 7-81 
 
 c 
 
 2-795 
 
 7-8 
 
 1280 
 
 320 
 
 a 
 
 20 
 
 400 
 
 25 
 
 6-25 
 
 f 
 
 3125 
 
 9-5 
 
 1021 
 
 256 
 
 c 
 
 22-36 
 
 500 
 
 20 
 
 5 
 
 (1 
 
 316 
 
 10 
 
 1000 
 
 250 
 
 f 
 
 25 
 
 625 
 
 16 
 
 4 
 
 a 
 
 3535 
 
 12-5 
 
 800 
 
 200 
 
 g 
 
 25-3 
 
 640 
 
 15-625 
 
 3-9 
 
 <■ 
 
 3-95 
 
 15-625 
 
 640 
 
 160 
 
 a 
 
 28-28 
 
 800 
 
 12-5 
 
 3-125 
 
 ./• 
 
 4-42 
 
 19 
 
 512 
 
 128 
 
 c 
 
 31-62 
 
 1000 
 
 10 
 
 2-5 
 
 g 
 
 4-472 
 
 20 
 
 500 
 
 125 
 
 f 
 
 35-36 
 
 1250 
 
 8 
 
 2 
 
 a 
 
 5 
 
 25 
 
 400 
 
 100 
 
 g 
 
 35-76 
 
 1280 
 
 7-8 
 
 1-95 
 
 c 
 
 5-59 
 
 31-25 
 
 320 
 
 80 
 
 a 
 
 40 
 
 1600 
 
 6-25 
 
 1-56 
 
 f 
 
 6-25 
 
 39 
 
 256 
 
 64 
 
 c 
 
 44-72 
 
 2000 
 
 5 
 
 1-25 
 
 g 
 
 6-32 
 
 40 
 
 250 
 
 625 
 
 f 
 
 50 
 
 2500 
 
 4 
 
 1 
 
 a 
 
 7-07 
 
 50 
 
 200 
 
 50 
 
 g 
 
 50-6 
 
 2560 
 
 3-9 
 
 •975 
 
 0 
 
 f 
 
 7-9 
 
 62-5 
 
 160 
 
 40 
 
 a 
 
 56-56 
 
 3200 
 
 3-125 
 
 •781 
 
 8-84 
 
 78 
 
 128 
 
 32 
 
 0 
 
 63-24 
 
 4000 
 
 2-5 
 
 •625 
 
 9 
 
 8-94 
 
 80 
 
 125 
 
 31-25 
 
 f 
 
 70-72 
 
 5000 
 
 2 
 
 •5 
 
 a 
 
 10 
 
 100 
 
 100 
 
 25 
 
 f 
 
 100 
 
 10,000 
 
 1 
 
 •25 
 
 Series / includes the unit apertures of the Zeiss systems. Series 
 and c are the same as in last table. 
 
Appendix. 
 
 161 
 
 (I). — Rules for Marking Apertures. 
 Classification by Exposure. 
 
 1. Square the ratio number to find relative exposure given in 
 column headed " Unit / 1." 
 
 2. Divide relative exposure by 16 to find U.S. number. 
 
 3. Divide relative exposure by 10 to find Dallmeyer number. 
 
 4. Divide relative exposure by 100 to find C.I. number. 
 
 5. If desired to adopt any other unit aperture, the exposure 
 number of any aperture can be found by dividing its relative ex- 
 posure by the relative exposure for the unit aperture selected. 
 
 Classification by Intensity. 
 
 6. Multiply reciprocal of relative exposure by 10,000 to find Zeiss 
 old system number, which is four times the value of the new system 
 number. 
 
 7. The general rule for any other unit aperture is ; — to find in- 
 tensity number multiply the reciprocal of the relative exposure 
 by the relative exposure for the unit aperture. 
 
 To find ratio number of an aperture marked by any of the above 
 systems, reverse the processes described in above rules. For 
 example, a U.S. number multiplied by 16 gives relative exposure 
 for unit aperture of /l, and the square root of the result is the 
 ratio number. Or divide 10,000 by the Zeiss old system number 
 to find the relative exposure for unit aperture /l, and take square 
 root of the result to find ratio number. 
 
 11 
 
162 
 
 Appendix. 
 
 (J). — Table op Angles. 
 
 Find the diameter of the circle that is to be subtended by the 
 angle, and divide by the focal distance of the lens from the plate. 
 The quotient (T) is equal to twice the tangent of half the angle, but 
 the value of the angle can be found very approximately from the 
 following table. 
 
 IF T IS 
 
 THE ANGLE 
 
 IF T IS 
 
 THE ANGLE 
 
 IF T IS 
 
 THE ANGLE 
 
 UNDER 
 
 IS UNDER 
 
 UNDER 
 
 IS UNDER 
 
 UNDER 
 
 IS UNDER 
 
 Narroi 
 
 i Angles. 
 
 •95 
 
 51° 
 
 2-3 
 
 98° 
 
 •8 
 
 17° 
 
 1 
 
 53° 
 
 2-4 
 
 100|° 
 
 •35 
 
 20° 
 
 Wide A ngles. 
 
 2-5 
 
 103° 
 
 •4 
 
 23° 
 
 11 
 
 58° 
 
 2-6 
 
 105° 
 
 •45 
 
 25 4° 
 
 1-2 
 
 62° 
 
 2-7 
 
 107° 
 
 •5 
 
 28° 
 
 1-3 
 
 66° 
 
 2-8 
 
 110° 
 
 •55 
 
 31° 
 
 1-4 
 
 70° 
 
 2-9 
 
 111° 
 
 •6 
 
 33|° 
 
 1-5 
 
 74° 
 
 3 
 
 112i° 
 
 •65 
 
 36° 
 
 1-6 
 
 77° 
 
 3-2 
 
 116° 
 
 Mid 
 
 Angles. 
 
 1-7 
 
 81° 
 
 3-4 
 
 119° 
 
 ■7 
 
 39° 
 
 1-8 
 
 84° 
 
 3-6 
 
 122° 
 
 •75 
 
 41|° 
 
 1-9 
 
 87° 
 
 3-8 
 
 124i" 
 
 •8 
 
 44 ° 
 
 2 
 
 90° 
 
 4 
 
 127° 
 
 •85 
 
 46i° 
 
 21 
 
 93° 
 
 4-3 
 
 130° 
 
 •9 
 
 48|° 
 
 2-2 
 
 95|° 
 
 4-7 
 
 134° 
 
 To determine the covering angle the diameter of the circle is 
 taken as equal to twice the distance from the principal axis of the 
 lens to the farthest corner of the plate. To determine the view 
 angle we take simply the diagonal of the plate as the diameter. 
 
 (K)— Diagonals of Stock Size Plates (Approximate). 
 
 ins. 
 
 x2 
 x 2k 
 x3j 
 x3| 
 x4 
 
 ins. 
 
 4 
 
 5 
 
 6i 
 
 ins. 
 6f x3| 
 6|x4f 
 7ix 5 
 8jx6i 
 9 x7 
 10 x 8 
 
 DIAGONAL. 
 
 ins. 
 7i 
 7f 
 
 io| 
 Hi 
 
 12i 
 
 ins, 
 12 x 10 
 15x12 
 18 x 16 
 20 x 16 
 22 x 18 
 24 x 20 
 
Appendix. 163 
 
 (L).— Depth Constants in Inches and Feet. 
 
 FOCAL LENGTHS IN INCHES. 
 
 o 
 
 rH 
 
 10,000 
 833-3 
 
 2500 
 208-3 
 
 1768-3 
 147-36 
 
 1250 
 104-16 
 
 b- X 
 rH CD 
 
 CO 
 
 oo 
 
 00 
 
 625 
 52-08 
 
 442-08 
 36-84 
 
 312-5 
 26-04 
 
 221-04 
 18-42 
 
 156-25 
 13-02 
 
 oo 
 
 •CO 
 
 o ob 
 o oo 
 ■* m 
 
 50 
 
 •CO 
 
 O cb 
 
 O CO 
 CO rH 
 
 1131-7 
 94-31 
 
 800 
 66-6 
 
 565-87 
 47-15 
 
 400 
 33-3 
 
 CO t- 
 31 in 
 ck cb 
 
 30 (M 
 CM 
 
 200 
 16-6 
 
 141-46 
 11-78 
 
 ■CO 
 O CO 
 
 o 
 
 rH 
 
 i— 
 
 •CO 
 
 O 00 
 
 o o 
 
 31 rH 
 
 1225 
 102-08 
 
 866-48 
 72-2 
 
 612-5 
 51-04 
 
 433-24 
 36-10 
 
 306-25 
 25-52 
 
 216-62 
 18-05 
 
 153-12 
 12-76 
 
 108-31 
 9-02 
 
 76-56 
 6-38 
 
 
 3600 
 300 
 
 O iO 
 O b- 
 
 31 
 
 636-6 
 53-05 
 
 450 
 37-5 
 
 318-3 
 26-52 
 
 225 
 18-75 
 
 159-15 
 13-26 
 
 b- 
 
 ip CO 
 CM 31 
 
 b- CO 
 
 in cp 
 
 OS CD 
 b- 
 
 56-25 
 4-68 
 
 5 
 
 2500 
 208-3 j 
 
 00 
 
 o 
 
 «Q CM 
 
 cm w 
 
 CD 
 
 442-08 
 36-84 
 
 312-5 
 26-04 
 
 221-04 
 18-42 
 
 156-25 
 13-02 
 
 110-52 
 9-21 
 
 CM rH 
 
 ■h in 
 h cb 
 
 CO 
 
 CM CO 
 
 in -* 
 
 m 
 
 39-06 
 3-26 
 
 
 2025 
 168-75 
 
 506-25 j 
 42-19 
 
 00 rH 
 
 © qp 
 
 30 31 
 lO CM 
 CO 
 
 253-12 
 21-09 
 
 179-04 
 14-92 
 
 126-56 
 10-54 
 
 89-52 
 7-46 
 
 30 b- 
 CM CM 
 
 cb in 
 
 CO 
 
 44-76 
 3-73 
 
 rH CO 
 CD CO 
 rH CM 
 
 CO 
 
 
 ■cp 
 o CO 
 O CO 
 CO rH 
 rH 
 
 •CO 
 O CO 
 O CO 
 
 rH 
 
 282-9 
 23-58 
 
 200 
 16-6 
 
 141-46 
 11-79 
 
 •CO 
 O » 
 
 o 
 
 70-73 
 5-89 
 
 •CO 
 
 o 
 m 
 
 CO 
 
 Cp cp 
 lb CM 
 CO 
 
 •CO 
 00 
 
 m c-i 
 
 CM 
 
 eo 
 
 O W 
 O b- 
 OS 
 
 225 
 18-75 
 
 159-15 
 13-26 
 
 112-5 
 9-37 
 
 79-57 
 6-63 
 
 56-25 
 4-68 
 
 39-78 
 3-31 
 
 28-12 
 2-34 
 
 19-89 
 1-65 
 
 14-06 
 1-17 
 
 1 ^ 
 
 •CO 
 
 1 8* 
 
 rH 
 
 00 
 
 o 
 m cm 
 
 CM 
 
 17-68 
 1-47 ! 
 
 rH 
 
 in o 
 
 CM rr 
 
 CO 
 
 30 b- 
 
 6-25 
 •52 
 
 CM CD 
 
 rH CO 
 
 cm co 
 
 —1 CM 
 
 CO 
 
 2-21 
 •18 
 
 1-56 
 •13 
 
 APER- 
 TURE 
 RATIO NO. 
 
 IH 
 
 rH 
 
 in 
 
 CO 
 
 in 
 
 00 
 
 11-31 | 
 
 CO 
 
 j 22-62 | 
 
 CM 
 CO 
 
 CM 
 
 in 
 
 r* 
 
 CO 
 
 The constant is found under the focal length and opposite the 
 aperture, the upper figure being the constant in inches, and the lower 
 one the constant in feet. The constants for any other apertures can 
 
1 64 Appendix. 
 
 be found by dividing the constants for/ 1 by the ratio number of the 
 aperture ; the constants for any other focal length by multiplying 
 the constant for a focal length of 1-in. by the square of the focal 
 length. Beyond giving the constants used in formula? 21 to 26 this 
 table serves other purposes. If we focus on infinity, the constant 
 is the focal distance of the nearest object in focus. If we focus 
 on an extra-focal distance equal to the constant, we obtain a 
 maximum depth of field from approximately half the constant dis- 
 tance up to infinity. The constant is then the hyper-focal distance. 
 
 (M). — Consecutive Extra-focal Depths for 
 Aperture /8 (Approximate). 
 
 
 
 FOCAL LENGTHS IN 
 
 INCHES. 
 
 
 
 
 
 1 
 
 3 
 
 4 
 
 
 5 
 
 6 
 
 7 
 
 8 
 
 10 
 
 O 03 
 
 *g 
 
 OS £ 
 w < 
 
 GC 
 
 GC 
 
 GC 
 
 GC 
 
 GC 
 
 
 
 GC 
 
 GC 
 
 GC 
 
 GC 
 
 w S 
 
 ft. 
 10417 
 •5208 
 •3473 
 •2604 
 •2084 
 •1736 
 •1488 
 •1302 
 •1157 
 •1041 
 •0947 
 •0868 
 
 ft. in. 
 9 4£ 
 4 8J- 
 3 ll 
 2 4 
 1 11 
 1 7 
 1 4 
 1 2 
 
 ft. in. 
 16 9 
 
 8 4 
 
 5 7 
 
 4 2 
 
 3 4 
 
 2 9| 
 
 2 4i 
 
 2 1 
 
 ft. in. 
 21 0 
 10 6 
 7 0 
 5 3 
 4 3 
 3 6 
 3 0 
 2 8 
 
 — 
 
 ft, in. 
 26 0 
 13 0 
 8 8 
 6 6 
 5 2 
 4 4 
 3 9 
 3 3 
 2 11 
 
 ft. in. 
 37 6 
 18 9 
 12 6 
 9 4 
 7 6 
 6 3 
 5 4 
 4 8 
 4 2 
 3 9 
 
 ft. in. 
 51 0 
 25 6 
 17 0 
 12 9 
 10 3 
 8 6 
 7 3 
 6 4 
 5 8 
 5 1 
 4 8 
 4 3 
 
 ft. in. 
 66 8 
 33 4 
 22 3 
 16 8 
 13 4 
 11 2 
 9 6 
 8 4 
 7 4 
 6 8 
 6 0 
 5 7 
 
 ft. in. 
 104 0 
 52 0 
 34 9 
 26 0 
 20 10 
 17 4 
 14 11 
 13 0 
 11 7 
 10 5 
 9 6 
 8 8 
 
 /8 
 
 /16 
 
 /24 
 
 /32 
 
 /40 
 
 /48 
 
 /56 
 
 /64 
 
 /72 
 
 /80 
 
 /88 
 
 /96 
 
 These consecutive depths are very approximate, and are only 
 applicable for aperture /8. Each vertical column is a complete 
 series for one particular lens. Similar series for any other lens can 
 be found by multiplying the figures in the first column by the square 
 of the focal length. Taking any three consecutive distances in one 
 vertical column, the lowest distance is the near, and the highest the 
 far extra-focal depth of field when focussing on the middle distance. 
 Every distance is also a hyper-focal distance for the aperture given 
 in the last column. 
 
Appendix. 
 
 (N). — Appkoximatk Infinity fob Lenses of 
 Vaeious Focal Lengths. 
 
 
 DISTANCE OF 
 
 FOCUSSING-SCKEEN BEHIND PRINCIPAL FOCUS. 
 
 FOCAL 
 
 
 
 
 
 LENGTH, 
 
 
 
 
 
 INCHES. 
 
 i to iu - 
 
 •rib in - 
 
 
 
 1 
 
 3 Tds. 
 
 7| yds. 
 
 15 yds. 
 
 30 yds. 
 
 2 
 
 11 
 
 28 „ 
 
 55 * „ 
 
 no „ 
 
 3 
 
 25 „ 
 
 63 „ 
 
 125 „ 
 
 250 „ 
 
 4 
 
 45 „ 
 
 113 „ 
 
 225 „ 
 
 450 „ 
 
 5 
 
 70 „ 
 
 175 „ 
 
 350 „ 
 
 700 „ 
 
 6 
 
 100 „ 
 
 250 „ 
 
 500 „ 
 
 1000 „ 
 
 7 
 
 136 „ 
 
 340 ,, 
 
 680 „ 
 J mile 
 
 i a en 
 looU ,, 
 
 8 
 
 178 ., 
 
 \ mile 
 
 1 mile 
 
 9f 
 
 264 „ 
 
 660 yds. 
 
 3 
 
 4 >> 
 
 1^ miles 
 
 "I 
 
 351 „ 
 
 | mile 
 
 1 
 
 2 „ 
 
 12* 
 
 434 „ 
 
 1085 yds. 
 
 1^ miles 
 
 2* „ 
 
 13| 
 
 525 „ 
 
 § mile 
 
 li „ 
 
 3 „ 
 
 10 
 
 700 „ 
 
 
 2 „ 
 
 4 „ 
 
 174 
 
 875 „ 
 
 \\ mile? 
 
 2 1 
 
 ^2 » 
 
 5 ■, 
 
 19§ 
 
 1056 ., 
 
 U „ 
 
 3 „ 
 
 6 „ 
 
 21" 
 
 1225 „ 
 
 :: 
 
 3* „ 
 
 7 „ 
 
 22* 
 
 1406 „ 
 
 
 4 „ 
 
 8 „ 
 
 24 
 
 1600 „ 
 
 
 4| „ 
 
 9 „ 
 
 25 
 
 1 mile 
 
 2i 
 
 5 „ 
 
 I 0 „ 
 
 28 
 
 \\ miles 
 
 *-U 
 
 •>4 „ 
 
 <* „ 
 
 13 ., 
 
 30 
 
 1* „ 
 
 
 71- „ 
 
 15 „ 
 
 33 
 
 
 4| „ 
 
 9 „ 
 
 is " 
 
 35 
 
 2 „ 
 
 5 „ 
 
 10 „ 
 
 20 „ 
 
 By focussing accurately on distances not less than those given, 
 we ensure that the focussing-screen is within j-^, ■^\ 1S , ass or > 
 nnnr from the true principal focus. 
 
INDEX. 
 
 {The Numbers refer to Sectional Paragraphs ; the Letters to the 
 Appendix. ) 
 
 SEC. 
 
 SEC. 
 
 Aberration, Astigmatic Spherical 
 
 
 Aperture, Measurement of . 25, 
 
 93 
 
 (Astigmatism) 53, 
 
 55 
 
 ,, Ratio .... 
 
 64 
 
 Chromatic . . 7, 
 
 57 
 
 ,, Ratio Number 
 
 65 
 
 , , Correction of (General 
 
 
 , , Systems of Marking 65, E 
 
 —I 
 
 Principles) 
 
 48 
 
 Astigmatism . . . .53, 55, 
 
 56 
 
 ,, Effect on Action of 
 
 
 ,, and Coma, Notes on . 
 
 56 
 
 Lenses . 
 
 59 
 
 Averaging Focus .... 
 
 70 
 
 ,, Lateral Spherical 
 
 
 Axial Rays .... 10, 
 
 22 
 
 (Coma) . 
 
 54 
 
 Axis Principal .... 
 
 18 
 
 ,, Longitudinal Spheri- 
 cal . . . 52, 
 
 53 
 
 
 43 
 
 , , Positive and Negative 
 
 49 
 
 Circles of Confusion . 
 
 73 
 
 ,, Primary 
 
 46 
 
 ,, Definition . 
 
 68 
 
 ,, Secondary . 
 
 '. 
 
 ,, ,, Illumination 
 
 07 
 
 ,, Spherical . 
 
 
 Collimator, Use of Lens as . 
 
 89 
 
 ,, ,, in Direct 
 
 
 
 6 
 
 Pencils . 
 
 52 
 
 
 56 
 
 ,, ,, in Oblique 
 
 
 Combining Lenses . . . 39-44 
 
 Pencils . 
 
 53 
 
 ,, Lens Surfaces 
 
 16 
 
 Achromats, Normal and Abnormal 
 
 57 
 
 ,, more than two Lenses 
 
 44 
 
 Actinic Light 
 
 7 
 
 Compromises, Essential . 
 
 50 
 
 Angle, Critical .... 
 
 12 
 
 Conjugate Extra-Focal Distances . 
 
 33 
 
 ,, of Definition 
 
 68 
 
 ,, Focal Distances 
 
 30 
 
 ,, ,, Deviation 
 
 10 
 
 „ Foci 
 
 29 
 
 ,, ,, Incidence ... 10 
 
 12 
 
 „ Principal Foci 
 
 35 
 
 ,, ,, Reflection 
 
 12 
 
 „ Symmetric Foci . 
 
 34 
 
 ,, ,, Refraction 
 
 10 
 
 Contact, Nodal, between Lenses . 
 
 40 
 
 ,, View and Covering 
 
 69 
 
 ,, Optical, ,, Surfaces * 
 
 12 
 
 Angles of Illumination . 
 
 67 
 
 Covering Power .... 
 
 71 
 
 ,, Subtending Circles . 
 
 J 
 
 Crossed Lenses .... 
 
 15 
 
 ,, Wide, Mid, and Narrow 
 
 70 
 
 ,, Nodes .... 
 
 86 
 
 Angular aperture ... 64, 71 
 
 ,, Separation 
 
 40 
 
 Aperture (Effective and Nominal) . 
 
 9 
 
 Crossing-Point .... 
 
 20 
 
 ,, and Exposure . 
 
 64 
 
 Curvature of Field . . . 21, 
 
 51 
 
 ,, Fractional Diameter of . 
 
 65 
 
 ,, „ „ Effect of Dia- 
 
 
 ,, Inconstancy of . 9, 64, 
 
 94 
 
 phragm upon 53, 54 
 
 167 
 
168 
 
 Index. 
 
 SEC. I 
 
 Definition, Angle and Circle of . OS j 
 Constant ... 78 
 ,, Volume ... 81 
 ,, ,i and Swing- 
 
 back. . 82 1 
 
 . Depth 72 
 
 ,, Constants . . . 75, L 
 ,, Measurement of ... 74 
 ,, Pseudo- .... 75 
 Depth of Definition . . 72, 78-80 
 ,, „ „ Axial . . 78 
 „ ,, ,, and Focussing 79 
 ,, ,, ,, and Scale of 
 
 Image . . 80 
 Depth of Distance . . . .72 
 „ „ Field . . . 72, 75-77 
 ,, „ „ Axial . . . 75 
 ,, ,, ,, Consecutive . 76, M 
 ,, ,, Maximum . . 75 
 „ ,, „ Variations in . 77 
 Depth of Focus . 72 
 
 Deviation . . . . 10, 14 
 Diagonals of Plates K 
 
 Diaphragm 9 
 
 ,, and Aberration . . 48 
 ,, ,, Coma . . .54 
 
 , , , , Curvature of Field 53, 54 
 ,, ,, Longitudinal 
 
 Spherical Aber- 
 ration . . 52 
 ,, ,, Secondary Plane of 
 
 Oblique Pencil 53 
 ,, Distortion . . 53, 54, 58 
 
 Dioptries 28 
 
 Direct Pencils . . .14 
 
 Dispersion 11 
 
 Doublet, Using half of . . .96 
 
 Enlargement and Reduction 62, C, D 
 Exposure . ... 64, D 
 Extra-Focal Distances ... 33 
 
 FieId Volume . ... .83 
 
 Flare 60 
 
 Flatness of Field .... 21 
 
 Focal Centre 26 
 
 ,, Distances .... 30 
 ,, Length Comparative . . A 
 , ,, and Exposure . 64 
 „ ,, and Power . 28, B 
 , , , . Measurement of . 91 
 
 
 SEC. 
 
 Focal Length of Combination 
 
 42, 
 
 44 
 
 ,, ,, To calculate 
 
 
 92 
 
 Foci Astigmatic 
 
 
 55 
 
 ,, Conjugate 
 
 
 29 
 
 ,, ,, Principal 
 
 
 35 
 
 ,, ,, Symmetric 
 
 
 34 
 
 
 
 27 
 
 ,, ,, to find 
 
 
 89 
 
 ,, Real and Virtual 
 
 
 8 
 
 
 
 9 9 
 
 
 
 79 
 
 ,, Excentric 
 
 
 81 
 
 ,, by Scale 
 
 84-87 
 
 ,, with non-Achromatic 
 
 
 Lenses 
 
 
 95 
 
 ,, with Working Aperture 
 
 52 
 
 ,, Scales, Marking of . 
 
 85, 
 
 SO 
 
 Formulae, Calculating Focal 
 
 
 Length . '. 
 
 
 92 
 
 ,, Combining Lenses 
 
 41 
 
 -44 
 
 ,, Consecutive Depth of 
 
 
 Field . 
 
 
 75 
 
 , , Depth of Definition 
 
 
 7S 
 
 „ „ Field 
 
 
 75 
 
 ., Extra- Focal Distances 
 
 
 33 
 
 ,, Focal Distances 
 
 
 30 
 
 ,, Scale of Image . 19, 
 
 01 
 
 62 
 
 Gauss Points (Nodes) 
 
 
 23 
 
 
 
 25 
 
 Hyperpocal Distance . 
 
 
 75 
 
 Illumination, Intensity of . 
 
 
 G3 
 
 ,, Uniformity of 
 
 66, 
 
 67 
 
 Images, Formation of . 
 
 
 17 
 
 ,, Lens .... 
 
 20, 
 
 22 
 
 ,, Pinhole 
 
 IS, 
 
 19 
 
 Infinity Approximate . 
 
 .85 
 
 ,N 
 
 „ Mark 
 
 
 85 
 
 Infra-red Light Waves . 
 
 
 7 
 
 
 
 03 
 
 ,, and Exposure 
 
 
 64 
 
 
 
 57 
 
 Law of Inverse Squares . 
 
 
 03 
 
 ,, ,, Reflection . 
 
 
 12 
 
 
 
 in 
 
 Lens, Effect on Compound Light 
 
 
 6 
 
 ,, ,, ,, Light Rays . 
 
 
 4 
 
 ,, ,, ,, Waves . 
 
 
 3 
 
Index. 
 
 169 
 
 
 
 
 Lens, Equivalent, Theory of 
 
 
 24 
 
 ,, Cardinal Points of 
 
 
 3G 
 
 ,, Image . 
 
 
 20 
 
 ,, ,, Scale of 
 
 
 22 
 
 , , Minor Points of . 
 
 
 26 
 
 „ Nodes of 
 
 
 23 
 
 ,, Surfaces of . 
 
 
 . 16 
 
 Lenses Achromatic 
 
 
 . 7, 57 
 
 ,, Adjustable . 
 
 
 . 100 
 
 ,, Anastigmatic 
 
 55 
 
 98, 104 
 
 ,, Aplanatic . 
 
 
 5 
 
 ,, Apochromatic . 
 
 
 7 
 
 ,, Bergheim . 
 
 
 . 100 
 
 ,, Casket 
 
 
 . 102 
 
 , , Cementing of 
 
 
 . 12 
 
 , , Centring of . 
 
 
 . 13 
 
 „ Collinear 
 
 
 46, 104 
 
 „ Concentric . 
 
 55 
 
 98, 104 
 
 „ Convergent. 
 
 
 . 14 
 
 , , Convertible 
 
 
 . 103 
 
 ,, Cooke . . .53, 
 
 54, 
 
 103, 104 
 
 ,, Divergent . 
 
 
 . 14 
 
 ,, Doublet 
 
 
 . 15 
 
 ., Effect of Combining 
 
 
 39 
 
 ,, Equivalent. 
 
 
 . 3S 
 
 , , Forms of 
 
 
 15 
 
 ,, Goerz . . • . 
 
 
 55 
 
 ,, Landscape . 
 
 
 7 
 
 , , Measurement of . 
 
 
 . 88 
 
 ,, Meniscus 
 
 
 15 
 
 „ Mid and Narrow Angle 
 
 . 70 
 
 ,, Negative 
 
 
 2, 4, 14 
 
 ,, Nomenclature of 
 
 
 . 104 
 
 ,, Orthoscopic 
 
 
 58 
 
 , , Patent Portrait . 
 
 
 . 100 
 
 ,, Periscopic . 
 
 
 . 15 
 
 ,, Piano . 
 
 
 15 
 
 ,, Portrait 
 
 
 . 99 
 
 ,, Positive 
 
 
 2, 4, 14 
 
 ,, Rapid . 
 
 
 . 97 
 
 ,, Rectilinear . 
 
 .58 
 
 97, 104 
 
 ,, Spectacle 
 
 
 . 95 
 
 ,, Stigmatic . 
 
 
 103, 104 
 
 ,, Telephoto . 
 
 
 . 101 
 
 ,, Theory of . 
 
 
 . 25 
 
 ,, Thin and Thick . 
 
 
 . 22 
 
 ,, Triplet 
 
 
 . 15 
 
 ,, View . 
 
 
 . 96 
 
 ,, Wide Angle 
 
 
 . 70 
 
 ,, Zeiss . 
 
 
 . 55 
 
 Light, Compound . 
 
 
 6 
 
 SEC. 
 
 Light, Lost by Reflection 
 
 
 12, 60 
 
 ,, and Optics . 
 
 
 1 
 
 „ Rays . 
 
 
 4 
 
 ,, Reversibility of . 
 
 
 . 10 
 
 ,, Theory of . 
 
 
 3, 6, 7 
 
 
 
 2 
 
 Luminosity of Object . 
 
 
 . 17 
 
 
 
 45 
 
 Meridional Focus . 
 
 
 . 55 
 
 Nodal Contact and Separation 
 
 . 40 
 
 
 
 . 25 
 
 ,, Space . 
 
 
 23, 36 
 
 Nodes, Coincident and Crossed 
 
 . 36 
 
 „ To find 
 
 
 . 90 
 
 
 
 
 ,, of Lens 
 
 
 23 
 
 „ Positions of, in 
 
 various 
 
 Types of Lenses 
 
 
 . 37 
 
 ,, Separated . 
 
 
 36 
 
 Nominal Aperture . 
 
 
 9 
 
 ,, Intensity . 
 
 
 . 64 
 
 ,, Ratio Aperture 
 
 
 . 64 
 
 Optical Centre 
 
 
 . 26 
 
 Optics, Geometrical 
 
 
 
 ,, and Light . 
 
 
 1 
 
 ,, Physical 
 
 
 3 
 
 Pencils of Light . 
 
 
 8 
 
 ,, Direct 
 
 
 . 14 
 
 ,, Oblique 
 
 
 . 53 
 
 Poles of Lenses 
 
 
 . 26 
 
 
 
 
 ,, Plane of Oblique Pencil 
 
 . 53 
 
 Principal Points 
 
 
 . 23 
 
 Power, Covering . 
 
 
 . 71 
 
 ,, Dispersive . 
 
 
 11 
 
 
 
 
 ,, Refractive . 
 
 
 . 10 
 
 Radial Focus 
 
 
 . 55 
 
 Reflection 
 
 
 . 12 
 
 
 
 
 Sagittal Focus 
 
 
 . 55 
 
 Scale of Image 
 
 19, 22, 
 
 61, 62 
 
 ,, and Depth of Definition. 
 
 . 80 
 
 ., and Exposure 
 
 
 64, D 
 
 Secondary Focus . 
 
 
 55 
 
 Plane of Oblique Pencil 53 
 
 12 
 
170 
 
 Index. 
 
 SEC. 
 
 Sign Convention, Popular . . 31 
 
 „ Standard . . 32 
 
 Spot, Flare 60 
 
 Stop -. 9 
 
 Surfaces of Lenses . . 16 
 ,, Spherical . . .13 
 
 Swingback and Definition . . 82 
 
 Tables in Appendix : 
 
 Comparative Focal Lengths . A 
 
 Corresponding Focal Lengths 
 
 and Powers B 
 
 Scale Table of Conjugate Dis- 
 tances C 
 
 Exposure and Scale D 
 
 Uniform System (U. S.)of mark- 
 ing Stops E 
 
 Dallmeyer System of mark- 
 ing Stops E 
 
 Continental System (C.l.) of 
 
 marking Stops G 
 
 SEC. 
 
 Zeiss System of marking 
 
 Stops II 
 
 Rules for Marking Stops . . I 
 
 Angles . . .... J 
 
 Diagonals of Plates K 
 
 Depth Constants L 
 
 Consecutive Depths of Field . M 
 
 Approximate Infinity . . N 
 Tangent Focus . . . .55 
 
 Test for Astigmatism ... 55 
 
 , . „ Chromatic Aberration . 57 
 
 ,, ,, Coma .... 54 
 
 ,, ,, Diaphragm Distortion . 58 
 
 „ „ Flare 60 
 
 ,, ,, Longitudinal Spherical 
 
 Aberration ... 52 
 
 Ultka-Violet Light Waves . . 7 
 
 Wave Theory of Light . . 2, 3, 6, 7 
 Working Intensity . . . .64 
 
 PAGES ON WHICH SECTIONAL PARAGRAPHS COMMENCE. 
 
 SEC. 
 
 PAGE 
 
 SEC. 
 
 PACE 
 
 SKC. 
 
 PAGE 
 
 SEC. 
 
 PAGE 
 
 1 
 
 
 27, 28 
 
 . ..37 
 
 
 76 
 
 79 .. .. 
 
 120 
 
 2 
 
 
 29 
 
 39 
 
 56 
 
 81 
 
 80 .. .. 
 
 121 
 
 3 
 
 7 
 
 30 . . 
 
 . ..40 
 
 57 
 
 82 
 
 81 .. .. 
 
 122 
 
 4 
 
 , . . 9 
 
 81 
 
 , 42 
 
 58 
 
 83 
 
 82 .. .. 
 
 124 
 
 
 10 
 
 32 
 
 43 
 
 59, 60 . 
 
 . . 86 
 
 83 .. .. 
 
 125 
 
 6 
 
 . , . 11 
 
 33—30 
 
 . ..45 
 
 61 
 
 92 
 
 84, 85 . . 
 
 .. 127 
 
 7 
 
 12 
 
 37 
 
 47 
 
 62 
 
 93 
 
 86 .. .. 
 
 128 
 
 S 
 
 . , , 13 
 
 38 
 
 49 
 
 63 . 
 
 , 95 
 
 87 .. .. 
 
 . 129 
 
 9 
 
 . . . 14 
 
 39, 40 
 
 . ..50 
 
 64 
 
 96 
 
 88, 89 . . 
 
 .. 133 
 
 10 
 
 . . , 15 
 
 41 
 
 ,,52 
 
 65 
 
 , . 100 
 
 90, 91 . . 
 
 .. 134 
 
 11, 12 
 
 . ..18 
 
 42 
 
 54 
 
 66 
 
 102 
 
 92 .. .. 
 
 135 
 
 13, 14 
 
 . ..21 
 
 43, 44 
 
 . ..56 
 
 67 
 
 103 
 
 93 .. .. 
 
 140 
 
 15 
 
 23 
 
 45 
 
 . ..57 
 
 68 
 
 105 
 
 94 .. .. 
 
 , 141 
 
 16 
 
 . 24 
 
 46 
 
 61 
 
 69, 70 . 
 
 . . 106 
 
 95 . . 
 
 143 
 
 17, 18 
 
 . ..27 
 
 47, 48 
 
 . ..62 
 
 71 
 
 , 108 
 
 96 .. .. 
 
 144 
 
 19 
 
 28 
 
 49 
 
 . ! 63 
 
 72 
 
 110 
 
 97 .. .. 
 
 , , 145 
 
 20, 21 
 
 . ..29 
 
 50 
 
 , 64 
 
 73, 74 . 
 
 .. 112 
 
 98, 99 . . 
 
 .. 146 
 
 22 
 
 30 
 
 51 
 
 66 
 
 75 
 
 , . 113 
 
 100 .. .. 
 
 .. 147 
 
 23 
 
 31 
 
 52 
 
 68 
 
 76 
 
 117 
 
 101 .. .. 
 
 148 
 
 24 
 
 32 
 
 58 
 
 71 
 
 77 
 
 , . 118 
 
 102, 103 . . 
 
 .. 150 
 
 25 
 
 33 
 
 54 
 
 74 
 
 78 
 
 , . 119 
 
 104 .. .. 
 
 151 
 
 26 
 
 34 
 
 
 
 
 
 
 
 Printed by Hazdl, Watson, & Viney, Ld„ London and Aylesbury.