OCULAR DEFECTS 
 
 &e. 
 
LONDON: PRINTED ur 
 
 SPOTTISWOOUB AND CO., NEW-STKEKT SQUARE 
 AN1> PAIILIAMENT STKEET 
 
THE THEORY 
 
 OF 
 
 OCULAE DEFECTS 
 
 AND OF 
 
 SPECTACLES. 
 
 TRANSLATED FROM THE GERMAN OF 
 
 DK. HEBMANN SCHEFFLER 
 
 ROBERT BRUDENELL CARTER, F.R.C.S. (ExAM.) 
 
 FELLOW OF THE ROYAL MEDICAL AND CHIRURGICAL SOCIETY, 
 COXSULTIXG SURGEON TO THIS GLOUCESTERSHIRE EYE INSTITUTIOX. 
 
 With PREFATORY NOTES and a Chapter of PRACTICAL INSTRUCTIONS. 
 
 LONDON : 
 
 LONGMANS, GKEEN, AND CO. 
 1869. 
 

c 
 
 PKEFACE. 
 
 EARLY in the past year, I received from Dr. Scheffler a 
 copy of his Theorie der Augenfehler und der Brille, then just 
 published, with a request that I would endeavour to make 
 arrangements for the appearance of an English version. 
 
 The labours of Professor Bonders, which were first made 
 known in this country in 1862, and which were fully laid 
 before the profession in 1864, in his great work on The 
 Anomalies of Refraction and Accommodation of the Eye, pro- 
 duced so great an advance in this department of scientific 
 inquiry as to lead to the impression, among many, that little 
 more remained to be achieved. The contrast between the 
 results afforded by the empirical employment of spectacles in 
 former times, and those afforded by their proper adaptation 
 subsequently, was certainly so great as to explain, if not to 
 justify, the general prevalence of such a belief. 
 
 In common, however, I think, with many other ophthalmic 
 surgeons, I had gradually become convinced that the subject was 
 not exhausted, and that the distinguished Utrecht Professor 
 had left some room for others, even in the field that he had 
 cultivated with such signal success. Cases every now and then 
 presented themselves in which the facts would not perfectly 
 harmonize with the doctrines, and in which there was mani- 
 festly present some additional unknown quantity, some disturb- 
 ing factor, of which the doctrines took no account. Spectacles 
 that seemed properly adapted to the requirements of the eyes 
 were found to become sources of discomfort ; and a fresh and 
 
VI PREFACE. 
 
 more careful testing of vision failed to furnish an explanation 
 of its cause. 
 
 Under these circumstances I was prepared to receive gladly 
 any additional information upon the subject ; and a first perusal 
 of Dr. Scheffler's treatise assured me that, even if he had not 
 removed my difficulties, he had at least pointed out the way 
 in which their removal should be attempted. The separate 
 examination and complete analysis of the several elements of 
 the visual act, and the introduction of the function of l appli- 
 cation,' were steps that seemed likely to explain the conditions 
 actually present in obscure cases, and to supply the unknown 
 factor of which I had felt the need. Especially I believed that 
 I saw through them a clue to the comprehension of some 
 perplexing questions arising out of the manifold varieties of 
 squint ; and the view put forth in the chapter on the * Op- 
 ponency of the Fields of Vision,' by which the nervous fibres 
 of the two eyes are regarded as having functions in some sense 
 analogous to those of the positive and negative poles of a gal- 
 vanic circuit, and as being able, by a sort of commutation, to 
 reverse the action and relieve each other by becoming, as it 
 were, anodes and cathodes by turns, seemed likely, if con- 
 firmed by further investigation, not only to explain many 
 important visual phenomena that are at present imperfectly 
 understood, but even to throw light upon the general purpose 
 of the duality of nervous organs. For a complete account of 
 these theories, and of the facts upon which they are founded, 
 the reader must be referred to Dr. Scheffler's earlier works, 
 the PTiysiologische Optik, and the Gesetze des raumlichen 
 Sehens, to which reference is so frequently made in the text 
 of the present treatise. 
 
 As regards the combination, in spectacles, of lenses and 
 prisms, I was aware that steps in this direction were taken, 
 
PREFACE. Vli 
 
 several years ago, by Dr. Giraud-Teulon, of Paris ; and that 
 his * decentred ' spectacles, as they were called, were sold 
 in London. On referring to his writings upon the subject I 
 find that his work was, to a certain point at least, parallel with 
 that of Dr. Scheffler ; and to Dr. Giraud-Teulon belongs, 
 as far as I know, the merit of having originated the idea. To 
 Dr. Scheffler belongs, I believe, the merit of having worked 
 out the particular ' orthoscopic ' combination ; and of having 
 investigated the forces that must be taken into account before 
 the combinations required for special cases can be correctly 
 determined. 
 
 The original work, as published in Germany, was carried 
 down only to page 167 of that now in the hands of the reader. 
 I suggested to Dr. Scheffler that it would be desirable to 
 supply detailed practical instructions for the examination of 
 eyes and the selection of spectacles ; and this task he was good 
 enough to undertake. The last seventy-three pages are devoted 
 to the subject, and were translated from the author's manuscript. 
 
 The test case, or the collection of lenses and prisms, re- 
 quired for carrying out the investigations, has demanded 
 careful consideration. As regards the primary element, the 
 lenses, it is well known that much dissatisfaction has been felt 
 with the irregular intervals between those in the test cases 
 commonly sold ; and that confusion has been produced by the 
 varieties of the so-called t inches ' in which focal lengths have 
 been expressed. At the last International Ophthalmological 
 Congress, held at Paris in 1867, a committee was appointed 
 to investigate these questions. Their report is not yet made 
 public, but I have reason to believe that they will advise the 
 general adoption of a metrical scale, and of a series of which 
 the unit, and the interval between each two steps, shall be a 
 lens having a focal length of 240 centimetres. This would 
 
Vlll PREFACE. 
 
 give a series of 48 convex and 48 concave lenses, proceeding 
 in regular gradation from 240 to 5 centimetres of focal length. 
 These lenses might be designated by their numbers, according 
 to the plan suggested by Dr. Zehender. The lens of 240 cm. 
 focal length would be No. 1, the unit of the series. The lens 
 of 120 cm. focal length would be No. 2, and would be equal 
 to two such units. The lens of 24 cm. focal length would be 
 No. 10, and would be equal to ten such units. The lens of 
 5 cm. focal length would be No. 48, and would be equal to 
 forty-eight such units. 
 
 As a practical question, however, it would net be necessary 
 to possess the whole of such a series ; because, if the lenses 
 were made with one plane side, it would be easy to supply any 
 breaks by combinations of two, put together in the testing 
 frame. A set of twenty pairs of convex lenses, and twenty 
 pairs of concave lenses, would be sufficient for every purpose ; 
 and the interrupted series given in the following table is that 
 which would, I believe, satisfy all requirements. The first 
 column contains the number of the lens, the second its focal 
 length in centimetres, the third its focal length in Paris inches, 
 the fourth its focal length in Prussian inches, and the fifth the 
 refracting angle (i.e. the measured angle between the sides), 
 expressed in English degrees, of the prism that, for eyes sepa- 
 rated by the ordinary interval, would form with it an ortho- 
 scopic combination. The convex and the conca,ve series would, 
 of course, be the same,; and the prisms could be used with 
 either, by reversing the direction of their bases in the testing 
 frame. In the series suggested it will be seen, by a glance at 
 the table, that the breaks can be supplied, between 6 and 24, 
 by the addition of No. 1 to the lower number ; between 24 
 and 30, by the addition of No. 1 or No. 2 ; between 30 and 
 34, by one of the first three numbers; between 34 and 40, by 
 one of the first five ; and between 40 and 48, by one of the 
 
PREFACE. 
 
 IX 
 
 first six, whose number, when added to the next lower num- 
 ber in the table, will make up the number required. 
 
 Number 
 of each. 
 Lens 
 in the 
 Series 
 
 Focal Length 
 in 
 
 Centimetres 
 
 Focal Length 
 in 
 Paris Inches 
 
 Focal Length 
 in Prussian 
 Inches 
 
 Angular Measurement 
 of the Prism 
 that forms with it an 
 ' Orthoscopic Com- 
 bination ' 
 
 1 
 
 240 
 
 88-66 
 
 91-75 
 
 1 29' 10" 
 
 2 
 
 120 
 
 44-33 
 
 45-88 
 
 2 58 23 
 
 3 
 
 80 
 
 29-55 
 
 30-58 
 
 4 27 37 
 
 4 
 
 60 
 
 22-16 
 
 22-94 
 
 5 56 54 
 
 5 
 
 48 
 
 17-73 
 
 18-35 
 
 7 26 14 
 
 6 
 
 40 
 
 14-78 
 
 1529 
 
 8 55 39 
 
 8 
 
 30 
 
 11-08 
 
 11-47 
 
 
 10 
 
 24 
 
 8-86 
 
 9-17 
 
 
 12 
 
 20 
 
 7-38 
 
 7-65 
 
 
 14 
 
 17-14 
 
 6-33 
 
 6-55 
 
 
 16 
 
 15 
 
 5-54 
 
 5-73 
 
 
 18 
 
 13-3 
 
 4-93 
 
 5-10 
 
 
 20 
 
 12 
 
 4-43 
 
 4-59 
 
 
 22 
 
 10-90 
 
 4-03 
 
 4-17 
 
 
 24 
 
 10 
 
 3-69 
 
 3-82 
 
 
 27 
 
 8-8 
 
 3-28 
 
 3-40 
 
 
 30 
 
 8 
 
 2-95 
 
 3-06 
 
 
 34 
 
 7-05 
 
 2-61 
 
 2-70 
 
 
 40 
 
 6 
 
 2-22 
 
 2-29 
 
 
 48 
 
 5 
 
 1-85 
 
 1-91 
 
 
 The angular measurement of the prisms, in degrees, minutes, 
 and seconds, as given in the table, has been calculated for 
 each focal length/, upon the index of refraction (l - 53) of the 
 glass actually used by Messrs. Paetz and Flohr, of Berlin, for 
 making testing prisms, and upon the supposition that d, the 
 interval between the centres of the eyes, has its average 
 value of 66 millimetres, according to the formula on page 6, 
 
 , and in the manner following: If 
 / 
 
 be 
 
 *(n-l) 
 
 the refracting angle of the prism, 
 
 m = the chord (to radius 1) of the subtending arc. 
 
 = 2 sin. t . 
 
sin. - = 
 
 PREFACE. 
 d 1 
 
 2 4 (-!)'/' 
 whence the value of sin. - can be determined, and thence - , 
 
 and therefore </>, be found from a table of sines. 
 
 Ex. 1. By natural numbers. 
 
 Let d = 66 millimetres; / = 2400 millims. ; 
 n = 1-53 /. n 1 = 0'53. 
 
 /. _ - == = 31-1320755 ; 
 
 4 (n - 1) 4 x 0-53 
 
 /. sin. ^ = 31>182Q755 = 0-0129717, 
 which will be found in the tables as sin. 44' 35 ". 
 .-. 1 = 44' 35", .-. 0=1 29' 10". 
 
 Ex. 2. The same by logarithmic computation. 
 
 Log. sin. |- = 10 + log. 66 log. 4 log. 0-53 loo-. 2400. 
 
 2 
 
 10+ log. 66 = . . . . 11-8195439 
 log. 4 =0-6020600 
 
 log. 0-53 = f-7242759 . . 0-3263359 
 
 /. 10 + log. 66 - (log. 4 + log. 0-53) . = 11-4932080 
 
 and log. 2400 . = 3-3802112 
 
 .-.log. sin. | . = 8-1129968; 
 whence from the table of logarithmic sines 
 
 | = 44' 35", and = 1 29' 10" as before. 
 
 Hence, to find the refracting angle required : 
 
 1. By natural numbers; divide 31-1320755 by the pro- 
 posed focal length of the lens in millims., look for the quotient 
 in a table of natural sines, and double the angle there given ; or 
 
PREFACE. XI 
 
 2. By logarithms; subtract the logarithm of the focal 
 length in millims. from 114932080, look for the remainder in 
 a table of logarithmic sines ; double the angle thus found will 
 be the angle required. 
 
 If it be desired to find the angle for any other breadth 
 between the eyes than 66 millimetres, or for glass of any other 
 refractive index than T53, this can be done from the com- 
 mencement, after the manner of either of the preceding 
 examples. 
 
 It is found in practice that the difficulties arising from aber- 
 ration and dispersion become insuperable, excepting by achro- 
 matic prisms, when an angular measure of more than 8 is 
 attained ; and therefore, for the purpose of orthoscopic com- 
 binations, and for the required testing, only the first five 
 prisms given in the table will be required, and it has not been 
 thought necessary to calculate the rest. But for other and 
 ordinary purposes, the test case should contain, of course, the 
 usual full set of prisms, from 1 to 24, the special ones 
 given in the table forming a separate series. Messrs. Weiss 
 and Son have designed an instrument for holding glasses, in 
 which the lenses and prisms may be combined ; and will be 
 prepared to supply the complete cases, from Paetz and Flohr, 
 and also the orthoscopic or other combination spectacles. 
 
 The phrase ' proportion of deviation,' used to denote the 
 quantity h or tan ;//, would have been better translated ' ratio 
 of deviation.' It is, in the original, ' AblenkungsverhciltnissJ 
 and is simply the goniometric function or ratio which ex- 
 presses the amount of deviation caused by the prism. 
 
 I put forth this little book with much hopefulness, fully 
 believing that practical use will demonstrate its value, and 
 that it will be found to contain the key to many difficulties. 
 The complete application of the author's principles is at 
 
Xll PREFACE. 
 
 present only possible within a limited range, on account of 
 the impediments mentioned above. If the progress of science 
 should furnish us with glass of higher refractive power, without 
 a corresponding increase of the dispersive power, these impe- 
 diments would be proportionately removed. 
 
 In conclusion, I have to thank the author for the care 
 with which he has replied to my inquiries upon various ques- 
 tions that have arisen from time to time ; and also to say that 
 I am indebted to the great kindness of the Eev, W. de Lancey 
 Lawson, of Fishponds, for the verification of all the formulae 
 in the work, as well as for many valuable suggestions and 
 much general assistance. Mr. Lawson has also favoured me 
 with the following explanatory notes, calculated to remove 
 difficulties that might otherwise perplex the reader at the very 
 threshold of his undertaking. 
 
 On the Optical Formulae in pages 6, 7 3 and 8. 
 
 The investigations in this work are founded on a few mathe- 
 matical formulae, presented in its early pages, which are either 
 supposed to be already known to the reader, or for which he is 
 referred to the author's earlier writings, which have not been 
 translated into English. For some of these, reference might 
 be made to well-known English books, where, however, they 
 would be found expressed in other characters, and, in some 
 cases, with other algebraical signs, than those used by Dr. 
 Scheffler ; while others are his own deductions from them, of 
 which we are here presented only with the results. It has 
 therefore been thought that it would be a boon to the English 
 reader if a short investigation of these were prefixed to the 
 translation, so as to render it intelligible, without reference to 
 other books, to any one possessed of an elementary knowledge 
 of algebra, geometry, and trigonometry. 
 
PREFACE. 
 
 Xlll 
 
 These investigations are necessarily founded on the well- 
 known law of refraction, that the ratio of the sines of the 
 angles which the incident and refracted rays make with the 
 normal, or perpendicular to the surface, at the point of refrac- 
 tion, is always constant for the same substance, and the same 
 kind of light. This ratio is called the index of refraction, and 
 is represented by n in the following pages, but by English 
 writers is usually expressed by p. It should also be observed 
 that the refracting angles of the prisms, and therefore the 
 angles of total deviation of the rays of light, are supposed to 
 be small ; so that in some cases their sines, tangents, and the 
 corresponding arcs and chords, to radius unity, may be sub- 
 stituted for each other without sensible error. 
 
 1. To find the focus of a pencil of rays after refraction 
 through a thin lens. 
 
 Let Q' Q /; be the axis of a lens A p p' B, and also that of a 
 pencil of rays proceeding from a point Q ; let Q p be another 
 ray of the pencil refracted by the first surface at P into a 
 
 Fig. a. 
 
 direction p p', produced backwards to Q' ; o, o' the centres, and 
 r 7 , r" the radii of the first and second surfaces ; and let A P be 
 supposed small, so that Q A, and Q' A are nearly equal to Q r, 
 Q' P respectively. 
 
 Let P Q = Q A = a ; /. o Q = a + r'. 
 p Q' = Q'A = a', /. o Q' = a'+ /. 
 
XIV PREFACE. 
 
 r sin. o P o sin. s P Q f since these are the ansrles of 
 
 Ihen = = n < n -, / ,. 
 
 sin. OPQ' sin. SPQ' I incidence and refraction. 
 
 But, in the triangles P o Q, P o Q', 
 
 p Q . a 
 
 sin. P o Q = - sin. o P Q = sm. OPQ, 
 
 QO a + r 
 
 sin. p o Q'= ~~ sin. o P Q' = -j- , sin. OPQ'; 
 
 Q'O a' + r' 
 
 Whence, since the angles P o Q, P o Q' are the same, we have 
 
 a a' . , 
 
 . sin. o P Q = -; sm. OPQ; 
 
 a + r a' + r' 
 
 or n . - -= j- - t ; or 
 a -f r a' + r 
 
 . % !j=,-z! M 
 
 a a! r 
 
 Then let the ray Q' P P' be refracted from the lens at p' in 
 the second surface ; it will pass out of the glass with an index 
 
 of refraction ; let it meet the axis in Q". 
 n 
 
 Let P' Q"= B Q" = x ; .-. o' Q"= x -f r"; o' Q' = a' - r", 
 since the thickness of the lens is supposed small. 
 
 It may be shown, by the same steps as in the former case, 
 that 
 
 1 a' x a' r" _ x + r" 
 
 n ' ~^- r " ~ ^+7' ' r H ' ~~d~ x 
 
 Hence, by adding [a] and [5] together, we have 
 
 . . 4 
 
 This expression, though found for a bi -convex lens, is per- 
 
PREFACE. XV 
 
 fectly general, and can be adapted to concave surfaces by 
 making one or both of the radii negative ; to plane surfaces 
 by making one or both of them infinite ; to parallel rays, 
 before or after refraction, by making a OT x infinite ; to con- 
 vergent incident rays by making a negative; to divergent 
 emergent rays by making x negative.* 
 
 To adapt it to rays parallel before refraction ; 
 
 Let a = oo , .% = ; and for x substitute/, as it will now 
 become the principal focal length of the lens ; ' 
 .*. = (n 1) ( - ; -f Tl J for a bi-convex lens. 
 
 = (n 1) ( 7 + -77 )> for a bi-concave lens, since 
 
 in the latter case r' and r" are both negative. 
 
 r ' r " 
 From the former f= , as in page 8 ; and, 
 
 by substituting this value in formula [1], we have 
 
 + = , for converging rays after refraction ; 
 
 CL X J 
 
 1 1 1 , ,. 
 
 = ; , for diverging rays. 
 
 tt X J 
 
 If r" is negative, and > r', then we have the convexo-con- 
 cave lens mentioned in p. 230, for which the formula becomes 
 
 yfyjl 
 
 f = 
 
 (n - 1) (r" - r')' 
 
 Again, from the formula 4- = , we have 
 a x f 
 
 a-/ -/ a 
 
 * Various assumptions have been made by different writers about 
 the signs of these quantities ; that of Mr. Coddington (Reflexion and 
 Refraction, 85) has been here adopted, as it makes the formulae 
 harmonize in this respect with those in the following pages. 
 
XVI 
 
 PREFACE. 
 
 where a 1 is substituted for a /; that is, the distance of the 
 original luminous point is measured from a point which lies at 
 a distance f from the front of the lens, or of the cornea of 
 the eye, to which this last formula, though with different 
 characters, is applied in pages 23 and 35. 
 
 The expression [1] is the foundation for the whole theory 
 of the lens ; but, where great accuracy is required, it admits 
 of a second and nearer approximation, and also a correction 
 for the central thickness of the lens, where that is too con- 
 siderable to be neglected. It is, however, sufficient for our 
 present purpose. 
 
 2. To find the total deviation of a ray, or axis of a pencil of 
 rays, passing through a thin prism, in terms of the base and 
 sides of the prism. 
 
 In Fig. a in the last article, suppose tangents to be drawn 
 
 Fig. b. 
 
 V 
 
 to the curves at p and p' ; then we shall have a figure like b, 
 representing a section of a prism by a plane perpendicular to 
 its edges ; and since at the points p, P' the faces of the prism 
 
PKEFACE. XV11 
 
 coincide with the curved surfaces, the direction of a ray 
 Q PP' Q" passing through these points will be the same for the 
 prism and the lens, and the equation 
 
 - + = (n-l)f - 7 + - ff ) will still hold good. 
 
 Let PN, DF, P'N' be drawn perpendicular to Q/'Q, and QP, 
 Q" p' be produced to K and I, meeting in i. Since the angles 
 at Q, Q" and D are small, we may consider approximately that 
 QN= QP a; Qf r N r =Q"p r =x m 1 PN = p / N / =e ; andDH = DE. 
 Then, the total deviation ^ = the exterior angle Q i K = angles 
 Q + Q /r . .*. tan. ^ = tan. (Q + Q") = tan. Q + tan. Q" (since 
 the angles are small), 
 
 
 
 . 
 
 Q N Q" N' \ a x J \ r f 
 
 ^ Y [a] 
 r" J 
 
 D . , e P'N' FH, .., ,. 
 
 J3ut .-= ---- = -- : and -=_ -~ -- by similar tri- 
 r P o D E r" P' o' D H 
 
 angles ; 
 
 /I IN 
 
 .\ e [_+__)= 
 \r f r" J 
 
 FE + FH HE 
 
 -= -- m. 
 
 D E D H 
 
 /. h = tan. \ = e n 1 + ~ m n ~~ 
 
 = e (n 1) ( + ~/7 J 
 
 where w = the ratio of the base to the side of the prism, or 
 = the chord, to radius unity, of the refracting angle of the 
 prism. 
 
 3. To find the relation between the lens and the prism 
 which form an orthoscopic glass as in p. 6. 
 
 Let the reader, after considering the explanation given in 
 pages 5 and 6, suppose b and &', the centres of the glasses 
 (Figs. 1 and 2), to be joined by a line b m b f cutting the frontal 
 axis af at right angles at m. 
 
 Then b m = V m =^_ = ~ nearly. 
 2 2 
 
XV111 PREFACE. 
 
 Let a = ba = b'a=ma nearly. 
 x= b e = b' e = TTZ e nearly. 
 a = the angle b a m = V a m. 
 (3 = the angle b em = b' e m. 
 Then, in Fig. 1 for the convex lens, 
 
 b m d r> b m d 
 
 tan. a = - = - ; tan. p = - = - -. 
 am 2 a em 2x 
 
 And in order that the prism may be adapted to the lens, we 
 have the angle of deviation $ = a /3, 
 
 /. fc = tan. ^ =tan.(a-/3) = ^^^^L^JL^ =tan.a-tan./3 
 ^ J 1 + tan. a tan, /3 
 
 = [_ _j; since the angles are small. 
 2 \ a x J 
 
 But, from formula [1] for the lens, we have -- 
 
 Q, X 
 
 == (n 1) ; where, since the emergent rays diverge, the 
 negative sign is given to x ; and, since the second surface is 
 plane, r"=<x> , /. = 0. 
 
 or, if the angle of the prism be required, 
 
 ; as in pages 6 and 7. 
 
 n - 1 2 (n - 1) / 
 
 The same investigation would apply to figure 2 for the con- 
 cave lens, except that /would then be negative, which would 
 make m also negative, which would imply that the base of 
 the prism must be turned in the contrary direction from the 
 frontal axis. 
 
 4. To find the distance from the centre of a larger lens, at 
 which smaller prismatic lenses may be cut eccentrically. 
 
 In Fig. b, if c be the point where the ray p p' crosses the 
 
PREFACE. 
 
 XIX 
 
 central line, and G H, L M be drawn at equal distances from 
 c and parallel to N' N, and the chords L G, M H be produced 
 to meet in I, we have the figure c, which represents a smaller 
 prismatic lens G H p M L p' cut out eccentrically from a larger 
 lens D D'. 
 
 Then, since p N is supposed approxi- 
 mately equal to p' N', the chords H M, L G 
 may be considered as parallel to the tan- 
 gents at P and p' ; the prism L I M will 
 have the same refracting angle at i as 
 that in 2, and the deflection of the ray 
 Q p p' Q" w iH n ot be altered. 
 
 Hence it may be shown, as in that sec- 
 tion, that, if F c e, 
 
 e ( ~r + -77 i = m ; " e = \ , r m ; 
 
 \ r r / r + r 
 
 as in p. 8 ; and, the other two formulae in 
 that page having been already proved, we 
 
 have 
 
 _ d 
 
 This lens, though made of a continuous 
 piece of glass, evidently consists, inform, of a prism interposed 
 between two plano-convex lenses. The reader can easily 
 conceive, or draw for himself, the figure representing a prism 
 interposed between two plano-concave lenses, where the base 
 of the prism will be turned from the centre of the larger lens, 
 and its sides will be tangents to its surfaces at p and p'. 
 
 It will be seen from this investigation, that any eccentric 
 portion of a lens must always combine the deflecting proper- 
 ties of a prism, with the focalizing properties due to its own 
 curved surfaces. 
 
XX PREFACE. 
 
 To express the prism, required in any case, in degrees 
 of its refracting angle. 
 
 If, instead of adapting a prism, to a lens of given focal length 
 f, it be required to find a prism of the convergence distance g, 
 since 
 
 m = -Ap and 7*=^, (page 70) 
 
 d 1 . d 1 
 
 or, by substituting g for /*, the same method may be applied, 
 or the same table used, to find the angle of the prism. 
 
 If ;//, the required angle of deviation of the prism, be given 
 in degrees, as in page 77, and be small, we may, without con- 
 siderable error, find the refracting angle ty from the equation 
 
 so that $, the refracting angle of the prism, will be very nearly 
 equal to double the angle of total deviation. 
 
 8 PRINCES STREET, HANOVER SQUARE : 
 January 1869. 
 
CONTENTS. 
 
 PAGE 
 
 PREFACE v 
 
 INTRODUCTION 1 
 
 ORTHOSCOPIC SPECTACLES 4 
 
 THE INDEPENDENCE AND THE INTERDEPENDENCE OP ACCOMMO- 
 DATION AND CONVERGENCE 17 
 
 DEFECTS OF ACCOMMODATION, OR ANOMALIES OF EEFRACTION 
 
 AND OF ACCOMMODATION 22 
 
 SPECTACLES FOR DEFECTS OF ACCOMMODATION AND EEFRACTION 49 
 
 DEFECTS OF CONVERGENCE, OR ANOMALIES OF DIRECTION AND 
 
 OF MOTILITY 61 
 
 SPECTACLES FOR ANOMALIES OF DIRECTION AND MOTILITY . 69 
 CONCURRENCE OF DEFECTS OF ACCOMMODATION WITH DEFECTS 
 OF CONVERGENCE. COMBINATION SPECTACLES . . . .78 
 
 DETERMINATION OF VISUAL STRAIN 82 
 
 ASTHENOPIA 86 
 
 GRAPHIC DELINEATION OF THE COMBINED DEFECTS OF ACCOM- 
 MODATION AND OF CONVERGENCE 88 
 
 THE KECIPROCAL INFLUENCE OF THE ACCOMMODATION AND THE 
 
 CONVERGENCE 91 
 
 APPLICATION 95 
 
 DEFECTS OF APPLICATION, OR ANOMALIES OF POSITION AND OF 
 
 NUTATION 103 
 
 SPECTACLES FOR ANOMALIES OF POSITION AND OF NUTATION . 110 
 
XX11 CONTENTS. 
 
 PAGE 
 
 THE CONCURRENCE OF DEFECTS OF ACCOMMODATION, CONVERGENCE, 
 AND APPLICATION . ... . . .,.111 
 
 UNILATERAL OCULAR DEFECTS 121 
 
 APPARENT SIZE AND DISTANCE .,'.". 125 
 
 PERCEPTION OF SIZE AND DISTANCE 133 
 
 ASYMMETRICAL OCULAR DEFECTS , 138 
 
 THE ASSISTANCE TO FIXATION BY THE DISPLACEMENT OF THE 
 BACILLI . ; , . 142 
 
 THE EEACTION OF OCULAR DEFECTS UPON ONE ANOTHER . .144 
 THE OPPONENCY OF THE FIELDS OF VISION . . . .152 
 THE MICROSCOPIC EXAMINATION OF THE KETINA BY MAX SCHULTZE 154 
 
 PRACTICAL RULES FOR THE DETERMINATION OF PROPER 
 SPECTACLES 168 
 
 1. ANOMALIES OF REFRACTION AND ACCOMMODATION . 169 
 
 2. ANOMALIES OF DIRECTION AND OF MOTILITY . . 190 
 
 3. ANOMALIES OF POSITION AND NUTATION . . .201 
 
 4. SPECTACLES FOR DISSIMILAR EYES . 205 
 
 5. PERISCOPIC COMBINATION SPECTACLES . . . .219 
 
 6. THE MANUFACTURE OF COMBINATION GLASSES . 233 
 
THEOEY 
 
 OF 
 
 OCULAR DEFECTS. 
 
 INTEODUCTION. 
 
 THE DETERMINATION of the proper spectacles, for any defect of 
 the eyes, is undoubtedly a matter of the greatest importance 
 in practical ophthalmology. This importance is increased, on 
 the one hand, by the number of patients requiring such 
 assistance a number that, throughout the civilised world, 
 may be reckoned by millions on the other, by the actual 
 injury inflicted by unsuitable glasses; and, lastly, by the 
 customary sale of most spectacles by persons who have no 
 adequate knowledge of the conditions of their utility. 
 
 Since the visual process is not an absolutely simple action, 
 but is compounded of several acts which are originally in- 
 dependent, and which, through the unity of the sensorium, 
 come secondarily to influence one another, it follows that, in 
 defects of vision, the determination of the necessary spectacles 
 can never be a perfectly simple procedure. It requires a due 
 consideration of all the ocular functions; and the neglect of 
 any one of them may produce a strain which will act injuriously, 
 and will impair the eyes ; while the proper spectacles, adapted 
 to correct the defect, would be preservative and beneficial. 
 For example, it is possible that a given pair of eyes, in order 
 to see clearly at a given distance, may exert only a moderate 
 
 B 
 
2 THEOKY OF OCULAR DEFECTS. 
 
 effort of accommodation, together with an excessive effort of 
 convergence ; and this discord between accommodation and 
 convergence may produce a painful strain, may prevent con- 
 tinued exertion of the eyes at the particular distance, and may, 
 by prolonged effort, aggravate the original defect. Spectacles 
 which remove this discord, and which in moderate tension 
 of the accommodation call for only a moderate degree of 
 convergence, would relieve all the inconvenience, and would, 
 consequently, exert a beneficial influence upon the organs of 
 vision. 
 
 But although we may set down the removal of disproportion 
 between the several elements of the visual act as an essential 
 principle in the selection of spectacles, yet in the practical 
 application of this principle it is easy to fall into error, unless 
 that which we call a disproportion be thoroughly understood. 
 In any particular case we must not affirm the existence of 
 disproportion between accommodation and convergence, be- 
 cause their relative tension differs from that of a perfectly 
 normal pair of eyes, or from the absolutely normal proportion 
 between the two functions ; since that proportion which is 
 most suitable for a given pair may be determined by their 
 special physiological and pathological conditions, or by the 
 nature of their defects. We have, therefore, generally speak- 
 ing, to consider an individual or relative normal proportion, 
 which cannot be deduced from completely normal eyes alone, 
 but only by the proper estimation of the influences of certain 
 abnormal conditions. 
 
 It is well known that perfectly normal eyes, by unusual 
 methods of exertion, may be brought into a state in which the 
 absolutely normal proportion between the elements of the 
 visual function becomes itself a cause of strain. 
 
 This state may even become more lasting ; and then the 
 conditions for the suitable or relative normal proportion are 
 already modified ; the eye has become accustomed to an ab- 
 normal proportion between its functions. Most defective eyes 
 are found to be permanently in such a condition ; in which a 
 
INTRODUCTION. 3 
 
 proportion, more or less deviating from 'the normal, is the one 
 most comfortable to them. 
 
 If it be necessary to quote any ophthalmological authority 
 for these facts, I may adduce the words of Profess cr Bonders, 
 in his work on the ' Anomalies of Befraction and Accom- 
 modation.' He says, on pages 124, 125, ' The myopic eye 
 has learnt to converge in a certain degree, without bringing 
 its power of accommodation into action in the same proportion 
 as the emmetropic eye. The hypermetropic eye, on the con- 
 trary, found itself obliged, even with parallel visual lines, to 
 put its power of accommodation on the stretch, and it has 
 brought itself so far in that respect, that it is no longer in a 
 position to become completely relaxed ; that at least on every 
 effort to see, the act of accommodation takes place involuntarily. 
 The use of positive or negative glasses has, even after the 
 lapse of a few hours, an influence on the range of accom- 
 modation of the emmetropic eye. The relative range of 
 accommodation is displaced in ametropic individuals when 
 they have for a long time worn correcting glasses.' 
 
 It therefore depends very much upon circumstances how 
 far an abnormal proportion between the functions, when once 
 established, should be respected ; and how far it should be 
 corrected by glasses. In my Physiologische Optik, and in 
 the supplement thereto, entitled Die Gesetze des raumlichen 
 SehenSj I have entirely assented to the principle of latitude in 
 reference to the modification of existing functional proportions ; 
 and I have given calculations and conclusions chiefly for those 
 ocular defects in which the functional proportion actually 
 present during unaided vision is the most suitable, and is to 
 be maintained. As far as the theory is concerned, it is im- 
 material whether the number of such cases be large or small ; 
 but the application of the theory in practice requires a careful 
 study of the cases in which the contrary will hold good, and 
 the more so, the more numerous they are. The most recent 
 observations, and especially those of Bonders and von Graefe, 
 seem to leave no doubt that cases of the latter kind, in 
 
 B2 
 
4 THEORY OF OCULAR DEFECTS. 
 
 which it is necessary to combat the actually existing pro- 
 portion in defective eyes, occur in considerable numbers. I 
 purpose, therefore, in the following pages, to place before 
 ophthalmologists a general mathematical analysis of ocular 
 defects, and of the principles on which spectacles for them 
 should be selected. I believe that the data thus furnished, 
 together with the admirable works of Donders and of von 
 Graefe, will not only contribute to an elucidation of the nature 
 of the visual processes, but will also furnish useful aids to 
 practical treatment. 
 
 ORTHOSCOPIC SPECTACLES. 
 
 Before entering upon the special subject of my inquiry, it 
 seems desirable to justify, by a short description, the com- 
 mendation that I have bestowed in my earlier writings upon 
 orthoscopic spectacles. I believe that these glasses, although 
 they may require modification in many instances, are yet the 
 most useful for large classes of ocular defects ; and that they 
 rest upon the only rational basis of spectacle construction for 
 binocular vision ; so that modifications of them should be 
 considered exceptions, rendered necessary by the peculiarities 
 of individual cases. 
 
 When any normal or abnormal pair of eyes is furnished 
 with ordinary convex or concave centric spectacles, no change 
 takes place in the direction of the visual lines, and none, there- 
 fore, in the convergence of the eyes; although the object 
 looked at changes its apparent distance on the visual line of each 
 eye. There is therefore a change in the accommodation, while 
 the convergence remains unaltered ; and it follows that the pro- 
 portion between the two functions is modified. There is no 
 longer a single object presented to the two eyes, but to each 
 eye its own object : the two no longer fixing a single point in 
 space, but two separated points. (It is evident that no double 
 
ORTHOSCOPIC SPECTACLES. 5 
 
 vision would be thus produced, since the two images fall upon 
 identical portions of the retinae.) By this the eyes are placed 
 in a condition which does not correspond with their state 
 during unassisted vision without glasses, since a single ob- 
 ject is then always presented to them. Under certain cir- 
 cumstances the modification so produced may be beneficial ; 
 
 Fig. 1. 
 
 Fig. 2. 
 
 but, in all cases in which the functional proportion maintained 
 during unassisted vision is the most convenient, the use of 
 common spectacles produces an undesirable visual strain. 
 
 The spectacles that I have called orthoscopic consist of a 
 combination of lenses and prisms, so arranged that the devia- 
 tion of the rays of light occasioned by the prism corresponds 
 accurately with the change in the visual distance occasioned 
 
6 THEORY OF OCULAR DEFECTS. 
 
 by the lens. Through such spectacles (Fig. 1 or Fig. 2), the 
 simple actual object a is displaced, by means of the lenses, to 
 the distances g d, g'd' ; and, by means of the prisms, in the 
 directions g e, g'e. There is presented to the eyes, therefore, 
 through the orthoscopic spectacles, instead of the single actual 
 object a, a single apparent object e ; and the eyes, looking at 
 a through the spectacles, are in precisely the same condition 
 as if they were looking at e without spectacles. Vision through 
 these spectacles is therefore, for every pair of eyes, whether 
 normal or abnormal, completely free vision. The spectacles 
 produce only an apparent change in the distance of the object, 
 and a consequent displacement of the region of accommoda- 
 tion ; with the result that the limits of the clearly visible space 
 are, according to the power of the glasses and the state of the 
 eyes, extended, contracted, or moved. 
 
 From the foregoing it follows that the orthoscopic spectacles 
 never produce strain. Although they may not diminish the 
 strain sometimes associated with the unaided use of the eyes, 
 they never increase it ; and usually, indeed, leave it wholly 
 unchanged. 
 
 It is of special importance that the proportion of the prism to 
 the lens, in an orthoscopic glass, should be determined not on 
 physiological, but solely on physical principles, and by a purely 
 mathematical calculation. Iff be the positive focal length of 
 a convex, or the negative focal length of a concave lens, d the 
 distance between the central points of the two eyes, n the 
 index of refraction of the glass, m the proportion of the base 
 of a prismatic or wedge-shaped glass to the length of its sides, 
 $ the angle of total deviation of the prism, h = tang. i|/, the 
 proportion of deviation of the prism, then the prism that 
 should be combined with the lens f to form an orthoscopic 
 glass is shown by the formula: 
 
 d 1 
 
 or, since we have h = m(n 1), by the formula, 
 
ORTHOSCOPIC SPECTACLES. 7 
 
 d 1 
 
 h = tang. $ = 2 -j- 
 
 For a convex lens the base of the prism must be turned 
 inwards, that is, towards the nose ; and for a concave lens it 
 must be turned outwards. 
 
 The above formula gives for every lens a determinate 
 prism, which forms with it an orthoscopic glass. This prism 
 is wholly independent of the distance of the object; an ortho- 
 scopic glass being orthoscopic not only for a given distance, 
 but for all distances. Its determination does not rest upon 
 physiological grounds, and it is therefore orthoscopic for any 
 eye, whether normal or abnormal. From hence it follows 
 that the orthoscopic glass is the rational or absolute basis for 
 the construction of spectacles for binocular vision. 
 
 A spectacle lens composed of a spherical and a prismatic 
 glass may be called, as a general term, a combination lens. 
 Such a lens becomes orthoscopic by reason of the proper value 
 of its prism. 
 
 It is evident that the prism and the convex or concave lens 
 need not be formed from separate pieces, but may be ground 
 from a single glass, and that the combination may be either 
 plano-convex and plano-concave, a and a' (Fig. 3), bi-convex 
 and bi-concave, b and &', or periscopic, c and c'. 
 
 . Fig. 3. 
 J c u/ k' 
 
 A) f ? 
 
 
 In 50 of the Physiologische Optik, I have also shown 
 that these glasses may be regarded as eccentric portions of 
 larger lenses, and may therefore be called eccentric glasses. 
 
8 THEORY OF OCULAR DEFECTS. 
 
 This is of special importance as regards their fabrication, since 
 the larger lenses, from which the eccentric portions are cut, 
 may be made in the ordinary way. It is shown in p. 99 of 
 
 Pig. 4. 
 
 Fig. 5. 
 
 the second part of the Physiologische Optik, that the distance 
 F c = e (Figs. 4 and 5) of the centre c of the eccentric glass A B 
 from the centre F of the larger lens, and that, when we take 
 r' and r" as the radius respectively of the upper and under 
 surface, 
 
 m. 
 
 If we substitute for the radii r" and r' the focal length of 
 the lens/, after the formula, 
 
 r' r" 
 
 and for the prism m the already ascertained value 
 
 d 1 
 
 we obtain, for the distance F c, the value 
 
 d 
 
 This result is highly important. It teaches that the dis- 
 tance of the centre c of the eccentric spectacle lens A B, from 
 
OKTHOSCOPIC SPECTACLES. 9 
 
 the centre F of the large lens out of which it is cut, must in all 
 cases be equal to the distance of the centre of the eye from 
 the axis of the nose, whatever may be the focal length of the 
 lens. 
 
 Furthermore, it shows that two companion eccentric lenses, 
 A B and A' B', may be cut from the single large lens B B', since 
 in such a lens there are two portions of which the centres c 
 and c' measure exactly the distance apart c c' =2e, p d, the 
 distance between the centres of the eyes. 
 
 If we consider further, that spectacle lenses have a breadth 
 of 1*3 Paris inch, and that therefore the large lens, for a 
 medium distance of 2*44 inches between the eyes, should have 
 a breadth of 3*74 inches, or, on account of waste in grinding, of 
 4 inches, the diagram in Fig. 6 will show that a large lens of 
 such dimensions would furnish three pairs of similar eccentric 
 lenses, or three pairs of orthoscopic spectacles. 
 
 The fabrication of eccentric glasses from larger lenses is on 
 more than one account important. The cost is thus very con- 
 siderably reduced. The large lens furnishes the eccentric 
 ones in perfect pairs, of identical material, with equal curva- 
 tures, and with similar prismatic forms always corresponding 
 to the curvature. 
 
 The glasses cut from such a large lens have always the focal 
 length of the large lens itself, excepting for the slight difference 
 due to spherical aberration. Without regard to this, it is 
 highly important in manufacture that an eccentric lens, cut 
 out of a large one in the manner described, is always ortho- 
 scopic, that is, that its convexity or concavity is always in 
 unison with its prismatic form, whatever may be the qualities 
 of the glass or its index of refraction. 
 
 If we allow, as in Fig. 7, the rays from a luminous point G 
 to pass through the symmetrically situated parts A B and A' B' 
 of the large lens D D', these rays will be reunited in a point H 
 of the axis G H ; and this reunion in a single axial point is the 
 characteristic of orthoscopic spectacles. 
 
 The defect from spherical aberration, above referred to, 
 
10 
 
 THEORY OF OCULAR DEFECTS. 
 
 amounts to this, that the two bundles of rays, passing from 
 the point G through the two portions A B and A' B' of the lens 
 
 Fig. 6. 
 
 D, do not unite accurately in one axial point H, but in two 
 lateral points J and j'. This defect increases with the curva- 
 ture of the lens, and may easily be shown in convex lenses by 
 covering the surface D D' by a diaphragm perforated so as to 
 
OBTHOSCOPIC SPECTACLES. 
 
 11 
 
 admit light only to the portions A B and A' B'. The rays from 
 G, passing through these portions, must be received upon a 
 screen, placed at the focal length of A B and A' B', as shown by 
 the sharp definition of the images J and j'. 
 
 Fig. 7. 
 
 If these two images do not exactly coincide, the portions 
 A B, A' B', in the same relative position that they occupy in the 
 large lens, will not be perfectly orthoscopic. 
 
 If the distance of the two foci, J and j', from the axis, be 
 only small, the error may be corrected by setting the two 
 glasses A B and A' B', after they are cut out, nearer together or 
 farther apart, until the foci unite. In this way we obtain 
 orthoscopic glasses for eyes of which the centres are separated 
 by a distance a little greater or less than the original distance 
 cc'. 
 
12 THEORY OP OCULAR DEFECTS. 
 
 If the distance between the two foci be more considerable, 
 or if the glasses are required for eyes separated by precisely 
 the distance c c', the foregoing method is not available ; and 
 it becomes necessary to modify the refracting angle in each 
 glass. It is, notwithstanding, always possible to cut the pair 
 of lenses required for orthoscopic spectacles from a single lens 
 of the given curvature. For when the distance of the centres 
 C c', of the portions to be cut out, is increased, the two foci 
 j j', when situated as in Fig. 7, also become still wider apart ; 
 but the distance between them does not increase in the same 
 degree as the distance between the openings A B, A' B'. Accord- 
 ingly, the distance between the openings in the diaphragm 
 may be increased until the distance apart of their central 
 points, c c', is too great by the distance apart of the foci 
 j j'. Then the lenses A B, A' B', when cut out, may be again 
 approximated, in the spectacle frame, by the same distance 
 j j' ; and in this way orthoscopic glasses are obtained for the 
 given distance C c' of the centres of the eyes. 
 
 As regards manufacture, it is highly important that this 
 accuracy can always be secured, without calculation, by simple 
 experiment with the diaphragm and screen upon the large lens. 
 
 For concave spectacles, which have no actual, but only vir- 
 tual foci, the proceeding must be somewhat modified. In 
 order to avoid repetitions, I will first refer to the means by 
 which glasses are to be tested. 
 
 It is desirable to have a simple method by which the ortho- 
 scopic character of any given pair of spectacles can be readily 
 proved. This method is furnished, for convex spectacles, by 
 holding them between any source of light and a screen, and 
 by moving them to and fro until they throw sharply-defined 
 images of light upon the screen. When these images are per- 
 fectly defined, they should exactly cover each other. If with 
 perfect definition their coincidence is incomplete, the spec- 
 tacles are not completely orthoscopic. 
 
 This method of testing can only be applied to convex glasses. 
 For concaves, which furnish only virtual images, the pro- 
 
ORTHOSCOPIC SPECTACLES. 
 
 13 
 
 ceeding given on p. 187 of Die Gesetze des raumlichen 
 Sehens, is to be recommended. In this, the rays from the 
 most distant possible luminous point G (Fig. 8), after passing 
 through the glasses A B, A' B', are received on two screens, L I/ 
 
 Fig. 8. 
 
 N' N' 
 
 and N N'. The resulting images are bright spots, having re- 
 spectively the diameters K L, K' I/, and M N, M' N'. These bright 
 spots are the bases of cones M H N, M' H ; N', having for their 
 apex the virtual focus H of the concave lenses. Beside these 
 bright spots there are two shadow spots, which form the bases 
 of cones having for their apex the luminous point G ; and 
 which, therefore, partly cover the bright spots when the screens 
 are near together, but are separate from them when the screens 
 
14 THEORY OF OCULAR DEFECTS. 
 
 are farther apart. If the spectacles are orthoscopic, the dia- 
 meter of the bright spots must increase, on the hinder screen, 
 in the same proportion as the breadth of the interval between 
 them ; that is, M' N', M N, must be larger than i/ K', L K, in the 
 same proportion as M M' is greater than K K' : since the rays 
 B L N, A K M, A' K' M', B' i/ N', have one and the same virtual 
 focus H as their point of convergence. 
 
 It would be a grave error to suppose that an oblique posi- 
 tion of common convex or concave centric lenses would give 
 to the spectacles an orthoscopic character. An oblique posi- 
 tion across the visual line has very little influence on the 
 course of the rays of light, either with common or with ortho- 
 scopic spectacles. But the orthoscopic glasses are very sensi- 
 tive to any rotation of the lenses upon the visual line, since 
 this will produce separation of the images of the object. It is 
 therefore necessary that the excentric lenses should be so fast- 
 ened in their frame that their thinnest and thickest points 
 should lie accurately in a horizontal plane, passing through the 
 centres of the eyes. 
 
 On account of the inconsiderable effect of an oblique position 
 of spectacle lenses, we may unhesitatingly place a meniscus, 
 or a concavo-convex glass (Figs. 9 and 10) c D, which has been 
 cut for an orthoscopic lens from the larger meniscus B B', in 
 the position c D, in which it is also periscopic. Moreover, as 
 the middle curve of a periscopic glass may be of any radius, so 
 two companion lenses for orthoscopic-periscopic spectacles may 
 be supplied by a smaller meniscus than B B'. 
 
 Orthoscopic spectacles, whether convex or concave, whether 
 of high or low power, have the remarkable peculiarity that 
 they may be used by any pair of eyes, normal, short-sighted, 
 or far-sighted, without aggravation. 
 
 It is self-evident that no pair of eyes can use these spec- 
 tacles at all distances, since the accommodation does not 
 admit of this. The rational use is confined within the limits 
 of clear vision dependent upon the eyes themselves and upon 
 the power of the lenses. At page 183 of Die Gesetze 
 
ORTHOSCOPIC SPECTACLES. 15 
 
 des rdumlichen Sehens, I have given these limits for normal 
 eyes, and for lenses of various power. For example, 
 
 Fig. 9. 
 
 D 
 
 Fi". 10. 
 
 normal eyes with orthoscopic convex spectacles of 100 inches 
 focal length, would not see clearly at a greater distance than 
 100 inches; with spectacles of 7-inch focal length, at a greater 
 distance than seven inches. And just as convex spectacles curtail 
 the farthest distance of the range of vision to the limit of their 
 own focal length, so they also curtail the middle distance, or 
 the range of clearest vision. Convex spectacles of 7-inch 
 focal length would, for example, curtail the middle distance to 
 four inches : that is to say, a pair of normal eyes, in order to 
 get the clearest possible image through such spectacles, would 
 require the object to be as near as four inches to them. 
 
 The concave orthoscopic spectacles have the contrary effect ; 
 and, for normal eyes, they increase the distance of the middle 
 visual region, and leave distant vision wholly unlimited. Con- 
 cave spectacles limit the region of vision, therefore, towards 
 the near point. Concave spectacles of 100 inches focal length 
 remove the point of clearest vision to a distance of ten inches ; 
 
16 THEORY OF OCULAR DEFECTS. 
 
 and those of 15 inches focal length, to a distance of 22 J 
 inches. Stronger concaves, of less than ten inches focal length, 
 merge the middle distance in infinity, and are no longer con- 
 venient for any distance whatever. 
 
 This limitation of the range of vision is, however, not an 
 effect of the prismatic form of the orthoscopic lenses, but solely 
 of their convexity or concavity. With common spectacles a 
 normal pair of eyes can scarcely see clearly at any distance ; 
 but such eyes, with orthoscopic spectacles, and within the 
 limits determined by their focal length, can see clearly and 
 without strain. 
 
 By convex spectacles the farther boundary of the range of 
 vision of normal eyes is contracted; and this contraction 
 occurs in a higher degree when they are placed before short- 
 sighted eyes. By concave spectacles the nearer boundary of 
 the range of vision of normal eyes is contracted, that is, it is 
 put farther away ; and this effect is more marked when they 
 are put before far-sighted eyes. It is obvious that convex 
 spectacles can seldom be useful to short-sighted eyes, concaves 
 seldom to far-sighted eyes ; but yet for these, as well as for 
 normal eyes, the statement remains entirely true that within 
 the limits of vision .prescribed by the form and power of the 
 lenses, every normal, or short-sighted, or far-sighted pair of 
 eyes may use any pair of convex or concave orthoscopic 
 spectacles with clearness and without strain. 
 
 The limitation of the range of vision which the use of 
 orthoscopic spectacles causes in normal eyes, must not be 
 regarded as if these spectacles were only adapted for vision 
 at some given distance. The limitation concerns only the 
 boundaries of the range; and, within these boundaries, the 
 spectacles are perfectly comfortable for every distance ; while 
 common spectacles, before normal eyes, are not comfortable 
 for any distance. 
 
 That orthoscopic spectacles can be worn comfortably by 
 normal eyes, is a fact important to many classes of work- 
 men. Watchmakers, engravers, and others, who require to 
 
ACCOMMODATION AND CONVERGENCE. 17 
 
 use convex glasses often and long at their occupation, now 
 employ a single magnifier. Without considering that the mag- 
 nifier occupies one hand because this evil can be obviated 
 by fixing it in the margin of the orbit it requires the use of 
 one eye only, and the suppression of the images received by 
 the other. Both conditions act injuriously upon the eyes. If 
 two lenses of the strength of the magnifier are set in a frame 
 orthoscopically, the above-named evils will be removed. A 
 normal pair of eyes can work with them as easily and con- 
 veniently as with unaided vision at the middle distance. 
 
 As with convex spectacles for near objects, so the orthoscopic 
 concave spectacles may be used for distance, as by military 
 men, and others, who wish to see the most remote objects as 
 clearly as possible, although diminished ; and thus to include 
 the deepest and widest range of vision in one view, or to change 
 quick movements over a large field into slower movements 
 over a small field. For example, nine-inch concave spectacles 
 bring infinitely distant objects into the middle range, and 
 therefore permit a continuous and exact fixing of the most 
 distant objects, and of the whole circle of the field of vision, 
 with as much comfort as in ordinary reading without glasses. 
 
 THE INDEPENDENCE AND THE INTERDEPENDENCE 
 OF ACCOMMODATION AND CONVERGENCE. 
 
 After we have in the foregoing learnt the principle of the 
 orthoscopic spectacles, and have convinced ourselves that they 
 in no way change the individual functional proportion between 
 accommodation and convergence, whether this proportion be 
 normal or abnormal, we may proceed to cases in which the 
 proportion is such that the eye does not feel permanently 
 
18 THEORY OF OCULAR DEFECTS. 
 
 comfortable ; and in which, therefore, the ordinary orthoscopic 
 glasses require a modification, or to be replaced by combina- 
 tion glasses suited to the condition. 
 
 A rational selection of spectacles requires a rational classifi- 
 cation of ocular defects ; and I will introduce this by first 
 considering the physiological pu; poses of accommodation and 
 of convergence, as well as the independence, and the con- 
 trasted interdependence, of these two acts of the visual process. 
 The accommodation is primarily governed by the effort to 
 obtain clearness of the visual impression, or of the retinal 
 images; the eyes accommodate for the object (if not prevented 
 by insurmountable impediments) exactly so much as to con- 
 centrate the cone of rays upon the retina. With regard to the 
 primary object of convergence, different views may be held. 
 In the present state of science the most prominent of these is 
 the assumption that convergence depends upon the effort to 
 obtain a single visual impression, or single vision ; since the 
 eyes, in the fixation of any object, take the degree of con- 
 vergence that is necessary in order to project the images upon 
 identical or corresponding portions of the retinas. I reserve 
 to myself to prove, in the sequel, that this assumption is 
 erroneous ; and at present only make the statement, which I 
 will hereafter show to be well founded, that the principal 
 purpose of convergence is to fix the object, and that single 
 vision depends chiefly upon a wholly different and independent 
 process, not yet sufficiently estimated in ophthalmology, and to 
 which, in my Gesetze des rdumlichen Sehens, I have given 
 the name of 'Application.' I postpone also the definition of 
 the idea of fixation until I come to the special consideration 
 of convergence ; and will at present only state as a fact that 
 the accommodation and the convergence are two primitive in- 
 dependent visual faculties, directed to the attainment of two 
 wholly distinct and definite purposes, through efforts after 
 wholly distinct and definite sensory impressions. By reason 
 of the original independence of these faculties, every pair of eyes 
 possesses the power to accommodate more or less with the same 
 
ACCOMMODATION AND CONVERGENCE. 19 
 
 convergence, and to converge more or less with the same ac- 
 commodation ; as also to exert accommodation and convergence 
 voluntarily, without the stimulus of light or of an external 
 object of vision, and to increase the degree of this voluntary- 
 exertion by practice. 
 
 Notwithstanding this original independence, the two faculties 
 pass into a secondary dependence upon one another; caused 
 either by congenital organization, or in consequence of their 
 usually coincident exercise, or as the result of an induction 
 framed by the sensorium, as a higher central apparatus do- 
 minating over all the visual processes. From the secondary 
 dependence (which in my opinion is the foundation of the 
 law of identity, when this is extended as set forth in Section 
 XXI. of my Gesetze des raumlichen Sehens) it follows that, in 
 voluntary convergence, even when the. eyes do not fix any 
 object, the accommodation, if unstrained, involuntarily keeps 
 pace with the convergence ; and, inversely, that in voluntary 
 accommodation without the stimulus of light, the eyes invo- 
 luntarily assume a corresponding degree of convergence. 
 
 So long as this secondary proportionate dependence between 
 accommodation and convergence, in regarding an object, is 
 such as corresponds to the normal relative proportion of the 
 functions in the pair of eyes concerned, the visual process is 
 carried on without strain and with entire completeness. So 
 soon, however, as the eyes are called upon for an abnormal 
 relative proportion, the visual process is attended by strain. 
 If this strain be small, it may be overcome without detriment 
 to the completeness of the visual impression ; and, in many 
 respects, the shifting of the bacillary layer, so often demon- 
 strated in my Physiologische Optik, and in the Supplement, 
 plays the part of a regulating power for the correction of the 
 disproportion. If, however, the strain exceeds a certain de- 
 gree, then one or other of the two functions, or both, will fail 
 in their duty ; in such a way that the eyes may accommodate 
 correctly, but converge falsely, so as not to see with their 
 poles, an effect that generally produces double vision ; or they 
 
 c2 
 
20 THEORY OF OCULAR DEFECTS. 
 
 may converge rightly, so as to see with their poles and singly, 
 but accommodate falsely, so as to see indistinctly. 
 
 In most cases, tinder such circumstances, the convergence 
 prevails, in consequence of the greater desire for single vision; 
 so that most pairs of eyes see indistinctly, but singly, rather 
 than double, and clearly. The contrary may, however, occur. 
 In either case there is the indication to regulate the functional 
 proportion in order to relieve the visual stra'n. This may be 
 done by proper combination spectacles, which I shall describe 
 in the following special researches into the different forms of 
 ocular defect. 
 
 It has been already observed that the combination spectacles, 
 formed of united lenses and prisms, are modified orthoscopic 
 spectacles ; and that they have the effect of dividing the actual 
 object into two apparently distinct images, which, however, do 
 not produce diplopia, since they fall upon identical portions of 
 the retinas. No spectacles neither the common spherical 
 (which are combination spectacles with the refraction of the 
 prisms=0), nor the common prismatic (which are combination 
 spectacles with lenses of infinite focal length), nor any possible 
 combination spectacles produce double vision directly. If 
 double vision occur during the use of spectacles it is but as a 
 consequence of increased disproportion between the efforts of the 
 independent ocular functions, and especially as a consequence 
 of transgression of the limit of the faculty of convergence. 
 
 With regard to the limits of the faculty of accommodation or 
 of convergence,the researches of Bonders and of von Graefe have 
 established that every pair of eyes possesses, for any given degree 
 of convergence, a certain play of accommodation (the relative 
 range of accommodation) ; and also for a given degree of accom- 
 modation, a certain play of convergence (the relative power of 
 squinting). Within the limits of such play the pair of eyes 
 can endure the already mentioned disproportion, even if not 
 without strain, or for a continuance. 
 
 Moreover, it is certain that every pair of eyes has absolute 
 limits for the accommodation (corresponding to the near- and 
 
ACCOMMODATION AND CONVERGENCE, 21 
 
 the far-point) and absolute limits to the convergence (corre- 
 sponding to the maximum and minimum convergence angle). 
 
 In the former results of the observations of Bonders and 
 von Graefe, I find a direct confirmation of the opinion stated 
 in my Physiologisclie Optik, and there based upon different 
 grounds, that the accommodation and the convergence may on 
 the one hand be regarded as two independent acts of the visual 
 process ; and on the other hand as acts materially dependent on, 
 or materially influenced through, a law of induction dominating 
 them both. If we grasp this as a fundamental principle of 
 vision, the above described results follow as necessary conse- 
 quences. On account of the independence of the two faculties, 
 either may vary while the other remains constant ; and, on ac- 
 count of their interdependence, every change in one produces 
 an effort to change the other also, and thus causes visual strain. 
 When this effort, or this strain, reaches a certain degree, the 
 independence of the two functions can no longer be maintained ; 
 that is, every independent variation of the one function, with 
 immutability of the other, is restrained within definite limits. 
 
 While the observations of Bonders and von Graefe furnish 
 confirmation of this principle of vision, the principle must itself 
 be regarded as the physiological ground of the phenomena, and 
 as conveying the explanation of them. 
 
 It is obvious that the condition of continuous vision without 
 strain is that the limits of the above relative ranges should be 
 avoided in both directions, and that nearly the middle point 
 between them should be preserved. This a priori conclusion 
 is entirely confirmed by the observations of ophthalmologists, 
 and forms an important element in the selection of proper 
 spectacles. 
 
22 THEORY OF OCULAR DEFECTS. 
 
 DEFECTS OF ACCOMMODATION,* OR ANOMALIES OF 
 REFRACTION AND OF ACCOMMODATION. 
 
 In order to specialize the ocular defects that are to be dis- 
 cussed, I commence with those of accommodation ; that is, with 
 the abnormal conditions which impair the distinctness of the 
 visual impression. We consider first the changes of accommo- 
 dation which an eye must undergo in order to accommodate in 
 succession for all points along a straight line of infinite length. 
 For this purpose, let c be the focal length, for parallel rays, 
 
 of the refracting system 
 of an eye, measured 
 from B (Fig. 11) the 
 
 ~A ~D anterior pole or apex of 
 
 the cornea. By a normal 
 eye we mean one in which the length of the axis B c, for 
 parallel rays, or in fixation of an infinitely distant object, is 
 exactly = c : so that the cone of light is united exactly upon 
 the retina c. We call this position of the retina the state of 
 
 * It will be observed that the author frequently uses the word accom- 
 modation in its ordinary or restricted sense, to express the action by 
 which the eye adapts itself for different distances ; and frequently in a 
 wider or more general sense, to express the actual refractive condition at 
 any given time. For although accommodation is an action, not a con- 
 dition, yet we speak of the condition of accommodation to express the re- 
 sult of that action. The refraction itself is also a condition of accom- 
 modation, that namely in which the action of accommodation is nil. 
 Hence the word accommodation may be used as a general term to include 
 the refraction ; and we may, for the sake of brevity, describe the ano- 
 malies of refraction and accommodation as ' defects of accommodation.' 
 The context will always clearly show in which sense the word is used in 
 any given passage, and I have thought it better to retain the author's 
 language than to alter it. TEAKS. 
 
DEFECTS OF ACCOMMODATION. 23 
 
 rest, in which the accommodation = 0. Let x be the distance 
 of any object D from the eye; or, more accurately, the distance 
 A D of the object D from the point A, which lies in front of the 
 eye at a distance A B = B c = c the focal length of the eye itself. 
 The distance B E of E, the crossing point of the rays of light, 
 
 from the cornea B, will be very nearly = c -f- , in which 
 
 x i 
 
 the nearly constant coefficient Jc is determined by the refrac- 
 tive indexes and the curvatures of the ocular media. The dis- 
 tance c E of this crossing point E from c, the point of rest of 
 
 k 
 the retina, is = . Accurately to accommodate for D the eye 
 
 must undergo changes, which if the curvatures and thickness 
 of its media remained unaltered would be expressed by an 
 
 Tc 
 
 elongation of its axis = . In actual fact the accommoda- 
 te 
 
 tion depends upon a simultaneous change in the curvatures, 
 the thickness, and the axial diameters of the refracting media ; 
 in which change the alteration of the curvatures of the crys- 
 talline lens plays the chief part. For the better display of 
 these changes we may reduce them all to an elongation of 
 the axis; that is, when we speak of an elongation of the 
 ocular axis, we mean thereby the sum of all the elementary 
 changes of curvature, thickness, and dimensions, which, in re- 
 spect of the position of the crossing point of the rays of light, 
 have the same effect that would follow a simple elongation or 
 simple shortening of the axis of the eye, if all its curvatures 
 and thicknesses remained unchanged. Hence it is possible to 
 
 k 
 
 take the value as the measure of the accommodation effort 
 x 
 
 required. 
 
 Let y be the effort of accommodation, which is necessary to 
 
 accommodate accurately for the distance x ; so that y = -. 
 
 x 
 
 This formula will apply not only to all positive values of x, 
 from = to 2 == + oo, that is, for the distances of all 
 
24 
 
 THEORY OF OCULAR DEFECTS. 
 
 objects in front of the point A, but also for all negative values of 
 x. A negative value of x indicates an object lying behind the 
 eye, and thus seems to involve an absurdity. This, however, 
 is not the case. The position in front or behind, or the geo- 
 metric place of the object, or generally the existence of an ex- 
 ternal object, is of no importance to the eye as a visual organ. 
 The eye sees only by virtue of physiological processes excited 
 within it ; and these depend entirely upon the nature of the 
 ether undulations or rays of light, which fall upon the retina 
 and call forth the sensory nervous process ; or, therefore, upon 
 the conditions which the rays of light assume in the vitreous 
 body. An anteriorly situated object D, corresponding to a 
 positive distance x = A D, 
 
 is to the eye only a bundle 
 of rays, which converge 
 towards the retina (Fig. 
 12). An object lying in 
 the point A, which cor- 
 responds to the distance 
 
 T, 
 
 or the value y = = -4- 
 
 oo , is to the eye only a 
 bundle of rays which pass 
 through the vitreous in a 
 parallel course towards 
 the retina (Fig. 13). An 
 
 object D' lying behind the point A, and therefore behind the 
 eye (Fig. 14), and corresponding to a negative distance A D' = 
 #', is to the eye a bundle of rays which in the vitreous body 
 diverge towards the retina ; that is, a bundle of rays which, 
 where they are outside the eye and uninfluenced by its refract- 
 ing media, converge towards a point D' lying behind A. 
 
 Hence both the positive and the negative values of x have a 
 decided bearing upon the function of the eye. A positive value 
 of X, that is, a luminous point lying in front of A in space, re- 
 
 Fig. 13. 
 
 Fig. 14. 
 
DEFECTS OF ACCOMMODATION. 25 
 
 Ic 
 
 quires an accommodative function - which tends to an elonga- 
 tion of the ocular axis. A negative valve of x = x', that is, 
 a luminous object which sends forth converging rays, tending 
 to meet in the distance x' behind A, requires a negative accom- 
 
 k Jc 
 
 modative function = . which tends to a shortening of 
 
 x ~ x' 
 
 the ocular axis. Whether the eye can exercise this negative 
 function or not (a normal eye cannot, that is, cannot shorten 
 its axis to less than the minimum length c) is wholly unimpor- 
 tant as far as the general law is concerned. The law expresses, 
 not the faculty of the eye to accommodate, or the power which 
 it actually exerts when affected by a given object, but that 
 which would be required of it, or which it must exert, in order 
 to accommodate for the object correctly. The latter, the re- 
 quired accommodation, is obviously the foundation of the ideal 
 law of visual function, and must be studied at the beginning. 
 As Jc is a coefficient which possesses in every eye a nearly 
 constant value, we may shorten the formula by substituting 1, 
 so long as we are examining the same visual organ, or are 
 comparing the accommodative power required under different 
 
 circumstances ; so that simply y = - may be used as the 
 
 x 
 
 measure of the accommodation required for an object lying at 
 the distance x. If we take x as the abscissa, and y as the 
 ordinate, of a curve, the law of required accommodation can 
 with the greatest clearness be graphically expressed. 
 
 We call the curve y = the curve of requirement for the 
 
 accommodation. This presents itself as an equilateral hyper- 
 bola (Fig. 15). The previously indicated point A, which lies 
 before the eye o at the focal length A o = c, is the point of 
 origin of the co-ordinates ; the axis of abscissas x' x is one, and 
 the axis of ordinates Y' Y is the other asymptote of the hyper- 
 bola. The branch E F G corresponds to the positive distances 
 
26 
 
 THEORY OF OCULAR DEFECTS. 
 
 A D = x and to the positive accommodation requirements 
 D F = ?/=-; and the branch E'F'G' corresponds to the negative 
 
 6 
 
 distances A D' = x =x f , and to the negative accommodation 
 
 Fig. 15. 
 
 F 
 
 E 
 
 
 KV 
 
 * 
 
 It" 
 
 or' D' o 
 
 iKf^L^ 
 
 
 y HW 
 
 requirements D' F' = y = ?/ = . Since for a normal eye 
 
 the focal length A o = c, the length of the axis of the eye itself, 
 or about = one inch, it follows that A, the point of origin of the 
 co-ordinates, must lie very near to and in front of the eye o. 
 
 If the eye were organized with absolute completeness, so 
 that it could perfectly realize the ideal plan of vision, and 
 accommodate exactly for every possible distance, it would 
 possess an accommodative faculty corresponding with the 
 hyperbolic curve of requirement. But no earthly organism, 
 no material system, and certainly not the eye, possesses such 
 an ideal completeness in the realization of a natural law. 
 The actual accommodation is only accurate for a certain 
 distance, the middle range of vision, or for a short departure 
 from that distance in either direction; and the actual accom- 
 
DEFECTS OF ACCOMMODATION. 27 
 
 modative faculty coincides with the accommodative require- 
 ment only over a certain short portion in the vicinity of the 
 point D, which is determined by the middle visual range A D, 
 and may be called the point of middle vision. For this point 
 D only, the ordinate D F is the measure of the actual accommo- 
 dation of the eye. On this and on that side of the middle 
 visual range, the actual accommodation does not attain to what 
 is required. The curve which illustrates the actual accommo- 
 dation and hence the accommodative effort or faculty, and 
 which we may term the facultative curve of accommodation 
 has the form of the dotted line B F c. The character of this 
 facultative curve consists in this, that it at first, over a certain 
 distance B K, runs parallel with the axis of abscissas, then 
 touches the curve of requirement at F, and at a point c falls 
 tangentially upon the axis of abscissae, and coincides with that 
 axis thenceforward. 
 
 For the visual distance A H, or the point H, which is the 
 near-point, the actual accommodation attains, therefore, its 
 maximum H K, which deviates only a little from the require- 
 ment, H j; but for still nearer objects remains almost the 
 same, and hence deviates from the requirement more and more. 
 Between the points H and c the deviation of the actual from 
 the required accommodation is inconsiderable. At the point c 
 the actual accommodation attains its minimum, which is = 0, 
 and which remains constantly = for all greater distances. 
 The deviation of the accommodation from the requirement is 
 inconsiderable for these greater distances, although the former 
 remains unchanged ; since the requirement itself is very incon- 
 siderable over this portion of the visual range, on account of 
 the approach of the hyperbola to its asymptote. The point c, 
 at the distance A c, is the far-point. 
 
 The prevailing doctrine is, that for a normal (emmetropic) 
 eye, the far-point is at infinite distance. This doctrine is not 
 accurate. It arose from the fact that for all great distances 
 only a small degree of accommodation is required, and hence 
 only a small degree is exerted. The principal criteria for 
 
28 THEORY OF OCULAR DEFECTS. 
 
 the near- and for the far-point are these, that the actual accom- 
 modation is at its maximum for the former, and at its minimum 
 ( = 0) at the latter. The accommodation is extinct for still 
 more distant objects, or for infinite distance; for which the 
 eye undergoes no further accommodative change, but remains 
 settled at its minimum axial length ; just as, for points within 
 its near-point, it undergoes no further change, and remains 
 settled at its maximum axial length. 
 
 It is hence not only my definition of the far- point, but my 
 definition of the near-point also, that differs from those com- 
 monly accepted. As a rule, people understand by the near- 
 point a point H', lying somewhat farther from the eye than H, and 
 for which the accommodation is yet completely clear and sharp ; 
 while I mean by the near-point H, the point at which the 
 accommodation attains its constant maximum. If, according 
 to common use, we accept for the near-point H', for which the 
 actual accommodation still coincides accurately with the re- 
 quired, we ought to take as the far -point c/, for which also the 
 actual and the required accommodation coincide. While H' lies 
 farther off than my near-point H, c' lies nearer than my far- 
 point c. The prevailing definition of near- and far-point not 
 only expresses, therefore, a different test from mine, but it is 
 for both points untrue ; a fault which, as well as those above 
 explained, causes error with regard to the limits of the weakest 
 accommodation. 
 
 I leave to practical ophthalmologists the accurate determi- 
 nation of the near- and far-points for the maximum and 
 minimum of accommodation, or for the limits of the accommo- 
 dative faculty. My near-point for a normal eye would be at 
 a distance A H = to about 3" ; and my far-point at a distance 
 of some hundred feet, or possibly A c = 2000" ; while the 
 middle visual distance A D = about 8". 
 
 To avoid misunderstanding I will observe that the incapacity 
 of the eye to accommodate for shorter distances than three 
 inches, and for longer distances than two thousand inches, 
 must not be supposed to imply incapacity to recognize shorter 
 
DEFECTS OF ACCOMMODATION. 29 
 
 and longer distances. The eye distinguishes such shorter and 
 longer distances very well, although it cannot accommodate 
 for them. As I have stated in my Physiologische Optik and 
 in the Supplement, and have shown by numerous facts, the 
 recognition of distance has nothing to do with the accommo- 
 dation ; and the accommodative faculty is not, as it is erro- 
 neously assumed to be in the physiology of the day, a measure 
 for this recognition. The accommodation has no other object 
 than the clearness of the visual impressions ; and the recog- 
 nition of distance rests upon the proportion of axial and of 
 lateral components in the bundle of rays, the undulations of 
 which strike upon and penetrate the bacilli of the retina : a 
 proportion which is directly felt, and the sensory impression 
 from which produces the conception of distance. In conse- 
 quence, the eye falsely accommodated may yet perceive cor- 
 rectly the distances of objects; and, since, the nerve substance 
 of the retinal bacilli is more sensitive than the motor apparatus 
 of accommodation, the former reacts for the determination of 
 distance beyond the limits of the accommodation as fixed by 
 the near-point and the far-point : that is, the eye recognises 
 the distance of nearer or of more distant objects, although it 
 can undergo no further change of accommodation with regard 
 to them. 
 
 If any phenomenon whatever is calculated directly to con- 
 firm my theory of the recognition of distance, and of the 
 independence between the judgment about distance and the 
 accommodation, it is the fact just stated, that the eye can 
 actually distinguish distances which lie on this side of its near- 
 point, and beyond its far-point, without undergoing any change 
 of accommodation with regard to them. 
 
 The range between the two limits of the strongest and the 
 weakest accommodation is measured by the length of the 
 ordinate A B. This ordinate forms a measure of the whole ac- 
 commodative faculty of the eye ; which is called, by Bonders, 
 the range or breadth of the accommodation. For the as- 
 sumption of the near-point at H', the actual accommodation, 
 
30 THEORY OF OCULAR DEFECTS. 
 
 H' j', coincides accurately with the required accommodation, 
 and gives the range H' j'. For my near-point H, the actual 
 range of accommodation A B = H K, and is somewhat less than 
 the required accommodation H J. 
 
 The two points H and c, which indicate the extreme limits 
 of the accommodative faculty, are important for many 
 purposes ; and, for others, the points H' and c' which indicate 
 the extreme limits of the coincidence of the actual with the 
 required accommodation, are of still greater importance. 
 Where it is necessary to distinguish these points from each 
 other, the former will be called the near- and far-point of the 
 accommodative faculty ; the latter, the near- and far-point of 
 accurate accommodation. 
 
 Having shown that the normal eye is characterized by the 
 above described hyperbolic curve of required accommodation, 
 and by the dotted facultative curve, we have next to deal 
 with the characteristics of abnormal or defective eyes. At the 
 first glance one would be inclined to assume that the essential 
 character of an eye with abnormal accommodation would be 
 shown by the special form of its facultative curve. This, 
 however, is not the case. The facultative curve of an 
 abnormal eye has always an abnormal form ; but the mos-t 
 important alteration is shown by the curve of requirement. 
 
 For example, a myopic or short-sighted eye has an axis too 
 long for its refraction. This shows a dioptric system which, 
 quite independently of its accommodative faculty, in the state 
 of rest of the retina, does not concentrate upon the retina 
 parallel rays, or those coming from infinite distance, but only 
 rays coming from a certain finite distance a. The required 
 accommodation, therefore, has not the value of when x = oo , 
 but when x = a ; that is, the curve of requirement has not the 
 axis of abscissas for its asymptote, but cuts this axis at the dis- 
 tance a. If we indicate the minimum axial length of the 
 short-sighted eye, which corresponds to the visual distance a, 
 by c ; then the distance from the cornea of the point of con- 
 vergence of a cone of rays, issuing from a luminous point at 
 
DEFECTS OF ACCOMMODATION. 
 
 31 
 
 the distance x, after refraction by the eye, will be = c + - 
 
 Off 
 
 _- . and hence the required accommodation y = = k 
 a x a 
 
 ) ; or, when we assume, as before, =1. y =. 
 
 x a) x a 
 
 If we make y + - = ?/'; so we perceive from the equa- 
 tion y = - that the curve of requirement of a short-sighted 
 
 tXf 
 
 eye is accurately congruent with that of an emmetropic eye, 
 but is vertically depressed by the depth , so that it is an 
 
 equilateral hyperbola E G, E' G' (Fig. 16), the horizontal 
 asymptote of which is not the axis of abscissae x'x, but a 
 
 horizontal line N' N, parallel to that axis, and at a depth 
 
 A M = - below it. This curve intersects the axis of abscissa? 
 a 
 
 in the distance A Q = a. The point of origin of the co- 
 
32 THEORY OF OCULAR DEFECTS. 
 
 ordinates, A, lies in front of the eye o in the distance 
 
 A = c ; and therefore not so far distant from the 
 
 short-sighted eye as the whole length of its axis, supposing 
 
 that c is > . 
 a 
 
 The facultative curve, which depicts the actual accommo- 
 dation for different visual distances, has a form B K Q, like the 
 preceding one : that is, it shows at its commencement B K, a 
 maximum of accommodation A B, which is attained for the 
 near-point H, and remains unchanged for all nearer distances. 
 It shows the most complete accommodation D F for the middle 
 distance A D, and this sinks to its minimum, 0, for the far- 
 point Q. The near-point H lies nearer to the short-sighted than 
 to the emmetropic eye ; and so does the far-point Q, the latter 
 being so near that it falls upon a proportionately strongly 
 curved portion of the hyperbola. For an object R, lying 
 beyond the far-point, the required accommodation R s = y 
 
 _ - _ - W ould be negative, since x > a, and therefore - is 
 < - An infinitely distant object would require the negative 
 accommodation y = - = =AM; that is, the cone of 
 
 GO Ct (JL 
 
 refracted rays would converge to a point at a distance = - in 
 
 front of the retina. 
 
 For the near- or far-point of the accommodative faculty, and 
 the near- or far-point of the exact accommodation, the same 
 difference holds good in the short-sighted, as in the emme- 
 tropic eye. The two far-points, however, fall more nearly 
 upon the same point Q, the more short-sighted the eye, or the 
 more abruptly the hyperbolic curve inclines to this point* 
 
 It is manifest that the short-sighted eye is distinguished 
 from the emmetropic by the position, rather than the form, of 
 the curve of requirement. They are also distinguished by the 
 
DEFECTS OF ACCOMMODATION. 33 
 
 form of the facultative curve : since, as a rule, both the near- 
 point and the far-point are nearer to the eye. It may, how- 
 ever, very well happen that the near-point and the far-point of 
 a short-sighted eye are at distances corresponding to those of 
 an emmetropic eye. If such be the case, the different position 
 of the curve of requirement would always serve to establish 
 the essential difference between them ; which consists in this, 
 that the short-sighted eye, on account of its great axial length, 
 requires a negative accommodation for distances beyond its 
 far-point, a requirement that it is unable to fulfil. 
 
 The common far-sighted eye is not distinguished from the 
 emmetropic by any special peculiarity. Although its axis is 
 very short, it is not too short to allow of parallel rays from 
 infinite distance being concentrated upon the retina. The 
 curve of requirement is the same hyperbola as for the emme- 
 tropic eye, having both the axes of co-ordinates as asymptotes. 
 
 The difference between the common far-sighted and the 
 emmetropic eye is expressed solely by the form of the facul- 
 tative curve. In consequence of the short axial length of 
 such an eye, its near-point H, and its point of middle distance 
 D (Fig. 17), recede farther from it than from an emmetropic 
 eye. The latter point, D, falls in a place at which the curve of 
 requirement has but a slight inclination towards the axis of 
 abscissae. The far-point, c, lies self-evidently far away, and 
 assumes proportions as in emmetropic eyes. Generally the 
 near- and far-point are much nearer together than in a normal 
 eye. 
 
 From the foregoing observations upon the peculiarities of a 
 normal, a short-sighted, and a far-sighted eye, it appears that 
 the -signs indicated do not convey any rational characteristic 
 of either ; that is, one based upon general laws. Essential 
 and non-essential, or specific and gradual differences, are 
 mingled together, and among them no principle appears. A 
 character based on principle can only be based upon the posi- 
 tion of the curve of requirement, and upon the form of the 
 
 D 
 
34 
 
 THEORY OF OCULAR DEFECTS. 
 
 facultative curve, and thus we have two separate points of 
 view from which the eyes may be considered. 
 
 We will turn first to the curve of requirement. This, for 
 every eye, is one and the same equilateral hyperbola. The 
 eyes are therefore to be distinguished, as regards the required 
 accommodation, only by the position of the hyperbola. This 
 
 position is determined by the position of the middle point 
 M of the hyperbola, with regard to the eye o, that is, to the 
 cornea of the eye (Fig. 16). It therefore has reference to two 
 definite distances : the vertical distance, A M, of the horizontal 
 asymptote, and the horizontal distance, o A, of the vertical 
 asymptote from the eye. Instead of the vertical distance A M 
 of the horizontal asymptote, we may take the abscissa M = 
 A Q as the determining distance. The latter is the clearer 
 method ; and we say that the position of the curve of require- 
 ment, with regard to the eye o, is determined by the distance 
 A Q of its point of intersection Q with the visual line o x, and 
 by the distance o A of the vertical asymptote from the eye o. 
 These two elements are determined in the following way, 
 
DEFECTS OF ACCOMMODATION. 35 
 
 through the refracting system of the eye o, or of any eye 
 
 (Fig. 18): We take 
 
 b = o B the focal length, or the distance of the focal point B 
 
 from the cornea o, for parallel rays, therefore, for rays 
 
 proceeding from infinite distance. 
 
 Fig. 18. 
 
 A a point in front of the cornea o, in the focal length o A 
 = o B. 
 
 a = A Q the distance from the point A to any luminous point 
 Q, from which the rays are accurately focussed upon the 
 retina, c, when in its state of rest : a distance that must 
 correspond with the visual distance, or, yet more pre- 
 cisely, with the accommodation distance. 
 
 c = o C the length of the axis of the eye in a state of rest, 
 therefore the minimum length of axis. 
 
 x = A D the distance of any desired luminous point D from A ; 
 or any desired visual distance. 
 
 y = c F the distance from c, the retina in a state of rest, of 
 the point F, at which rays coming from D are focussed ; 
 therefore the required accommodation for the visual 
 distance x. 
 
 I = o F the distance of the point F from the cornea o. 
 
 k a nearly constant coefficient, determined by the refraction 
 
 of the eye. 
 According to general optical laws, we have : 
 
 c = b+ - b = c~- 
 
 a a 
 
36 THEORY OF OCULAR DEFECTS. 
 
 If we assume the value Jc for the unit of length by which all 
 the magnitudes, <z, 5, c, #, y, , are measured, we may then, 
 in all the formulas, suppress the quantity k. 
 
 The curve of required accommodation is always one and the 
 
 same equilateral hyperbola, y = Ic ( - J whatever may 
 
 \x dj 
 
 be the values of the magnitudes a, b, c. According to these 
 last values, the eyes are classified in the manner following : 
 
 1. When Q, the point at which the curve of requirement 
 cuts the visual line, lies at infinite distance ; and when, there- 
 fore, a = oo , b = c, that is, the focal point of the eye falls on 
 the retina in its state of rest, and the vertical asymptote lies at 
 the distance c, the length of the ocular axis, in front of the 
 cornea ; so that an object at the distance c, that is, very near 
 the eye, would require the infinitely strong accommodation 
 y = oo . The equation of the curve of requirement would 
 
 here be simply y = - (Fig. 15). 
 
 This eye is the so-called emmetropic ; and the condition 
 a = oo , that is, an infinite visual distance, is the distin- 
 guishing sign of emmetropia. 
 
 2. When Q, the point of intersection of the curve of require- 
 ment, is situated in positive finite distance, that is, at the finite 
 distance a in front of the eye ; when, therefore, the rays from 
 an object at the distance a are united upon the retina in a state 
 of rest, then the focal length of the eye is 
 
 The focal point falls, consequently, within the eye ; and an 
 infinitely distant object requires a negative accommodation 
 
 k 
 y = . The vertical asymptote falls in the distance b = 
 
 c ; and is therefore very near the eye. The equation of 
 
 the curve of requirement is #=&(- - ) t This gives 
 
 \x a/ 
 
 an 
 
DEFECTS OF ACCOMMODATION. 37 
 
 equilateral hyperbola, the horizontal asymptote of which lies 
 
 j, 
 at the depth below the eye ; that is, an hyperbola depressed 
 
 to the extent -(Fig. 16.) 
 
 I call this eye the hypometropic ; and the condition of a 
 positive finite value for a, or a positive visual distance, is the 
 criterion of hypometropia. (Hypometropia is not, as assumed 
 by Bonders, in his work on the anomalies of refraction and 
 accommodation, the equivalent of short sight ; and we shall see 
 in the sequel that brachymetropia, the expression used by 
 Bonders, is not so expressive as hypometropia.) 
 
 3. When Q, the point of intersection of the curve of require- 
 ment, is situated in negative finite distance, and therefore 
 behind the eye at the distance a a' ; and when, there- 
 fore, rays which converge towards this point behind the eye, 
 are so refracted in the eye, as to be united on the retina in its 
 
 state of rest, then the focal length of the eye is b = c + - . 
 
 a' 
 
 The focal point falls, therefore, behind the eye ; and an infi- 
 nitely distant object, lying in front of the eye, requires the 
 
 positive accommodation y = k ( - + -, 1 =-,' The verti- 
 
 \co a'J a' 
 
 Jc 
 
 cal asymptote falls at the distance I = c + ; and therefore 
 
 in front of the eye, at a distance which is greater than the 
 ocular diameter c. The equation of the curve of require- 
 ment is 
 
 This describes an equilateral hyperbola, the horizontal asymp- 
 
 k 
 tote of which lies above the eye at the height -, ; that is, an 
 
 Ic 
 hyperbola raised to the extent (Fig. 19). 
 
38 
 
 THEORY OF OCULAR DEFECTS. 
 
 This eye is the so-called hypermetropic, and the condition 
 of a negative finite value for a = a', that is, a negative 
 visual distance, is the characteristic sign of hypermetropia. 
 
 The hypermetropic eye has the peculiarity that it may 
 positively accommodate for rays of light which converge to- 
 wards points behind the eye ; since y = k f - + -,) would 
 
 \x o, J 
 
 be positive for all negative values of x = x' for which 
 x' > a'. 
 
 Since both hypometropia and hypermetropia are deviations 
 from emmetropia; so they both fulfil the condition which 
 Bonders has designated ametropia. 
 
 The curve of required accommodation for the emmetropic, 
 the hypometropic, and the hypermetropic eye, that is, for any 
 given eye, is therefore shown by one and the same equation, 
 
 y = k (- - J in which the quantity a -is for the first in- 
 \x a/ 
 
 finite, for the second positive, and for the third negative. 
 It follows, from the foregoing, that emmetropia, hypo- 
 
DEFECTS OF ACCOMMODATION. 39 
 
 metropia, and hypermetropia, are determined exclusively by 
 the refractive power of the eye ; or, if we so please, by the 
 length of the ocular axis ; or, generally, by the structure of the 
 eye, and not by its inner vation or accommodative function. 
 If the length of the ocular axis, in the state of rest, is equal to 
 the focal length for parallel rays, the eye is emmetropic. If 
 the ocular axis be greater than the focal length, the eye is 
 hypometropic. If the ocular axis be less than the focal length, 
 the eye is hypermetropic. Hypometropia and hypermetropia 
 are therefore true anomalies of refraction, not anomalies of 
 accommodation ; and the position of the hyperbolic curve of 
 requirement forms the rational sign for the anomalies of refrac- 
 tion, so that it may be called the curve of refraction. 
 
 The power to vary the accommodation, which we call 
 the accommodative faculty, and which depends upon inner- 
 vation, furnishes totally different criteria, which depend not 
 upon the curve of requirement, but upon the facultative 
 curve. The essentials which come into consideration with 
 regard to this power are, first, the maximum of accommo- 
 dation, which corresponds to the greatest attainable axial 
 length of the eye, and determines the near-point, since the 
 eye, for all greater requirements of accommodation, retains 
 this maximum of axial length, or remains insensitive to the 
 greater requirement ; secondly, the minimum of accommoda- 
 tion, which corresponds to c, the smallest attainable axial 
 length, and determines the far-point, since the eye remains 
 insensitive to, or retains its shortest axial length notwithstand- 
 ing, all demands for a weaker accommodation. 
 
 The interval between this maximum and minimum corre- 
 sponds to the absolute range of accommodation. The pecu- 
 liarity of an eye in question is therefore determined by two 
 things, which we may express either by the maximum and 
 the minimum axial length, or the near-point and the far- 
 point, or the near-point and the range of accommodation, or 
 the far-point and the range of accommodation. 
 
 The general form of the facultative curve, which hereafter 
 
40 THEOHY OF OCTJLAK DEFECTS. 
 
 may also be called the accommodative curve, is, between the 
 near-point and the far-point, nearly the same as that of the 
 hyperbolic curve of requirement. On this side of the near- 
 point it changes abruptly into a horizontal ; and at the far- 
 point it falls into the horizontal visual line, having gradually 
 left the curve of requirement at some distance before. If we 
 pj g> 20. ta ^ e tne sudden bend- 
 
 . ing at the near-point 
 to be an acute angle, 
 
 K. B 
 
 L' 
 
 and assume, therefore, 
 
 f that the facultative 
 ..: &JL . 
 
 curve is at first, and 
 
 X' AH G t C X 
 
 until it enters the 
 
 curve of requirement, 
 horizontal, that it follows the curve of requirement from hence 
 to near the far-point, and there blends with the horizontal 
 visual line, so we may exhibit the accommodative faculty of 
 the eye, as in Fig. 20, by two horizontal lines K'K and L'L, 
 which both lie above the visual line x'x. The first, K'K, which 
 corresponds to the maximum of the accommodative faculty, 
 gives the near-point, H, by its intersection. K, with the hyper- 
 bolic curve of requirement. The second, I/L, which usually 
 lies close above the visual line, gives, by its intersection, L I? 
 with the curve of requirement, the far-point, C 1? of exact ac- 
 commodation ; which deviates but little from C, the far-point 
 of the accommodative faculty. (See definition before given.) 
 
 On this basis we may classify the anomalies of accommoda- 
 tion, which are expressed by the form of the facultative curve, 
 in the manner following : 
 
 1. When the total accommodative faculty, and therefore the 
 distance R B between the maximum and the minimum of ac- 
 commodation, corresponds to the normal range of accommoda- 
 tion, we call the eye broad- sighted, or europic (Fig. 21). 
 
 2. When the total accommodative faculty or the range of 
 accommodation R B is small, the eye is narrow-sighted or 
 stenopic. Under this head there are two forms to be distin- 
 
DEFECTS OF ACCOMMODATION. 
 
 4] 
 
 guished. First, that in which the maximum of accommodation 
 A B is small as compared with the normal standard, while the 
 minimum. A R corresponds to the normal minimum, and is 
 small (Fig. 22). This stenopic eye, of which the narrow range 
 of accommodation lies deep, is deep-sighted or bathopic. 
 
 Fig. 21. 
 
 ..R 
 
 Fig. 22. 
 
 Fig. 23. 
 
 ..R 
 
 Bathopia exists in an eye in which the power of reaction 
 soon ceases, when the focus of the cone of rays is distant from 
 the position of rest of the retina. The chief character of this 
 eye is weakness, with sufficient irritability or sensitiveness. 
 
 3. When the maximum AB of the stenopic eye is as large as 
 the normal maximum, and the minimum A R is large as com- 
 pared with the normal, so that the range of accommodation RB is 
 narrow, but placed high, then the eye is high- sighted or hypsopic 
 (Fig. 23). 
 
 Hypsopia exists in an eye which is not sufficiently sensitive 
 to react on a cone of rays the focal point of which is only a little 
 distant from the retina in a state of rest; which therefore 
 requires a considerable stimulus in order to set its apparatus 
 of accommodation in action. This eye is strong, or powerful, 
 but insensitive. 
 
 If the eye be emmetropic as regards its refraction, we 
 may characterize europia, bathopia, and hypsopia by saying 
 that for the europic eye the near-point will be near and the far- 
 point distant, for the bathopic eye the near-point will be 
 remote, and for the hypsopic eye the far-point will be near. 
 
 In general it must be stated that neither the anomalies of 
 refraction alone, nor the anomalies of accommodation alone, 
 furnish a sufficient criterion for short-sightedness or far-sighted- 
 ness ; and we cannot say either of a hypotropic or hypertropic, 
 
42 
 
 THEORY OF OCULAR DEFECTS. 
 
 of a bathopic or a hypsopic eye, that it is short or far-sighted 
 without further data. Short-sightedness and far-sightedness are 
 not simple or primitive defects ; but are the results of combi- 
 nations of different states of accommodation and refraction ; 
 and we may convince ourselves that both these defects may be 
 occasioned in several ways. 
 
 Since both for refraction and accommodation there are two 
 chief anomalies, besides the normal state ; that is, three chief 
 conditions of each, so the combinations between each state of 
 refraction and each state of accommodation produce on the 
 whole the following nine cases : 
 
 I. 1. Emmetropia, combined with europia, gives a normal 
 eye, with near near-point and remote far-point (Fig 24). The 
 character of this eye we may callfull-sightedness. 
 
 Fig. 24. 
 
 Fig. 25. 
 
 I. 2. Emmetropia, combined with bathopia, gives common 
 far sight or presbyopia (Fig. 25), in which the eye can accom- 
 modate for infinite distance. 
 
 I. 3. Emmetropia, combined with hypsopia, gives a short- 
 sightedness (Fig. 26) that differs essentially from the common 
 form, since the power of accommodation is gradually lost 
 .beyond the far-point, but returns again for greater distances ; 
 eo that the eye can see very distant objects with sufficient 
 
DEFECTS OF ACCOMMODATION. 
 
 43 
 
 distinctness, and therefore can see at once at very short and at 
 very long, but not at middle distances.* 
 
 Fig. 26. Fig. 27. 
 
 Fig. 28. 
 
 Fig. 29. 
 
 II. 1. Hypometropia, combined with europia, gives common 
 short sight (Fig. 27). 
 
 * The peculiarities of the German language have permitted the author 
 to coin German names for all these nine combinations, and also for the 
 analogous defects of convergence presently to be described. These names, 
 of which Klammswhtigkeit and StrengwinkligJceit may be taken as ex- 
 amples, are not translatable into English to any useful purpose, and 
 they are therefore omitted. TRANS. 
 
THEORY OF OCULAR DEFECTS. 
 
 II. 2. Hypometropia, combined with bathopia, produces, 
 according to circumstances, either short sight or presbyopia 
 (Fig. 28). If the hypometropia be of high degree, there Avill 
 be myopia with a narrow range of accommodation. If the 
 hypometropia be of small degree, there will be presbyopia. 
 Such presbyopia is distinguished from the common form by 
 this, that the eye cannot accommodate for infinite, but only for 
 a finite distance ; and that therefore it can only see objects that 
 
 Fig. 30. ' 
 
 lie between two definite finite limits. This peculiarity is well 
 known as the characteristic of a particular class of over- 
 sightedness. 
 
 II. 3. Hypometropia, combined with hypsopia, gives a 
 short-sightedness (Fig. 29) which differs from the common 
 form in this, that the power of vision is gradually lost beyond 
 the far-point, and returns again at a certain definite farther 
 distance, without being extended from this last to greater dis- 
 tances ; BO that the eye can see at a very small distance, and 
 also at some greater finite distance. 
 
 III. 1. Hypermetropia, combined with europia, gives accord- 
 
DEFECTS OF ACCOMMODATION. 
 Fig. 31. 
 
 45 
 
 ing to circumstances far-sightedness or common over-sighted- 
 ness. If the hypermetropia be slight in degree, there will be 
 far-sightedness, distinguished from the common form by this, 
 that the eye uses yet some effort of accommodation in looking 
 
 Fig. 32. 
 
46 
 
 THEORY OF OCULAR DEFECTS. 
 
 at infinite distance. If the hypermetropia be of high degree, 
 there will be over-sightedness or hyperopia (Fig. 31), in which 
 the eye sees clearly no object in front of it, but is able to react 
 on rays which converge towards points lying behind it. 
 
 III. 2. Hypermetropia, combined with bathopia, gives, when 
 of very slight degree, that form of far-sightedness which is 
 described in the foregoing paragraph. When the hyperme- 
 tropia is more considerable, it produces over-sightedness, as in 
 
 Fig. 33- 
 
 Fig. 33, in which the eye cannot see clearly at any positive 
 distance. 
 
 III. 3. Hypermetropia, combined with hypsopia, gives, when 
 slight, short-sightedness, as in Fig. 34 ; and, when considerable, 
 far-sightedness, as in Fig. 35, distinguished from the common 
 form by this, that the eye cannot see in infinite distance, but 
 only between two definite boundaries. In both cases the eye 
 may only be able to accommodate for rays that converge 
 towards points situated behind it. 
 
 It must here be remarked that, when the stenopia or small- 
 sightedness reaches the highest degree, so that the range of 
 
DEFECTS OF ACCOMMODATION. 
 Kg. 34. 
 
 47 
 
 accommodation R B is reduced to nothing, or the two lines I/L 
 and K'K (Fig. 20) coincide, the eye can then only accommodate 
 for a single distance, and therefore has either only a near-point 
 or only a far-point, which, according to circumstances, may be 
 
 Fig. 35. 
 
43 THEORY OF OCULAK DEFECTS. 
 
 situated either before or behind it. Such an eye may yet see 
 with sufficient clearness an object that corresponds with the 
 null point of its accommodative power, and therefore that is 
 situated at the distance at which the visual line is intersected 
 by the curve of requirement. 
 
 The state of an eye deprived of its lens, the aphakia of 
 Bonders, fulfils this condition of extreme stenopia ; since such 
 an eye has almost entirely lost its power of accommodation, 
 and can only see clearly at one determinate distance, or has 
 its range of accommodation nil. 
 
 "We may, however, remark, that a narrowing of the range of 
 accommodation is in general associated with its displacement. 
 Fig. 35 shows that by a narrowing of the range of accommoda- 
 tion the points of intersection of the facultative curve with the 
 curve of requirement, that is, the near- and the far-point, or one 
 of them, must be displaced. 
 
 Which among the foregoing ocular defects will be the most 
 common in actual practice, and which the most unusual, can 
 only be determined by practical ophthalmology. The informa- 
 tion will be interesting ; but the principles of our classification 
 will be in no way affected by it. 
 
 The description of these defects by Bonders, in his most 
 recent work upon the anomalies of the refraction and accom- 
 modation of the eye, contains a much smaller number of charac- 
 teristic states, and moreover puts together, as identical, defects 
 between which I recognise an entire difference. Hypsopia is 
 not described as a special failure of accommodation ; and hence 
 there is no account of the defects thence arising. Moreover, 
 Bonders has been led, by his non-recognition of hypsopia, to 
 describe short sight and hypometropia as equivalent conditions ; 
 and to declare that myopia is the opposite to hypermetropia, 
 and presbyopia not the opposite to myopia, but a normal con- 
 dition of an emmetropic eye with its range of accommodation 
 diminished by age. I believe I have shown in the foregoing 
 that only in the single case II. 1 is myopia the defect described 
 by Bonders ; that as a general thing myopia does not coincide 
 
SPECTACLES FOR DEFECTS OF ACCOMMODATION. 49 
 
 with hypometropia, and does not depend upon a simple anomaly 
 of refraction, but may be a result of very different anomalies of 
 refraction and accommodation. Presbyopia, too, is the defect 
 described by Donders only in the single case I. 2 ; and does 
 not generally depend upon a simple anomaly of accommoda- 
 tion, but may arise from different anomalies of refraction and 
 of accommodation. I do not doubt that an unprejudiced 
 estimate would admit the foregoing general principles to a 
 share of recognition. 
 
 SPECTACLES FOR DEFECTS OF ACCOMMODATION AND 
 REFRACTION. 
 
 After the foregoing exposition of fundamental principles, the 
 question about the optical aids for the defects described can be 
 rationally discussed. 
 
 It is at once manifest that no such aids can be given to the 
 anomalies of accommodation. Defects depending upon inner- 
 vation, or upon irritability or insensitiveness, cannot be set 
 aside ; and the range of accommodation of the eye cannot be 
 altered. We can by external means, especially by the addi- 
 tion of refracting media, modify the optical state of the eye, 
 and therefore its refracting power, but not its nerve power, or 
 its sensitiveness to rays of light. Doubtless diminution of light 
 will diminish, and increase of light will increase, the refraction 
 of the eye ; but these means would not alter the limits of 
 sensitiveness, and, even if under certain circumstances they 
 might do so, the effect would be feeble, not capable of defini- 
 tion by measure and numbers, only very imperfectly attainable 
 by optical means, such as clouded glasses, and best to be 
 realized by placing the eyes in a brighter or darker visual 
 space, so that we may wholly disregard them in the mathema- 
 tical determination of spectacles. 
 
 A stenopic eye therefore is and remains stenopic, and cannot 
 
 E 
 
50 
 
 THEORY OF OCULAR DEFECTS. 
 
 Fig. 36. 
 
 be changed into one that is europic ; we can never confer 
 upon it a great range of accommodation. 
 
 As regards, on the contrary, the anomalies of refraction, the 
 state of refraction of an eye may obviously be changed in either 
 direction and may be rendered normal. By combination with 
 
 a fitting lens the focal 
 length of any eye may 
 be altered to any de- 
 sired condition, and 
 may thus be normal- 
 ized. In the figures 36 
 and 37 : 
 
 Let / be the focal 
 length A F of a glass 
 
 Fig. 37. lens, and let it be posi- 
 
 tive for a convex and 
 negative for a concave. 
 We call this lens 
 shortly the lens/ 
 
 Let x be the actual 
 distance A x of a lu- 
 minous point from this 
 lens. 
 
 Let z be the distance 
 A z of the point to 
 which the rays con- 
 verge after having been refracted in passing through the lens, 
 and therefore the distance of the optical image of the former 
 point, or of the visible object, likewise measured from the lens. 
 The distances x and z must be positive when they are before 
 the lens, and negative when they are behind it. 
 
 Between the quantities/ a?, z, there exists the relation 
 
 JL = ll or ll 1 
 
 Z X /' ' x ' ' ~Z f ' 
 
 If we imagine the lens to be exactly in the point already de- 
 
SPECTACLES FOR DEFECTS OF ACCOMMODATION. 51 
 
 signaled A, and therefore at the same distance in front of the 
 eye as the length of its natural focal distance b, it follows, since 
 we have measured the distance of the object from this point, 
 that the optical effect of the lens will be to transform the 
 actual visual distance x into the apparent visual distance z. 
 Hereby therefore the eye is in no way changed in its function ; 
 the change affects solely the point of convergence of the ex- 
 ternal rays, which in a physiological sense is identical with the 
 object, since the eye does not react on the actual space, but on 
 the incident rays; and the space is but our objective conception 
 of the subjective light affection. Hence the formulae already 
 given for the curve of required accommodation retain their ap- 
 
 plicability, since z may be put in the place of a;, or = -- - f 
 
 z x f 
 
 in the place of This gives for the curve of requirement, 
 x 
 
 when the ordinate is expressed by y 1 , the equation 
 
 This equation gives ah equilateral hyperbola which is 
 completely congruent with the former, obtained from 
 
 y = lc ( _ V and differs from it only in position, while 
 \x a) 
 
 the ordinate of the new curve, y' = y -, differs from the 
 
 lc 
 ordinate of the former curve only by the constant term . 
 
 We call the former curve of requirement the natural, and 
 this one the artificial. 
 
 If we take the constant quantity i + = , we have 
 
 y'=k (L__l\. The quantity a' = I I = -^- indi- 
 \x a'J - + a+/ 
 
 a f 
 cates, therefore, the distance of that luminous point Q which, 
 
 2 
 

 
 52 THEORY OF OCULAR DEFECTS. 
 
 by the intervention of the lens /, has its rays accurately united 
 upon the retina in its state of rest ; while a is the natural visual 
 length, a' must be the artificial visual length. 
 
 From the preceding it is clear that a convex or positive lens 
 f depresses the hyperbolic curve of requirement by the depth 
 
 k 
 
 -, while a concave or negative lens / = f elevates this 
 
 Jc 
 curve by the height or that the convex lens moves the 
 
 terminal point Q of the visual distance a in a direction towards 
 
 1 
 the eye, since it changes this visual distance to a' = i f . 
 
 + /' 
 
 while a concave lens moves the terminal point in a direction 
 away from the eye, since it changes the visual distance to 
 
 1 
 a' = I i 
 
 / 
 The above-named terminal point Q of the visual distance is 
 
 the point of intersection of the hyperbolic curve of require- 
 ment with the visual axis. The hypometropia, which is the 
 fundamental basis of common short sight, and through which 
 the hyperbola lies too low, or the terminal point of the visual 
 length at a finite distance in front of the eye, is therefore re- 
 moved by a concave lens ; while the hypermetropia, which 
 forms the basis of common over-sightedness, and in which the 
 hyperbola is too high, or the terminal point of the visual length 
 is behind the eye, is neutralized by a convex lens. 
 
 Complete removal of the hypometropia or hypermetropia re- 
 quires the placing of the terminal point Q of the visual length 
 in infinite distance, or that a' = GO . This requirement is ful- 
 filled by a lens of which the focal length / = a ; therefore 
 by a lens, of which the focal length is equal to the negative 
 value of the natural visual distance a. For a hypometropic 
 eye, for which a is positive, / must therefore be negative, that 
 is, the lens must be concave ; but for a hypermetropic eye, for 
 
SPECTACLES FOR DEFECTS OF ACCOMMODATION. 53 
 
 which a is negative, / must be positive, that is, the lens must 
 be convex. 
 
 We see that the convexity or concavity of the lens is the 
 means by which the hypermetropia or the hypometropia, the 
 anomaly of refraction, may be set aside completely. By means 
 of a proper lens every eye without exception can be rendered 
 completely emmetropic; since the hyperbolic curve of require- 
 ment can be restored to its normal position, in which the hori- 
 zontal asymptote is the visual line. 
 
 It should still be observed that it is a necessary condition 
 for this result that the lens should be placed exactly in the 
 point A, and therefore at the distance of the natural focal length 
 b of the eye. This distance is nearly equal to the diameter 
 c of the eye ; although for hypometropic eyes it is somewhat 
 less, and for hypermetropic eyes somewhat more, than this 
 diameter. In the construction and fitting of spectacle frames 
 this requirement should be remembered. 
 
 It is manifest that the convexity or the concavity of the lens 
 is no general or essential means of aid as against the short, the 
 far, or the over-sightedness, since these defects do not depend 
 unconditionally upon a simple anomaly of refraction, but, as a 
 rule, upon a combination of an anomaly of refraction with an 
 anomaly of accommodation ; combinations which may, as we 
 have seen, be of various kinds. 
 
 It is also clear, since spherical lenses and external aids gene- 
 rally act by changing only the refraction, and not the accom- 
 modation, that as a rule a short-sighted, or a far-sighted, or an 
 over-sighted eye cannot be rendered normal, or changed into one 
 that is at once emmetropic and europic. When we arrive at 
 the question how to improve the vision of an abnormal eye for 
 actual use, we shall find that this can only be done within cer- 
 tain limits of distance, and in a degree more or less complete. 
 
 It is first, however, to be observed that every europic or 
 broad-sighted eye, that is, every eye with a normal range of 
 accommodation, may be rendered completely normal by a 
 spherical lens, since in such a case we have only to remove the 
 
54 THEORY OF OCULAR DEFECTS. 
 
 hypometropia or the hypermetropia. Hence the common short 
 sight or myopia, II. 1, of a hypometropic europic eye, and the 
 common over-sight or hyperopia, or that far-sightedness which 
 occurs in both the cases under III. 1, may respectively be 
 completely neutralized by a concave or a convex lens. 
 
 In every other ocular defect which has stenopia or small- 
 sightedness as its foundation, we can only displace the hyper- 
 bolic curve of requirement in such a manner that it will fall 
 within the range of accommodation for the desired visual dis- 
 tance, or will cover a certain portion of the facultative curve. 
 For this the above figures (24 to 35) furnish the necessary 
 guidance. The nine conditions given require the following 
 remarks : 
 
 I. 1. The full-sighted eye requires no lens. 
 
 I. 2. If the emmetropic far-sighted eye, with small deep 
 lying accommodation, requires to use near vision, its hyperbola 
 must be depressed. This is done by a convex lens. But with 
 this the eye loses its faculty of seeing at infinite distance, since 
 the depressed hyperbola cuts the visual line at a finite distance. 
 Such an eye sees best near at hand with a convex lens, and best 
 at distance without a lens. 
 
 I. 3. The emmetropic short-sighted eye, with small but high 
 lying range of accommodation, requires, for vision at moderate 
 distance, that its hyperbola should be considerably raised by a 
 comparatively strong concave lens. With this, however, it can 
 see only within narrow fixed limits, and loses the power pecu- 
 liar to it of vision at infinite distance. 
 
 II. 1. The common short-sighted eye, which is hypometropic 
 and europic, is rendered completely normal by elevation of its 
 hyperbola by a concave lens. 
 
 II. 2. The weak-sighted eye, which is hypometropic, with 
 narrow deep lying accommodation, and suffers, according to 
 circumstances, from short or from far sight, requires, when it 
 is short-sighted, the elevation of its hyperbola by a concave lens 
 for vision at a distance ; and when it is far-sighted, the de- 
 pression of its hyperbola by a convex lens for near vision. In 
 
SPECTACLES FOR DEFECTS OF ACCOMMODATION. 55 
 
 both cases it only acquires the power of vision between certain 
 limits, determined by the power of the lens. 
 
 II. 3. The short-sighted eye, which is hypometropic, and has 
 a narrow high lying range of accommodation, requires for 
 vision at greater distance the elevation of its hyperbola by a 
 concave lens, but obtains thus only the power of vision at cer- 
 tain distances, which the power of the lens determines. 
 
 III. 1. The common over-sighted eye, which is hypermetropic 
 and europic, and cannot see clearly at any positive distance, 
 may be rendered completely normal by depression of its hy- 
 perbola by a convex, usually by a rather strong convex, lens. 
 Also when such an eye is far-sighted, it can be rendered nor- 
 mal by a convex lens. 
 
 III. 2. The hypermetropic eye with narrow deep lying range 
 of accommodation, which according to circumstances may be 
 far-sighted or over-sighted, requires for vision at a positive dis- 
 tance a depression of its hyperbola, and therefore a convex 
 lens, but thus obtains vision only between certain limits which 
 the power of the lens determines. 
 
 III. 3. The hypermetropic eye with narrow high lying ac- 
 commodation, requires, according as it is near-sighted or far- 
 sighted, for vision respectively at a distance and near, a concave 
 or a convex lens. It thus obtains vision only between certain 
 limits, which the power of the lens determines. 
 
 The preceding results are interesting, as showing that short- 
 sightedness is always relieved by a concave, far-sightedness by 
 a convex, and over-sightedness by a proportionately strong 
 convex lens ; and further that the short-sightedness II. 1, and 
 the far-sight and over-sight III. 1 of an europic eye, can be 
 removed completely, but all other defects only within certain 
 limits, determined by the power of the lens ; so that the eyes 
 II. 1 and III. 1 are enabled to see clearly at every distance by 
 one and the same lens, while all other eyes require different 
 lenses for different distances. 
 
 For the ophthalmic surgeon it is important to know the 
 most simple methods which are adapted to discover, first, the 
 
 I 1 
 
6 THEORY OF OCULAR DEFECTS. 
 
 kind and degree of any occurring anomaly, and, secondly, the 
 lens that will produce any given effect. The foregoing in- 
 quiries point out the way to this end. 
 
 With regard to any given ocular defect, the anomaly of 
 refraction has first to be ascertained ; and it is necessary to 
 discover whether the eye be ernmetropic, and, if not, in what 
 degree it is hypometropic or hypermetropic. If the eye 
 receives the luminous impression of an infinitely distant object 
 without any exercise of accommodation, and sees the object 
 clearly, then it is emmetropic. If it receives the rays from an 
 infinitely distant object certainly without effort of accommoda- 
 tion, but without obtaining a clear image, then it is hypo- 
 metropic. If it requires some effort of accommodation in 
 order to obtain a clear image of an infinitely distant object, 
 then the eye is hypermetropic, or possesses the power to 
 accommodate for rays that converge towards a point lying 
 behind it. 
 
 The degree of the hypometropia or hypermetropia is most 
 easily determined by the experimental application of convex 
 or concave lenses of different powers. If we find by such 
 experiments that / is the lens (i.e. the focal length of the lens) 
 with the aid of which the eye sees an infinitely distant object 
 with clearness and without accommodative effort ; so that the 
 lens I makes the artificial visual distance a' infinitely large, 
 we have, from the formula, 
 
 --. 
 I 
 
 The natural visual distance a is therefore equal to the nega- 
 tive value of the focal length I of the above indicated lens. If 
 the eye requires a concave lens I = I', it is then hypome- 
 
 tropic in the degree ~ = /7 If it requires a convex lens /, 
 
 it is then hypermetropic in the degree - _-. 
 
 a I 
 
SPECTACLES FOR DEFECTS OF ACCOMMODATION. 57 
 
 This lens, / = a, which makes the artificial visual dis- 
 tance infinitely long, and serves to determine the degree of 
 hypometropia or of hypermetropia, is also that which com- 
 pletely removes this hypometropia or hypermetropia, if such 
 complete removal be in contemplation. We call this lens Z, 
 briefly the emmetropic lens. 
 
 To determine the anomaly of accommodation the question 
 arises whether the eye is europic ; and, if not, whether the 
 stenopic eye is bathopic or hypsopic, and in what degree. 
 The determination is most readily accomplished by the aid of 
 the above described emmetropic lens /. Furnished with this, 
 the eye is to be regarded as equivalent to one that is em- 
 metropic ; so that its curve of requirement has the equation 
 
 k 
 y = -. If we determine, in this condition, that is to say, with 
 
 00 
 
 the aid of the emmetropic lens, the distance n of its near- 
 point of exact accommodation (H', in Fig. 15), and the 
 distance w of its far-point of exact accommodation (c', in 
 Fig. 15), and if we call these points the artificial near- and far- 
 points of exact accommodation, then the corresponding ordi- 
 nates which indicate the limits of/ the exact accommodation 
 
 k k 
 
 will be respectively and , and the range of accommo- 
 dation, which shows the actual accommodative power of the 
 
 eye, will be k( l _~Y The coefficients i, I, 1 _ i 
 \n wj n w n w j 
 
 can be taken as measures for the degree of the anomaly of 
 
 accommodation. If be smaller than the normal range, 
 
 n w 
 
 then the eye is more or less stenopic. If we indicate the 
 normal range of accommodation of an europic eye by - , since 
 
 ft is the distance of its near-point ; then the degree of stenopia 
 
 1 
 
 of the given eye is expressed by the quantity = - 
 
 ' 1 1 ft(w n)' 
 
 n w 
 
58 THEOKY OF OCULAR DEFECTS. 
 
 If - be nearly null, then the eye is as deep-sighted as a 
 normal eye. The greater -, the more high-sighted is the 
 eye ; so that expresses the degree of the hypsopia. 
 
 We know already that the europic eye, for which is = 0, 
 
 and - = , can be rendered completely normal by a lens. 
 n ft 
 
 For such an eye, therefore, the already described emmetropic 
 lens I is the means by which its defect of refraction is wholly 
 set aside. 
 
 For every stenopic eye only a certain movement of the 
 curve of refraction (hyperbolic curve of requirement) can be 
 
 accomplished, by which an- 
 other part of the visual line 
 comes into the territory of 
 vjx accommodation. The con- 
 
 [f\ --- T\_ 
 
 dition for the required lens 
 may therefore be formu- 
 lated by saying that an ob- 
 
 ^ _ _^____ ject lying at a given dis- 
 
 tance must be brought into 
 
 a given part of the range of accommodation. In order that the 
 eye may be exerted continuously, this given part must fall 
 
 nearly in the middle of the range of accommodation -- 
 
 n w 1 
 
 and must, therefore, nearly correspond to the value 
 
 If, in order to make the condition somewhat more general, 
 we state the requirement to be that an object lying in the 
 visual distance s shall correspond with that portion of the 
 range of accommodation which makes the part i of the whole, 
 
SPECTACLES FOR DEFECTS OF ACCOMMODATION. 59 
 
 so that, therefore, when, in Fig. 38, for the near-point H, 
 
 A B = -, and for the far-point c, A R = , so that the 
 n w 
 
 range of accommodation R B = , an object lying in the 
 
 n w 
 
 distance A D = s shall correspond with the portion T of the 
 range of accommodation, for which 
 
 R T = I . R B = if ), 
 
 \n wj 
 consequently, 
 
 w \n w) n w 
 
 The latter condition involves that the abscissa x in the curve 
 
 of refraction corresponds to the ordinate ?/ = Icl - + ~ l ). 
 
 \n w J 
 
 Suppose the eye furnished with any lens/, the equation of 
 
 the curve of refraction is y = Tc( _._]. In this, a shows 
 
 \x f a) 
 
 the natural visual distance, which is = /, when I expresses 
 the emmetropic lens. The equation is, therefore 
 
 Our last condition requires that x = s, whence y = 
 
 jfcfl _ * + \ = Jcf- + *H!V it is therefore fulfilled by 
 \8 f I) \n w J 1 
 
 / s I \n w 
 
 If the power of accommodation desired for the given object 
 
 be taken as one half, so that t = -J, then = {- - 
 
 s * 
 
 w 
 
CO THEORY OF OCULAR DEFECTS. 
 
 Hence the lens/, adapted to produce the desired effect, may 
 be determined from the easily observed magnitudes /, w, and 
 w, and the given values s and , by a very simple calculation, 
 in any case however complicated. 
 
 We must repeat that in these formulas n, the distance of the 
 near -point, and w, the distance of the far-point, are taken not 
 from the naked eye, but from the eye furnished with the 
 emmetropic lens Z = a. If n^ and w^ indicate the distances 
 of the near- and far-points from the naked eye, of which the 
 
 curve of requirement has the equation y = Jc( ) == 
 
 \x a} 
 
 k[- H ), there will obviously be, between the magnitudes 
 \x I J 
 
 w, w, and TC!, w l7 the relation 
 
 =_=+ and =_=+ 
 
 n n l a n } I w w l a w l t 
 
 If, therefore, there should be an opportunity of learning the 
 values of n lt w lt and a without the application of the em- 
 
 metropic lens, the values of - and for the formulae may be 
 
 easily calculated. The difference is always = -- 
 
 n l w l n w 
 
 that is, equal to the absolute range of accommodation. 
 
 Example. Given an eye which obtains an infinite visual 
 length by a concave lens I = 40 (Paris inches) ; it is there- 
 fore hypometropic, and has a natural visual length a= Z=40 
 inches. With this lens the artificial near-point is found at a 
 distance n = 3 inches ; and the artificial far-point at a dis- 
 tance w = 15 inches. The range of accommodation is there- 
 
 fore --- = _ _ = YV = K? J and the eye is mode- 
 n w 3 lo 15 3J 
 
 rately stenopic, and being = - it is somewhat high- 
 sighted. If it be desired that this eye should clearly see 
 
DEFECTS OF CONVERGENCE. 61 
 
 objects at a distance of 20 feet, or s = 240 inches, with the 
 use of half its accommodation, therefore, with i = ^, it will re. 
 quire a lens which is determined by 
 
 = _ _ . M __^L -1 
 
 ~ 15; : 240 = 4' 
 
 . _ 
 
 / 240 40 2 15; 240 4}' 
 
 and which is, therefore, the strong concave lens/= 
 
 DEFECTS OF CONVERGENCE, OR ANOMALIES OF 
 DIRECTION AND OF MOTILITY. 
 
 The preceding chapters refer exclusively to monocular 
 vision. The common function of the two eyes certainly in no 
 way changes the fundamental principles of refraction and 
 accommodation for each eye alone; but yet, in binocular 
 vision, the function of convergence has to be taken into con- 
 sideration as an especial factor which, on account of the 
 reciprocal interdependence of all the visual functions in the 
 general visual plane, so far influences the effect as to determine 
 the degree of strain, or even the very possibility of using the 
 vision at all. 
 
 In principle the convergence is independent of the accom- 
 modation, but each function exerts an acquired influence over 
 the other. Hence this influence first obtains a definite degree 
 in a more or less considerable deviation from the normal actions. 
 We leave this degree, and generally the influence of the two 
 functions, for a time out of account ; and consider first the 
 convergence, in its original independence, as the act which is 
 performed solely for the sake of fixation, or in order to guide 
 the two retinal images to the poles of the retinae. 
 
 The physiological stimulus, which excites this function of 
 convergence, depends entirely upon the production of the 
 
62 THEORY OF OCULAR DEFECTS. 
 
 equilibrium, to which the nerve affections of the two retinal 
 images forcibly constrain the spherical organs of the binocular 
 visual apparatus ; as mechanical forces, which act upon the 
 circumference of a sphere, rotate the sphere and cause it to 
 assume a definite position. 
 
 In monocular vision the single eye is constrained to place 
 itself with its pole in the optical centre of intensity of the 
 retinal image ; that is, upon that point of this image, in which 
 lies the centre of intensity of the affection of the nerve pro- 
 duced by the objective stimulus of light. In binocular vision 
 the two retinse, connected through the sensorium into an 
 united physiological apparatus, concur in such a manner that 
 the pole of each strives to attain to the centre of intensity of 
 its retinal image ; and this concurrence produces primarily the 
 convergence which is only secondarily influenced through other 
 functions. (See the Gesetze des rdumlichen Sehens.) 
 
 The convergence has therefore the object of guiding the 
 optical centre of intensity of the luminous image, or the centre 
 of intensity of the objective nerve affection, to the physiolo- 
 gical pole of each eye, lying in the yellow spot. This pole 
 forms the most sensitive point of the retina, and is the sensa- 
 tional centre of the eye. With the affection of the pole is 
 therefore united the most perfect definition and the most 
 complete sensibility, and therefore also the clearest conscious- 
 ness of the visual impression. I believe that this subjective 
 straining after definition and clear consciousness of the sen- 
 sory impression is identical with the previously described 
 tendency to turning of the eyes ; that this tendency constitutes 
 the fundamental element of fixation ; and that, therefore, the 
 function of convergence is directed principally to the fixation, 
 not to the single vision, and that the single vision is, in a 
 normal pair of eyes, only secondarily united with correct con- 
 vergence, or dependent upon it ; and that the singleness of the 
 visual impression rests primarily upon a perfectly distinct 
 function, of which I shall say more in the sequel. The fully 
 conscious vision with the pole of the eye, is also the so-called 
 
DEFECTS OF CONVERGENCE. 
 
 63 
 
 direct vision ; and we may therefore describe direct vision as 
 the purpose of convergence. 
 
 If the two eyes B and c (Fig. 39), of which the distance 
 apart of their central points B c = d, are directed to a point 
 lying at the distance A D = a?, then each eye must describe an 
 arc which corresponds to the angle EBD = BDA = ^^D c = 0, 
 that is, the angle of inclination of the ocular axis B D to the 
 
 Fig. 39. 
 
 axis of the face A D. In two eyes of equal convergence this 
 angle is equal to half of B D c, the angle of convergence of 
 the two eyes. At first sight, therefore, the magnitude of the 
 angle ^ would appear to be a measure for the effort of con- 
 vergence of one eye. If we consider, however, that this effort 
 would be infinitely great when the angle = 90, it will 
 appear that not the angle itself, but its tangent, furnishes 
 this measure. Consequently we may express the necessary 
 effort of convergence, or the convergence requirement y by 
 
 the value of tan. d>, therefore by ... If now *; indicate a 
 
 M^O 
 
 certain constant factor, given by the physiological nature of the 
 eye, we obtain for the convergence required from each eye, for 
 binocular vision at the distance a?, simply the formula 
 
 y 
 
 - 
 
 or, if we make, 
 
64 THEORY OF OCULAR DEFECTS. 
 
 If we consider the visual distance x as the abscissa, and the 
 required convergence y as the ordinate of a curve which we call 
 the curve of required convergence, this curve, as well as the 
 curve of required accommodation, will form an equilateral 
 hyperbola which differs from the latter only by the constant 
 coefficient &', therefore by the length of its axes, and in 
 the fact of A, the point of commencement for the axes of 
 co-ordinates of the curve of convergence, lying in the centre 
 of the line B c that unites the centres of the two eyes, or 
 at the root of the nose ; while the point of commencement 
 of the axes of co-ordinates for the accommodation curve lies 
 about the length of its own diameter in front of each eye. 
 This difference of position of the point of commencement, 
 from which the visual distance x is measured, is so trifling 
 that it may be neglected ; and we may assume that the mag- 
 nitude x represents both the hyperbolic curve of accommo- 
 dation and the hyperbolic curve of convergence for the same 
 visual distance from the face. 
 
 A normal pair of eyes requires no effort of convergence for 
 infinite distance, or for parallel position, when for x = oc , 
 y is = 0. I call such a pair of eyes engonic (caring more, in 
 this expression, for analogy with ' emmetropic' than for gram- 
 matical logic). The criterion for such a pair of eyes is, that 
 they are parallel in the absence of effort, or in the state of 
 rest of their converging apparatus. When we do not con- 
 sider a pair, but one eye only, then the criterion of engonia is 
 that the eye, during the state of rest of its convergence, shall 
 be parallel to the axis of the face. 
 
 The anomalies of convergence, like those of refraction and 
 of accommodation, fall into two great classes, of which the first 
 rests upon the condition or structure of the converging ap- 
 paratus, and the second upon its irritability, sensitiveness, 
 
DEFECTS OF CONVERGENCE. 65 
 
 innervation, or functional state generally. We will, in the 
 first place, consider the former the anomalies depending upon 
 the structure of the converging apparatus. They are such as 
 affect the direction of convergence of the eyes in the state of 
 rest of the converging apparatus, and they may be called the 
 anomalies of ocular direction. The following three forms 
 require to be considered. 
 
 1. The normal state of ocular direction, or engonia, has 
 been already characterized. In it the curve of requirement is 
 
 TJ 
 
 y = _, which we call the curve of direction. There are two 
 
 anomalies to be considered. 
 
 2. The first anomaly of ocular direction affects a pair of 
 eyes which, in the state of rest of the converging apparatus, 
 converge towards a positive finite distance e, or which are 
 convergent in the state of rest. We may call such eyes 
 hypogonic. For them the curve of required convergence is 
 expressed by the equation 
 
 since, when the eye B leaves the state of rest of the convergence, 
 in order to fix, or to see directly, an object at the distance x, it 
 describes the angle D B D' (Fig. 39) which forms the difference 
 between the angles A D B and A D' B ; the tangent of which, on 
 account of the presumed smallness of this angle, is with suf- 
 
 ficient nearness proportional to the value - This curve, 
 
 like the curve of required accommodation of a hypometropic 
 eye, forms a depressed hyperbola, which cuts the visual line in 
 the distance A Q = e (Fig. 16). In order to accommodate for 
 the infinite distance x = <x> , or in order to assume a direc- 
 tion parallel to the frontal axis, this eye requires a negative 
 
 converging effort y = 
 & 
 
 We call the distance e the converging distance of a pair of 
 eyes. While the infinite converging distance is the sign of an 
 
 F 
 
66 THEORY OF OCULAR DEFECTS. 
 
 engonic pair, the positive converging distance is the sign of a 
 hypogonic pair. 
 
 3. The second anomaly of ocular direction is represented by 
 a pair of eyes which are divergent in the state of rest, and are 
 directed to a point Q lying behind them in a negative distance 
 e = e'\ so that they have a negative converging distance. 
 We may call them Tiypergonic. Their equation is 
 
 e'J 
 
 This, like the curve of required accommodation of a hyperme- 
 tropic eye, gives an elevated hyperbola, which cuts the visual 
 line behind the head in the distance A Q = e e' (Fig. 19). 
 To be directed to infinite distance, x = GO , such eyes require 
 
 k' 
 a positive effort of convergence y = . 
 
 Hypogonia and hypergonia, being both deviations from 
 engonia, may be called anengonia. 
 
 It is apparent that the curve of requirement of engonic, of 
 hypogonic, and of hypergonic eyes, that is to say, the curve 
 of convergence requirement of any eyes whatever, is expressed 
 
 by one and the same equation, y = k'f -- ) in which the 
 
 \x ej 
 
 quantity e is for the first infinite, for the second positive, and 
 for the third negative. 
 
 Like the faculty or power of accommodation, so also the 
 faculty of convergence is for every eye restrained within two 
 limits which determine the facultative curve. The anomalies 
 of the faculty of convergence are in truth anomalies of 
 motility, wherefore the facultative curve of convergence may 
 also be called the curve of motility. The eye has a near-point 
 and a far-point of convergence, and is not able to attain to 
 any stronger or to any weaker degree; so that it sees objects 
 in front of its near-point and behind its far-point, not with its 
 pole ; that is to say, not directly or with complete fixation. 
 A consequence of this is deviation of the ocular axis from the 
 
DEFECTS OF CONVERGENCE. 67 
 
 visual line, or squinting ; and this squinting in the use of both 
 eyes, or in binocular vision, is attended by diplopia, since the 
 retinal images in the two eyes fall upon different portions of 
 the retinae. 
 
 In Figs. 21 to 23, let the ordinate AB represent the near- 
 point, and the ordinate A R the far-point of the convergence ; 
 then by the difference R B of these two ordinates is given the 
 quantity called by von Graefe the squint capacity ; and which, 
 by analogy to the range of accommodation, may be called the 
 range of convergence. On the basis of the different values of 
 A R and A B there are the following three cases to be considered. 
 
 1. The normal state of motility, which may be called 
 eurogonia (Fig. 21). In this state the range of convergence 
 R B is of normal breadth, and A R is so small that it may be 
 considered nearly = 0. 
 
 The near-point of the convergence of the eurogonic may 
 nearly coincide with the near-point of the accommodation of 
 the europic eye ; but in general it seems to me to be some- 
 what nearer. 
 
 The two anomalies of motility are characterized by a small 
 range of convergence R B (Figs. 22 and 23), and may be 
 described as stenogonia ; and also, 
 
 2. As bathogonia, when, as in Fig. 22, A R is very small. 
 
 3. As liypsogonia, when, as in Fig. 23, A R is proportionately 
 large. 
 
 If the eye, as regards its direction, is engonic, then euro- 
 gonia, bathogonia, and hypsogonia are thus characterized. In 
 the eurogonic eye the near-point of convergence is near and 
 the far-point distant ; in the bathogonic eye the near-point is 
 distant ; and in the hypsogonic eye the far-point is near. In 
 general, however, no single one of these anomalies of motility, 
 and no single one of the anomalies of direction, determines the 
 possible detects of convergence. 
 
 We may find all the defects of convergence by combining 
 the three states of direction with the three states of motility. 
 This gives a system of nine conditions, accurately corresponding 
 
 F2 
 
68 THEORY OF OCULAR DEFECTS. 
 
 with the foregoing system of defects of accommodation. The 
 figures 24 to 35 display the corresponding defects of conver- 
 gence, if we put the hyperbola for the curve of direction, and 
 the dotted lines for the limits of motility. It will here be 
 sufficient to indicate the nine conditions briefly. They are as 
 follows : 
 
 I. 1. Engonia with Eurogonia, or the condition of the 
 normal eye (Fig. 24). 
 
 I. 2. Engonia with Bathogonia (Fig. 25). 
 I. 3. Engonia with Hypsogonia (Fig. 26). 
 II. 1. Hypogonia with Eurogonia (Fig. 27). 
 II. 2. Hypogonia with Bathogonia (Fig. 28). 
 II. 3. Hypogonia with Hypsogonia (Fig. 29). 
 
 III. 1. Hypergonia with Eurogonia (Fig. 30 or Fig. 31). 
 
 III. 2. Hypergonia with Bathogonia (Fig. 32 or Fig. 33). 
 
 III. 3. Hypergonia with Hypsogonia (Fig. 34 or Fig. 35). 
 
 The immediate optical effect of the defects of convergence 
 consists in this, that the defective eye, of which the conver- 
 gence is abnormal, does not assume the position of fixation as 
 towards the object, but squints at it, so that the retinal image 
 of the object does not fall on the pole of the eye. If the conver- 
 gence be too great, the squint is convergent, so that the image 
 of the object falls on the inner side of the pole. The eye, there- 
 fore, projects the object too far outwards. In binocular vision, 
 with two similar eyes in this condition, there will be diplopia 
 with homonymous images. If, on the other hand, the conver- 
 gence be too feeble, the squint will be divergent, the retinal 
 image will fall on the outer side of the pole, and the eye will 
 project the image too far inwards. In binocular vision with 
 two similar eyes in this condition, there will be diplopia with 
 crossed images. 
 
 If we have to deal with two similar eyes, then the kind and 
 the degree of the defect of convergence, namely, the conver- 
 gence distance e, and the range of convergence, may be deter- 
 mined by the double images. But with unlike eyes the 
 defective projection of each must be observed singly, which 
 may be done as well in binocular as in monocular use. 
 
SPECTACLES FOR DEFECTS OF CONVERGENCE. 
 
 69 
 
 SPECTACLES FOE ANOMALIES OP DIRECTION AND 
 MOTILITY. 
 
 With regard to the optical means of affording relief to the 
 defects of convergence, it is manifest that by such means only 
 the anomalies of ocular direction can be remedied, and that 
 those of motility cannot even be influenced. The rational 
 resource against the anomalies of direction is the use of the 
 common plane prismatic spectacles. Their action is shown 
 by the figures 40 and 41. We call a prism, as in Fig. 40, 
 
 Fig. 40. 
 
 B 
 
70 THEORY OF OCULAR DEFECTS. 
 
 positive, which has its base inwards, or to the nasal side, and 
 which, therefore, refracts the rays inwards, and consequently 
 produces an apparent displacement of the object outwards, or 
 homonymous double images. We call a prism negative, when, 
 as in Fig. 41, its base is turned outwards, so that it refracts 
 rays outwards, and produces an apparent displacement of 
 the object inwards, or crossed double images. If 
 
 \b be the angle of total deviation DBG = BGAof the prism, 
 so that ^ is positive for a positive prism and negative for 
 a negative, 
 
 g the convergence distance A G of the prism, that is, the 
 distance from the eye A, in which a ray parallel with the 
 visual line strikes that line after being refracted by the 
 prism ; so that g is positive for a positive prism and nega- 
 tive for a negative, and -^ is equal to the angle of inclina- 
 tion of the refracted ray towards the visual line, 
 
 h = = tan. \fr the refracting or deviating proportion of the 
 
 *9 
 prism (since d = B c, as before, the distance between the 
 
 centres of the eyes, or - = A B the distance of one centre 
 ft 
 
 from the median line) so that h is positive for a positive 
 
 prism and negative for a negative, 
 x the actual distance A x of a luminous point x from the 
 
 eye (or rather from the point A, in which the prism is 
 
 placed), 
 z the distance A z of the point z in which the ray refracted 
 
 by the prism cuts the visual line, so that B z is the line of 
 
 sight of the apparent object. 
 
 The refracted ray z B inclines towards the unrefracted ray 
 x B at the angle z B x t//, and consequently the angle 
 
 Z B X = AGB = DBG = t//. 
 
 The angle B x A = D B x, which the ray B x, proceeding 
 from the actual object at the distance a?, forms with the visual 
 line A x, is consequently equal to the sum of the angles z B x ;//, 
 and D B z = A z B, formed by the ray proceeding from the 
 
SPECTACLES FOR DEFECTS OF CONVERGENCE. 71 
 
 apparent object at the distance , and the visual line. Hence 
 we have BXA = BGA + BZA. Since now the tangents of 
 
 these three angles are respectively , , -, we obtain a 
 
 ' 2x 2g 2z 
 
 known trigonometrical formula. 
 
 JL A. 
 
 -5L. _J5 
 2x A A* 
 
 2g ' 2z 
 
 Considering that the quantity . - is always very small 
 
 in practice, much less than 1, we may omit it from the deno- 
 minator on the right side of the equation. This gives the 
 
 simple expression - = | , or = - , which accurately 
 x z g z x g 
 
 coincides with the analogous formula for lenses. 
 
 When we now, in our equation for the requirement curve of 
 
 the convergence, have put z in the place of x, or = 
 
 in the place of, we obtain as the equation for the curve of 
 x 
 
 required convergence for eyes furnished with prismatic spec- 
 tacles, when we take y' as the ordinate, 
 
 x g e 
 This equation again gives an equilateral hyperbola which is 
 
 congruent with the former one expressed by # = &'(- ) , 
 
 \x ej 
 
 and is only distinguished from it by position : while the ordi- 
 nate of the new curve, y' = y , differs from the ordinat 
 
 k r 
 of the former curve only by the constant term . 
 
 \j 
 
 Here also we call the former curve of requirement the 
 natural, and the present the artificial. 
 
72 THEORY OP OCULAR DEFECTS. 
 
 If we take the constant quantity _ + _=; we have 
 
 ee' 
 
 The quantity e' = ~l = -- indi- 
 
 ff 
 cates, therefore, the distance of that luminous point Q which, 
 
 with the aid of the prismatic spectacles #, sends its rays in the 
 state of rest of the eyes to corresponding portions of the retinae. 
 While e was the natural convergence distance, e' must be the 
 artificial convergence distance. 
 
 From the foregoing it is clear that a positive prism g de- 
 
 Jc 1 
 presses the hyperbolic curve of requirement by the depth , 
 
 while a negative prism g = g' raises this curve by the 
 
 k' 
 height . ; or, that positive prismatic spectacles displace the 
 
 y 
 
 terminal point Q of the convergence distance e in a direction 
 towards the eyes, since they change this convergence distance 
 
 1 
 into e' = I 1, and that negative prismatic spectacles dis- 
 
 ~e ~ff 
 place the terminal point of the convergence distance away 
 
 1 
 from the eyes, since they change it into e' = 1 1 
 
 j / 
 
 Hence hypogonia, in which the hyperbola lies too deep, is 
 to be corrected by negative glasses (bases outwards as in 
 Fig. 41), and hypergonia, in which the hyperbola is too 
 high, by positive glasses (bases inwards as in Fig. 40). A 
 complete removal of these anomalies requires the placing of 
 the terminal point of the convergence distance at infinity, or 
 that e' = oo . This condition is fulfilled by the prismatic 
 spectacles g = e, therefore by those spectacles of which the 
 convergence length is equal to the negative value of the con- 
 vergence distance e of the pair of eyes. For a hypogonic eye, 
 for which e is positive, the prism g must be negative ; and for 
 
SPECTACLES FOR DEFECTS OF CONVERGENCE. 73 
 
 a hypergonic eye, for which e is negative, the prism g must be 
 positive. 
 
 A prism g is best determined by its proportion of deviation 
 
 7i, which, as above stated, is = . The prism which entirely 
 
 2g 
 
 removes the hypogonia or hypergonia must, therefore, have 
 the proportion of deviation 
 
 , d d 
 
 h =lg- 'IT 
 
 By prismatic spectacles, therefore, the anomalies of ocular 
 direction can be completely neutralized, and every eye ren- 
 dered completely engonic. The anomalies of motility, on the 
 contrary, cannot be altered by any optical means. In practice 
 we can therefore only displace the curve of requirement of a 
 pair of eyes not completely eurogonic, by means of prismatic 
 spectacles, so far as the limits of the curve of motility (the 
 squint range) will allow, with reference to the part of the range 
 of vision which it is desired to render accessible. For this 
 purpose, in the already specified nine cases, the methods to be 
 observed are so completely analogous to those for the anoma- 
 lies of refraction and accommodation, and will be so readily 
 apparent from the Figs, 24 to 35, that any detail would be 
 superfluous. 
 
 Upon the determination of the degree of an occurring ano- 
 maly of position or motility, in a concrete case, and upon the 
 determination of the prismatic spectacles for a given effect, we 
 have to remark as follows. 
 
 If the pair of eyes receives the luminous impression from an 
 infinitely distant object without any exertion of convergence, 
 and sees the object singly, then it is engonic. If the pair of 
 eyes receives the impression of an infinitely distant object with- 
 out any effort of convergence, but in a convergently squinting 
 position, so that it sees homonymous double images, or projects 
 the object too far outwards, then it is hypogonic. If the pair 
 of eyes requires a certain effort of convergence for single vision 
 
74 THEORY OP OCULAR DEFECTS. 
 
 of an infinitely distant object, and therefore, without such 
 effort, sees with a divergent squint and with crossed double 
 images, then it is hypergonic, or possesses the faculty of con- 
 verging towards points lying behind it, and therefore of di- 
 verging forwards. 
 
 If it be found by experiment that p is the convergent dis- 
 tance, or q the proportion of deviation, of the prismatic spec- 
 tacles with the aid of which the pair of eyes, without any effort 
 of convergence, in the position of rest, sees infinitely distant 
 
 objects singly; if therefore the prism p = , or q = ( ~ 
 
 2q 2p 
 
 renders the artificial convergence distance infinite ; then, from 
 the formula 
 
 1 l l 
 
 = -, 
 
 e p 
 
 The natural convergence distance e is therefore equal to the 
 negative value of the convergence distance p of the above- 
 named prism. Hence, if the pair of eyes requires negative spec- 
 
 tacles p p'j it is hypogonic in the degree = , = -r- 
 
 If it requires positive spectacles p, then it is hypergonic in the 
 
 , 1 1 2q 
 
 degree - = = ~~. 
 e p d 
 
 These spectacles, p = e, which render the artificial con- 
 vergence distance infinite, and which serve to determine the 
 degree of hypogonia or of hypergonia, are precisely those which 
 remove this hypogonia or hypergonia completely, when such 
 removal is desired. We call the spectacles p briefly the en- 
 gonic spectacles. 
 
 An anomaly of motility is most easily discovered by the aid 
 of these engonic spectacles. When furnished with them, the 
 
 eyes are to be regarded as engonic ; so that their curve of re- 
 
 j./ 
 quirement has the equation y In this condition we de~ 
 
 OG 
 
 termine the distance n of the near-point of accurate conver- 
 
SPECTACLES FOR DEFECTS OF CONVERGENCE. 75 
 
 gence (H', in Fig. 15), and the distance w of the far-point of 
 accurate convergence (c 7 , in Fig. 15), and call these points the 
 artificial near- and far-points of the accurate convergence. Then 
 
 the corresponding ordinates and are the limits of the 
 
 n w 
 
 accurate convergence ; and, for the range of convergence or 
 squint range, which expresses the true convergence faculty of 
 
 the pair of eyes, we have k' ( --- ] . The coefficients - , 
 
 \n w J n 1 
 
 _, _ -j may be taken as measures for the degree of ano- 
 w n w 
 
 maly of motility. If _ be smaller than the normal 
 n w 
 
 range of convergence, then the eyes are stenogonic. If is 
 nearly 0, then the eyes are as deep-angled as a normal pair. 
 The greater the more high-angled the eye ; so that displays 
 
 the degree of hypsogonia. 
 
 Only an eurogonic pair of eyes can be rendered completely 
 normal by prismatic spectacles. For a stenogonic pair the 
 displacement of the curve of requirement can only produce a 
 more or less favourable effect. In order to give the eyes en- 
 durance in vision, their motility must only employ half of the 
 squint range. In general the conditions of endurance may be 
 formulated thus, the part i of the squint range being employed, 
 and the ordinate of the curve of requirement for the point of 
 fixation having, therefore, the value 
 
 Let us imagine a pair of eyes furnished with any prismatic 
 spectacles g ; then the equation of the curve of requirement is 
 
76 THEORY OF OCULAR DEFECTS. 
 
 In this e represents the natural convergence distance, which 
 is = p, when p represents engonic spectacles. The equa- 
 tion is therefore, 
 
 \se ~9 ~P 
 
 If it be desired that through these spectacles an object lying 
 at the distance s should be seen with the indicated degree of 
 effort of convergence, then, x being = s, 
 
 9 P 
 The condition is fulfilled by the spectacles 
 
 g s p \n w 
 
 If it be desired to express the prisms not by their conver- 
 gence lengths g and p, but through their proportions of devia- 
 
 tion h and q ; since g = and p = , we have 
 , d d i 1 - 
 
 If for the given object one half of the faculty of convergence 
 is to be called into play, so that i = -, then 
 
 Hence the prismatic lens h, for the effect desired, may be 
 determined, for any given case, by the easily observed magni- 
 tudes g, n, and w, and the given values of s, i, and d. 
 
 The distance d between the centres of the eyes has in adults 
 a mean value of GG"" 11 * or 2'44 Paris inches ; so that, in inches, 
 
 is = 1-22. 
 
SPECTACLES FOR DEFECTS OF CONVERGENCE. 77 
 
 If instead of the proportion of deviation A, it is desired to 
 determine the prisms by the angle of deviation i// in degrees, 
 we have then only to consider that 
 
 180 
 y = - h 57-306 . h degrees, 
 
 h = -^ + = 0-01745 . ^ 
 
 Between the values n and w, which indicate the position of 
 the near- and far-points of convergence for the eyes furnished 
 with the engonic spectacles p, and the values of n^ and w x , 
 which indicate the position of the near- and far-points for the 
 naked eyes, the proportions hold good which are given on page 
 60 for the near- or far-points of accommodation. 
 
 Example. 
 
 Suppose a pair of eyes that obtains an infinite convergence 
 distance by positive prismatic spectacles with a proportion of 
 
 deviation q = '01 52 5, or a convergence distance p = = 
 
 1 .00 
 
 - 80 ; then such eyes are hypergonic, and have the 
 
 natural convergence distance e = p = 80 ; that is, they 
 diverge in the state of rest of the converging apparatus. Under 
 the application of these spectacles they have an artificial near- 
 point of convergence n = 10 inches, and an artificial far-point 
 w = 100 inches. The convergence or squint range is there- 
 fore - = = = : and the eyes are ste- 
 
 n w 10 100 100 11' 
 
 nogonic and deep-angled. 
 
 If it be desired that these eyes should see an object singly 
 at a distance of 20 feet, or s = 240 inches, with the exercise 
 
 of half their convergence, or with i= - ., then they will require 
 spectacles, 
 
THEORY OF OCULAK DEFECTS. 
 
 = 0-01525+ g| _ ^g + _M = - 0-04677 
 
 Hence negative prismatic spectacles (with bases outward) 
 
 of the proportion of deviation O04677 = ~yj or of the 
 
 r*f 
 
 d 1*22 
 
 convergence distance g = = 26 inches, 
 
 Ah ' 
 
 will be necessary. 
 
 CONCURRENCE OF DEFECTS OF ACCOMMODATION 
 
 WITH DEFECTS OF CONVEKGENCE. COMBINATION 
 
 SPECTACLES. 
 
 By inclusion of the normal conditions we have now become 
 acquainted with nine chief states of accommodation, and with 
 nine chief states of convergence. Any one of the former may 
 be united with any one of the latter. This gives a total of 
 81 possible states, one of which is normal, while the other 
 80 constitute defects. All these may be grouped, according 
 to what has gone before, into a synoptical system, in which the 
 chief anomalies of refraction, of accommodation, of direction, 
 and of motility, form the leading points. It is wholly un- 
 necessary to devise names for the possible cases, since the 
 preceding mathematical analysis shows that, together or sepa- 
 rately, they require only special combinations of general and 
 independent formulae, of which one expresses the accommoda- 
 tion and the other the convergence. These formulae, and the 
 magnitudes contained in them, will determine the individual 
 special cases, both in kind and in degree, with far more 
 certainty than -any verbal descriptions, and will also allow the 
 limits between the two chief anomalies to be unequivocally 
 
COMBINATION SPECTACLES. 79 
 
 defined. I shall therefore content myself by setting forth some 
 important general results. 
 
 An entirely normal pair of eyes is at the same time emme- 
 tropic, europic, engonic, and eurogonic ; that is, has its visual 
 and converging distances infinite, and its range of accom- 
 modation and convergence of normal extent and of the lowest 
 depth. 
 
 A pair of eyes with only a defect of accommodation, but no 
 defect of convergence, can only use common spherical or 
 centric spectacles. If, therefore, a short-sighted, or far-sighted, 
 or over-sighted pair of eyes possesses a normal position and 
 convergence, they can only be aided by common convex or 
 concav-e spectacles ; the use of prisms would be irrational and 
 directly injurious. 
 
 A pair of eyes with only a defect of convergence, can only 
 use common plane prismatic spectacles. The combination 
 with lenses would be ineffectual. 
 
 An europic pair of eyes can be rendered completely normal 
 by spherical glasses ; and an eurogonic pair by prismatic 
 glasses. 
 
 A pair of eyes at once europic and eurogonic, can be 
 rendered completely normal by combination spectacles ; that is, 
 by the union of the required lenses with the required prisms. 
 
 If for an europic and eurogonic pair of eyes, the visual 
 distance is equal to the converging distance, that is, if the 
 eyes are in the same degree hypometropic as hypogonic, or 
 in the same degree hypermetropic as hypergonic, the neces- 
 sary combination spectacles will be normal or orthoscopic. 
 In any other case this will not be so ; and it may happen that 
 either concave or convex lenses may have to be combined 
 with positive or with negative prisms. 
 
 An europic pair of eyes would be rendered normal by the 
 emmetropic lenses / = a, and an eurogonic pair by the 
 
 engonic prisms p = e, or q= - . An europic and also 
 
 "2. 6 
 
 eurogonic pair would therefore be rendered normal by the 
 
80 THEORY OF OCULAR DEFECTS. 
 
 combination of I = a, and^ = e. If the visual distance a 
 of these eyes be equal to their converging distance e, then we 
 
 have I = p = a, or q = __=_. The latter pro- 
 portion, however, is that for normal orthoscopic spectacles. 
 (See Gesetze des raumlichen Sehens, s. 177, z. 1.) 
 
 Whatever may be the combination of accommodation defect 
 with convergence defect, the necessary combination spectacles 
 may always be determined through the lens / and the prism </, 
 according to the above two formulas. In order, namely, that 
 the pair of eyes may see clearly, singly, and continuously, and 
 therefore with the corresponding portion i of accommodation 
 and of convergence effort, an object lying at the distance s, it 
 is necessary that, 
 
 i = 1 + 1 . . ( ' 
 
 / . i U 
 
 and = H - 
 
 g s p 
 
 In both these formulas s has the same given value : I ex- 
 presses the emmetropic lens, and p the engonic prism, i the 
 accommodation effort, and i' the convergence effort. The two 
 quantities n and w in the first formula are the artificial near- 
 and far-point of accommodation ; and the two quantities n' 
 and w' in the second formula are the artificial near- and far- 
 point of convergence. If the near- and far-point of accom- 
 modation have respectively the same position as the near- and 
 far-point of convergence, so that n = n 1 and w = w', and if we 
 desire equal accommodation and convergence effort, so that 
 i = i 1 , then subtraction of the two foregoing equations gives 
 the proportion 
 
 / 9 I P 
 
 This proportion between the lens and the prism is indepen- 
 dent of the distance s of the object, and also of the near- and 
 
COMBINATION SPECTACLES. 81 
 
 far-points, and is conditioned only by the visual distance I and 
 the converging distance p. If these two are infinite, the eyes 
 
 being einmetropic and also engonic, we have - = 0, 
 
 J U 
 
 therefore / = g = . The latter condition characterizes 
 
 *- ft 
 
 normal orthoscopic spectacles, and teaches that an emmetropic 
 and engonic eye, with an equal anomaly of accommodation 
 and of motility, or in an equal degree stenopic and stenogonic, 
 has need of a normal orthoscopic lens f, g, determined by 
 the degree of the anomaly and by the given visual distance s. 
 
 If a common short-sighted or hypometropic eye, which 
 possesses a somewhat depressed accommodation hyperbola, has 
 long worked without glasses, it will have its hyperbola of 
 convergence also more or less depressed, and the spectacles 
 required will more or less approach the orthoscopic. But, 
 if exertion has not yet brought about this condition, then only 
 common concave lenses will be required. 
 
 A common far-sighted eye, the defect of which consists in a 
 small deep-lying range of accommodation, while the hyperbolic 
 curves of requirement, of accommodation, and of convergence, 
 and also the range of convergence, are entirely normal, requires, 
 for near vision, not only a depression of the hyperbolic accom- 
 modation curve, but also an equal depression of the hyperbolic 
 convergence curve, since, if the latter be not also lowered, the 
 eye will exert for near vision a feeble effort of accommodation 
 with a strong effort of convergence, which will occasion strain. 
 Consequently, far-sighted eyes almost always require normal 
 orthoscopic spectacles, not common convex spectacles. 
 
 The two last results, namely, that a common far-sighted 
 (i. e. bathcpic) eye requires an orthoscopic glass, and a com- 
 mon short-sighted (i.e. hypometropic) eye requires, on the 
 contrary, a common spherical glass, are of special importance 
 on account of the great frequency of these conditions. For 
 the far-sighted eye, this rule has no exception when the eye 
 has become bathogonic by use, that is, has acquired a small 
 
82 THEORY OF OCULAR DEFECTS. 
 
 range of convergence. For the short-sighted eye, however, 
 an exception occurs continually, as soon as the eye has become 
 hypogonic by use, that is, has acquired a shortened con- 
 vergence distance ; since then the spherical lens must more 
 approach the orthoscopic. 
 
 The great difficulty which exists, according to Donders, in 
 the treatment of myopia, seems to me to depend essentially 
 upon the mischievous application of common spherical glasses, 
 and to admit of removal by the addition of properly determined 
 prisms. 
 
 It should be observed that, by the foregoing arrangement, 
 the visual process is so regulated as to place the effort of 
 accommodation at the part indicated by i of the absolute range 
 
 of accommodation - , and the effort of convergence at 
 
 the part i' of the absolute range of convergence f . To 
 
 do this seems very important when the sharpest possible vision 
 is required, as in the use of the eyes about common employ- 
 ments, such as reading, drawing, and the like. If, however, 
 spectacles are required for other purposes, as for the most con- 
 tinuous possible vision at great distances, then the relative ac- 
 commodation and convergence range come in the place of the 
 absolute. They are given by the course of the two facultative 
 curves, and I shall speak more fully of them in the sequel. 
 
 DETERMINATION OF VISUAL STRAIN. 
 
 If we furnish a pair of eyes with any given combination 
 spectacles, /, g ; then the two foregoing formulas enable us to 
 determine the strain of accommodation and of convergence 
 with which the eyes see, clearly and singly, an object situated 
 at the distance s. This strain is to be estimated by the values 
 of and i'. For these we have, through the first and second 
 formula, 
 
DETERMINATION OF VISUAL STRAIN. 83 
 
 1 + 1-1-1 1-fl 1 J- 
 
 , and i<= 
 
 n w n' w' 
 
 If the tension of the accommodation and the convergence 
 be eqMal, the equality of these values of i and i' gives a rela- 
 tion between the lens /arid the prism g. 
 
 From the same formulae it may also be ascertained with 
 what tension the unaided pair of eyes will see at the distance 
 s. Unaided eyes are equivalent to the lens/ = oo , and the 
 prism g = oo , so that we obtain, 
 
 1+1 1 1 + 1 1 
 
 w 
 
 If we imagine a pair of short-sighted and broad- sighted 
 eyes, with complete faculty of accommodation and no defect 
 of convergence, so that w; = oo , ^ = co , and w' = oo ; then, 
 
 For these eyes the artificial near-point of accommodation co- 
 incides very nearly, and the artificial near-point of convergence 
 exactly, with the natural ; so that the two quantities n and 
 n' have about the normal value. The quantity I = a ex- 
 presses the negative value of the visual distance ; and has 
 therefore a negative and finite value, so that, since n is nearly 
 
 I-IW.JL d{ , = ^ = ;+ . 
 s a J a 8 a 
 
 For a completely normal pair of eyes a = oo , and therefore 
 
 G 2 
 
84 THEORY OF OCULAR DEFECTS. 
 
 From this it follows that a pair of common short-sighted 
 eyes, in order to see at the distance s, must make the same 
 convergence as a normal pair; and, at the same time, a 
 smaller effort of accommodation. If the object be in front of 
 the middle visual distance of a normal eye, s is < 2 n, and 
 
 therefore - > -. In this case, where il > -, the short-sighted 
 
 eye must therefore exceed the measure of convergence that is 
 adapted for continuous vision ; while its effort for accommoda- 
 tion, i = , has not yet reached this measure. 
 
 s a, 
 
 The quantity i is the proportion of the faculty of accommo- 
 dation actually exerted, to the range of accommodation shown 
 
 by ~ ; and the quantity i' has the same significance with 
 
 n w 
 
 regard to the range of convergence. When i is nearly = 
 
 - the eye has the greatest accommodation endurance, and 
 
 2 
 
 the visual strain is therefore as small as possible. We will 
 assume generally that the least visual strain corresponds to the 
 amount of accommodation i = j, and to the amount of con- 
 vergence i f = /. The measure of this visual strain is then 
 the difference respectively between i and j, and between i' and 
 /. If we indicate this exact visual strain respectively by 
 and ', then we have = i j, and ' = i' /. 
 
 If or ' be positive, then too strong an effort is demanded 
 from the eye ; and, if or ' be negative, too weak an effort is 
 demanded. When i has the value 1, and therefore = 1 j, 
 the highest endurable positive tension is attained ; and, when 
 t = and hence = j t the highest endurable negative 
 tension is attained. 
 
 The quantities i, j, and are proportional numbers, which 
 give a certain proportional part of the range of accommodation 
 
 -- for convergence -J For many considerations 
 this proportional part is of less importance than the absolute 
 
DETERMINATION OF VISUAL STEAIN. 85 
 
 value of the portion of the range of accommodation expressed 
 by it. If in a small-sighted eye (of small range of accommo- 
 dation) i is actually = j, and therefore = 0, and when 
 therefore actually no visual tension exists, yet such an eye does 
 not possess the endurance of an europic eye, since, in the 
 former the required effort of accommodation lies nearer to its 
 limits on either side than in the latter. A normal endurance 
 requires that the effort of accommodation should be sufficiently 
 far above its lower limit, and sufficiently far below its higher 
 limit. Hence the difference between the actual effort of 
 accommodation, and the limits of accommodation, appears to 
 be a measure for the endurance of the eyes ; and we have to 
 consider this endurance, which we express by a, in two 
 opposite relations. The difference between the maximum of 
 accommodation and the actual accommodation is the en- 
 durance <*!, in the exertion, the difference between the actual 
 and the minimum accommodation is the endurance et 2 , in the 
 relaxation, of the accommodating apparatus. We have, for the 
 eye furnished with the lens/, 
 
 ! 1\ 1 1 11 
 
 n -\f l ' 1 ^ l l 1 l 
 
 = (I Of ) = ~ -T+-F 
 ' \n w J n s I f 
 
 and for the naked eye, or for =0, 
 
 ./I 1\ 1 1 1 
 
 a 2 =z( ) = _ + -- 
 \n w J s I w 
 
 ,.,.,/ 1 IN 1 1 1 
 a, = (1-zM --- )=_- --- _ 
 \ n w J n s I 
 
 Complete endurance requires that the two values a l and a 2 
 should have values corresponding to those in an europic 
 eye. This condition is obviously not generally to be fulfilled ; 
 and the foregoing formulae enable us to estimate the difference 
 between the endurance of a defective eye and that of a 
 
86 THEORY OF OCULAR DEFECTS. 
 
 normal eye, or between the endurance of an eye in normal 
 and in abnormal use, since for normal use i will be = j in the 
 formulas, and for a normal, that is, broad-sighted or europic 
 eye, n and w have the normal values. 
 
 In the foregoing we have treated the quantities n and w, 
 which belong to the near- and far-points, as constants, since we 
 
 have taken - as the absolute maximum, as the absolute 
 n w 
 
 minimum, and - - as the absolute range of the accommo- 
 n w 
 
 dation. If it be wished to pursue more exactly the facts of 
 visual strain and endurance, then we may understand by - 
 
 the relative maximum, by the relative minimum, and by - 
 
 the relative range of accommodation which corresponds 
 
 with the convergence for the visual distance s, under the 
 spectacles Z, and which we shall hereafter consider more 
 closely. In this way we obtain the formulae of the relative 
 visual strain, and the relative endurance for a determinate 
 distance of vision. 
 
 Besides these relative values, the absolute values of n and w 
 always retain their special importance ; since they enable us to 
 ascertain what absolute effort the eye must make for vision at 
 this distance. 
 
 ASTHENOPIA. 
 
 The last preceding inquiries into visual strain and endurance 
 lead to the knowledge that a stenopic eye cannot have the en- 
 durance of one that is europic ; since the actual effort of accom- 
 modation, even when so regulated by spectacles that i has its 
 most favourable value j, lies too near the limits of the range. 
 
ASTIIENOPIA. 87 
 
 This inconvenience is especially likely to occur in a hyper- 
 metropic eye, since such a one, even when europic, cannot 
 employ the lower portion of its range of accommodation, on 
 account of the elevation of the curve of requirement. But 
 the europic eye may always be supplied with the normal 
 endurance by proper spectacles, since these will depress the 
 curve of requirement. It is only the stenopic eye that cannot 
 be relieved of the infirmity, although we can raise its 
 endurance to the maximum attainable, by the spectacles that 
 render the value i = j. 
 
 The above described deficiency in endurance, which, as an 
 insuperable defect, arises only from stenopia, but which may 
 occur also in other cases, as in hypermetropia, constitutes the 
 essence of the abnormalities described in ophthalmology under 
 the name of asthenopia. 
 
 If the defect be in the accommodation, we have the so- 
 called accommodative asthenopia. If the defect be in the con- 
 vergence, for which z", j and , in the foregoing formulae, must 
 be replaced by z v ,/ and ', we have the so-called muscular 
 asthenopia. 
 
 From the foregoing it appears, therefore, that asthenopia 
 does not, as assumed by Bonders, arise solely from hyperme- 
 tropia ; that it is in principle the result of too small a range 
 of accommodation ; but that hypermetropia must necessarily 
 tend to produce asthenopia, even when the eye possesses a 
 large range of accommodation. 
 
 It appears, further, that the asthenopia of an europic eye, 
 and hence that of most hypermetropes, may be completely re- 
 lieved by spectacles ; but that this can never be done when 
 the evil depends upon a narrow range of accommodation ; 
 although in these cases a regulation of the accommodation 
 between its extreme limits may be attempted by spectacles, 
 and will be useful. 
 
 Lastly, it follows that the common far-sighted eye, since it 
 possesses too small a range of accommodation, must constantly 
 experience more or less asthenopia from near vision through 
 
88 THEORY OF OCULAR DEFECTS. 
 
 convex spectacles ; and hence that a far-sighted person, with 
 even the best spectacles, cannot have the endurance for near 
 vision of a short- sighted person or an emmetrope. 
 
 If, for example, in a far-sighted emmetropic eye, for which 
 
 the emmetropic lens I = oo , or r = 0, the near-point be at 
 the distance n = 50", and the far-point so distant that without 
 
 much error we may make w = oo or - =0, then the absolute 
 
 w 
 
 range of accommodation is only about the sixteenth part of 
 50 
 
 the range f J of a very europic eye. If, therefore, the eye in 
 
 question be enabled to see near at hand by convex spectacles, 
 and if the visual strain be made as small as possible, still the 
 difference between its actual accommodation and the extreme 
 upper or lower limit of its accommodative power, or the 
 value of a l and ct 2 , is only about 1-1 6th of the corresponding 
 difference for an europic eye. The endurance of the far- 
 sighted eye supposed will therefore be much less than that of 
 the europic eye. 
 
 GEAPHIC DELINEATION OF THE COMBINED DEFECTS 
 OF ACCOMMODATION AND OF CONVEEGENCE. 
 
 The concurrence of the different defects of accommodation 
 and convergence may be displayed by a single graphic delinea- 
 tion, which contains the two curves of requirement and the 
 two facultative curves. For this purpose, we may make the 
 two coefficients, k and k' each 1, and may 'therefore repre- 
 sent the hyperbolic curve of refraction by the equation 
 
 y = , and the hyperbolic curve of direction by the equa- 
 tion y = - - ; while the ordinates, limiting the curves of 
 
GRAPHIC DELINEATION. 
 
 accommodation and of motility, will be respectively indicated 
 by -and ; and -. Thus, in Fig. 42, the hyperbola, 
 
 drawn with a dark line, represents the curve of required ac- 
 commodation, and the dark dotted lines represent the limits 
 
 Fig. 42. 
 
 of the accommodative faculty. The hyperbola drawn with a 
 faint line represents the curve of required convergence, and 
 the faintly dotted lines represent the limits of the converging 
 faculty, in both cases for any given pair of eyes. If the two 
 hyperbolas coincide, then the eyes are in equal degrees hypo- 
 metropic and hypogonic, or hypermetropic and hypergonic, 
 and therefore require normal orthoscopic spectacles, which in 
 any other case they do not. 
 
 This simultaneous graphic delineation of the require- 
 ment and the facultative curves of accommodation and of 
 convergence furnishes a very complete picture of the pecu- 
 liarities of the eye, and a guide to the determination of the 
 proper spectacles, when we consider that the requirement 
 curves, that is, the hyperbolas, may be elevated or depressed 
 in any required degree, respectively by spherical or by pris- 
 
90 
 
 THEORY OF OCULAR DEFECTS. 
 
 matic glasses ; while the horizontal limits of the facultative 
 curves, which express the absolute ranges of accommodation 
 
 and of convergence, are 
 immovable, and that 
 from these fixed limits 
 the values of t and i 1 in 
 the foregoing formula? 
 are to be determined. 
 
 With the help of the 
 graphic delineation, the 
 spectacles, to produce 
 any desired effect, may 
 very easily be geome- 
 trically constructed in 
 the following manner : 
 Suppose that we 
 have to deal with the eye represented in Fig. 43, in which 
 the dark hyperbola E G represents hypometropia with the 
 range of accommodation R B, and the faint hyperbola E' G' 
 represents hypergonia, with the range of convergence R' B'. 
 Both curves are laid down according to the formulae for k = ; 1 
 and Tc' = 1, and respectively according to the formulae y = 
 
 __ and y = , in which a indicates the (in this case 
 
 x a x e 
 
 positive) accommodation distance, and e the (in this case nega- 
 tive) convergence distance. 
 
 It is required that the eye shall see clearly an object at the 
 distance A D, with the accommodation effort A c, and the con- 
 vergence effort A c'. 
 
 If we draw through D the vertical ordinate D K, and through 
 C and c', the horizontals c F, c' F', to F, F', the points of inter- 
 section with the ordinate D K, then the hyperbolic curve of 
 accommodation must pass through the point F, and the hyper- 
 bolic curve of convergence through the point F' ; so that the 
 former curve E G must be elevated by the height H F, and the 
 latter curve E' G', depressed by the depth K F', which is to be 
 
ACCOMMODATION AND CONVERGENCE. 91 
 
 accomplished respectively by a negative or concave lens, and 
 by a positive prism. 
 
 If H F = - and K F 7 = -, then / is the focal length of the 
 
 required concave lens, and g the converging length of the 
 required positive prism, of which the proportion of deviation 
 
 h = -, and the angle of deviation \Js = 57*306 . h degrees. 
 
 c/ 
 
 It is evident that this method is very simple ; and it may be 
 much facilitated by using, for the equilateral hyperbola, which, 
 in all cases, has the same outline for accommodation and for 
 convergence, a form which shows also the position of the rect- 
 angular asymptotes, and which can easily be so arranged that 
 the vertical asymptote remains in the line A B, while the 
 anterior or posterior branch of the curve passes through the 
 experimentally determined terminal point of the accommo- 
 dation or convergence distance. 
 
 The arrangement of the spectacles will then only be de- 
 pendent upon the measured determination of the points c and 
 c', which indicate the exertion that is to be demanded from 
 the accommodation and the convergence. For determining 
 these, the limits of the absolute range of accommodation R B, 
 and of the absolute range of convergence R' B', are not ex- 
 clusively authoritative, but serve for general guidance. 
 
 THE EECIPKOCAL INFLUENCE OF THE ACCOMMODA- 
 TION AND THE CONVERGENCE. 
 
 It has been observed in an earlier chapter, that accommoda- 
 tion and convergence are faculties originally independent, but 
 that they influence each other secondarily, or by induction. 
 This influence is displayed, when the normal proportion be- 
 tween accommodation and convergence is either fallen short of 
 or exceeded, in the first instance as visual strain. So long as 
 
92 THEORY OF OCULAR DEFECTS. 
 
 the disproportion does not exceed a certain degree, the pair of 
 eyes will see clearly and singly, but with strain. When this 
 certain degree is exceeded, either one or the other function 
 will no longer be discharged ; and there will be either single 
 vision with an imperfect image, or double vision with clear 
 images, so long as the apparatus, still in activity, does not 
 reach the absolute limit of its power. 
 
 To what degree an eye can bear this visual strain between 
 accommodation and convergence, for the different degrees of 
 each, that is, between what limits, for a given convergence, the 
 accommodation can be raised above, or lowered below, the 
 normal standard, or between what limits, for a given accommo- 
 dation, the convergence may be increased or diminished, might 
 be expressed after a series of observations, in the form of a 
 general law, which, in particular cases, would be more or less 
 modified by individual conditions. I believe that such an 
 average of the relative accommodation and convergence limits 
 would lead to the result, when y is a given convergence, and 
 therefore the ordinate of the hyperbolic convergence curve for a 
 definite convergence distance x, that the limits of accommoda- 
 tion with visual strain may be shown by two ordinates of the 
 
 values m y and ?., and that the coefficient m will, for all values 
 m 
 
 of y, be nearly constant. If, for example, we find that for 
 any given convergence the normal accommodation may, with 
 
 strain, be increased by one-sixth ; so that m 1 -f- = - ; 
 
 6 6 
 
 then the accommodation may, for any convergence, range from 
 
 If this supposition were confirmed, the limits of the relative 
 accommodation would be shown by two equilateral hyperbolas 
 of which one, with the ordinates m y, would be above the re- 
 quirement curve of the convergence, while the other, with the 
 
 ordinates - , would be below it. Inversely, the limits of rela- 
 m 
 
 tive convergence would be shown by two similar hyperbolas, 
 
ACCOMMODATION AND CONVERGENCE. 93 
 
 one of them above, and the other beneath, the requirement 
 curve of the accommodation. 
 
 Hence, for an emmetropic and engonic eye, for which the 
 requirement curve of the accommodation coincides with that 
 of the convergence, as shown, in Fig. 44, by the hyperbola 
 drawn with a continuous line, the limits of the relative ac- 
 commodation (or convergence) are shown by the two dotted 
 hyperbolas. The figure teaches that 
 the eye, in convergence for the dis- 
 tance A D, may exert its power of 
 accommodation from a mini-mum 
 D F" to a maximum D F'. The dis- 
 tance F" F' denotes the relative range 
 of accommodation, or of convergence 
 respectively, for the distance of ac- 
 commodation or of convergence A D. 
 
 The absolute limits, A R and A B, 
 of the accommodation or convergence, or the absolute range 
 of accommodation or convergence, R B, would be very little, 
 or not at all, influenced by the foregoing, since they correspond 
 to the extreme limits of either faculty. 
 
 The foregoing purely theoretical doctrine finds its confirma- 
 tion in the observations of Bonders. The graphic delineation 
 given in his work, On the Anomalies of Refraction and Accommo- 
 dation, of the relative range of accommodation of an emmetropic 
 eye, for different conditions of convergence, shows that, so long 
 as the object is not too near either to the near- or the far-point, 
 the positive part of the relative range of accommo.dation may 
 increase to double what is normally required, and the negative 
 part may sink to one half of this amount. Hence m = 2. 
 The normal accommodation, y, may, with strain, be intensified 
 
 to 2 y or relaxed to ^. 
 
 In Fig 44, in the neighbourhood of the near-point, the 
 upper boundary line of the relative accommodation passes, 
 earlier than the lower boundary line, into the horizontal corre- 
 sponding to the limit of the absolute accommodation. Near 
 
94 
 
 THEORY OF OCULAR DEFECTS. 
 
 the far-point, on the other hand, the lower boundary of the 
 relative accommodation reaches, earlier than the upper bound- 
 ary, the horizontal corresponding to the lower absolute limit. 
 Hence the boundary lines of the relative accommodation for an 
 emmetropic eye follow the course of the dark line in Fig. 45, 
 which accurately coincides with the observations of Bonders. 
 For a hypometropic eye, the figure (46) is nearly the same 
 as that for an emmetropic eye. For a hypermetropic eye, 
 
 Fig. 45. 
 
 Fig. 46. 
 
 however, we obtain the essentially 
 changed Fig. 47. These results of 
 theory again correspond to the ob- 
 servations of Donders. 
 
 The form of the limiting curves of 
 the relative convergence, in given 
 accommodation, is precisely like that 
 of the limiting curves of relative ac- 
 commodation in given convergence. 
 For an abnormal position of the curve of requirement of one 
 ocular faculty, e.g. for a depressed accommodation hyperbola, 
 the form of the limiting curves of the other faculty, that is, of 
 the relative convergence, would be more or less modified. 
 But since the absolute limits of the other faculty are immov- 
 able, so it would always be possible, by the associated course 
 of the requirement curves, and the absolute limits of the 
 facultative curves, for both faculties, to decide, according to 
 page 91, upon the localisation to be effected by spectacles. 
 
APPLICATION. 
 
 95 
 
 APPLICATION. 
 
 According to the prevailing doctrine, when an eye B (Fig. 48) 
 fixes upon an object D, it looks at this object in the direction 
 of its visual line B D (the slight deviation of the visual line, 
 B D, going to the physiological pole, or yellow spot, from the 
 geometrical axis of the eye or of the cornea, is in this respect 
 unimportant). This doctrine rests upon an error. 
 
 Fig. 48. 
 
 If the left eye saw the object in the direction BD, and the 
 right eye the same object in the direction CD, that is, if the 
 geometric lines B D and c D were the sources of the sensory im- 
 pression which we call direction, the two eyes would receive 
 two totally different impressions, which must necessarily lead 
 to two different conceptions or opinions of the direction. For 
 since the object would appear to each eye in a different direc- 
 tion, it could not appear to the two together in the same direc- 
 tion. It could not therefore be seen as a single object ; but 
 must necessarily appear double. 
 
 But, in reality, an object fixed by the two eyes appears 
 single, and therefore in one single direction. This fact is quite 
 
96 THEORY OF OCULAR DEFECTS. 
 
 sufficient for the inference that the right and left eyes do not 
 see the object in the directions CD and BD, or that the visual 
 lines are not the apparent directions. 
 
 The circumstance that the two visual lines, c D and B D, meet 
 in one and the same point of space, is, for the visual organ, 
 wholly without significance. All the demonstrations of phy- 
 siologists, which refer to the coincidence of the relations of 
 absolute space with the phenomena of visual appearances, are 
 brain cobwebs. That two straight lines intersect in a point, 
 and that hence a point can be determined by the direction of 
 two straight lines, is a pure reflection, or exercise of the under- 
 standing. But the eye is not an organ of intelligence ; it is 
 an organ of sense. The eye does not think, does not reflect, 
 does not conclude, does not judge ; the eye discharges a totally 
 different function; it sees, that is, it reacts upon the ether 
 affections of the rays of light, or it feels the vibrations of these 
 rays. There is no absolute space, and no mathematical exten- 
 sion for the eye, but only for the understanding ; the conditions 
 of vision rest only in the affection by the rays of light of the 
 nervous apparatus of the retina : what is actually present ex- 
 ternal to the eyes, whether there is an actual object or not, 
 whether space exists as something objective, and whether points 
 and lines exist therein, and behave in such or such a manner, 
 whether there is an above and a below, whether the retinal 
 image is erect or inverted, of all this the eye knows nothing ; 
 since knowing is not an affair of the sensorium, but of the 
 understanding, whence, on the other hand, seeing and feeling 
 are not the business of the understanding but of the sensibility. 
 Certainly the sensational faculty, by means of the organic unity 
 of the brain, arouses an abstract faculty ; and inversely the 
 abstract faculty, as attention, intelligence, inquiry, can influence 
 the sensational faculty, just as the will also can influence, call 
 forth, and sustain this faculty ; but this influence is something 
 merely accessory, which does not in the least affect the dis- 
 tinctness, in principle, of the sensational and the abstracting 
 processes. At present there prevails, with regard to the sensa- 
 
APPLICATION. 97 
 
 tional and the intellectual processes, much needless confusion, 
 arising from the use of the same words, such as bulk, distance, 
 light, colour, and so forth, to express both the direct sensory 
 impressions, and the intellectual judgments founded upon them. 
 
 The first condition for single vision is, that the apparent 
 directions for the two eyes should have no deviation from each 
 other, that they should therefore be parallel. But this alone 
 would not suffice for single vision ; and its fulfilment would 
 allow the left and right eye to see the object separated, as at 
 D' and D". There is yet a second condition to be fulfilled, that 
 both apparent directions should have a common point of origin. 
 In fact, in vision with both eyes we see the object D in a direc- 
 tion A D proceeding from the forehead or root of the nose A, 
 and therefore in the frontal axis. 
 
 If A B D c be the circle of the horopter, then the centre A of 
 the arc between the two eyes will be the exact point of origin 
 of the frontal axis; and, for an object E, placed obliquely in 
 front of the face, A E shows the oblique frontal axis, and the 
 common apparent direction in which both eyes see the object. 
 
 These facts about the apparent direction I have, as I believe, 
 in 5-8 of the Gesetze des rdumlichen Sehens, conclusively 
 established. It follows therefrom that the apparent direc- 
 tion A E undergoes a deviation from the actual direction B E 
 and c E for either eye, in which the angle B E A = c E A, and 
 therefore, in a normal pair of eyes, is equal to half of the 
 angle of convergence EEC. Generally, for a normal or abnor- 
 mal pair, the deviation of the apparent direction from the 
 actual corresponds to the angle of inclination of B E or c E, the 
 visual line of the eye concerned, towards the frontal axis A E. 
 
 In the place referred to I have also shown that this deviation 
 of the apparent from the actual direction for each single eye is 
 not dependent upon the participation of the other eye in the 
 visual act ; but that it occurs even in a greater degree when 
 the other eye is excluded. Hence the deviation appears to be 
 a sensational process in every eye, analogous to accommoda- 
 tion and to fixation, which may certainly be in some degree 
 
 H 
 
98 THEORY OF OCULAR DEFECTS. 
 
 influenced by these functions, just as theyreciprocally influence 
 one another, but which is originally independent. Hence I 
 believe that the visual faculty which is subservient to the 
 knowledge of the direction of objects requires to be distinguished 
 by a special name ; and I have called it the Application. 
 
 The Application, since it affords a definite sensational feeling, 
 must rest upon a definite nervous process, and must therefore 
 have a definite material basis. In the Gesetze des raumlichen 
 Sehens I have not declared this basis ; but now it is so clear 
 to my mind, that I may do so. I believe that the mobility of 
 the bacillary layer of the retina, of which, from the deduc- 
 tions in my Physiologische Optik and Gesetze, there can no 
 longer be any doubt, forms the basis of the Application. 
 
 The bacilli, as terminal points of the optic nerve fibres, are 
 the organs in which the physiological process excited by the 
 luminous ether vibrations occurs ; the various concussions and 
 other affections which excite this process being felt in a manner 
 wholly analogous to that in which sensitive organs feel mecha- 
 nical concussions, pressure, traction, shock, caloric, and other 
 effects, and producing conditions of the organism which we, 
 by means of the consequent affection of the brain through the 
 sensorium, call by such names as distance, direction, lumi- 
 nosity, colour, and so forth. We call the axial (or parallel 
 with the axis of the bacillus) components of the luminous 
 concussion the direction of the object. Since these components 
 lie in the direction of the bacillary axis, the direction of the 
 percipient bacillus is also the direction in which we feel the 
 luminous impression, that is, in which we see objects. If we 
 assume that the bacilli are perpendicular to the retina, and that 
 the retina is spherical, we must see the object in a line of 
 direction perpendicular to the retina, and this line must pass 
 through the central point of the sphere. It has been hitherto 
 believed that the bacilli stand perpendicular to the retina, and 
 that the retina, at least in the vicinity of the pole, is spherical. 
 Both these suppositions are only true approximatively and at 
 times. 
 
APPLICATION. 99 
 
 In 23 of the Physiologische Optik I have already remarked 
 on the deviation of the retina from the spherical outline at 
 the blind spot, where the entrance of the optic nerve forms 
 a papillary elevation. Here the direction of the bacilli is not 
 through the centre of the sphere ; and this deviation from the 
 spherical outline affords an explanation of the hitherto erro- 
 neously interpreted phenomena produced by the incidence of 
 rays of light upon the blind spot. This coincidence of the 
 phenomena with our theory is a strong confirmation of the 
 latter ; especially for this, that we feel the direction of the ob- 
 ject in the direction of the bacilli. 
 
 The mobility of the bacilli has two forms ; first, the shifting 
 of the bacilli in the retina, from which may be explained 
 many remarkable phenomena that have led to a variety of 
 contradictory opinions ; secondly, the change of inclination of 
 the bacilli with reference to the perpendicular to the retina. 
 From this I have already, in the Gesetze, p. 54, and in 28, 
 explained the peculiar phenomena of single and double vision 
 after strabismus operations, and the remarkable antipathy 
 against single vision observed by von Graefe. (A. f. 0. Bd. 1, 
 p. 117 et seq.) 
 
 It is the latter, the change of inclination of the bacilli towards 
 the retina, which we will now call the nutation of the bacilli, 
 on which the application, and therefore the recognition of 
 apparent direction, depends. It constitutes a faculty, and is 
 associated with an effort, precisely as the accommodation and 
 the convergence or fixation. The striving after single vision, 
 or after union of the visual impressions of the two eyes, is the 
 immediate stimulus to the exercise of this faculty, precisely 
 as the striving after clearness of impression, or after concentra- 
 tion of the cone of rays upon the retina, is the occasion of 
 accommodation, and as the striving after a definite and certain 
 recognition of the impression, by guiding its optical centre of 
 intensity to the pole of the retina, is the occasion of conver- 
 gence. No ophthalmologist can have failed to notice that, in 
 many cases, such as in looking through prismatic spectacles or 
 
 H2 
 
100 THEORY OF OCULAR DEFECTS. 
 
 into a stereoscope, on account of ' the abnormal proportion of 
 functions demanded from the eyes, the object at first appears 
 double, but that the two images, after first being approximated 
 to a certain degree, and when an effort is made to fix their 
 corresponding parts, rush together with remarkable force to 
 cover and blend with one another. In this movement of the 
 double images, during complete rest of the eyes, I see a direct 
 proof of the assumption that the apparent direction is variable, 
 that it depends upon material changes in the eyes, and 
 that it requires a definite measure of nervous power to be 
 exerted. The process I judge to be as follows. 
 
 When the eye B (Fig. 49) has fixed and accommodated for 
 the object D, the retinal image is sharp, and falls upon the 
 
 physiological pole ; so 
 that the object is seen 
 accurately and clearly. 
 If the polar bacillus re- 
 tains its original direc- 
 tion E F, towards the 
 centre of the eye, the 
 object D will appear 
 to be in the direction 
 A 1) E D. If the same condi- 
 
 tion obtains in the other 
 
 eye, the object will appear to the second eye in another di- 
 rection, and a double object will be seen. The tendency to 
 union of the two visual appearances produces a movement and 
 a nutation of the bacillus E F, so that it assumes a position B F 
 parallel to the frontal axis AD. By means of the nutation by 
 the angle E F B = E D A the object now appears in the direction 
 B F D' of the bacillus B F, and therefore in a direction parallel 
 to the frontal axis. By means of the movement of the basal 
 extremity E of the bacillus to B, over the arc E B, the point of 
 origin of the line in which the eye sees the object, appears dis- 
 placed towards the forehead in a direction towards A. The nu- 
 tation and the movement together have therefore the effect that 
 
APPLICATION. lai 
 
 the object D appears to the one eye in the frontal axis A D. 
 Since the same applies also to the other eye, so, by means of 
 the application, the object, in a binocular visual act, is seen as 
 a single object, notwithstanding the convergence of the two 
 eyes. 
 
 If the second eye does not participate in the visual act, then 
 certainly the striving after the union of two visual impres- 
 sions would not be present. A person born with only one eye 
 would have less demand for application, and would perceive 
 the object more in the direction of the visual line E D of his 
 single eye. But in monocular vision the faculty of application 
 is not wholly in abeyance. It is highly probable, when 
 the eye turns from its state of rest, in which the optic nerve 
 has the position G H, to assume the indicated position of con- 
 vergence, in which the nerve necessarily acquires the position 
 G' H', that this change is attended by a traction upon all the 
 nerve fibres, which must produce in all of them, and 
 therefore in all points of the retina, an affection which in- 
 fluences the position and direction of all the bacilli. This 
 affection, since it is a common angular rotation, must be nearly 
 the same for all bacilli, and must displace each for nearly 
 the same distance E B, and incline each to nearly the same 
 angle B E F with the normal position. In consequence of the 
 equal displacement, the forehead A remains the common point 
 of origin for the lines of direction of all objects seen at once, 
 
 Pig. 50. Fig. 51. 
 
 and the frontal line A D remains for any single object D the 
 line of common apparent direction for that object. When the 
 eye rightly completes the application, all the bacilli retain, for 
 
102 THEORY OF OCULAR DEFECTS. 
 
 every convergence, the same direction B F that they assume in 
 the state of rest of the eye ; if therefore Fig. 50 shows the di- 
 rection of the bacilli when at rest, Fig. 51 will represent their 
 direction for any state of convergence. 
 
 In so far as the second eye is inactive in this convergence, 
 the tendency to the union of two different visual impressions 
 is wanting, the application will therefore not follow in a com- 
 plete degree ; it will rather exist only as a secondary or induced 
 action of the convergence and the accommodation, and will 
 consequently produce a more or less false projection. The eye 
 exerting insufficient application will see the object too far to- 
 wards the inner side ; that is, the left eye would project the ob- 
 ject too much to the right, the right eye too much to the left. 
 An excessive application would produce the contrary effect 
 a false projection to the outer side. 
 
 The foregoing views certainly bear a hypothetical character, 
 but yet they are the necessary fruits of the theory, and since 
 this theory is based upon simple and rational considerations, 
 and since it explains, without being strained, the most compli- 
 cated and peculiar phenomena, I hold them to be correct, even 
 although I cannot adduce experimental proof of the material 
 changes, for which the means of a clinique and a physiological 
 laboratory are wanting to me. But the above-mentioned part 
 of my theory derives an essential confirmation from the work 
 of Max Schultze, recently come to my knowledge, Zur 
 Anatomic und Physiologie der Retina. 
 
 According to his new and extremely careful microscopical 
 examination of the retina, the arrangement of the cylindrical 
 bacilli and of the flask-shaped cones, as well as their union 
 with the choroid and with the external granular layer of the 
 retina, are of such a kind that a movement of these, the nerve 
 elements perceptive of luminous impressions, seems to be prac- 
 ticable and as if designed. The wrinkles on the external at- 
 tachments of the cones signify that these parts have to undergo 
 actual changes of form during life, and in Plate III. of that 
 work, in Fig. 12, there is a representation of the bacillary layer 
 
DEFECTS OF APPLICATION. 
 
 103 
 
 of a diseased eye, in which the bacilli have assumed a curvi- 
 linear form. 
 
 I purpose, in a later portion of this treatise, after the theory 
 of application has been completely laid down, to make some 
 further comments upon Schultze's observations. 
 
 DEFECTS OF APPLICATION, OR ANOMALIES OF POSITION 
 AND OF NUTATION. 
 
 We proceed now to a more close examination of the mathe- 
 matical theory of Application and its anomalies. The basis of 
 this theory rests upon the following positions. 
 
 In the normal eye, the bacilli in a state of rest stand per- 
 pendicular, or in a radial direction to the retina, and are 
 directed towards the central point of its sphere. They possess 
 the power of nutation outwards, as shown in Fig. 51, and they 
 perform this nutation, when the eye is made to converge in- 
 wards, so that they form the same angle outwards as the eye 
 itself inwards. The angle of nutation is therefore equal to 
 the angle of convergence of 
 the axis of the eye towards 
 the frontal axis ; and hence, 
 in two normal eyes, is equal 
 to half of the angle of conver- 
 gence of the ocular axes to- 
 wards each other. In conse- 
 quence of this, each bacillus, 
 in convergence, retains its ori- 
 ginal direction with regard to 
 the frontal axis. 
 
 The angle of nutation, or the unilateral angle of conver- 
 gence, or rather the tangent of this angle, is the measure of 
 
 Fig. 52. 
 
104 THEORY OF OCULAR DEFECTS. 
 
 the effort of application. The application or nutation has, 
 like the accommodation and the convergence, its tipper and 
 lower limits, which correspond respectively to its near-point 
 and its far-point. Above the upper limit or maximum of the 
 application, which is the near-point, the eye cannot apply ; 
 and, when an object is brought within this limit, it is seen 
 double, and with crossed images, which we call negative double 
 images. When, in the convergence of the left eye B upon the 
 object D (Fig. 52), the bacillus B does not perform a com- 
 plete nutation, so as to attain the position B D' parallel to the 
 frontal axis A D, but only reaches the direction B E', then the 
 line A F', drawn from the forehead parallel to B E', will be the 
 apparent direction of the object D for the left eye. Hence this 
 eye will see the object D at F', or too far inwards, or to the 
 right ; and the right eye under the same conditions would see 
 the object D in the direction c F" at F", too tar inwards or to 
 the left. 
 
 Beyond the lower limit or minimum of application, which 
 gives the far-point, the stimulus to application is too weak to 
 excite reaction. The bacilli remain, therefore, in their radial 
 position as regards the retina, and this produces the effect that 
 the apparent direction deviates inwards, since it remains 
 parallel to the line B D. When under this weak stimulus no 
 displacement of the bacillus B occurs, the point of origin of the 
 apparent direction does not reach the centre A of the forehead; 
 and hence there follows a small degree of diplopia, which 
 depends upon the difference of the points of origin of the ap- 
 parent directions for the two eyes, and which gives crossed 
 images very near together. 
 
 According to these fundamental considerations, the defects 
 of application, like those of accommodation and convergence, 
 fall naturally into two chief classes. The first class depends 
 upon the structure of the eye ; that is, upon an abnormal, or 
 non-radial, position of the bacilli upon the retina in a state of 
 rest. We call these the anomalies of position. The second class 
 depends upon the innervation, or upon the applicative faculty, 
 
DEFECTS OF APPLICATION. 
 
 105. 
 
 the power to apply accurately, and we call these the anomalies 
 of application or of nutation. 
 
 1. We take first the anomalies of position, and we call an 
 eye of normal structure, in which the bacilli have a radial 
 direction in the state of rest, and, therefore, in vision at infinite 
 distance, an enklitic eye. 
 
 Since the measure of the effort reqiiired for accurate appli- 
 tion is the tangent of the angle of inclination of the nutation 
 of the bacilli, and since this angle coincides with the angle of 
 convergence of the ocular axis towards the frontal axis, the 
 formulas for the required application y, in a given visual dis- 
 tance a?, have the same form as those already given for the 
 convergence or fixation requirement ; since in the new for- 
 mulae only the constant coefficient &', which here also may be 
 taken as unity, would have a different value corresponding to 
 
 the application. The curve of requirement of the application 
 
 u 
 is therefore an equilateral hyperbola y = . 
 
 cc 
 
 2. The anomalies of position, which are characterized by 
 deviation of the bacilli from the radial position in a state of 
 rest, stand opposed to enklisis as aklisis. Two opposite forms 
 require consideration. 
 
 If the bacilli in the state 
 of rest deviate outwards 
 from the radial position, 
 as in Fig. 51, we call the 
 eye hypoJclitic. Such an 
 eye, in convergence upon 
 an object D (Fig. 53) lying 
 at a given finite distance, 
 A D = e, requires no effort 
 of application, since the ap- 
 parent direction B D' is al- 
 ready parallel to the frontal axis A D. For such an eye the 
 
 = &' ( _ - J , is a depressed hyperbola, 
 
 Fig. 53. 
 
 D 
 
 requirement curve, y 
 
106 
 
 THEORY OF OCULAR DEFECTS. 
 
 as for the hypometropic and hypogonic eyes. This curve cuts 
 the axis of abscissae in the positive finite distance e, which we 
 call the application distance. The application for infinite 
 
 Fig. 54. 
 
 distance would require a negative application, which the eye 
 cannot accomplish. The left eye B (Fig. 54) would therefore 
 see the infinitely distant object D in the direction A E', parallel 
 to B D', and the right eye c would see the same object in the 
 direction A E" ; that is, the two eyes would see double, with 
 homonymous or positive double images, when accurately fixed 
 and accurately accommodated. As a rule, the tendency to 
 single vision would prevail over the straining after sharpness 
 and clearness, and the eyes would forsake the parallel direction 
 and converge so far as to see singly, although less clearly. 
 
 The hypoklitic eyes, therefore, for visual distances greater 
 than their application distance, will have convergent strabis- 
 mus in binocular vision, but will see singly and will project 
 correctly. In monocular vision, on the contrary, they will fix 
 the object, and therefore will not squint, but will project falsely 
 to the outer side. 
 
 Strabismus with single vision hence appears to be the 
 immediate consequence of the defect of application, and the 
 action of the anomaly of position. 
 
 The application distance e of a hypoklitic eye is that at 
 which the eye, without squinting or strain, sees singly and 
 
DEFECTS OF APPLICATION. 
 
 107 
 
 clearly ; supposing that it does not suffer from any defect of 
 accommodation or of convergence. 
 
 We must observe that a hypoklitic eye, in the fixation of 
 objects which lie nearer than its far- point of application, does 
 not squint ; and that it sees these objects clearly, with less 
 effort of application than an enklitic eye. 
 
 3. If the bacilli in the state of rest deviate inwards from 
 the radial position, we call the eye hyperklitic. Such an eye 
 would require no effort of application for rays which, as in 
 Fig. 55, were convergent towards a point D, placed at a given 
 
 Fig. 55. 
 
 Fig. 56. 
 
 negative distance A D behind the head. For such an eye the 
 curve of requirement has the same form as for the hypoklitic 
 eye, when we place A D = e = ~ e' as a negative application 
 distance. 
 
 The application for infinite distance requires from the hy- 
 perklitic eye (Fig. 56) a positive application, which the eye 
 may accomplish if the requirement be not too great, which 
 will depend upon the degree of the hyperklisis. If the eyes 
 can accomplish nutation to the directions B D' and c D," the 
 apparent directions will be parallel to the frontal axis. If the 
 eyes cannot accomplish so much, there will result, in bino- 
 cular vision, if the tendency to single vision preponderates, a 
 divergent strabismus ; so that the apparent directions A E' 
 
108 THEOKT OF OCULAR DEFECTS. 
 
 and A E" fall into the frontal axis. Hence we obtain diver- 
 gent strabismus, with single vision and incomplete clearness, 
 as effects of a high degree of hyperklisis in binocular vision at 
 any distance. In monocular vision the hyperklitic eye will 
 fix any object lying before it, and therefore will not squint, 
 but will falsely project the object inwards. 
 
 Like the faculties of accommodation and of convergence, the 
 faculty of application is confined within two boundaries, which 
 correspond to the absolute near- and far-points of application, 
 and inclose the absolute range of application. 
 
 1. The normal state of the application or nutation faculty 
 may be called euroMisis. In the euroklitic eye the range of 
 application is of normal extent. The near-point of application 
 lies very near, and the far-point distant. For visual distances 
 which lie between the two, the eye applies correctly, and 
 therefore does not squint. For objects within the near-point, 
 the application does not attain the required degree, and is 
 therefore too weak ; the eye will therefore project too far in- 
 wards, and the pair of eyes will receive crossed double images. 
 If the tendency to single vision prevails, the eye must incline 
 outwards, in divergent strabismus. 
 
 The same must also occur in looking at objects beyond the 
 far-point ; since here, also, the application is too weak. In 
 general the divergent strabismus for very distant objects can 
 be only inconsiderable. 
 
 The anomalies of application form stenoHisis^oxihe anomalies 
 of a small range of application, and they fall into two kinds. 
 
 2. The first anomaly of nutation is bathoklisis, in which 
 the small range of application lies deep. 
 
 3. The second anomaly of nutation is hypsoklisis, in which 
 the small range of application lies high. 
 
 The combination of these three forms of nutation faculty, 
 with the already described three states of position dependent 
 upon the structure of the eye, furnishes once more nine 
 varieties, of which one is the normal state, and the other eight 
 are defects of application. 
 
DEFECTS OF APPLICATION. 109 
 
 These defects of application form exact parallels to the 
 already described defects of accommodation and convergence. 
 They combine the different states of the infinite, the positive 
 finite, and the negative finite application distance e, with the 
 
 different states of the range of application , and the 
 
 n w 
 
 position of the near- and far-points of application, that is, the 
 
 values of - and . 
 n w 
 
 The specializing of these nine cases, after the analogy of the 
 foregoing, is easy, and may be left to the reader. These 
 defects produce manifold phenomena of strabismus with single 
 vision, and with greater or less clearness of the visual impres- 
 sion. By their concurrence with defects of convergence is 
 naturally produced strabismus with double vision. The 
 direction which the eye will assume in any given case, or for 
 any particular visual distance, is easily to be determined, 
 under the supposition that the tendency to single vision will 
 preponderate over the tendency to fixation and to accommo- 
 dation, and by remembering that this position must be of such 
 a kind that the direction of the bacilli shall be parallel to the 
 frontal axis, without the bacilli being called upon for a nega- 
 tive application, or for a positive application transcending the 
 boundary of the range of application. Hence it may befall 
 that the same eye may have convergent squint for some visual 
 distances and divergent squint for others ; and that the degree 
 of squint may vary with the visual distance. 
 
 We must further mention that the squinting of an eye with 
 defective application will occur only during binocular vision, 
 since it can only exist in consequence of a preponderance of 
 the effort after single vision over the efforts after clearness and 
 after direct vision, and under the supposition* of a sufficient 
 motility. In vision with one eye only the effort after single 
 vision, and therefore the above concurrence, ceases entirely ; 
 the eye then fixes rightly (sees directly with its pole), and ac- 
 commodates rightly, but projects falsely. Since the squinting 
 
110 THEORY OF OCULAR DEFECTS. 
 
 with single vision in binocular vision is dependent upon se- 
 veral contingencies, which sometimes may not be fulfilled, it 
 is best to determine the defects of application not in binocular, 
 but in monocular vision, when there will be no squinting but 
 a false projection, and by observation of the latter. 
 
 It is to be considered that, in binocular vision, there is the 
 tendency always present to fix with one eye and to see indi- 
 rectly with the other. 
 
 Between the squinting and the projection arising from 
 defects of convergence and from defects of application there 
 exists an essential difference. For a defect of convergence the 
 squint, since there is a defect of fixation present, is indepen- 
 dent of binocular or of monocular vision, and is always com- 
 bined with the same false projection. In monocular vision, 
 however, the squint may be avoided by a corresponding ob- 
 lique position of the head (see Gesetze des raumlichen Sehens, 
 25). In binocular vision the squint is always associated 
 with double images. In defects of application, on the con- 
 trary, the sqiiint occurs only during binocular vision, and is 
 then not combined with false projection, and therefore not 
 with diplopia, but only with indirect vision. In monocular 
 vision the squint ceases, and there is fixation or direct vision 
 associated with false projection. 
 
 The manner in which an association of defects of application 
 and of convergence would modify the degree and kind of the 
 strabismus, the projection, and the diplopia is easy to be 
 perceived. 
 
 SPECTACLES FOR ANOMALIES OF POSITION AND 
 
 OF NUTATION. 
 
 
 
 The optical aids for the diminution of the defects of appli- 
 cation are prismatic spectacles. Since these deflect the rays 
 of light, and therefore change the actual into an artificial 
 visual direction, they elevate or depress the hyperbolic require- 
 
CONCURRENCE OP DEFECTS. Ill 
 
 ment curve of the application, and may therefore completely 
 counteract hypoklisis and hyperklisis. By prismatic spectacles, 
 therefore, anomalies of direction are to be relieved. The 
 peculiar facultative or nutation anomalies are not to be cor- 
 rected by optical means. With these, as with those of motility 
 and accommodation, spectacles can only act by raising or 
 depressing the hyperbolic curve of requirement, so that the 
 power exerted shall be made to fall at a fitting mean between 
 the boundaries of the range. 
 
 The determination of application spectacles is based upon 
 the same formulae that have been given for convergence spec- 
 tacles. The spectacles p = e, which render the artificial 
 application distance infinitely great, are the enklitic spectacles. 
 By the help of these may be found by experiment, for the 
 
 above formulae, the values of- and , the near- and far-points, 
 
 n w 
 
 and of , the range of the application. 
 n w 
 
 THE CONCURRENCE OF DEFECTS OF ACCOMMODATION, 
 CONVERGENCE, AND APPLICATION. 
 
 Each of the nine conditions of the accommodation may be 
 associated with any one of the nine conditions of convergence, 
 and with any one of the nine conditions of application, in 
 any eye. This gives 9x9x9 = 729 different cases ; of 
 which one is the normal condition, and the others form an 
 assemblage of ocular defects, which, according to the foregoing 
 methods, may be divided into synoptical groups, and every one 
 of which may be experimentally determined. 
 
 A graphic delineation of the three curves of requirement, 
 and of the three facultative curves, on the same diagram, gives a 
 convenient and complete view of the combined peculiarities of 
 the eye. 
 
 The visual strain, which may also thus be recognised, and 
 
112 THEORY OF OCULAR DEFECTS. 
 
 which follows the fixation of any object, has three sources : first, 
 a misproportion between accommodation and convergence; 
 secondly, a misproportion between accommodation and appli- 
 cation ; and, lastly, a misproportion between convergence and 
 application. 
 
 The graphic delineation shows most easily the way in which 
 the curves of requirement may be raised or depressed, so as to 
 bring the ocular functions to convenient places between their 
 absolute limits. 
 
 The relative faculty or range of each single function comes 
 into consideration in relation to each of the other two ; but in 
 the determination of spectacles it is the absolute limits of each 
 faculty with which we are in most cases concerned. 
 
 Since for the defects of convergence and of accommodation 
 there exists only a single optical means of help, the prismatic 
 spectacles, it is manifest that, when these two defects exist to- 
 gether, in different degree and kind, only one of them can be 
 relieved by spectacles. Under such circumstances, therefore, 
 if we were limited to optical means, we must either entirely 
 neglect one of the defects, or take some mean between their 
 requirements. In either way would be proved the inefficiency 
 of art to regulate the visual processes, were there not a third 
 and independent means within our reach. This third means 
 is furnished by a surgical operation. 
 
 The direction of convergence of the eyes can be regulated by 
 the displacement, forwards or backwards, of the ocular muscles. 
 The operation upon the muscles fulfils, therefore, the same 
 purpose as the prismatic spectacles, as regards the defects of 
 convergence ; and I am of opinion that it should be considered 
 as the special or proper remedy for such defects. While 
 accommodation and application are really sensory processes, 
 since they require the direct activity of the sensory apparatus 
 of the eye, convergence is essentially a motor function, proper 
 to the ocular muscles. We have, therefore, for the two defects 
 of sensational function, the two optical remedies, the lens and 
 the prism ; and for the defect of motility the mechanical remedy 
 of the operation. 
 
CONCURRENCE OF DEFECTS. 113 
 
 It is highly interesting to observe that this mechanical 
 remedy cannot be otherwise applied, in principle, than the 
 two optical remedies. Just as, by the lens and the prism, we 
 can only remove an anomaly depending upon structure, that 
 is, one of refraction or of position, and not one that depends on 
 innervation or function, that is, not one of accommodation or of 
 nutation ; so only an anomaly of direction, and not an anomaly 
 of motility, can be removed by operation. By the lengthening 
 or shortening of a muscle, or by its displacement backwards or 
 forwards, we can in general only adjust a defective direction of 
 the eye ; that is, we change a hypogonic or hypergonic eye into 
 one that is engonic, and thus restore the hyperbola to its normal 
 place. But, generally, we cannot in this way enlarge a small 
 range of convergence, or increase the motility of the eye ; that 
 is, we cannot change a bathogonic or hypsogonic eye into one 
 that is eurogonic ; although in exceptional cases effects of such 
 a kind may be produced. 
 
 Although the range of convergence cannot generally be 
 altered by operation, yet still the operation, for stenogonic eyes, 
 fulfils the same purpose as the lens or the prism for eyes of 
 small range of accommodation or of application. This depends 
 upon the fact that every elevation or depression of the curve 
 of requirement produces a displacement of the range of con- 
 vergence ; since, with every such elevation or depression, the 
 points of intersection of the requirement and the facultative 
 curves are altered. The operation is therefore a remedy for 
 defects of convergence in the same way that a lens or a prism 
 is a remedy for defects of accommodation or of nutation. 
 
 I think it is important, in practical ophthalmic surgery, to 
 lay especial stress upon the different object of the surgical 
 operation according as it is intended to relieve an abnormal 
 convergence distance e, or an abnormal range of convergence 
 
 . : since a clear appreciation of this difference must 
 
 n' w' 
 
 essentially modify the proceeding in the two cases. As a 
 hypometropic or hypermetropic eye, notwithstanding that it 
 
114 THEORY OF OCULAR DEFECTS. 
 
 has full power of accommodation, requires a lens in order to 
 see at certain distances; so a hypogonic or hypergonic eye, 
 notwithstanding its large range of convergence, or complete 
 motility, requires an operation in order to raise or depress its 
 hyperbolic curve of requirement. And as, farther, an emme- 
 tropic, but stenopic eye, that is, an eye with a narrow range of 
 accommodation, requires a lens to bring its accommodation 
 effort to a convenient part of this narrow range ; so an engonic 
 eye, one therefore with no defect of direction, when it has a 
 narrow range of convergence, requires an operation in order 
 to bring its convergence effort to a convenient part of this 
 narrow range. By consideration of the special object, in 
 either case, the kind of operation must be determined. 
 
 The aim of the operator should always be a definite eleva- 
 tion or depression of the curve of requirement. For elevation 
 it is necessary that the eye should be turned more outwards, 
 or rendered more divergent ; for depression, that it should be 
 turned inwards, or rendered more convergent. The degree 
 may be learned from the graphic delineation of the require- 
 ment and the facultative curves. 
 
 Self-evidently, in concurrence of defects of convergence with 
 defects of accommodation and of application, the conjoined 
 conditions must all be taken into account ; which may be done, 
 most readily, by examining the three requirement and three 
 facultative curves in the figure already recommended. 
 
 It is manifest that, in complicated cases, it may be necessary 
 to use all three remedies, the operation, the lens, and the prism ; 
 and that the drawing in question will show in what way to 
 combine them. 
 
 The effort which, in the combined exercise of the three chief 
 faculties, any one of them has to make, we will endeavour, by 
 the aid of Fig. 57, to set forth in the following manner. 
 
 Let L be an object lying in the frontal axis AL, and looked 
 at by the left eye B ; and let B G be parallel to the frontal axis. 
 If the eye accommodate for the distance A K, the condition cor- 
 responds to a case in which the object is placed in the line K M, 
 
CONCURRENCE OF DEFECTS. 
 
 115 
 
 perpendicular to A K. A K is the distance of accommodation, 
 and BKA is the angle of accommodation. If the eye converge 
 towards the point D on the frontal axis, then A D is the conver- 
 gence distance, and BDA = GBDis the angle of convergence. 
 
 Fig. 57, 
 
 The eye therefore does not fix the object, but squints inwards ; 
 does not see the object with the polar bacillus, or directly, but 
 with an inner bacillus >, which is in a line L c b, leading from 
 L through the centre c. If the eye applies in such a degree 
 that the polar bacillus B makes the angle of nutation E B D 
 with the ocular axis or the frontal axis, or that the percipient 
 bacillus b makes the same angle with the luminous ray b L ; 
 then if A F be drawn parallel to B E, and the angle D B H be 
 made = E B G, we have angle BHA=GBH = EBD = BJA, 
 and this latter angle, which is equal to the angle of nutation 
 E B D, is the angle of application, and the distance A H is the 
 application distance. 
 
 The tangents of the angles of accommodation, of convergence, 
 and of application, or the reciprocals of the accommodation, 
 the convergence, and the application distances, measure the 
 respective functional efforts. 
 
 If the luminous object be situated in the visual line or 
 ocular axis B D, that is, if the light be received by the polar 
 
 i 2 
 
116 THEORY OF OCULAR DEFECTS. 
 
 bacillus B, the object would appear at F, in the direction A F, 
 parallel to B E. But, since the laterally situated bacillus b is 
 affected, the apparent direction is in the line A M, which forms 
 with A F the angle M A F = L c D. The deviation of the apparent 
 direction A M from the frontal axis A L, or the angle M A L, is 
 therefore the sum of the angles F A L = E B G, and M A F = L c D. 
 The directions of the polar bacillus B, and of the affected 
 bacillus &, or the directions B E and b e, intersect under the 
 angle BN& BC& = LCD; since these bacilli, as regards the 
 radii of the eye CB and c b, have equal nutation e bi, = E B D. 
 Hence the apparent direction A M is parallel to the direction b e 
 of the affected bacillus b : or the direction be of the bacillus b, 
 affected by the object L, shows the direction, commencing 
 from the forehead, in which this object appears to the eye. 
 The effort of application is determined either by the nutation 
 of the polar bacillus B to the ocular axis or visual line B D, or 
 by the nutation of the affected bacillus b to the ray of light b L. 
 The apparent direction A M deviates from the ray of light b L by 
 the angle of nutation or application MPL = e#L = EBD. Since 
 this angle of application is also equal to the angle B H A, so is 
 the effort of application of the affected bacillus b to be mea- 
 sured by the same angle of application B H A, or by the recipro- 
 cal of the application distance A H, as the effort of application 
 of the polar bacillus B ; which also is manifest from the nature 
 of the case. 
 
 The eye under consideration squints inwards, or with, con- 
 vergent squint, as regards the object L, and therefore does not 
 see it directly, but with a lateral bacillus. This defect of con- 
 vergence or fixation does not determine a false projection or 
 , double'vision. The latter occurs, but by reason of the devia- 
 tion of the .apparent direction A M from the actual A L. The 
 application is here in fault. Since the direction b e of the 
 affected bacillus is not parallel to the frontal axis, but deviates 
 outwards, so the eye falsely projects the object outwards, to- 
 wards M, or experiences in binocular vision an outward (posi- 
 tive or homonymous) double image. 
 
CONCURRENCE OF DEFECTS. 117 
 
 It is apparent that, notwithstanding false convergence and 
 false application, a correct projection, and therefore single 
 vision, may occur. This is the case whenever the direction of 
 the affected bacillus is parallel to the frontal axis. The squint- 
 ing eyes then see singly. 
 
 Inversely, it is apparent that a correctly converging or fixing 
 eye may project falsely, and may therefore receive a double 
 image ; and that this happens whenever the direction of the 
 affected bacillus deviates from that of the frontal axis. 
 
 In monocular vision with the eye B, and therefore when the 
 other eye is covered, the object L affords a stimulus only to 
 accommodation and fixation, not to application, since no double 
 images are present, and consequently there is no tendency to 
 unite them. The eye will then accommodate and fix as well as 
 possible, and the application will only follow the accommodation 
 by induction. But these two chief faculties are united by so 
 energetic an inductorial tension that, when there is no objective 
 stimulus to the application, the accommodation, that is, the 
 effort of accommodation, seems to correspond to one. 
 
 I believe that, even in anomalies of accommodation and ap- 
 plication, the induced tension between the two functions is very 
 strong ; so that both, in common vision, even when an ob- 
 jective stimulus would call upon them for different efforts, and 
 therefore also in binocular use of the eyes, in general differ 
 but little from each other : that is to say that the application 
 imder given accommodation, or the accommodation under 
 given application, will vary only within narrow limits. These 
 relative limits for accommodation and application are certainly 
 far more narrow than the relative limits for accommodation 
 and convergence, or for application and convergence. 
 
 The last proposition is of special importance with regard to 
 the artificial regulation of the visual processes. It does not 
 forbid that the absolute range of accommodation in an eye may 
 be essentially different from its absolute range of application ; 
 or that the eye may accommodate in a degree totally different 
 from that in which it applies. It only expresses that the accom- 
 
118 THEORY OF OCULAR DEFECTS. 
 
 modation effort and the application effort stand in a seemingly 
 close relation, so that each is powerfully influenced by the 
 other. 
 
 In the binocular use of the eye B the objective stimulus to 
 accommodation and fixation is associated with the objective 
 stimulus to single vision, or to the union of two images. This 
 stimulus is in general very strong. But, since the induced 
 tension between accommodation and application is likewise very 
 energetic, so the tendency to single vision, as a rule, obtains its 
 fulfilment not by a change in the application or nutation of 
 the bacilli, which would involve also a change in the accom- 
 modation and impaired clearness, but by a change in the con- 
 vergence, so that the fixation or direct vision with the polar 
 bacillus is abandoned. The great range of the muscular func- 
 tion of convergence facilitates this deviation, and the result is 
 clear and single, but indirect, vision. For this purpose the eye 
 B becomes more or less convergent, or moves so far inwards or 
 outwards that the affected bacilli of the two eyes have a com- 
 mon direction, and therefore for two similar eyes the direction 
 of the frontal axis A L, while the accommodation and the nuta- 
 tion, and therefore the angle E B D or e b L, remain unchanged. 
 This requires, for the proportions given in the figure, a farther 
 rotation inwards by the angle M A L = e b G* Hence the ocular 
 axis B D comes into the direction B Q, which deviates from the 
 direction B H by the angle QBH = eNE=Bc = DCL, and 
 for which the angle of convergence B Q A is = G B Q = e B D 
 = EBD + eNE = EBD + DCL, the sum of the nutation 
 angle E B D, and of the distance from the pole B c 5 = DCLof 
 the first affected bacillus. After this convergence the affection 
 falls naturally upon a more internally situated bacillus. 
 
 If we proceed from the condition in which the eye B, in 
 
 * Strictly speaking, there is no angle e b G in the construction. The 
 angle intended is that between the lines e b and B G. If another letter, 
 as r, had been set at their point of meeting, the angle would have been 
 e r G, but the figure would have been inconveniently crowded. The 
 same may be said of the angle c B D. TRANS. 
 
CONCURRENCE OF DEFECTS. 119 
 
 monocular vision, fixes the object L, then direct vision occurs 
 when the bacillus b coincides with the polar bacillus, so that 
 the point D turns towards L, and the angle DCL = BC& = 
 e N E = 0. The angle of nutation is then equal to E B L, 
 and we have for the angle of application the value B H A = E B L. 
 In order to unite the homonymous double images in binocular 
 vision, the eye turns by the angle E B G ; that is, so far inwards 
 that its axis moves from the direction B L to the direction B Q, 
 for which the angle of convergence B Q A = E B L, that is, 
 is equal to the angle of nutation. Hence follows single, but 
 indirect vision, with a bacillus which has from the pole the an- 
 gular distance EBG = EBL GBL = EBL BLA, which 
 is equal to the difference between the angle of nutation or con- 
 vergence E B L and the angle of fixation BLA. 
 
 The preceding examination gives also a simple means of de- 
 termining the actual state of application of an eye. This can 
 be accomplished in several ways. 
 
 Let L be the point in space (Fig. 58) which app.ears single 
 to the two eyes B and B', when they converge towards the point 
 D. Draw from L, through the centres of the eyes c and c', the 
 
 Fig. 58. 
 
 rays L I and L b' ; and then the affected bacilli, b and ', must 
 have the direction of the frontal axis A L. If, therefore, the 
 lines b e and b' e' be drawn parallel to A L, e b L and e' b' L 
 will show the angles of application or nutation, since these 
 
120 
 
 THEORY OF OCULAR DEFECTS. 
 
 angles correspond to the inclination of the affected bacilli, 
 b and &', to the radii b c and b' c'. Every point in the lines 
 b L and b f L would appear to be in the frontal axis A L, and the 
 points lying in each of the two lines would therefore coincide. 
 
 Fig. 59. Fig. 60. 
 
 -M 
 
 --M 
 
 Next, let the object L appear double to a pair of eyes con- 
 vergent towards the point D; and let it be seen (Fig. 59) by the 
 eye B at M, and by the eye B' at M'. If we draw from L, 
 through the ocular centres c and c', the rays L b and L b' and 
 through the affected points of the retinae b and b f , the lines b e 
 and b' e' parallel to A M and A M' ; then these lines show the di- 
 rections of the affected bacilli, and e b L and e' b' L are the 
 angles of application. All points on the line b L would appear 
 to be in the direction A M ; and all points on the line b' L would 
 appear in the direction A M'. 
 
 Again, let the two objects L and L ; (Fig. 60) appear blended, 
 at the point M of the frontal axis, to the pair of eyes convergent 
 upon the point D. If we draw from L and i/ through the ocu- 
 lar centres c and c', the rays L b and i/ ', and through the af- 
 fected points of the retinae b and >', the lines b e and b' e' 
 parallel to the frontal axis A M, then e b L and ef b' L' will be 
 the angles of application. All points on the lines b L and b' L' 
 would appear to be in the frontal axis, and would coincide. 
 
UNILATERAL DEFECTS. 121 
 
 UNILATERAL OCULAR DEFECTS. 
 
 With regard to any of the already described defects of accom- 
 modation, convergence, and application, it may happen that both 
 eyes are not alike ; so that any one of the already-mentioned 729 
 conditions of the left eye may coexist with any one of these con- 
 ditions in the right. This gives 729 x 729 = 531,441 possible 
 states of the binocular organs of vision, of which one is normal, 
 and the others form over half a million of different defects. 
 
 The condition of each single eye may be determined from 
 the foregoing for each of the three visual functions. As regards 
 the convergence and the application, the frontal axis bears the 
 same relation to each eye that is borne, in the case of two 
 similar eyes, by the line bisecting the angle of convergence. 
 We may, therefore, by covering one eye, determine for the 
 other the degree of hypometropia or hypermetropia, of hypo- 
 gonia or hypergonia, of hypoklisis or hyperklisis, or its accom- 
 modation distance, its convergence distance, and its application 
 distance; as also the degree of bathometropia or hypsome- 
 tropia, of bathogonia or hypsogonia, of bathoklisis or hypso- 
 klisis, or the accommodation range, the convergence range, 
 and the application range, in magnitude and in position ; and 
 may thus obtain a complete and clear picture of the condition 
 of the whole organ, which, by the often recommended graphic 
 delineation of the requirement hyperbolas and the facultative 
 curves, is not only clear of comprehension, but contains all 
 the necessary data for a rational conclusion upon the kind 
 and degree of the necessary aids. 
 
 It is then only requisite to consider whether it is desirable 
 to treat each of such eyes as an independent organ, without 
 reference to the other; or whether the interdependence of the 
 two eyes has any influence upon the selection of remedies. 
 
 Bonders objects, in general, to the use of two different glasses 
 for two different eyes (p. 565 of his work), and he generally 
 
122 THEORY OF OCULAR DEFECTS. 
 
 employs like glasses for such eyes, and considers any departure 
 from this rule to he exceptional. As a reason for his practice 
 he asserts that two unlike eyes are accustomed to their dissimi- 
 larity, and are therefore brought, by dissimilar glasses, into a 
 new and inconvenient state. If this rule were well founded, it 
 would not do to supply any myope of equal eyes with ordinary 
 concave spherical spectacles ; but, on principle, with orthoscopic 
 spectacles only. For we must in that case consider that two 
 equally short-sighted eyes have become accustomed to their 
 abnormal functional proportion between accommodation and 
 convergence, and that they require glasses which respect this 
 abnormal proportion. Such glasses are not the ordinary, but 
 the orthoscopic only. 
 
 He, therefore, who attaches so much importance to habitual 
 abnormal proportion that he prescribes, on principle, similar 
 glasses for dissimilar eyes, must also, on principle, prescribe 
 orthoscopic glasses for similar eyes ; and, vice versa, he who 
 prescribes, on principle, ordinary spectacles for similar eyes, 
 must also prescribe dissimilar glasses for unlike eyes. The 
 general recommendation of ordinary glasses for like eyes, and 
 of similar glasses for unlike eyes, appears to me to be a con- 
 tradiction in principle. 
 
 Generally, however, I think that the question is not to be 
 decided by general principles, but solely by actual facts. 
 Where any given proportion of function has become completely 
 habitual, it is self- evident that it should be respected ; although 
 whether the habit has attained the presumed degree is always 
 a priori doubtful, and cannot be determined by a bare assump- 
 tion. 
 
 The actually existing condition of the eyes, and the 
 functional proportion suitable to them, are the essential ques- 
 tions in every rational endeavour for the determination of the 
 suitable remedies. When these data are determined by ex- 
 periment, the solution of the problem follows without any 
 further hypothesis. 
 
 If the function of one eye were wholly uninfluenced by the 
 
UNILATERAL DEFECTS. 123 
 
 other, then the proper aid for each single eye could be deter- 
 mined by the foregoing. But the functions of the two eyes, 
 although originally independent, influence each other as much 
 as any two chief functions in a single eye as much as accom- 
 modation and convergence. A binocular vision without strain 
 corresponds, therefore, to a certain definite functional propor- 
 tion between the two eyes, which is individual, and can be 
 determined by experiment. A deviation from this proportion 
 produces strain ; and, when the deviation oversteps a certain 
 degree, binocular vision becomes impossible, and the active 
 visual function becomes a burden to one or to the other eye, 
 so that the general incompleteness of monocular vision is 
 produced. 
 
 According to the observations of Bonders, the reciprocal 
 influence exerted between the two eyes appears to be very 
 considerable. Bonders says ' we are unable to equalize even 
 a small difference in the refraction of the two eyes by accom- 
 modation, so inseparably is the accommodation effort of one 
 eye bound up with that of the other.' This result, how- 
 ever, was chiefly obtained by experiments upon normal eyes, 
 different glasses being applied to each ; and, according to 
 my judgment, it does not therefore follow that there is any 
 necessity for applying like glasses to unlike eyes. While the 
 unlike glasses produce, for like or normal eyes, a dispropor- 
 tion which the visual organs resist, and by which the fore- 
 going observations are explained ; like glasses would maintain, 
 in unlike eyes, the already existing disproportion, which 
 unlike glasses would set aside. 
 
 Before proceeding further, I should observe that the ex- 
 planation given by Bonders of the resistance of the two eyes 
 to unequal efforts does not seem to me to be correct. Bonders 
 explains that unequal glasses are less comfortable than similar 
 ones, chiefly on the ground that, when the distances for clear 
 vision are made equal, the retinal images of the two eyes are 
 not equal, but of different sizes. But, with any lens, the 
 visual angle of the apparent object is so nearly equal to that 
 
124 THEORY OF OCULAR DEFECTS. 
 
 of the actual, and the distance of the crossing point of the 
 principal rays from the retina varies so little, that the size of 
 the retinal image does not change in any marked degree ; and 
 the size cannot therefore be the source of difficulty. The 
 fact mentioned by Donders, that through one lens we may see 
 an object large and small, and also that without spectacles the 
 same object sometimes appears larger or smaller to the left eye 
 than to the right, when the two eyes are unlike, does not 
 establish his conclusion about the presence of unequal retinal 
 images. As I have shown in my Physiologische Optik, and 
 in the Gesetze, the apparent size of an object does not depend 
 entirely upon the size of the retinal image, but is the result of 
 several processes, especially of those which impart a knowledge 
 of distance. 
 
 We shall obtain a sure guide to the determination of the fit 
 spectacles for dissimilar eyes, when the part taken by each eye 
 in the visual act, at any visual distance, is discovered by a 
 series of trials, and graphically delineated. The institution of 
 such complicated and extended experiments may be difficult 
 in practice ; and would, I believe, generally be unnecessary. 
 I believe that the establishment of the absolute lengths and 
 ranges of the three chief functions, in monocular vision, will 
 be sufficient for the rational selection of spectacles, when the 
 following points are recognized : 
 
 a. Is the exertion or relaxation of a chief function of one 
 eye united, in binocular vision, with exertion or relaxation of 
 the same function in the other? 
 
 b. Does the absolute maximum or minimum of a chief 
 function of one eye correspond with the absolute maximum or 
 minimum of the same function in the other ? 
 
 c. Does a definite degree of a chief function of one eye, 
 measured by the position, or ordinate, of a given point in the 
 absolute range of this function, correspond to the same degree 
 of the same function in the other eye ? 
 
 The first two points need but little comment; the third, 
 however, since it concerns the binocular visual strain, and 
 
APPARENT SIZE AND DISTANCE. 125 
 
 affects the comfort of the two eyes in their common action, 
 may require more careful consideration ; although this strain, 
 even in cases of marked inequality of the two eyes, may 
 easily be of small amount, and hence affords only an approxi- 
 mative standard for the regulation of vision. 
 
 In consideration of the difficulty, ascertained by ophthal- 
 mologists, of combining visual impressions of unequal size, it 
 seems to me that an essential condition for the proper 
 spectacles is that they shall make the object appear of equal 
 size to each eye. According to the above remarks upon 
 the apparent size of objects, the fulfilment of this condition 
 requires not only equal retinal images, but something more 
 also, to the consideration of which we will proceed. 
 
 APPARENT SIZE AND DISTANCE. 
 
 In the Physiologische Optik, and in the Gesetze des raum- 
 lichen Sehens, I have shown that apparent distance rests 
 principally upon the perception of the components of the ether 
 undulations in the conical pencil of rays that penetrates 
 the bacillary layer ; that this apparent distance is, therefore, 
 primitively independent of the accommodation and of all other 
 ocular functions, especially of binocular vision; that it is 
 recognized in false accommodation and in monocular vision ; 
 and that we can, at the same time, and in the same state of 
 accommodation, recognize very different distances, and there- 
 fore see stereoscopically; and that also each single eye sees 
 stereoscopically. 
 
 But, as all the ocular functions secondarily influence one 
 another, so the other functions have a certain influence over 
 the recognition of apparent distance. The influence arises 
 thus : that any effort of accommodation, or of convergence, or 
 
126 THEORY OF OCULAR DEFECTS. 
 
 of application, modifies, in a certain degree, the sensitiveness 
 of the perceptive nerve- substance of the bacilli, and produces 
 a modified perception. 
 
 I have also shown that it is one and the same fundamental 
 process which leads to a judgment of apparent distance, and 
 to a judgment of the size or width of the arc of the radial 
 elementary unit of the nervous substance in the retina ; and 
 that therefore the induced or secondary influences, which 
 modify the perception of distance, change, in the same degree, 
 the perception of the size of the latter elementary arc. Hence 
 the apparent size of an object is proportional to the product 
 of the size of the retinal image (the sum of all the affected 
 retinal units) and of the apparent distance, 
 
 As regards the secondary influence of the ocular functions 
 upon the apparent size and distance, this influence is exerted 
 first by the accommodation, and secondly by the application. 
 
 In the Gesetze I have ascribed the latter influence chiefly 
 to the convergence, since I have there supposed the application 
 to be normal, or coincident with the convergence. In a more 
 rigid separation of the two functions, which in abnormal con- 
 ditions is generally necessary, I think the influence must be 
 ascribed entirely to the application, which is a nerve-function, 
 and not to the convergence, which is a muscular function. 
 In Fig. 57, for the position A D of the ocular axis or visual line 
 as regards the frontal axis A K, let 
 
 a be the distance A L of the actual object L, or of the optical 
 image of this object given by a lens, since such an image 
 .always plays the part of an actual object; 
 
 a the visual angle of this object ; 
 
 r = a a, the actual diameter of the object; 
 
 b the distance A K, for which the eye is accommodated, or 
 the accommodation distance ; 
 
 /3 the tangent of the angle of accommodation B K A ; 
 
 c the distance A H, for which the eye is applied, or the 
 application distance ; 
 
 d the distance between the centres of the eyes; 
 
APPARENT SIZE AND DISTANCE. 127 
 
 7 = -, the tangent of the angle of application or nutation 
 
 EBD = BJA = BHA= GBR; then according to the Gesetze 
 (p. 117), we have the apparent distance of the object 
 
 a ' = y- = !' 
 
 and the apparent size or the apparent diameter of the object 
 
 /== JL ^ 
 
 " 7 ' " b ' 
 
 (If instead of the angle with the frontal axis, we have the 
 double angle, that is the angle of the convergence and accom- 
 modation directions of two eyes supposed to be alike, this does 
 not affect the result ; and, moreover, under the supposition of 
 the angles being small, we have put the angles themselves for 
 their tangents.) 
 
 From these formula? it appears that too strong an accom- 
 modation increases the apparent size and distance, that too 
 weak an accommodation diminishes them ; that too strong 
 application (eventually convergence) diminishes the apparent 
 size and distance, too weak application increases them. 
 
 Moreover, the apparent size and distance are not determined 
 by one of these two efforts alone, but by the ratio of one to the 
 
 other, or by the quotient = ~ . jf ? therefore, the eyes are 
 
 rightly accommodated for an object (which may be known by 
 the clearness of the image), while at the same time they are 
 falsely convergent, and consequently applying falsely (which 
 may be known by double vision), so in too strong a con- 
 vergence the object appears too small, and in too weak 
 convergence too large. False accommodation without false 
 application, and therefore without double vision, is much more 
 difficult with naked eyes. Too strong an accommodation 
 usually entails a yet stronger application, so that the object 
 does not appear enlarged, but diminished. 
 
 When the eye is rightly accommodated, and therefore I = a, 
 
128 THEORY OF OCULAR DEFECTS. 
 
 but falsely applied, the apparent distance a! is = c, that is, 
 equal to the application distance. 
 
 When the accommodation and application efforts are equal, 
 so that b = c, whether they are both too strong or both too 
 weak, the object always appears of right size and at the right 
 distance. 
 
 It is important to mention that the quantities b and c or /: 
 and y refer to the efforts of accommodation or applicatior 
 actually exerted ; which have their insurmountable limits o 
 value in the boundaries of the range of the accommodation 
 or the application. 
 
 I believe that these formulas, with their consequences, which 
 I have found to be in unison with the actual appearances 
 ( 19 and 20 of the Gesetze), deserve the attention of 
 physiologists ; since they afford a clue to many remarkable 
 phenomena. 
 
 An important result of the second formula is that the 
 apparent size of an object is wholly independent of a its dis- 
 tance ; and that hence an object of a given size, r, always 
 appears equally large, at whatever distance it may be placed, 
 so long as the accommodation and the application (or con- 
 vergence) are normal, or equally strong. Only when the 
 accommodation or the application, or both, become abnormal, 
 as is the case on this side of the near point or beyond the far 
 point, does the apparent size vary. 
 
 If, in the formula for apparent size, we substitute for the 
 diameter of the object r, its value a a, then the apparent size 
 will be 
 
 r' = - a a = -- a a = a a', 
 y b 
 
 If therefore an object retains a constant visual angle a, its 
 apparent size r will vary with its apparent distance a'. 
 
 If therefore we look at an object through orthoscopic con- 
 cave spectacles, which change the values of b and c in equal 
 proportions, but make a smaller, and therefore diminish the 
 
APPARENT SIZE AND DISTANCE. 129 
 
 apparent distance a', while they leave the visual angle a 
 unchanged, the object will appear nearer and smaller. Ortho- 
 scopic convex spectacles cause the object to appear more 
 distant and larger. 
 
 If we look at the object through common centric concave 
 spectacles, which diminish b and a, and also b and r, in equal 
 proportions, but leave c and a unchanged, then the object 
 appears in unaltered size and distance. The same applies to 
 vision through common convex spectacles. But here, how- 
 ever, the express condition must be asserted, that the supposed 
 abnormal proportion between accommodation and application 
 actually exists ; that therefore b and c have actually the 
 assumed values. Generally, for reasons immediately to be 
 stated, the supposed conditions are not accurately fulfilled ; and 
 we can therefore only say that the common spectacles respec- 
 tively magnify or diminish less than the orthoscopic. We 
 may further assert that such spectacles magnify or diminish 
 less in binocular vision than in monocular ; as may easily be 
 shown by trial. 
 
 The grounds on which the above described effects, as a rule, 
 are only observed in a slight degree, rest on the observation 
 already made, p. 117, that accommodation and application 
 have between them a very energetic induced concord, which 
 keeps them nearly coincident, both in monocular and binocular 
 vision. On account of this concord, which makes itself felt as 
 strained vision as soon as the proportion between the two 
 
 functions becomes abnormal, the quotient - retains a nearly 
 
 b 
 
 constant value. Since, under these conditions, a and r 
 may vary, the apparent size and distance change almost 
 as much as in using orthoscopic spectacles, but some- 
 
 / 
 
 what less, on account of the smaller change of - The 
 
 special process is as follows. On first looking through the 
 concave spectacles the accommodation distance b is smaller, 
 while the application distance c remains nearly the same. 
 
 K 
 
130 THEOKT OF OCULAR DEFECTS. 
 
 y 
 
 Hence the fraction ^ is too large. But, since the distance a 
 and the diameter r are diminished in proportion to the power 
 of the spectacles, the values of a and r will be diminished 
 
 notwithstanding the small enlargement of - ; that is, the com- 
 mon concave spectacles will diminish, but not in the same 
 degree as the orthoscopic. Inversely, the common convex 
 spectacles magnify somewhat less than the orthoscopic. These 
 results coincide accurately with those attained by observation. 
 If an object be looked at through common positive pris- 
 matic spectacles (bases inwards) there will immediately be 
 positive or homonymous double images. To unite these 
 images the eyes turn inwards, or converge more strongly, 
 so that they give up fixation with their poles, and see in- 
 directly. If, at the same time, the accommodation b remains un- 
 changed, and the application is increased, so that c is smaller ; 
 then, since a and r remain unchanged, the object must appear 
 nearer and smaller. But, as the induced link between accom- 
 modation and application keeps those functions nearly in 
 
 accordance, the proportion - really changes but little, and 
 
 thus no considerable difference in apparent size and distance 
 is observed. The special process is as follows. Since, on first 
 looking through the glasses, the accommodation distance b 
 remains the same, but the application distance c is larger, 
 
 
 
 the fraction T will be somewhat too large ; that is to say, the 
 
 homonymous images will be not quite so far apart as in a 
 normal relation between accommodation and application. The 
 
 subsequent convergence does not change the value of -, and 
 
 b 
 
 therefore leaves the object by so much too large and too remote 
 as - is too great. Inversely, negative prismatic glasses (with 
 
APPARENT SIZE AND DISTANCE. 131 
 
 bases external), produce a slight diminution and approxima- 
 tion of the object. This result also is in accordance with 
 observation. 
 
 Since the effect of apparent enlargement or diminution is 
 dependent upon the efforts of accommodation and application 
 actually exerted, and since these efforts have, for abnormal eyes, 
 according to the standard of their accommodation and appli- 
 cation curves, totally different values from those for normal 
 eyes, and can be changed by regulating these curves, so it is 
 manifest that this effect, on looking through the same spec- 
 tacles, will be totally different, for an abnormal and for a 
 normal pair of eyes. For example, a hypometropic (myopic) 
 eye requires, for the finite distance of its far point, scarcely 
 any effort of accommodation, but a considerable effort of 
 application. With its concave lens the effort of accommoda- 
 tion is increased, or b rendered smaller, and consequently - 
 
 larger, since the application c remains unchanged. Since 
 without the lens a strain exists between accommodation and 
 application, which is relieved by the lens, so in fact the fraction 
 
 T- is considerably larger, nearly in the degree by which b is 
 diminished. In the same degree is increased the distance a of 
 
 /> 
 
 the object, and hence - a retains its former value. For the 
 
 myopic eye, therefore, concave spectacles do not diminish 
 objects as they do for the normal eye, a result which is 
 familiarly known in practice. 
 
 An important confirmation of this theory of apparent size 
 and distance is furnished by the appearances of stereoscopic 
 pictures and of tapestry patterns. 
 
 Let us assume that the two eyes B and B', as in Fig. 61, are 
 looking at the objects L and L'. The eye B fixes the object 
 L and the eye B' the object L', so that both see with the polar 
 bacillus, B or B', and assume a position of convergence to- 
 wards the point D lying behind L and L'. If now the two 
 
132 
 
 THEORY OF OCULAR DEFECTS. 
 
 Fig. 61. 
 
 objects appear in one place M, and therefore in the frontal 
 axis A M, the parallels to A M, B e and B' e', will be the direc- 
 tions of the affected 
 polar bacilli, and there- 
 fore eB D will be the 
 angle of application y, 
 and A D, the application 
 distance c. On account 
 of the rays coming from 
 L, the bacillus B seeks 
 to accommodate for 
 the distance B L = A M 
 = a of the object L, 
 therefore to make b = a. But the strong induced link 
 between accommodation and application produces a deviation 
 of the accommodation towards the value of the application, c ; 
 that is, b will be greater than a, but smaller than c. The 
 
 r* 
 
 expression -r a, for the apparent distance of the object M, in 
 
 which L and L/ coincide, will therefore, on account of c > b, 
 produce an apparent removal and enlargement of the object L. 
 
 If, as in Fig. 62, the left eye B 
 fixes the right-hand object L, and the 
 right eye B' the left hand object L' ; 
 then, since both objects are seen 
 singly, the construction remains the 
 same. The application distance c is 
 then A D, and therefore much smaller 
 than before. The accommodation 
 distance b deviates from B L or A M 
 = a, towards c, and is therefore less 
 than a, but remains greater than c. 
 
 s> 
 
 The expression r a for the apparent 
 
 distance of the object M, in which L and L' coincide, shows 
 therefore, on account of c < , that this object will appear 
 nearer and smaller than the object L. 
 
 Fig. 62. 
 
 M 
 
PERCEPTION OF SIZE AND DISTANCE. 133 
 
 Since the vision as in Fig. 61 produces an enlargement, and 
 the vision as in Fig. 62 a diminution, the diminution in 
 transition from the former to the latter must be still more 
 remarkable, and this is experimentally found to be the case. 
 
 From the foregoing it follows that the apparent size and 
 distance may be regulated, for each eye singly, by the 
 formulae 
 
 and that we may therefore always obtain an equality of the 
 visual impression in the two eyes, which, for binocular vision, 
 is certainly of importance. 
 
 PERCEPTION OP SIZE AND DISTANCE. 
 
 In the foregoing we have established the formula a' ==. -a 
 for the apparent distance a'. In this - is the actually exerted 
 
 accommodation effort, and - the actually exerted application 
 
 effort. From the circumstance that the formula does not ex- 
 plicitly contain the convergence effort, it does not follow that 
 the apparent distance is wholly uninfluenced by the conver- 
 gence. Since all the ocular functions act more or less upon 
 each other by induction, so has the convergence also more or less 
 influence upon the rest. We say, however, that the conver- 
 gence influences the process on which the recognition of distance 
 rests, not directly, but only through its effect upon the ac- 
 commodation or application. An abnormal convergence would 
 modify the values of b and c, and is therefore contained, by 
 
 Sk 
 
 implication, in the formula - a. 
 o 
 
 An increased effort of any one visual function tends always 
 
134 THEORY OF OCULAR DEFECTS. 
 
 to produce a more or less increased effort of all the rest, or to 
 arouse a tendency to such effort. This tendency, even when 
 not followed by any external consequences, is to be considered 
 as an increase of the particular effort, with which we are here 
 concerned. 
 
 Since an abnormal convergence influences not only the ac- 
 commodation #, but also, and in a nearly equal degree, the 
 application c, it only tells, in the above formula, upon the 
 
 proportion - of these two functions ; and thus the general in- 
 fluence of the convergence upon the apparent distance is, in 
 most cases, very inconsiderable. 
 
 It has been observed that the induction between accommo- 
 dation and application is far more energetic than that between 
 either of these faculties and convergence. In whatever degree, 
 therefore, one of the two magnitudes, b or c, is influenced, the 
 other will nearly follow. Accommodation and application can 
 more easily emancipate themselves from convergence than 
 from each other. 
 
 While b and c, in the expression -, a for the apparent dis- 
 tance, indicate two functions or efforts, a, as the actual distance, 
 indicates a mathematical quantity. For the eye, however, 
 such a quantity, as a pure matter of the understanding, has no 
 significance. The eye recognizes, or feels, only sensory nerve 
 processes ; and what the understanding, by its special function, 
 abstracts from such impressions, is no affair of the visual 
 function, but only of the power of abstraction. Hence, in any 
 physiological formula, the quantity a must have its physio- 
 logical significance, and this must rest upon a material process. 
 
 In the Physiologische Optik, and in the Gesetze, I have 
 shown that a is the proportion of the rectangular components 
 of the luminous concussions given by the conical pencil of 
 rays to the nervous substance of the bacilli. Distance. is there- 
 fore immediately felt by the bacilli, from the concussions that 
 they sustain. If the sensibility of the bacilli were altered, the 
 
PERCEPTION OF SIZE AND DISTANCE. 135 
 
 sensation itself would be altered also. We may assume that 
 this sensibility may so far be influenced by the accommodation 
 and the application, that the elastic state of the bacilli may, 
 through those functions, undergo a change. Since, therefore, 
 accommodation and application may modify the state of 
 elasticity of the bacilli, or their sensitiveness to luminous 
 oscillations, they may influence the sensation itself, or the 
 apparent distance. 
 
 It is obvious that the sensitiveness of the bacilli, like the 
 functional reaction of every other part of the visual apparatus, 
 must have its upper and its lower limit. A proportion of 
 components, which exceeded a certain amount, would still be 
 only attended by the maximum of sensation ; and a propor- 
 tion which fell below a certain amount, would no longer pro- 
 duce any reaction of the bacilli. No reaction implies no 
 function, and no sensation. It would be erroneous, however, 
 to assume that such a state answers to a null or infinite value. 
 The null, the nothing, and the infinite, are pure ideas, not 
 perceptions of sense. The perceptions of sense, as such, can 
 exist only in a perceptible degree, as a definite maximum or 
 minimum. In so far, therefore, as the bacilli experience a 
 sensation of distance, it can be only of finite distance, lying 
 between a certain maximum and minimum. In other words, 
 all objects nearer to the eye than a certain point, the near 
 point of perception, must appear to be at the distance of this 
 near point ; and all objects beyond a certain point, the far 
 point of perception, must appear to be at the distance of this 
 far point, not at infinite distance. 
 
 If we, for brevity, call the quantity a the perception of dis- 
 tance, then this will have a minimum a { for the near point, 
 and a maximum a 2 for the far point. By means of the influ- 
 ence of the accommodation and the application, the apparent 
 
 perception will be a' ~ -a. The accommodation and the ap- 
 plication cannot alter the absolute limits of perception, so that 
 a l and a 2 indicate also the minimum and the maximum of a'. 
 
136 THEORY OF OCULAR DEFECTS. 
 
 In normal accommodation and application, for which we 
 have - = 1, the apparent distance, a', corresponds always to 
 
 the perception a. Within the imagined limits, this perception 
 is the true expression of the actual proportion of the com- 
 ponents of the luminous vibrations, and the object therefore 
 appears at the proper distance ; while within the near point 
 the object appears at the distance a^ farther than it is, and 
 beyond the far point, at the distance 2 , nearer than it is. 
 
 On approaching the boundaries a l and 2 , the apparent 
 distance deviates a little gradually from the actual, until the 
 former becomes stationary at a l or at 2 . Hence we have 
 for the perception a facultative curve, which, like those 
 already considered, is parallel to the axes of abscissas at its ex- 
 tremities, and accurately corresponds to the curve of require- 
 ment at its central part, especially near the terminal points of 
 the middle visual distance, deviating gradually from the re- 
 quirement curve elsewhere. 
 
 s* 
 
 The expression for the apparent size of the object, r' = - r 
 
 is not purely physiological, since r represents a mathematical 
 quantity, namely, the diameter of the object. It first assumes 
 
 a physiological form as r' = a - a, in which b and c repre- 
 sent the accommodation and application, a the perception, and 
 a the quantity of the sensitive nerve points lying in the 
 diameter of the retinal image, corresponding to the visual 
 angle of the object. Hence we may call a the perception of 
 the visual angle. 
 
 Since -a is equal to the apparent distance, we have also 
 
 r f = aa' that is, in abnormal accommodation and application, 
 when the apparent distance deviates from the real distance, 
 the apparent size always corresponds to the product of the 
 visual angle and the apparent distance; or, in other words, 
 by the influence of accommodation and application, size and 
 distance are increased or diminished in equal proportion. 
 
PERCEPTION OF SIZE AND DISTANCE. 137 
 
 Between the boundaries of perception, but not too near to 
 these boundaries, an object appears, in normal accommodation 
 and application, of its proper size, r' = a a = r. An object 
 of given size, r, appears, therefore, within any moderate dis- 
 tance equally large. Hence the remarkable fact is explained 
 that an object, lying at moderate visual distance, does not 
 appear to change its size, even when moved considerably 
 nearer or farther. 
 
 On this side of the near point of perception an object of 
 constant diameter, r, appears of the size r' = a. a l ; and there- 
 fore too large. Beyond the far point it appears of the size 
 r = a a 2 ; and therefore too small. Beyond these limits, 
 since a l and a 2 remain constant, the apparent size varies as 
 the visual angle a. This result also is supported by ex- 
 perience. 
 
 An object which everywhere retains the same visual angle, 
 a, and which is therefore increased by distance, and diminished 
 by approximation, since it always fills a definite visual angle, 
 appears of proper size at moderate distances, because r' = a a 
 = r, and therefore increases proportionately as the distance. 
 But on this side of the near, and on that side of the far point 
 of perception, the same object appears always of the constant 
 size a flj or a a 2 > because r' = a a, or r 7 == a 2J while a, a\, 
 and a 2 , all remain the same. 
 
 Since a, as a physiological quantity, shows the magnitude of 
 the retinal image, so it is influenced by the distance from the 
 retina of the crossing point of the chief rays. If we call this 
 distance = />, and the visual angle of the object = 0, we have 
 a = p 0. So long as the object (for the naked eye the actual 
 object, and for an eye with a lens the optical image of the 
 object) is not too near, that is, not nearer than four inches, the 
 variation of the visual angle 0, measured from the crossing 
 point of the chief rays, that can be produced by displacing 
 this crossing point, is so trivial that it may be left out of ac- 
 count ; and, therefore, <f> may be assumed to be constant. 
 With the distance of the object, and, therefore, also with the 
 use of a concave or convex glass, only the distance p of the 
 
138 THEORY OF OCULAR DEFECTS. 
 
 crossing point varies ; and because a = p ^>, this variation 
 produces proportionate enlargement or diminution of the retinal 
 image. The displacement of the crossing point, in conse- 
 quence of a change in the distance of the object, or of the 
 luminous point, is not considerable, even in the case when the 
 eye maintains its refraction. But the change of accommoda- 
 tion that attends upon the change of distance, so nearly 
 equalizes this small displacement of the crossing point that it 
 may be disregarded, and p also may be assumed to be nearly 
 constant. Hence it follows that the retinal image does not 
 change its size a, by the use of spectacles, in any noteworthy 
 degree ; and that we may always substitute the visual angle 
 of the object as a measure for this size.* 
 
 ASYMMETRICAL OCULAR DEFECTS. 
 
 In all the abnormal states hitherto mentioned, it has been 
 supposed that the individual eye is symmetrical with regard 
 to its axis. We meet, however, with defects that are asym- 
 metrical. They may depend either upon structure or upon 
 innervation or function, and they may affect any of the three 
 chief functions, the accommodation, the application, or the 
 convergence. This consideration furnishes the scheme for the 
 systematic division of asymmetrical defects. 
 
 We will consider first the asymmetrical defects of the 
 accommodation. These consist of an incomplete concentration 
 of the pencil of rays proceeding from a luminous point ; that 
 is, of an aberration. Aberration may arise from two different 
 causes first, from a want of symmetry of the eye in the dif- 
 ferent meridional planes passing through its axis, which istastig- 
 matism ; and secondly, from a want of symmetry of the eye in 
 its conical radial surfaces, that is, in the different conical sur- 
 
 * The remaining paragraph of this chapter in the original is devoted 
 to the correction of an error in the Physiologische Optik ; and has no 
 interest except in relation to a passage quoted from that work. TRANS. 
 
ASYMMETRICAL OCULAR DEFECTS. 139 
 
 faces of which the centre of the eye is the apex; and this 
 formation I have called * Aspherismus ' in the Gesetze des 
 raumlichen Sehens. 
 
 Astigmatism may have its basis in the structure of the eye ; 
 and it is this form which has been especially investigated by 
 Bonders. It may also be functional; that is, the eye may 
 in accommodation become asymmetrical in the different meri- 
 dians, or astigmatic. 
 
 Astigmatism may arise from defects of different organs, from 
 an asymmetrical curvature of the cornea or of the lens, from 
 an asymmetrical density of the lens or of the vitreous body, 
 and so forth. According to the researches of Bonders, the 
 asymmetric curvature of the cornea is the most frequent cause ; 
 and in this there are almost always two meridians, of maximal 
 and minimal curvature, at right angles to each other. 
 
 The optical means of relief in astigmatism is the cylindrical 
 convex or concave lens, which equalizes the refraction or 
 accommodation in all the meridians. 
 
 With such a cylindrical lens a spherical lens may be com- 
 bined, in so far as there is any defect of accommodation beyond 
 the astigmatism, and with these two a prism also may be united, 
 if there be any defect of convergence or of application. The 
 prism must be in the middle ; so that it may have on one side 
 a piano-cylindrical lens, on the other side a piano-spherical. 
 
 Aspherism also may depend either upon structure or func- 
 tion. The curvature of the cornea is the chief cause of this 
 defect. If the marginal part of the cornea or of the lens is too 
 strongly curved (or the marginal part of the lens too dense), 
 there is an over-curved eye ; but if these marginal parts are 
 too little curved (or wanting in density), there is an under- 
 curved eye. 
 
 For the relief of aspherism there is the aspheric lens, de- 
 scribed in the Gesetze, p. 144, which is convex-concave, with 
 a hyperbolic form for the over-curved eye, and an ellipsoid 
 form for the under-curved eye. The characteristic of this lens 
 is that its external and internal curvatures are equal at the 
 
140 THEORY OF OCULAR DEFECTS. 
 
 apex, while the section of the internal curve differs from that 
 of the external. 
 
 The asymmetrical defects of convergence may be either ano- 
 malies of position or of motility. They include the cases of 
 oblique and distorted ocular direction, and the abnormal mus- 
 cular efforts which move the eyes in other than the natural 
 direction of convergence. 
 
 The optical remedy for these defects is a prism placed ob- 
 liquely. But in general we know, from the foregoing, that the 
 rational means of relief is not optical, but a surgical operation 
 on the ocular muscle. 
 
 For this operation a knowledge of the laws of normal ocular 
 movement, and a correct judgment about the normal and ab- 
 normal action of the various muscles is highly important. In 
 18 of the Gesetze I have shown that the hitherto received 
 laws of ocular movement are erroneous, and I have there, 
 as well as previously in the Physiologische Optik, given the 
 actual laws of movement. Moreover, in the Physiologische 
 Optikj 14 and 15, I have calculated the effect of every 
 normal or abnormal muscular action, and have described, under 
 the name of 'ocularium,' a very convenient instrument for 
 teaching a knowledge of these actions. 
 
 The asymmetrical defects of application, again, may be either 
 structural or functional, or may be anomalies either of position 
 or of nutation. 
 
 From these defects arise the most various cases of false pro- 
 jection to the left, to the right, upwards, downwards ; and also 
 the most various cases of abnormal strabismus. Moreover, an 
 abnormal direction or nutation of the bacilli towards different 
 points of the retina may produce apparent distortion of the 
 object, apparent obliquity of position, and apparent changes of 
 size, as well as other deformities. (See Phys. Opt. 75.) 
 
 If the defect consist of a uniform deviation towards a 
 meridian, it may be corrected by a straight or oblique prism. 
 Irregular defects would require glasses of irregular form. 
 Defects depending upon a displacement of the retina (so- 
 
ASYMMETRICAL OCULAR DEFECTS. 141 
 
 called incongruence) upon a displacement of the lens, upon an 
 obliquity of the lens to the ocular axis, upon a curvature of the 
 retina deviating from the spherical form, which give a false 
 direction to the chief rays, or change their apparent direction, 
 may, on account of this change of direction, be numbered with 
 the defects of application ; and may be corrected as far as 
 practicable, partly by glasses, partly by operations. 
 
 It does not form part of the plan of this work to follow 
 farther the asymmetric and irregular ocular defects. If we 
 consider that any of the former symmetrical defects may in any 
 eye be combined with any of the asymmetrical defects, and 
 that in each of the two eyes any such combination may exist, 
 we see that the number of the possible conditions of the bino- 
 cular visual apparatus amounts to an incalculable legion. But 
 the systematic classification arranges this legion in groups, and 
 allows a rational treatment of all to appear possible. In reality, 
 there is no absolutely normal pair of eyes ; and we may say 
 that each one possesses all the defects that can be combined, 
 only in a greater or less degree. 
 
 The mathematical formula suffices for the general expression 
 of any condition, the normal as well as the abnormal. The 
 formula knows no specific, but only gradual distinctions ; and 
 the special values, which are assumed by the different quantities 
 between the limits of and co , not only characterize the 
 above-mentioned legion of cases, but the practically infinite 
 number of intermediate steps and transitions with more cer- 
 tainty than could be attained by any nomenclature. 
 
 Moreover, the general formula points out the remedy for 
 each defect. A glass formed by measurement, or a measured 
 surgical operation, is able to equalize every ocular defect based 
 upon the structure of the organ ; and it is only the functional 
 or facultative limits, which depend upon individual nerve 
 force, that cannot be altered by external means. These limits 
 form, therefore, definite and immovable halting-places for the 
 artificial regulation of the visual processes. 
 
 In conclusion, we should observe that defects of both eyes, 
 
142 THEORY OF OCULAR DEFECTS. 
 
 symmetrical in each and alike in both, and which therefore 
 constitute symmetrical defect of the combined organs, yet do 
 not always occasion symmetrical function. The tendency to 
 the most complete possible satisfaction of the three chief func- 
 tions in any eye preponderates, not uncommonly, over the 
 tendency to an equal participation of both eyes in the visual 
 act ; and produces an asymmetrical discharge of function by 
 the entire apparatus. Consequently a pair of eyes, compelled 
 to squint by some defect of convergence or of application, does 
 not usually look symmetrically at the object, so that both eyes 
 may squint in an equal degree. More frequently the object is 
 fixed by one eye, which therefore completely satisfies the 
 desire for fixation by seeing with the polar bacillus ; while the 
 other eye deviates from the point of fixation by so much the 
 more. 
 
 THE ASSISTANCE TO FIXATION BY THE DISPLACEMENT 
 OF THE BACILLI. 
 
 The fixation, which fulfils the purpose of bringing the pole of 
 the eyeball to the affecting ray of light, and thus of producing 
 direct vision with the polar bacillus lying in the yellow spot, 
 is accomplished by the ocular muscles, and is, if we consider 
 only the acting apparatus, a motor function, the function of 
 convergence. But the impulse to this function manifestly 
 proceeds from the optic nerve ; that is, the cause of the mus- 
 cular movement must be a sensational process. In fact, we have 
 seen already (pp. 61-2) that the tendency to preserve the 
 optical equilibrium directly occasions the ocular rotation. 
 
 When this is recognized it is not difficult to assume that, 
 when the convergence, for any reason, does not attain the end 
 towards which it is directed, and therefore does not bring the 
 pole of the eyeball to the luminous ray, a tension may continue 
 on the retina, and may constantly strive to move the polar 
 bacillus into the desired position. This tendency would have 
 
DISPLACEMENT OF THE BACILLI. 143 
 
 more or less result; that is, the polar bacillus, and after it 
 in a diminishing degree the surrounding bacilli of the yellow 
 spot, would be moved towards the ray of light ; thus, notwith- 
 standing the false convergence or the strabismus, direct vision, 
 or fixation by the polar bacillus, being brought about, so long 
 as too great a movement of this bacillus is not required. 
 
 "We obtain hence, for further consideration, the very im- 
 portant conclusion that in convergent strabismus the polar 
 bacillus tends inwards, towards the entrance of the optic 
 nerve ; and that in divergent strabismus it tends outwards, 
 away from the entrance of the nerve. 
 
 Just as the two polar bacilli in binocular vision strive 
 after the points that receive the light, so the corresponding 
 bacilli of the two retinae strive after equal affections, and, in 
 vision in material space, they move in the manner explained 
 in 12 and 13 of the Gesetze, so that, notwithstanding 
 the incidence of the rays of light upon different points of 
 the retina, in 'consequence of this movement identical bacilli 
 are influenced by similar rays. 
 
 The movement of the corresponding or identical bacilli to 
 similar rays of light arises, as in the polar bacilli, from the 
 effort after optical equilibrium in the two eyes, under the 
 stimulus of the images formed on both retinae by the luminous 
 vibrations and the nervous processes. 
 
 That, notwithstanding this co-operation of identical retinal 
 fibres, we need not imagine any specific function of single 
 vision, and that therefore the common belief in a law of 
 identity is founded on error, is fully shown in 21 of the 
 Gesetze. The singleness of the binocular visual impression 
 is no special quality, but only a result of the coincidence 
 of two impressions. For the coincidence of the apparent 
 visual directions of these two impressions there is nothing 
 necessary, according to our theory of application, but simi- 
 larity of direction of the affected bacilli ; whether these are 
 identical or different is of no moment ; but, by means of the 
 force subservient to the tendency to optical equilibrium, 
 identical bacilli are drawn towards the spots equally affected. 
 
U4 THEORY OF OCULAR DEFECTS. 
 
 THE REACTION OF OCULAR DEFECTS UPON ONE 
 ANOTHER. 
 
 Although, as already frequently set forth, the three chief 
 functions of the eyes, accommodation, application, and con- 
 vergence, are originally independent ; yet it is necessary, for 
 the easy exertion of any one of them, that the other two 
 should bear a certain proportion to it. This proportion forms 
 that fundamental law of the visual function, or of the scheme 
 of vision, which, in 21 of the Gesetze, I have declared as the 
 proper law of identity. The consensus of the ocular muscles, 
 and the law of the ocular movements, form only a special por- 
 tion of this law, since in free movement the exertion of in- 
 dividual muscles must bear a definite proportion to each 
 other. 
 
 If this normal proportion does not exist, visual strain is the 
 result. The several ocular functions, in like manner, secon- 
 darily influence one another ; but this influence does not display 
 itself in any external effect, that is, in any irregularity of the 
 individual functions, until the visual strain exceeds by a certain 
 degree the relative limits of function. Hence the originally 
 independent chief functions come to stand in a relation of 
 acquired dependence upon one another. 
 
 The earlier chapters contain all the materials necessary for 
 the experimental determination of the actual condition of the 
 individual organs and their functions, and of the means by 
 which the act of vision may be regulated, and the visual strain 
 (for given conditions) relieved. They contain also the key to 
 the explanation of many phenomena, interesting in themselves 
 and important to ophthalmic surgeons, which depend upon the 
 acquired influence, or upon the mutual reaction of the respec- 
 tive functions in states of abnormal proportion. Of this, several 
 illustrations have already been given, and some others must be 
 briefly noticed. 
 
THEIR REACTION. 145 
 
 The general basis for the explanation .of this reaction is 
 furnished by the three following positions : 
 
 1 . Too great an increase or diminution of any single function 
 produces an abnormal increase or diminution of the others. 
 
 2. The interdependence between the functions of the ocular 
 nerves -that is, between accommodation and application, is 
 far closer than that between either of these and the con- 
 vergence, which is a function of the muscles. 
 
 3. If the motor function of convergence should fall short of 
 the attainment of its object, then the effort after fixation, 
 which occasions convergence, produces a tendency to movement 
 of the polar bacilli, and of the corresponding bacilli generally. 
 This tendency is very energetic, but only produces the extreme 
 result of a sufficient movement when the requirement does not 
 exceed a certain degree. If this degree be greater than the 
 power of movement of the bacilli, there arises a sensible 
 tension or visual strain, which more or less influences the 
 accommodation or the application. 
 
 Let us examine, on these data, the frequent divergent stra- 
 bismus of the hypometropic, and the very frequent convergent 
 strabismus of the hypermetropic. Donders has found these forms 
 of squint so frequently in the respective anomalies of refraction, 
 that he cannot doubt the causal connection between them. 
 He explains them, however, upon different grounds ; the diver- 
 gent squint of the short-sighted by insufficiency of the muscles, 
 or of motility ; and the convergent squint of the hypermetropic 
 by abnormal effort of the accommodation. I believe these ex- 
 planations to be erroneous ; that the cause of the strabismus 
 must be sought elsewhere, and that for both forms it is the same. 
 
 The myope must certainly, in near vision and with small 
 effort of accommodation, exert a strong convergence ; and it 
 is possible that for this his muscles may be insufficient. If 
 this were only so, and no other cause were present, his squint 
 would necessarily be united with double vision. Such, how- 
 ever, is not the case ; and, therefore, the insufficiency cannot 
 produce the strabismus. 
 
 L 
 
146 THEORY OF OCULAR DEFECTS. 
 
 The same applies to the convergent squint of the hyperme- 
 tropic, who, with slight convergence, is called upon for a 
 
 considerable effort of accommodation. If the latter induced 
 the squint by occasioning too much convergence, the single 
 vision would be inexplicable. 
 
 According to my conviction, the strabismus depends, in 
 both cases, not upon the comparatively lax interdependence of 
 the accommodation and the convergence, but upon the far 
 closer interdependence of the accommodation and the appli- 
 cation. 
 
 When the myope fixes the near object D (Fig. 63), he exerts 
 only a small amount of accommodation. For single vision he 
 would require the great nutation E B D, = B D A, in order to 
 render the bacillus B parallel to the frontal axis A D. If the 
 function of application be in its normal state, then so extreme 
 an application is impossible under a feeble accommodation ; 
 and the bacilli B and B' must therefore apply in less degree, as 
 in the directions B F and B'F'. If we draw A G parallel to B F, 
 and A'G' parallel to B'F', it becomes manifest that diplopia with 
 crossed images must be produced. The craving after single 
 vision compels the eyes, with small effort of accommodation 
 and a corresponding degree of application, to roll outwards 
 and to assume the position of divergent squint. 
 
THEIR REACTION. 
 
 147 
 
 When the hypermetrope fixes the distant object D (Fig. 64), 
 his eyes are called upon for a strong effort of accommodation. 
 With this the feeble application E B D, = B D A, cannot be com- 
 bined. The bacilli B and B' apply, therefore, in a greater 
 degree, and assume the directions BF and B'F'. If we draw 
 A G parallel to B F, and A'G' parallel to B'F', it becomes mani- 
 
 Fig. 64. 
 
 fest that diplopia with homonymous images must be produced. 
 Hence the craving after single vision compels the eyes, with 
 strong accommodation and a corresponding degree of applica- 
 tion, to roll inwards, and to assume the position of conver- 
 gent squint. 
 
 In this way the very common divergent squint of the short- 
 sighted, and the convergent squint of the hypermetrope, may 
 be explained upon one and the same principle, and without 
 disregard of the single vision ; while the explanation affords 
 also an important support to my theory of the application. 
 
 It is also manifest that, when a defect of application coin- 
 cides with a defect of accommodation, strabismus may be 
 absent, or may even occur in a direction contrary to the usual 
 one ; of which latter condition v. Graefe has seen many ex- 
 amples in the short-sighted. If the hypometropic or hyper- 
 
 L2 
 
148 THEORY OF OCULAR DEFECTS. 
 
 metropic eye be in the same degree hypoklitic or hyperklitic, 
 no strabismus will be produced. 
 
 The common concave spectacles remove the disproportion 
 between accommodation and application in the myope; and 
 common convex spectacles remove this disproportion in the 
 hypermetrope. By the use of proper spectacles, therefore, the 
 strabismus of myopic and of hypermetropic eyes is relieved. 
 
 The relief of divergent strabismus is equivalent to an in- 
 crease of the convergence, and the relief of convergent strabis- 
 mus is equivalent to a decrease of the convergence. A normal 
 pair of eyes must assume a convergent squint on looking 
 through common concave spectacles, and a divergent squint 
 on looking through common convex spectacles. In the former 
 case the increased accommodation, produced by the apparent 
 approximation of the object, is attended by increased applica- 
 tion, that is, by diplopia with homonymous images, which are 
 united by the convergent squint. In the latter case the exact 
 reverse will hold good. 
 
 Hence it follows that experiments with common centric spec- 
 tacles, in order to determine the limits of relative accommoda- 
 tion in a given convergence, can lead to no accurate results, 
 since the supposed degree of convergence is not accurately 
 maintained. 
 
 When a pair of normal eyes squint in consequence of look- 
 ing through common centric spectacles, such eyes no longer fix 
 accurately, or see directly with the polar bacillus, but indi- 
 rectly. The same thing occurs in the unaided vision of myopic 
 and hypermetropic eyes. These cease to see directly, since they 
 squint ; or, rather, when they squint. 
 
 In all these cases of incomplete fixation, and, therefore, in 
 strabismus generally, the already described movement of the 
 bacilli becomes of very great importance. If the degree of 
 strabismus required to afford single vision be such as not to 
 deviate too far from the position of fixation, the polar bacil- 
 lus may so move as to come into the visual line, and may 
 receive the incident ray. We have then fixation, or direct 
 
THEIR REACTION. 149 
 
 vision with the displaced polar bacillus, notwithstanding the 
 strabismus ; a result that certainly merits special consideration. 
 It affords the seeming paradox of fixation under false con- 
 vergence. 
 
 It must be expressly remarked that the movement of the 
 polar bacillus, which takes place for the sake of fixation, has 
 no influence or effect upon the strabismus, which takes place 
 for the sake of the application, or of single vision. The polar 
 bacillus, that moves into the incident ray, assumes the same 
 nutation as the bacillus that its movement displaces. 
 
 The above-described movement of the polar bacillus natu- 
 rally occurs only in binocular vision ; since, in the use of one 
 eye only, there can be no obstacle to fixation. 
 
 Hence it follows that the movement cannot be observed 
 through the ophthalmoscope ; since the use of this instrument 
 requires that the axis of the observed eye should come into 
 the visual line of the observer. But in eyes in which, on ac- 
 count of their structure or of the nature of their employment, 
 the displacement of the bacilli takes place frequently and con- 
 tinuously, this movement comes to be observed by the produc- 
 tion of certain permanent conditions. 
 
 Upon this ground the question of the movement of the polar 
 bacilli, and of corresponding bacilli generally, acquires much 
 importance in many ocular defects, and especially- in short sight. 
 
 We have seen that an eye, which is short-sighted by reason 
 of hypometropia, makes an effort of application or nutation of 
 the bacilli outwards, that is disproportionately strong with 
 regard to its accommodation, and that thus, in order to pre- 
 serve single vision, it assumes a divergent squint ; while, for 
 the sake of preserving fixation notwithstanding the squint, the 
 polar bacillus moves outwards until it either comes into the 
 visual line, or attains the limits of its range in that direc- 
 tion. We have here a ready explanation of all the abnormal 
 conditions of the retina and of the choroid by which short 
 sight is so frequently attended (see Bonders); such as the 
 following : 
 
150 THEORY OF OCULAR DEFECTS. 
 
 1. The atrophy, stretching, and thinning of the choroid. 
 Since the pole B of the eye, Fig. 65, lies at a small distance 
 from, and external to, the entrance of the optic nerve, and 
 since the polar bacillus B dips by its external extremity into 
 the pigment of the choroid, the movement of the bacillus to b 
 must produce an unnatural stretching of the choroid between 
 A and b. 
 
 Fig. 65. 
 
 2. The separation of the choroid from the optic nerve, and 
 the degeneration of the choroid chiefly around the nerve, espe- 
 cially on the outer side A, that is, the establishment of the 
 atrophic ' crescent.' 
 
 3. The extension of the retina and the stretching of its 
 vessels between A and b. 
 
 4. The diminution of the diameter of the optic papilla in 
 the direction of b A ; since a displacement of .the bacillus B 
 away from A must produce a secondary pressure against the 
 papilla. 
 
 5. The separation of the retina from the choroid in conse- 
 quence of the great nutation of the bacilli, by which the latter 
 may be torn from the choroid. 
 
 6. The gradual increase of distance between the yellow spot 
 and the papilla. 
 
 7. The general morbid condition into which all the mem- 
 branes and the whole eye may be brought by such abnormal 
 strivings. 
 
THEIK REACTION. 351 
 
 8. The increase of the short sight, as an effect of the morbid 
 condition, in this way, that the thinning and tearing of the 
 membranes promotes a further elongation of the ocular axis. 
 This action takes place more rapidly in slight myopia, where 
 the membranes still retain considerable extensibility, than in 
 more extreme cases, when these membranes have already been 
 reduced by atrophy to hard tissues. 
 
 Hence, we must assume, not, as Bonders supposes, that 
 myopia depends upon a condition which itself determines the 
 development of atrophy ; but, that the atrophy is the direct 
 result of the myopic refraction, in consequence of the employ- 
 ment of the myopic eyes without sufficient preservative means. 
 
 It is further manifest that any means by which the above- 
 mentioned forms of tension, and therefore the tendency to 
 movement and excessive nutation of the bacilli, can be re- 
 lieved, will be preservative against the progressive increase of 
 the disease, and of the shortness of sight. Such relief is chiefly 
 to be sought from proper spectacles. Orthoscopic spectacles 
 are insufficient for this purpose, since their only action is to 
 bring distant objects within the near range of vision, and they 
 do not correct the disproportion between accommodation and 
 application. If there be only an anomaly of refraction pre- 
 sent, so that the shortness of sight depends solely upon hypo- 
 metropia (which may be determined in the manner formerly 
 shown), the disproportion in question may be relieved by cen- 
 tric concave spectacles. 
 
 It seems, however, to be indicated, for the relief of this evil, 
 to unite with the concave lens a prism with its base turned 
 inwards, by which the bacilli will be caused to move and bend 
 in the opposite direction, or towards the papilla. 
 
 It is self-evident that hypermetropia will exert upon the 
 membranes an effect precisely opposite to that of short sight. 
 The bacilli will be pressed towards the papilla. From this 
 can arise only a thickening of the membrane, which does not 
 appear to be a source of disease. In consequence of such a 
 change, atrophy could only be expected at the ora serrata ; 
 
152 THEORY OF OCULAR DEFECTS. 
 
 but, as the distance from the pole to this region is consider- 
 able, and as only the terminations of the nerve fibres and of 
 the vessels would be affected, it is not easy to imagine that 
 any hurtful influence could be exerted. Possibly, however, 
 the thickening of the ocular membranes may explain the di- 
 minished acuteness of vision so frequently seen in the hyper- 
 metropic. (See Bonders.) 
 
 If we were to follow into all details the action of one abnor- 
 mal ocular function upon the remaining functions, we should, 
 doubtless, arrive at an explanation of many other phenomena. 
 
 Among such we might thus elucidate the degeneration of 
 many ocular muscles ; and, in turn, the reaction of this dege- 
 neration, and of abnormal motility, upon the sensory functions. 
 
 By going a step farther, and including in the inquiry the 
 position and movements of the head, we should come to un- 
 derstand the tendency to assume certain attitudes, which have 
 the effect of diminishing, more or less, a visual strain tnat is 
 felt when the head is erect. (See Gesetze, &c., 25.) 
 
 THE OPPONENCY OF THE FIELDS OF VISION. 
 
 In conclusion I desire to set forth briefly the doctrine de- 
 veloped in the Phys. Optik, and in the Gesetze, and, as I be- 
 lieve, completely established by actual phenomena, with regard 
 to the character of the opponency between the fields of vision 
 of the two eyes. 
 
 Of two corresponding nervous fibres only one is always see- 
 ing ; that is, only one of the two immediately evokes the visual 
 phenomena caused by the incident ray. The other, however, is 
 not devoid of function, for the visual process is carried on in the 
 chiasma and the sensorium by both ; but one takes in a cer- 
 tain sense the active part of a positive pole, and the other the 
 passive part of a negative pole. Even when one eye is covered 
 
OPPONENCY OF THE FIELDS OF VISION. 153 
 
 or blinded, this process is not interrupted, but is only so far 
 limited for the useless eye, that the action of the sensorium is 
 propagated only more or less deeply along the corresponding 
 fibre. 
 
 In binocular vision two corresponding fibres or bacilli take 
 the active part alternately, each for a time, and as a conse- 
 quence of the fatigue of its fellow. Consequently only one of 
 the two fixes, in the full sense of the word, and with complete 
 functional activity ; and this one is in a certain degree relieved 
 from the visual act when the other takes it up. Hence the 
 negative fibre, in consequence of the continuance of the visual 
 process, remains accommodated and applied with more or less 
 exactness. 
 
 It does not therefore follow that, of the fibres receiving a 
 particular retinal image, all those of one eye shall be active, 
 and those of the other passive, at the same time, so that any 
 extended object would always be seen and completely fixed by 
 only one eye at a time. Bather, of the sum total of the fibres, 
 a certain portion of those of one eye and a certain portion of 
 those of the other are in simultaneous activity ; so that both 
 eyes see at the same time, but each a different part of the ob- 
 ject. The active fibres of each eye vary incessantly, so that 
 sometimes these, sometimes those, now more, and now less, are 
 called into play. 
 
 I believe that the proofs I have advanced leave no room for 
 doubt with regard to these facts. For such evidence I must 
 refer to the works already cited, and will only here remark that 
 the doctrine laid down explains the instances observed by oph- 
 . thalmologists, in which one eye evidently did not originate any 
 visual impression, but in which its participation in binocular 
 vision essentially aided this impression. 
 
 Moreover we may conclude that under abnormal conditions, 
 where there is a strain of the convergence or of the application, 
 such as would be produced by strabismus or by the lack of 
 fixation, and where, therefore, all the bacilli of the retinal image 
 of each eye are striving immoderately after fixation, all the 
 
154 THEORY OF OCULAR DEFECTS. 
 
 bacilli of one eye may in such an effort obtain a preponder- 
 ance over all the corresponding bacilli of the other eye, and 
 consequently first the one eye, and then the other, will exclu- 
 sively assume the fixation of the whole object. 
 
 Hence, of two similar eyes, it is usual for only one to squint, 
 while the other is rightly directed ; and the two fix alternately. 
 Especially is- this the case in high degrees of strabismus, when 
 a correction by movement and nutation of the bacilli is no 
 longer possible. 
 
 In dissimilar eyes, moreover, it is usual for the same one al- 
 ways to assume the fixation, and the other the negative part. 
 
 In fixation with one eye, if the deviation of the other should 
 exceed a certain degree, so that very different bacilli receive 
 the image, then the participation of the latter becomes so small 
 that it is scarcely more functionally active than an eye that is 
 covered. 
 
 In the Gesetze, 24 and 25, I have entered more fully 
 into the varieties of strabismus ; but the views there stated, 
 in 23 to 28, upon squinting and upon ocular defects in 
 general, require modification and correction in many respects, 
 through the investigations laid down in the present work, and 
 in consequence of a more strict separation of the convergence 
 from the application. 
 
 THE MICROSCOPIC EXAMINATION OF THE RETINA. 
 By Max Schultze. 
 
 The already-mentioned researches of Max Schultze upon the 
 retina are not only of great anatomical interest, but they afford 
 also many points of support to my theory of vision, and these 
 I purpose to set forth. 
 
 The bacillary layer of the human eye, and of all examined 
 eyes of vertebrata, consists of cylindrical rods, Fig. 66, and of 
 flask-shaped cones, Fig. 67. The distribution of these rods and 
 
MICROSCOPIC EXAMINATION OF THE RETINA. 
 
 155 
 
 cones forms, in every zone encircling the pole of the eye, a 
 regularly designed figure, and varies with the place of the 
 zone, so that the rods preponderate towards the ora serrata, 
 
 and the cones towards the pole, while the human eye, in its 
 yellow spot, contains cones only. But the nearer they are to 
 the pole so much the more do the cones (while still having the 
 general anatomical characters of cones) lose their flask shape, 
 and approach the cylindrical outline of the rods. 
 
 The eyes of animals living in a very subdued light, as of 
 owls, bats, moles, rats, rabbits, and fish, contain almost exclu- 
 sively rods; and the eyes of animals living in bright light, as 
 birds, lizards, and serpents (which, to use Schultze's expres- 
 sion, play in the sunshine) contain almost exclusively cones. 
 
 Schultze assumes that the rods and cones are nerve elements, 
 and the terminal portions of nervous fibres, as may be anato- 
 mically proved by the structure of their extremities that are 
 turned towards the retina, as at c. He assumes also, as I my- 
 self have done, that they are the perceptive organs of luminous 
 impressions, and upon the above described distribution of the 
 two kinds, and upon the exclusive presence sometimes of one, 
 sometimes of the other, he founds the conclusion that rods and 
 cones alike possess the faculty of receiving these impressions. 
 
 It is of great importance that Schultze confirms, in every 
 case, without exception, the discovery previously made by 
 Krause, that every rod or cone is sharply divided into two 
 
156 THEORY OF OCULAR DEFECTS. 
 
 portions by a perfectly plane surface, as at b. The outer portion, 
 a b, penetrates the choroidal pigment, and the inner portion, 
 b c, is in contact with the membrana limitans externa. The 
 homogeneous substance of the outer portion is distinguished 
 from the substance of the inner portion by a higher degree of 
 refractive power and also by chemical peculiarities, shown by 
 treating it with reagents. Under the action of acids the pre- 
 viously homogeneous substance of the inner portion is coagu- 
 lated into a finely granular mass, while that of the outer portion 
 is not affected. The outer portion of a rod is cylindrical, that 
 of a cone is conical. 
 
 On these facts Schultze rests the hypothesis that the plane b 
 is the surface on which the luminous rays are concentrated in 
 exact accommodation, and on which they form, by reflexion, 
 the retinal image, and that through this reflexion the visual 
 impression exists. To this hypothesis I cannot assent. Re- 
 flexion, as the turning back of luminous rays from a smooth 
 surface, is, on the one hand, a purely geometrical conception, 
 which has no bearing upon a physiological nervous function. 
 On the other hand, if we suggest a functional participation of 
 the reflecting surface, then reflexion, as an action perpendicular 
 to this surface, is not adapted to produce an affection with 
 which a sensation of the system of ether oscillations represented 
 by the cone of rays must be combined. Upon sensibility to 
 these oscillations, and especially upon sensibility to the propor- 
 tion between the lateral and the axial components of the lumi- 
 nous affections, depends, in my opinion, the recognition of the 
 relations of objects in space, a recognition for which accurate 
 accommodation is required. In order to feel this system of vi- 
 brations accurately and completely, the nervous substance must 
 itself be regularly and thoroughly agitated by the luminous 
 cone. This is accomplished by the penetration of the luminous 
 cone into the outer portion of the retinal rod or cone, and the 
 different refractive power of the outer portion is an important 
 aid to the sensation. For a body to be sensitive to a system 
 of luminous oscillations, it must in the first place be accessible 
 
MICBOSCOPIC EXAMINATION OF THE RETINA. 157 
 
 to light, that is, it must be transparent, as the outer portion of 
 the rod or cone actually is. A transparent substance, however, 
 by reason of its very transparency, would only be feebly sti- 
 mulated by rays of light passing through it, but this stimula- 
 tion would be increased if the substance had a different refrac- 
 tive power from its limiting medium (see the Exposition of the 
 Action of Light in 2 of the Phys. Opt.). The stimulus 
 would be still more increased if the portion b a consisted of a 
 substance the refractive power of which varied from b to a, al- 
 though the investigations of Schultze do not show whether this 
 is actually the case. The insertion of the outer portion a b 
 into a layer of dark pigment would have chiefly the effect of 
 preventing the passage of the penetrating rays of light into 
 neighbouring rods, and thus of preventing disturbing luminous 
 action in those rods. Hence this pigment does not form the 
 foil of a reflecting surface, but surrounds the whole outer por- 
 tion of the rod or cone as an enclosing sheath, which, in many 
 animals assumes the form of a tuft of hairs increasing in a 
 direction forwards from the choroid. 
 
 Schultze has observed that the conical outer portion, a 5, of 
 every cone, is more or less wrinkled together. I conclude 
 from this that such portion possesses, during life, the power of 
 lengthening and shortening. Such a power, according to my 
 theory of the method of recognising distance, would be of 
 great importance, if we assume that the division at b forms the 
 elementary nervous unit, and that this division, by prolonga- 
 tion or by shortening of the outer portion, is so arranged that 
 the point of the luminous cone falls accurately upon the point 
 a of this portion (see especially the 19th section of the 
 Gesetze). 
 
 If, in this prolongation and shortening of the outer portion, 
 the plane b retains nearly its fixed position, there must in 
 strong accommodation be an extension, and in weaker accom- 
 modation a drawing together, of the outer portion. The 
 wrinkling when accommodation is relaxed, and therefore after 
 death, would thus be a natural consequence. 
 
158 THEOEY OF OCULAR DEFECTS. 
 
 The variation in the size of the nervous unit, in consequence 
 of accommodation, that is required by my theory, need not, 
 since we are concerned with sensations only, be an external or 
 mechanical variation of the section b. The size in question 
 can be changed, as regards our sensation, only by the varying 
 internal tension of different states of accommodation ; although 
 a mechanical change of the thickness of the cone at b would 
 conduce to accuracy of sensation. According to the above 
 cited section of the Gesetze, a diminution of size at b must 
 be associated with increased accommodation, or with a pro- 
 longation of the outer portion of the cone ; and an increase of 
 size at b with diminished accommodation or shortening of the 
 outer portion ; since such changes would correspond with the 
 unchanged quantity of the contents of the outer portion. For 
 such changes, moreover, the flask-like shape of the cones 
 would be favourable, if we assume that the point a of the 
 outer portion remains fixed in the choroidal pigment ; and the 
 plane b moves forward towards the vitreous body in stronger 
 accommodation, and backwards away from it in weaker ac- 
 commodation. 
 
 The latter hypothesis, however, of the mobility of the plane 
 b, requires the surrender of the former one, of the mobility of 
 the point a of the conical outer portion ; since we cannot 
 assume that both extremities of this portion are movable. It 
 is possible, and even most probable, that the point a is firmly 
 fixed in the pigment ; and that therefore the place in which 
 the bundle of rays must be concentrated in exact accommoda- 
 tion remains nearly the same ; while the outer portion a b is 
 .lengthened in consequence of accommodation effort by the 
 movement forwards and the contraction of the surface &, and 
 shortened as the accommodation relaxes by the movement 
 backwards and the expansion of the surface. 
 
 Whether the cylindrical rods, which show no marks of 
 wrinkling, possess the same extensibility, cannot at present be 
 determined. If they are not extensible and contractile, this 
 would not imply that the recognition of distance by them is 
 
MICROSCOPIC EXAMINATION OF THE RETINA. 159 
 
 impossible, but only that it is incomplete. In fact, the recog- 
 nition of the distance of objects seen indirectly, and the rays 
 from which fall upon peripheral parts of the retina, where 
 rods predominate, is incomplete ; and it is very probable that 
 animals living in twilight, and possessing chiefly rods, do not 
 require so exact a recognition of distance as those living in a 
 bright light, and possessing chiefly cones. Some of the latter 
 class, as birds, are called upon to scan a considerable range of 
 space. 
 
 What then is the significance of the difference between the 
 rods and the cones ? In order to reply to this question it is 
 necessary further to describe the peculiarities that have been 
 brought to light by Schultze's investigations. 
 
 The difference in question has reference partly to the union 
 of the outer portion with the retina, partly to the union of the 
 inner portion with the choroid. The outer portion of a rod 
 penetrates deeply into the choroidal pigment, and appears to 
 pass almost through it ; while the pointed outer portion of a 
 cone does not penetrate so deeply, and appears completely to 
 terminate in the pigment. The inner portion of a rod sends a 
 single, extremely fine, nervous filament c d, through the mem- 
 brana limitans to the intergranular layer d d, upon which this 
 filament terminates by a small enlargement d. The inner 
 portion of a cone sends a larger nervous cord, consisting of a 
 bundle of single fibres, to the intergranular layer ; and this 
 bundle there divides into its constituent filaments, which pene- 
 trate into the tissue of the layer. 
 
 From the compound structure of the cones, Schultze forms 
 the conjecture that they chiefly subserve sensations of colour ; 
 and that the rods subserve sensations of proportionate intensity 
 and position in space. The former conjecture rests upon the 
 hypothesis of Young and Helmholtz, that particular filaments 
 are sensitive to red light, others to yellow, and others again to 
 blue. Such an hypothesis certainly requires bundles of fibres 
 for the recognition of different colours ; since a single fibre 
 would react to a single colour only. 
 
160 THEORY OF OCULAR DEFECTS. 
 
 The above-named hypothesis stands upon the same level 
 with the hitherto generally diffused doctrine of the identity of 
 corresponding retinal elements, so far that they both ascribe to 
 the elements in question a mysterious quality not explicable by 
 a natural law. In the ' law of identity ' this quality consists 
 of the faculty of single vision ; and in the hypothesis of Young 
 and Helmholtz, it consists of the faculty of reacting only on a 
 single colour, and certainly in a perfectly inexplicable manner. 
 The latter hypothesis is, indeed, nothing else than the substitu- 
 tion, for the problem of colour sensation, of a problem still 
 more difficult ; since it does not come a step nearer to explain 
 the material process of sensation as regards that colour for 
 which a particular filament is supposed to be adapted, and 
 gives, therefore, no help as regards the principal phe- 
 nomenon ; while it adds to the chief problem two new 
 problems, by requiring an explanation of the specific dif- 
 ferences of the red, yellow, and blue colour processes, and an 
 explanation also of the strange circumstance that these three 
 colours should each require a specific process, while the 
 intermediate colours requires only a simple mingling of the 
 specific processes. 
 
 After the microscopic researches of Schultze, however, we 
 need no longer seek for the refutation of this hypothesis in 
 principles ; it is best refuted by the anatomy of the eye. For 
 while the cones consist of bundles of filaments, the rods are 
 single, and could, therefore, according to the hypothesis, per- 
 ceive only a single colour, and that a simple one, as red ; but 
 could never perceive a mixture, such as white. But beyond 
 the area of the yellow spot the rods greatly predominate. 
 Notwithstanding this, we recognise all colours in every part 
 of the field of vision ; and hence it follows irresistibly that a 
 single nervous filament may conduct each and every colour 
 process as well as a compound one ; and that the hypothesis 
 referred to is devoid of foundation. 
 
 In 65 of my Phys. Opt. I have ascribed the colour 
 process to a chemical or material oscillation, which may take 
 
MICROSCOPIC EXAMINATION OF THE RETINA. 161 
 
 place in every single nerve filament. This view seems hitherto 
 to have escaped observation ; but I do not doubt that it will 
 compel assent, and that the conception of chemical oscillation 
 will play a great part in natural science. The observations of 
 Schultze afford much support to this conception. For the 
 inner portion, b c, of a rod or cone is shown, by its becoming 
 a granular mass when treated by acids, to consist of a chemically 
 compound substance, such as my notion of the nature of the 
 colour process requires. This circumstance suffices to lead 
 me to the assumption that the inner portion of each rod or 
 cone is the nerve element by which the feeling of colour is 
 received. 
 
 According to the foregoing, either the rod or the cone is a 
 percipient organ for the recognition of relations in space and 
 of colour. The former faculty is seated in the outer portion, 
 a b, and depends upon the relative elasticities of the ether and 
 the nerve substance ; and the latter is seated in the inner 
 portion, b c, and depends upon the chemical constitution of its 
 substance, which is irritated by the rapid vibrations of the 
 light. (Phys. Opt., 65.) 
 
 A third function, which Schultze. following Aubert, has 
 called the sense of light, has no existence. This assumed 
 sense of light was to measure the intensity of the oscillations. 
 But for this purpose no other organ would be required than 
 those which habitually react on these oscillations ; and their 
 intensity would be felt through the sensitiveness or irritability 
 of these organs. 
 
 We have thus arrived at no explanation of the different 
 terminations of the outer and inner portions of the rods and 
 the cones, as regards space and colour. Upon this I venture 
 to express the following conjecture. I conceive that the blood 
 and the nervous substance, like the copper and zinc of a bat- 
 tery, furnish the two poles necessary to afford a physiological 
 nervous current. The deeper penetration of the rods towards 
 the bloodvessels of the choroid appears to me to produce a 
 more intimate contact with these vessels. The more intimate 
 
 M 
 
162 THEORY OF OCTJLAIl DEFECTS. 
 
 the contact, the more sensitive or excitable would be the ner- 
 vous element. An increased sensitiveness would appear to be 
 desirable for the elements more remote from the yellow spot, 
 since these receive a weaker and less complete luminous im- 
 pression ; and hence in such situations the rods prevail. For 
 the same reason animals living in a feeble light are furnished 
 chiefly with rods. In the yellow spot, the irritability given by 
 a copious blood supply might be hurtful and perplexing ; since 
 the external stimulation by light is more intense, and an ex- 
 cess of blood might diminish the accuracy of perception. On 
 that account the less deeply penetrating cones prevail in this 
 portion, and cones alone are found in animals living in bright 
 sunshine ; in which also these cones sometimes contain turbid 
 corpuscles, adapted to diminish the external light. 
 
 Furthermore the less firm insertion of the cone points into 
 the choroid would increase the mobility of the cones on every 
 side. This is more important for the elements lying in the 
 yellow spot, and for the percipient elements of animals living 
 in sunshine generally, on account of the need of more exact 
 accommodation, than for elements that are feebly irradiated, 
 and therefore less adapted for accurate vision. For such the 
 exact accommodation is less essential, and may be sacrificed in 
 order to obtain the firmness of position that is on other grounds 
 desirable. 
 
 As regards the continuation of the inner portions in nervous 
 filaments, the intergranular layer d c?, into which these filaments 
 penetrate, plays an important part. This layer forms a fibrous 
 network, to which the single rod fibres are loosely applied by 
 a bulbous termination ; while the bundles of cone fibres, after 
 their division into single filaments, appear to extend into it. 
 From this intergranular layer other single fibres penetrate in- 
 wards, towards the vitreous body, as far as the so-called mole- 
 cular layer. The latter forms a second inextricable network 
 of nerve fibres, unequally thicker than the intergranular layer. 
 From the molecular layer spring, lastly, the fibres of the optic 
 nerve layer, which proceed to the papilla. This layer consists of 
 
MICEOSCOPIC EXAMINATION OF THE RETINA. 163 
 
 thicker cords, and of fine filaments. The thick cords are pro- 
 duced by several fibres of the molecular layer, uniting in a 
 ganglion cell, and proceeding forwards from this cell as a 
 compound fibre ; but the fine filaments appear to pass through 
 no ganglion cell, and to proceed directly from the molecular 
 layer to the optic nerve layer as single threads. 
 
 The two fibrous webs, of the intergranular layer and of the 
 molecular layer, next attract our attention. By these webs is 
 accomplished a manifold contact of all the nerve filaments, and 
 also the formation of two concentric organs, by which the ner- 
 vous function discharged by a single perceptive element may 
 communicate certain actions over the whole of the retinal cup, 
 to the ora serrata, and to all points of this circumference. In 
 my earlier works I have already expressed the opinion that 
 every rod or cone tries to accommodate for itself. But such 
 an accommodation, on account of the spherical shape of the 
 eye, would not be practicable unless all parts of the spherical 
 mass, lens, vitreous, iris, and cornea, were affected by it. 
 Hence every element must stand with the rest in a state of 
 organic union ; and such a union may be accomplished by 
 the above-named two layers, in which the filaments entwine. 
 I further conjecture that the first or intergranular layer, the 
 more external, controls the changes of the pupil ; and that the 
 second, the thicker molecular layer, which is more internal or 
 nearer the vitreous, controls the communication for accommo- 
 dative purposes between the lens and the vitreous humour. 
 This conception would very well explain the account given by 
 Schultze of the development of the eye of the fowl. 
 
 The elements which are adapted for the most exact ac- 
 commodation, which would be injured in their function by too 
 much light, and hindered by too little, that is to say, the cones, 
 must hence require an especially extended contact with all the 
 filaments, and must be able to act in the most energetic pos- 
 sible way upon the pupil. We here obtain a plausible ground 
 of explanation for the division of each cone fibre into several 
 
 M2 
 
164 THEORY OF OCULAR DEFECTS. 
 
 filaments in the intergranular layer. The single filaments of 
 the rods would act far less energetically upon the pupil. 
 
 The union between the intergranular and the molecular 
 layers appears to be effected by independent filaments, which 
 are not direct continuations of the filaments of either the rods 
 or the cones. But the cones would, notwithstanding, act upon 
 the molecular layer, and hence upon the accommodation of the 
 lens and the vitreous body, more energetically than the rods, 
 in consequence of possessing more numerous points of contact 
 with the intergranular layer. 
 
 The division, interruption, contact, reunion, and intertwining 
 of all the nervous filaments in the different layers, appears to 
 me to give important support to my doctrine of all nervous 
 sensations : which is that these sensations occur not from a 
 bare external excitation, but also from a subjective perception 
 in the peripheral nerve elements ; and that therefore the form 
 or way of nervous conduction to the brain may be immaterial, 
 since the peripheral nervous system is itself nothing else than 
 an extension of the brain over the organism. 
 
 According to this conception, the corresponding or identical 
 nervous fibres of the two eyes need not necessarily be in direct 
 union with one another. The account of the corresponding 
 retinal points, according to my view of the law of identity, 
 as given in 21 of the Gesetze, retains, nevertheless, its 
 entire applicability. By the reciprocal contact of the fibres 
 an organic union of all fibres is produced. In this way an 
 influence upon the fibres of one eye by those of the other is 
 already rendered possible. An external luminous stimulus, 
 which falls upon a fibre of one eye, produces certain optical 
 forces or tensions, determined by the direction of the nervous 
 point concerned. From the participation of the brain, as the 
 common central organ, a reflex action upon the other eye may 
 be occasioned, and would be concentrated upon the correspond- 
 ing point therein ; thus leading to normal binocular vision, as 
 much as when the second eye is equally impressed by the ex- 
 ternal light, and the affection of the corresponding point is 
 
MICROSCOPIC EXAMINATION OF THE KETINA. 165 
 
 produced. This reaction between the two eyes may also very 
 suitably illustrate the positive character of the nervous current 
 in the fibre of one eye, and the negative character of the 
 current in the fibre of the corresponding point of the other, as 
 my theory of opponency requires. Furthermore, the unfulfil- 
 ment of the law of identity would entail the impulse to the 
 movement of the fibres of the corresponding point for the 
 attainment of single vision. 
 
 If we think of the system of all the sensory fibres of the 
 visual organs, which have their peripheral extremities in the 
 rods and cones, and their common central organ in the brain, 
 and which on their way to the brain unite, divide, and touch, 
 at many points, and if, for the sake of a clearer conception, we 
 identify the nervous with a galvanic current, it will be mani- 
 fest that such a current, when passing between a given rod and 
 the central point in the brain, must take a perfectly determi- 
 nate course through the maze of fibres, namely, the course of 
 the smallest resistance to its conduction. We may regard this 
 course as the direct nervous union between the rod and the 
 brain ; and by means of this union two corresponding rods 
 form an anatomically united whole. 
 
 As regards the movement and the nutation of the rods and 
 cones, required for single vision and for fixation, the observa- 
 tions of Schultze do not enable us to construct these movements; 
 but that they are possible may be inferred from many con- 
 siderations. 
 
 Among these may first be remarked the wrinkling of the 
 outer portions of the cones, which renders their prolongation 
 and contraction probable. But the possibility of such a change 
 of length alone implies the possibility of a change of place of 
 the moveable part. If the point a remains fast in the pigment, 
 and the base b moves ; the latter cannot simply move forwards 
 and backwards, but must also be displaced along the periphery, 
 so that the plane of the important section b may obtain a dif- 
 ferent inclination to the radius of the eye. The membrana 
 limitans would offer no impediment to such a movement, 
 
160 
 
 THEORY OF OCULAR DEFECTS. 
 
 Fig. 68. 
 
 since it is not a closed membrane, but is perforated by all the 
 cones. 
 
 The flask shape of the cones seems to allow of their expan- 
 sion and contraction, and thus permits the movement of the 
 cones, and of the rods leaning against them. 
 
 The connecting tissue found by Schultze in all the retinal 
 layers, and affording an arboriform support to all the filaments 
 and cells, appears to render possible a movement and change 
 of shape of all nerve filaments without mechanical strain upon 
 them. 
 
 The Fig. 12 in Plate III. of Schultze's drawings, shows that 
 the rods themselves can bend. This figure is taken from a 
 diseased eye, in which the bending has become permanent ; 
 
 but it allows us to conclude that 
 bending is possible in a sound eye 
 also. 
 
 The form of the several ele- 
 ments and layers at the yellow 
 spot is also of special importance. 
 According to Fig. 68 the cones 
 in the yellow spot, where they are 
 nearly cylindrical, are so much 
 longer than elsewhere that they 
 cause the membrana limitans ex- 
 terna, c, to curve inwards towards 
 the vitreous body. And since all 
 the other layers of the retina 
 are here very thin, so that the 
 membrana limitans interna curves 
 outwards towards the choroid, 
 and forms the fovea, the cone 
 
 filaments c e?, which unite the cones with the intergranular 
 layer d, assume a very oblique direction with regard to the 
 radius of the eye. It is plain that the elongation of the cones, 
 and the oblique direction of their fibres, must both consider- 
 ably facilitate the movement and nutation of the cones. The 
 
MICROSCOPIC EXAMINATION OF THE RETINA. 167 
 
 same arrangement, with the inclination of the rods outwards, 
 would also render a movement outwards easily possible, such 
 as is required for the normal visual act. For this purpose we 
 must consider the point of the outer portion of the rod or cone 
 to be firmly fixed in the choroidal pigment, and the inner base 
 to be moveable. Such a movement would then be accom- 
 plished by the contractility of the retina, especially of its inter- 
 granular layer. 
 
 Moreover the observation that the outer portion of a rod or 
 cone may be easily separated from the inner portion, in the 
 plane >, by mechanical injury, has a significance for my theory. 
 Since, according to this theory, myopia occasions an abnormal 
 movement and inclination of the rods, it is unquestionable that 
 a stronger tendency to separation at b would be occasioned by 
 such movement if the outer portion were fixed in the choroidal 
 pigment and only the inner portion were moveable. Hence 
 the theory explains the frequent detachment of the retina in 
 myopia; and, inversely, the frequency of the detachment 
 confirms the theory. 
 
 If, in conclusion, we reflect that the degree of movement 
 and of inclination necessary to an accurate visual impression, 
 depends upon the distance of the object ; and that the visual 
 act is not directed to a single point in space, but to an extended 
 object, or to a great part of the visual field, which, at the same 
 time, presents very different degrees of depth, it will be mani- 
 fest that complete vision of a large object will not require an 
 equal movement and nutation of all rods ; but that there must 
 rather be a power for different points of the retina to apply 
 differently. If the bacillary layer were furnished in all parts 
 with similar and easily moveable elements, especially with 
 cones, a tendency to movement manifested in any part, as, for 
 instance, at the yellow spot, and in any direction, would be 
 propagated over the whole layer, and would carry with it the 
 elements which for accurate vision were required to assume a 
 weaker or a stronger movement, or to remain at rest. I con- 
 ceive that the division of the elements of the bacillary layer 
 
168 THEORY OF OCULAR DEFECTS. 
 
 into rods and cones, is an important aid to the regulation of 
 the application over the whole retina. It would seem as if 
 the cones, which are easily moveable, and liable through their 
 strong fibres to be directly moved from the intergranular 
 layer, would find in the rods, which are firmly inserted into 
 the pigment, and incapable of being moved from the inter- 
 granular layer, a passive impediment which certainly, on 
 account of its elasticity, would yield somewhat to the lateral 
 push of the cones ; but yet which would prevent the unchecked 
 propagation of this push over large portions of the retina. 
 
 Hence the difference in the structure of the two kinds of 
 perceptive elements, and also the regular distribution of the 
 two kinds in the retina, may be reasonably explained with 
 reference to the regulation of the function of application. 
 
 PRACTICAL RULES FOR THE DETERMINATION OF 
 PROPER SPECTACLES. 
 
 The determination of the spectacles necessary for any defec- 
 tive eye presupposes a knowledge of its optical condition 
 depending upon its structure, and of its functional condition 
 depending upon its innervation. The discovery of these con- 
 ditions is effected by suitable physical experiments. It is 
 manifest that these experiments may be conducted after 
 different methods, and by the aid of different kinds of appa- 
 ratus ; and that the practical value of each procedure will not 
 entirely depend upon the objective peculiarities of the method 
 itself, but also upon the subjective habitude and skill of the 
 observer ; so that many ophthalmic surgeons may maintain 
 the superior value of certain tests and accessories, on no other 
 ground than that furnished to themselves by their own custom 
 and experience. In this book, which has for its chief object 
 to lay down the fundamental principles of a rational spectacle 
 
PRACTICAL RULES FOR SPECTACLES. 169 
 
 construction, it would lead us too far if we attempted to describe, 
 and to follow into the smallest details, all the possible methods 
 of examination ; but it may be useful to commence the descrip- 
 tion of selected methods by some general observations, intended 
 practically to elucidate the theories set forth in the earlier 
 chapters, to remove, as much as possible, certain sources of 
 error, and to assist the observer, who wishes to apply these 
 theories, in the conduct of a rational method of inquiry, 
 
 1. Anomalies of Refraction and Accommodation. 
 
 We will, in the first place, consider the Anomalies of Ke- 
 fraction and Accommodation. 
 
 The anomalies of refraction require for their comprehension 
 the determination of the refraction of the eye in the state of 
 repose of the accommodation. This refraction may be such 
 as to constitute emmetropia, hypometropia, or hypermetropia. 
 The distinction rests upon a knowledge of the visual distance, 
 or, more definitely, of the accommodation distance a ; that is, 
 in the knowledge of the distance of the luminous point, the 
 rays from which are accurately united upon the retina, when 
 the accommodation of the eye is at rest. In the determination 
 of this distance a there are, therefore, two conditions to be 
 fulfilled : first, the accurate concentration of the rays upon 
 the retina ; and, secondly, that the accommodation of the eye 
 should be at rest. Of the fulfilment of the first condition, the 
 clearness or sharpness of the visual impression of the patient 
 affords a direct, and, generally speaking, a sufficient criterion. 
 When, however, the patient can form no adequate judgment 
 upon such a point, it is evident that only those methods of in- 
 vestigation can be used which are independent of his testimony; 
 for example, the examination with the ophthalmoscope. As 
 regards the second condition, the sensation which attends upon 
 an accommodation effort, and which, in the experiment, should 
 sink to nil, affords some evidence. This evidence, however, 
 is in general uncertain, and hence the point must be deter- 
 
170 THEORY OF OCULAR DEFECTS. 
 
 mined without the direct co-operation of the patient. To 
 paralyse the accommodation by means of atropine affords to 
 the observer the most trustworthy proof of complete repose of 
 the function. In examination with the ophthalmoscope, since 
 the eye of the patient does not fix any object, we may generally 
 assume that its accommodation is nearly relaxed ; but this 
 assumption is not entirely trustworthy, since the stimulus of 
 the light reflected from the mirror, or some object in the 
 patient's field of vision, or even an involuntary effort, may 
 produce some exertion of the accommodation. Every method 
 of investigation, whatever it may be, must be so arranged that 
 the question whether the accommodation be in repose shall 
 not be left to the decision of the patient, but to that of the 
 observer. 
 
 That for the determination of the state of the refraction 
 two conditions must be fulfilled, is a requirement of the 
 theory. The physiological possibility of such fulfilment is 
 quietly presupposed. It may be questioned, however, whether 
 the co-existence of the two conditions is always attainable in 
 practice ; and whether physiological laws do not, under certain 
 circumstances, render such a co-existence impossible. We 
 may imagine that the exercise of the conscious visual function, 
 that is, the co-operation of the intellectual power of perception 
 with the sensational power of seeing may, in many persons, 
 never permit complete repose of the accommodation to occur ; 
 for that, in such persons, when they see with consciousness, 
 and therefore clearly, a certain degree of action of the ac- 
 commodation is always exerted, as a result of the induced in- 
 fluence of the brain upon the visual apparatus ; and that when 
 their accommodation is completely relaxed, they no longer 
 possess the power of seeing with consciousness, and therefore 
 clearly. The phenomena of hypermetropia certainly render 
 it probable that in many persons such an influence as between 
 seeing and perception is exerted by the brain ; and it is es- 
 sential not to lose sight of the probability. 
 
 Where such an influence prevails, and we assume that it is 
 
PRACTICAL RULES FOR SPECTACLES. 171 
 
 more or less present in all persons, the eye, in repose of its 
 accommodation, does not possess the faculty of conscious or 
 clear vision. If, therefore, it were desired to experiment in 
 this state of repose, any proceeding would be futile which 
 required the testimony of the patient with regard to the 
 clearness of the visual impression, in the degree in which the 
 apparatus of accommodation had not its full sensitiveness. For 
 such a purpose the use of atropine is an indispensable accessory. 
 By it the accommodation is paralysed ; and therefore the 
 patient is enabled to receive a clear visual image in complete 
 quiescence of the function. Moreover, the examination with 
 the ophthalmoscope, during which, as the eye is not engaged in 
 seeing, the accommodation may be relaxed, will in many cases 
 answer the purpose with sufficient accuracy. 
 
 It must be asked, however, for the special purpose of 
 determining the required spectacles, which is the more im- 
 portant, the absolute repose of the accommodation, in which 
 the eye can no longer see consciously, or the state of minimal 
 accommodation which is necessary to clear vision. For many 
 purposes a complete knowledge of the eye, and therefore a 
 knowledge of both these conditions, is to be desired ; but for 
 the purpose of prescribing spectacles it appears to me that the 
 condition of absolute repose is wholly unimportant (since, in 
 this condition the eye is unable to see) ; and that the exercise 
 of a certain minimal accommodation is a much more essential 
 requirement. Hence, in my opinion, the spectacles actually 
 required for the present time can only be determined under 
 the supposition that the eye can. exert no smaller degree 
 of accommodation than the presumed minimum. And, since 
 it is possible, and therefore probable, that this minimum, 
 when it is unusually large, may be gradually reduced by the 
 continued use of suitable glasses, so, after the lapse of an 
 uncertain time, the investigation may require to be repeated, 
 and fresh glasses to be prescribed, according to the results 
 then yielded. The knowledge of the refraction of the eye 
 in absolute repose of its accommodation is, therefore, for the 
 
172 THEORY OF OCULAtt DEFECTS. 
 
 immediate purpose, not an essential requirement ; and the 
 knowledge of the refraction in the state of minimal accom- 
 modation is much more important. 
 
 Since, for certain phenomena of hypermetropia, we have 
 generalized the customary expressions, manifest, latent, and 
 total, we may call the refraction of the state of minimal 
 accommodation the manifest refraction, and that of absolute 
 repose of the accommodation the total refraction. Further, we 
 may call the distance of the luminous point, the rays from 
 which are accurately united upon the retina in the state of 
 minimal accommodation, so that therefore the point is clearly 
 seen, the manifest accommodation distance-, and the distance 
 of the point the rays from which are united upon the retina 
 in absolute repose of the accommodation, and which therefore 
 cannot be clearly seen, the total accommodation distance. The 
 difference between the maximum and the minimum accom- 
 modation is the manifest accommodation range ; the difference 
 between the minimum and nil is the latent accommodation 
 range ; and the difference between the maximum and nil is 
 the total accommodation range. It is self-evident that the 
 total range is the sum of the manifest and of the latent. 
 
 Moreover it follows that, as well in hypometropia as in 
 hypermetropia, the manifest part is to be distinguished from 
 the latent part, and either of them from the total hypometropia 
 or hypermetropia ; since, for example, manifest hypometropia 
 
 of the degree , is the defect which corresponds to a posi- 
 
 a \ 
 
 tive manifest accommodation distance a l ; while total hypo- 
 metropia of the degree is the defect which corresponds to a 
 positive total accommodation distance a ; and between them 
 
 the degree of hypometropia is latent. 
 
 a a l 
 
 Manifest emmetropia is displayed by an eye which unites 
 parallel rays upon its retina in the state of minimal accom- 
 modation; and which therefore is the subject of total hyper- 
 
PRACTICAL RULES FOR SPECTACLES. 173 
 
 metropia. Again, an eye is totally emmetropic which unites 
 parallel rays upon its retina in absolute repose of its accom- 
 modation; and which therefore displays manifest hypome- 
 tropia. Since no eye in common use exhibits its total defect, 
 but only the manifest part, so every eye which appears to be 
 emmetropic is in a certain degree hypermetropic ; and the 
 actual or totally emmetropic eye appears to be hypometropic. 
 An apparently hypometropic eye is, totally, less hypometropic 
 than it seems, and may even actually be hypermetropic ; and 
 a totally hypometropic eye appears more hypometropic than 
 it is. An apparently hypermetropic eye is totally more 
 hypermetropic ; while a totally hypermetropic eye appears 
 less hypermetropic than it is, and may even appear to be 
 hypometropic. 
 
 The most simple means of determining the manifest accom- 
 modation distance a l is afforded by causing the patient to look 
 with the naked eye at an object at various distances. The 
 greatest distance at which the eye sees the object clearly and 
 sharply is the value of a v This procedure is only applicable 
 to hypometropes, who have a positive accommodation distance. 
 For manifest hypermetropes, who in minimal accommodation 
 can only unite convergent rays upon the retina, and who have, 
 therefore, a negative accommodation distance, the same method 
 may be made available by the employment of a concave 
 mirror, since the curvature of such a mirror, and the distance 
 of the actual object which the hypermetrope sees clearly re- 
 flected, permit of a conclusion as to the backwards situated or 
 negative distance of the apparent object. 
 
 In surgical practice this simple method is often actually 
 employed, by the aid of Snellen's or similar test objects. 
 There are, notwithstanding, many objections to it; among 
 others, the difficulty of arranging for observation, at great 
 distances, objects of suitable size and illumination, and of 
 measuring these distances. A trial by the aid of glass lenses, 
 and of objects lying within the moderate range of distance 
 afforded by a consulting room, is in most cases to be desired ; 
 
174 THEORY OF OCULAR DEFECTS. 
 
 particularly as it is equally applicable to hypometropia and 
 hypermetropia. 
 
 The method by furnishing the eye of the patient with 
 lenses may be conducted according to two different plans. 
 We may proceed either by causing a variable object to be 
 looked at through the same lens, or the same object through 
 various lenses. The first of these methods is in reality nothing 
 more than the already mentioned fixation procedure with the 
 naked eye ; since the case of the naked eye is only a special 
 form of the addition of a constant lens. While, therefore, 
 under certain circumstances, the constant lens with varying 
 objects may render good service, yet the method is in the 
 same way and degree incomplete as the trial with the naked 
 eye. 
 
 It is a more useful and convenient method, and applicable 
 in a far larger number of cases, to cause the eye to fix a 
 constant object, placed at a known distance, through different 
 lenses of known focal length. Such an object retains, not- 
 withstanding the change of lenses, a constant retinal image, 
 occasions therefore no considerable change in the acuteness of 
 vision, requires no measurement of distances, and can be used 
 as well for the investigation of hypermetropia as of hypo- 
 metropia. It is self-evident that, in certain cases and for 
 special purposes, a change of object may be desirable, either 
 as regards its size, shape, colour, or distance, in order ac- 
 curately to investigate all the peculiarities of an eye, or in 
 order to prescribe spectacles for some particular employment, 
 as for reading, or for out-door use. This, however, involves 
 no departure from the general principle of the method, the 
 fixing a constant object with varying lenses. 
 
 In general, even with similar eyes, it is necessary to examine 
 each alone, not both together, since the binocular visual act, by 
 the co-operation of the convergence and of the application, may 
 exert a considerable induced influence upon the accommoda- 
 tion, and may thus much disturb the state of the refraction. 
 
 To carry out the investigation it is necessary to possess a 
 
PRACTICAL RULES FOR SPECTACLES. 175 
 
 set of lenses from the most powerful concave to the most power- 
 ful convex. We consider these lenses as forming a regular 
 series, commencing at the left hand with the highest concave, 
 having a plane glass in the middle, and the highest convex at 
 the right. The focal length / of these lenses commences on the 
 left with the smallest negative value, and passes through that 
 of the plane glass, + oo , to the positive values, thus repre- 
 senting a double series of numbers decreasing from left to right, 
 as 10, 50, 100, +00, 100, 50, 10; and the inverted 
 values forming a simple increasing series of numbers, as 
 
 - iV - i - TW ' i5o. &> ro ; so that we can cal1 every 
 
 lens of the whole series, which is placed farther to the right 
 than another, the more powerful of the two. The object for 
 fixation may be placed at any desired distance x. When the 
 whole series of lenses, beginning with the concave of the 
 greatest focal distance have been tried, the lens of highest 
 power / (that is, the lens of the focal length / which lies 
 farthest towards the right hand end of the series of convex 
 lenses) through which the eye still sees the object clearly, is 
 to be selected. The apparent distance z of the object is then 
 determined (page 50) by the formula 
 
 m f 
 
 and it is plain that this value of z shows the manifest accom- 
 modation distance. For example, if the object be placed at a 
 distance x = 10 inches, and if it be found that / = 15 inches 
 is the most powerful convex lens through which the object is yet 
 
 clearly seen, we have - = =_; and therefore a 
 z 10 15 oO 
 
 hypometropic eye of the manifest accommodation distance of 30 
 inches. If f = 8 inches were the most powerful convex lens, 
 
 then = -- = - ; and the eye would therefore 
 
 be hypermetropic, with a manifest accommodation distance ot 
 40 inches. If it were found that/ = 50, the concave lens 
 
176 THEORY OF OCULAR DEFECTS. 
 
 lying nearest to the convex series, and therefore the weakest of 
 the concave lenses, was the strongest through which the object 
 
 could still be clearly seen, then _ = + = j 
 
 and the eye would be strongly hypometropic, with a manifest 
 accommodation distance of 8^- inches. 
 
 The procedure is of the simplest kind when the fixation ob- 
 ject is placed at infinite distance, or, what is sufficient for all 
 practical purposes, at very great distance. Then x = oo , and 
 
 consequently - = - or z = /, so that the manifest ac- 
 commodation distance z is equal to the negative value of the 
 focal length of the strongest lens through which the object can 
 still be clearly seen. It is self-evident that, according to the 
 acuteness of vision of the eye, an object that subtends a 
 suitable visual angle must be selected. 
 
 The accommodation distance, or the value of z, has been in- 
 dicated in the earlier chapters by a, and the lens/ by /, and 
 we have called this lens, I = a, the emmetropic lens, since 
 it renders the accommodation distance of the eye infinite, and 
 therefore entirely neutralizes the hypometropia or hypermetro- 
 pia. In this the total accommodation distance was referred to, 
 and we called the manifest accommodation distance a l ; the lens 
 f found by the method just described, l\ = a j is there- 
 fore that which renders the manifest accommodation distance 
 infinite, and it may be called the manifest emmetropic lens. 
 We have already observed that the manifest accommodation 
 distance is that which has to be decided for the selection of the 
 spectacles immediately required, and it is therefore very im- 
 portant that this distance should be ascertained, independently 
 of the latent, in every case. The total accommodation distance 
 a may be discovered in the same manner, when desired, by 
 first paralysing the accommodation by the action of atropine. 
 
 We may now proceed to consider the anomalies of accom- 
 modation. The function of accommodation, which may con- 
 stitute Europia, Bathopia, or Hypsopia, is determined by the 
 
PRACTICAL RULES FOR SPECTACLES. 177 
 
 positions of the natural near point H and the natural far point 
 c l (Fig. 38), or by the position and extent of the natural range 
 of accommodation R B, or by the position and extent of the 
 natural territory of accommodation H Cj. The accommodative 
 function of the eye corresponds to the curve K L of the hyper- 
 bola in Fig. 38 ; the territory of the accommodation is the dif- 
 ference of the abscissae, and the range of accommodation is the 
 difference of the ordinates, of the same portion of the curve. 
 
 The extent and position of the range of accommodation R B, 
 meaning by this as we do the difference between the maximum 
 and the minimum of the accommodation effort, is completely 
 independent of the hyperbolic curve of requirement. But since 
 when this curve varies, the segment K L of the curve, which 
 corresponds to the near and far points, and to the territory 
 of accommodation H C 1} also varies, the relation between the 
 range and the territory of the accommodation changes. In order 
 to establish a fixed relation between these two quantities and 
 to simplify the formula for calculating the range of accommoda- 
 tion, R B, from the distance between the ends H and c l of the 
 territory of accommodation, we have in Fig. 38 furnished the 
 eye with its emmetropic lens, so as to consider the horizontal 
 asymptote of the hyperbola as situated in the fundamental 
 axis A Cp When thus H and c l indicate the artificial near and 
 far points, or the near and far points for the eye furnished with 
 the lens, and A H is taken = n, and A c l = w, we have A B' = 
 
 , and A R = , and therefore, for the range of accommo- 
 n' w 
 
 dation, R B = . 
 n w 
 
 The distances n and w of the artificial near and far points 
 
 bring, therefore, by means of the values and , the 
 
 w n w 
 
 existing degree of stenopia, or of bathopia or hypsopia, to 
 immediate recognition. 
 
 This determination of the range of accommodation requires 
 the putting back the horizontal asymptote of the requirement 
 
178 THEORY OF OCULAR DEFECTS. 
 
 hyperbola to the basic axis of the system of co-ordinates, from 
 which the accommodation effort y is measured as a vertical 
 ordinate. For this putting back, the totally emmetropic lens 
 is required. If this has been discovered by the aid of atropine, 
 and thus the total accommodation distance a been made known, 
 
 the necessary elevation or depression by the distance can 
 
 be accomplished. But if the observer confines himself to ex- 
 perimenting upon an eye in its natural condition, a remains 
 unknown, and then only the manifest accommodation distance 
 a lt and the manifest emmetropic lens Z L = a l can be em- 
 ployed. By the employment of this lens the horizontal asym- 
 ptote is not replaced in the basic axis A c l5 but in the horizontal 
 R L, which lies higher than the basic axis by the unknown 
 value A R of the minimal accommodation. In fact, with this 
 lens, the manifest accommodation distance is rendered infinite. 
 
 We obtain w = oo , or = ; and for A H = n, the ordi- 
 w 
 
 nate R B, = , is equal to the range of accommodation. 
 n 
 
 When the apparatus of accommodation works in such a 
 manner, that the tension of the accommodation quickly sinks to 
 nil when the manifest accommodation distance is exceeded, the 
 facultative curve of accommodation will assume, beyond the 
 far* point, a form like that of the dotted line in Figs. 26, 29, or 
 34. When, however, as will probably most frequently be the 
 case, the eye in looking beyond its far point continues over a cer- 
 tain space to exert its minimal accommodation, the facultative 
 curve of the accommodation will cut the curve of requirement, 
 at the far point, in a horizontal line, as in Fig. 20 in the portion 
 L! L. The last supposition, namely, that in vision beyond the 
 far point the eye still exerts for a space the minimal accommo- 
 dation, requires, as a general supposition, to be considered in 
 the experiments. 
 
 It will easily be seen from the foregoing that, in this book, 
 the manifestness, the latency, and the totality, which have been 
 regarded by practical ophthalmologists as peculiarities of 
 
PRACTICAL RULES FOR SPECTACLES. 179 
 
 hypermetropia, are considered in a more general way, and as 
 appertaining to all ocular defects ; since they are referred to the 
 simple general principle of the maximum, the minimum, and 
 the nil of the accommodation. In Figs. 20 to 23, and in Fig. 
 38, R B is the manifest, A R is the latent, and A B is the total 
 accommodation distance. 
 
 In all cases, the operation with the manifest emmetropic lens 
 /! = a n is completely sufficient for the determination of the 
 spectacles then immediately necessary ; since the judgment 
 thereupon (as, in any visual act, the accommodation may be 
 assumed to be within its possible range from A R to A B), may 
 be as well founded upon the hyperbola which has its horizontal 
 asymptote in the line R L, as upon that which has its asymptote 
 in the line AC r 
 
 We have called the near and far points of the unaided eye 
 the natural, and the near and far points of the eye furnished 
 with a lens, the artificial. It is in many cases impossible to 
 investigate the unaided eye ; and we must generally, therefore, 
 conduct our experiments with the help of lenses. A lens / 
 (of the focal distance /) changes the natural into the artificial 
 near and far points in such a way that when n l and iv l are the 
 distances of the natural, and n and w the distances of the 
 artificial points, according to page 60, 
 
 1+1 
 
 1=1+1 
 
 w w l f 
 or inversely, 
 
 = l _ 
 n, n f 
 
 (2); 
 
 (3). 
 
 _ 
 
 w l w f 
 
 By the first two formulae (2) we find the artificial near and 
 far points for the lens/, when the natural points have been dis- 
 covered by direct experiment. By the next two formula? (3) 
 we find the natural points when the artificial, with the lens/, 
 
 N2 
 
180 THEORY OP OCULAR DEFECTS. 
 
 have been noted. Hence we can, from the one, easily arrive at 
 a conclusion about the other. Under the supposition that, as a 
 rule, all experiments must be conducted with the aid of a lens, 
 it will be found a matter of great convenience to use always 
 that lens which has already served to determine the anomaly 
 of refraction ; so that, therefore, its corresponding artificial 
 near and far points may establish the values of n and w. If 
 then it be desired to ascertain the natural near and far points, 
 or the values of n l and w ly the formulae (3) will suffice for 
 this purpose. For the selection of the necessary spectacles, 
 however, such a calculation is not needed ; and, for the for- 
 mulae of this book, it is sufficient to know the artificial near 
 and far points with the manifest emmetropic lens, since the 
 thus observed magnitudes n and w enter immediately into 
 these formulae. 
 
 As regards, first, the natural far point, or that point for the 
 fixation and clear vision of which the minimal accommoda- 
 tion is required, its distance, w ly is equal to the manifest ac- 
 commodation distance a v If, therefore, as in the foregoing, ^ 
 be the manifest emmetropic lens, we havewj = a^ l l , and 
 from the formulae (2), we find, for the distance of the artificial 
 far point, the already stated result, 
 
 = -f = 0, or w = oo ; 
 w M! Zj 
 
 that is to say, this lens renders the manifest accommodation 
 distance infinitely great, or entirely corrects the manifest 
 ametropia. 
 
 The distance n of the artificial near point for the eye fur- 
 nished with this lens, that is, of the point at which the eye, 
 through this lens, sees still clearly, but with the extreme 
 effort of accommodation, may be easily ascertained by simple 
 approximation of an object. If, however, instead of such 
 approximation, and direct measurement of the distance n of 
 the near point, it be preferred still to fix an infinitely distant 
 object, then the value of n may be determined by the aid of 
 lenses. For a concave lens I', added to the emmetropic lens 
 
PRACTICAL RULES FOR SPECTACLES. 181 
 
 /!= a l produces an apparent approximation of the object ; 
 since it forms an optical image of this object at the distance 
 V in front of the emmetropic lens. If therefore /' be the 
 strongest concave lens, or, in the whole series of concave and 
 convex lenses, the weakest lens, through which the eye, fur- 
 nished with the emmetropic lens, can see clearly an infinitely 
 distant object, then ^ n is the distance of the artificial 
 
 near point, and is the range of accommodation.* 
 n 
 
 If it be wished in these experiments to employ only a single 
 lens / 2 , which is certainly to be preferred; then, since this one 
 
 lens has to supply the two lenses ^ and T, r -=y +,-. , and, 
 
 / 2 /! / 
 
 consequently, 1= -1 = 1-1. 
 
 t 7L 62 ^J 
 
 If, therefore, / 2 be the weakest lens, in the combined series of 
 concave and convex lenses, with which the eye can still see 
 clearly an infinitely distant object, then the distance n gives 
 
 the artificial near point by the formula = , which 
 
 n / 1 / 2 
 
 shows directly the range of accommodation. 
 
 It may also be remarked that, with the manifest emmetropic 
 
 lens, every eye appears completely bathopic, since =0; 
 and any hypsopia present remains latent. The degree of the 
 actual stenopia is indicated by the value of - , which shows 
 
 * The words strongest and weakest in the text are exactly translated. 
 The power of a lens is expressed by -, , the reciprocal of its focal 
 
 length. The strongest lens, therefore, is that in which - is greatest or/ 
 
 least ; the weakest lens has /greatest. But, when speaking of a whole 
 series of lenses, convex and concave, the author calls that convex lens 
 
 the strongest which has + - the greatest ; and that concave lens the 
 weakest which has - greatest, TEANS. 
 
182 THEORY OF OCULAR DEFECTS. 
 
 also the range of accommodation. If has the normal value 
 
 n 
 
 of J or , then the eye is europic. 
 The formula 
 
 i-l+l-fl+LzJ) , (4), 
 
 /"" J U w J 
 
 from page 59, is manifestly applicable to a hyperbola that 
 has its horizontal asymptote in the line R L, when we make 
 I = /! the manifest emmetropic lens, n the distance of 
 
 the artificial near point for this lens, and therefore the 
 
 range of accommodation R B, and assume the distance w of the 
 artificial far point to be infinitely great, so that = 0. Hence 
 this formula assumes the simple aspect, 
 
 H+K- <*> 
 
 and serves to determine the lens /which enables the eye to see 
 objects at the given distance s, with the degree of accommoda- 
 tion effort indicated by i. 
 
 When a normal eye, having its near point at a distance of 
 three or four inches, reads by preference at a distance of nine 
 
 O A 
 
 or ten inches, it has i = - =. OS, or i = =. 0'44 of its ac- 
 
 i U t7 
 
 commodation power. Under the supposition that a certain part 
 of the power of accommodation remains latent, or that the eye 
 cannot relax its effort below a certain minimum, the overplus 
 R T of the middle effort above the minimal effort A R, forms a 
 yet somewhat smaller part of the manifest range of accommo- 
 dation R B, that is, of the difference between the maximum and 
 the minimum of the accommodation. For the purpose of 
 
 reading we may therefore take i to be nearly equal to - ; and, 
 
 o 
 
 when we make s=10 inches, the formula becomes, 
 1111 ,. 
 
PRACTICAL RULES FOR SPECTACLES. 183 
 
 The formula (5) serves for the determination of the lens/, 
 under the supposition that the eye should see at a given dis- 
 tance s, with a given accommodation effort i. The limits of the 
 range of vision, to which the eye may attain with this lens, 
 are found from the same formula, by making at one time s = 0, 
 and then s = 1. The first substitution, s = 0, gives the greatest 
 visual distance s l by the form 
 
 1 _ 1 .. 1 (7) 
 
 ^~~7 ~h 
 
 and the second substitution, 8=1, gives the smallest visual 
 distance, s 2 , by the form 
 
 (8) 
 
 It is worthy of note that the difference --- is always = - 
 
 s 2 s } n 1 
 
 which is also the state of refraction l { of the eye, with whatever 
 lens /it may be supplied. 
 
 If the lens/ be 'required not for a middle distance s, and a 
 middle accommodation t, but in order to give vision at a given 
 maximal distance s l with a minimal accommodation effort 
 f = 0, then from formula (7) we have 
 
 >-r+7 <>' 
 
 / s l l l 
 
 and, if the lens /be determined under the supposition that the 
 eye shall see with the maximal accommodation effort i = 1, in 
 the given minimal distance s 2 , then, from equation (8), 
 
 1=1 + 1 * . (10). 
 
 / a J, n 
 
 If it be desired to know what part i of its manifest range of 
 accommodation an eye, supplied with any lens /, exerts in 
 looking at any distance s, or with what accommodation effort 
 it sees at the distance s, the equation (5) furnishes the value 
 of i. It gives, 
 
184 THEORY OF OCULAR DEFECTS. 
 
 The naked eye, which is equivalent to a lens / = oo , sees 
 therefore at the distance s with the effort 
 
 From the foregoing it appears that, for the determination of 
 the anomalies of refraction, only a single value need be experi- 
 mentally ascertained ; namely, the manifest emmetropic lens 
 /i ; that is, the strongest lens through which the eye is able to 
 see clearly an infinitely distant object. For the anomalies of 
 accommodation, likewise, only a single observation is required ; 
 namely, the distance n of the artificial near point for this 
 manifest emmetropic lens. 
 
 When these two magnitudes, ^ and w, are found, we have all 
 the data required to calculate, by equation (5), the power of the 
 lensy that will give clear vision at the given distance s, with 
 the prescribed accommodation effort i ; or, by equation (9), the 
 lens that will enable the eye to see at the given extreme distance 
 Si ; or, by equation (10), the lens that will give clear vision at 
 the given smallest distance S 2 . The quantity ^ indicates 
 the manifest visual distance, therefore the degree of manifest 
 
 ametropia, and the quantity - indicates the manifest range 
 
 H/ 
 
 of accommodation, or the degree of stenopia from which the 
 eye suffers. 
 
 If, for example, we find that an eye sees clearly in the dis- 
 tance with the concave lens ^= 30, and that with this lens 
 it sees also at a near point of which the distance n = 5 inches, 
 then, because l l = 30, the eye is hypometropic, and because 
 
 = -- , it is stenopic. The lens with which this eye will see 
 n 5 
 
 at a distance 5 = 9 inches, with an effort i equal to one-third 
 of its accommodation, is, according to equation (5), 
 
PRACTICAL RULES FOR SPECTACLES. 185 
 
 11111 
 
 / 9 30 3x5 yO' 
 therefore the concave lens / required is = -. If the eye re- 
 
 yu 
 
 quires vision at the maximal distance of ^ = 240 inches, then, 
 by equation (9), 
 
 != JL .!: ,JL --L nearly 
 / 240 30 240 34 *' 
 
 and therefore the concave lens _ is required. If it be de- 
 sired to see with the maximal accommodation effort at the 
 distance s 2 = 3 inches, then, from equation (10), 
 
 I = 1 _ JL_ _i=- JL 
 /~3 30" "5 10' 
 
 and a cpnvex lens/= 10 would be required. 
 
 If the eye be furnished with the convex lens/= 20, then it 
 will see, at a distance s = 8 inches, according to equation (11), 
 with the effort, 
 
 = 5f! JL _M = JL 
 
 V8" 30" 20/24' 
 
 At the same distance, according to equation (12), the naked eye 
 would see with the effort, 
 
 30 y 24 
 
 If it were found by experiment that an eye could see clearly 
 in distance with a convex lens / } = 30, and that, when furnished 
 with this lens, its near point was at a distance n = 5 inches, 
 such an eye, because l\ = 30, would be hypermetropic, and, 
 
 because = , it would be stenopic. Vision at s = 9 inches, 
 n 5 
 
 with t = of its accommodation, would require, by equa- 
 tion (5), 
 
186 THEORY OF OCULAR DEFECTS. 
 
 a convex lens / = 13 ; vision at the maximal distance 
 s^ == 240 inches, by equation (9), 
 
 11131 
 
 a convex lens / = 27 ; and vision, at the minimal distance 
 s 2 = 3 inches, by equation (10), 
 
 7 = ~3 + "30 ~~5 = (r 
 
 a convex lens/= 6. 
 
 If the same eye were furnished with a concave lens/ = 24, 
 it would see, by equation (11), at the distance s = 40 inches, 
 with the effort, 
 
 1 
 
 The naked eye, by equation (12), would see at the same dis- 
 tance with the effort, 
 
 It has already been shown that the distance n of the artifi- 
 cial near point may be deduced from observation of the weakest 
 lens, / 2 , through which the eye is still able to see clearly an 
 infinitely distant object. Hence, the anomaly of refraction is 
 determined by the strongest lens 1 19 and the anomaly of ac- 
 commodation by the weakest lens Z 2 , through which the eye 
 can see clearly at distance. The quantity Zj expresses the 
 manifest visual distance, that is, the degree of manifest ame- 
 
 tropia ; and the quantity = _ _ , the manifest range of 
 
 HI fcj / 2 
 
 accommodation, or the degree of stenopia, from which the eye 
 suffers. By means of the two lenses, l } and Z 2 , the five for- 
 mulas, (5), (9), (10), (11), and (12), assume the valuable forms, 
 
PRACTICAL RULES FOR SPECTACLES. 187 
 
 7 = i + i ' ' ' '< 15 >' 
 
 1 + L- _L 
 
 z - = A_^i__/ t g (16)> 
 
 (17), 
 
 which will be at once understood when it is observed that, in 
 Fig. 38, , = U, and 1 - ,' = 1?. 
 
 R B R B 
 
 If, for example, l\ = 15 were the strongest, and 1 2 = 15 the 
 weakest lens, with which an eye could see clearly in distance, 
 such an eye, because l^ = 15, is hypermetropic ; and, 
 
 because I =-L_ !_= J, - ( *} = A = 1 it is 
 r* /! / 2 15 V 15y 15 ?l' 
 
 strongly stenopic. 
 
 In order that it may see at nine inches distance, with an 
 accommodation effort i = ^, by equation (13), 
 
 1 _1 2 1 
 
 _ 
 9 3x15 3xT5 15~7i' 
 
 a convex lens/ = 7-^ will be required. 
 
 In order to see without effort at a maximal distance s l = 240 
 inches, by equation 14, 
 
 a convex lens/ = 14 will be required. 
 
 In order to see with the full effort of accommodation at a 
 minimal distance s l = 3 inches, by equation (15), 
 
188 THEORY OF OCULAR DEFECTS. 
 
 1 L 
 
 _ 
 
 T5~~T5 
 
 a convex lens / = 3 J must be employed. 
 
 If this eye be supplied with a concave lens / = 30, it 
 will see, by equation (16), at a distance s = 100 inches, with 
 the effort 
 
 i=l!> ( 1 ' 1 ' l N 33 
 
 2 " VlOO + 15 ' 30 J 40 
 
 Without a lens, at the same distance, by equation (17), it 
 would see with the effort 
 
 30 . 15 / 40 
 
 If Z L = 10 were the strongest, and Z 2 = 30 the weakest lens, 
 with which an eye could still see clearly in distance, such an 
 eye, because l l = 10, is hypermetropic. Its range of 
 
 accommodation = = = - It is there- 
 
 n Zi Z 2 10 30 15 
 
 fore highly stenopic. 
 
 If /! = 30 were the strongest, and Z 2 = 10 the 
 weakest lens, then the eye, because Z = 30, is hypome- 
 
 tropic. It has the range of accommodation = - 
 
 Q^ ( - ) = --= ; and is therefore stenopic in the same 
 oO \ 10 / lo 
 
 degree as the preceding. 
 
 We may next observe that, when it is not desired to ascer- 
 tain all the peculiarities of the eye, but only to select the 
 necessary spectacles for a given distance s, this may be done 
 immediately by the following method. 
 
 The object being placed at the given distance s, ascertain 
 by trials the strongest and the weakest lens, I' and Z 7 , through 
 which it can still be clearly seen in monocular vision. Then 
 choose the required lens by the formula 
 
PRACTICAL RULES FOR SPECTACLES. 189 
 
 in which i may have a value between J and . The smaller 
 the difference between the two lenses I' and I", the more im- 
 portant is it to make z = ^, and therefore 
 
 It is manifest that, in the latter cases it is often practicable 
 to determine the lens /, that produces the smallest visual strain 
 at the given distance, by direct experiments. 
 
 If it were desired, for example, to discover a suitable lens/", 
 for reading a particular writing at a given distance s, nothing 
 could be more simple and satisfactory than such experi- 
 ments. 
 
 Lastly, it should be remarked that, when, in these observa- 
 tions, no infinitely distant object of sufficient size and illu- 
 mination is attainable, so that some object at a known finite 
 distance x is employed, the trials can be conducted in the 
 following manner. We ascertain the strongest and the weakest 
 lens, Xj and X 2 through which the eye still sees this object 
 
 clearly. Since the lens would apparently displace the object 
 
 to infinite distance, the lens \ l produces precisely the same 
 effect as the two lenses x and Zj and therefore 
 
 = +, whence 
 
 Xj X li 
 
 I --_L JL 
 
 /! Xj X 9 
 
 I __ _! _ !_ 
 
 / 2 X 2 X ' 
 
 If we substitute these values for and - , in the formulas 
 
 tj 1 2 
 
 (13) to (15), we shall obtain the equations from which the lens 
 / is to be determined for the several conditions. For example, 
 the equation (13), for determining the lens with which the eye 
 will see at the given distance s, with the effort t, would become 
 
190 THEORY OF OCULAR DEFECTS. 
 
 f 
 
 f 8 x \ l A 
 
 When, however, the object is distant at least twenty feet, = 
 240 inches, - is less than - , and therefore so small that it 
 
 may be disregarded, and the distance treated as infinite, unless 
 the most extreme accuracy of calculation be desired. 
 
 2. Anomalies of Direction and Motility. 
 
 It has already been shown that the object of Convergence is 
 fixation, or direct vision with the poles of the eyeballs ; and 
 that the object of application is single vision, or the production 
 of similar apparent visual directions. It has also been shown 
 that the defects of convergence, which constitute the anomalies 
 of direction and motility, are rationally treated not by optical, 
 but by surgical means ; and that the defects of application, on 
 the contrary, which constitute the anomalies of position and 
 nutation, are to be relieved by prismatic spectacles. But as, 
 notwithstanding, the prismatic spectacles are also applied to 
 diminish defects of convergence, it must be borne in mind 
 that they can only completely fulfil this purpose when they do 
 not come into conflict with the application. And, since the 
 present object is to determine the prism required for any defect 
 of convergence, it is well to simplify the examination by the 
 following preliminary assumptions : 
 
 a. That the two eyes are alike in their refraction and 
 
 accommodation. 
 
 b. That the accommodation is not modified either by the 
 
 convergence or the application. 
 
 c. That the application is undiminished and unimpeded. 
 
 All fundamental principles, as regards manifestation, latency, 
 and totality, which have been laid down under Refraction, 
 apply also to ocular direction. There is a minimum of mo- 
 
PRACTICAL RULES FOR SPECTACLES. 191 
 
 tility, below which the action of convergence, or the motility 
 in conscious fixation, does not descend. The motility lying 
 between this minimum and the state of absolute rest remains 
 latent ; and only that manifests itself which lies between the 
 maximum and the minimum, and constitutes the range, or, 
 when we consider the distance of the corresponding far and 
 near points of the convergence, the territory of convergence. 
 If we possessed a means of paralysing the motility, as atropine 
 paralyses the accommodation, we should then be in a position 
 to study the state of rest of the convergence. In the absence 
 of such means, it is possible that we might attain to such a 
 state of rest by the suppression of all visual function ; but we 
 abstain from the attempt, because it is not necessary for the 
 practical determination of the spectacles actually required. 
 For this purpose we remain at the stand-point assumed with 
 regard to the lens required for the accommodation. 
 
 If the defective convergence of the two eyes were unequal, 
 each eye would certainly require its especial prism. If, how- 
 ever, the sum of the two prisms be divided equally between 
 the two eyes, there will be produced only a false projection. 
 If we overlook this, or if we have it in view to correct the 
 false projection by a measured division of the prisms, we may 
 always operate (in examining) with equal prisms before the 
 two eyes; and this is desirable in order to facilitate the 
 experiments. 
 
 For conducting these experiments a series of negative and 
 positive prismatic spectacles are required. At the extreme 
 left of the series is the strongest negative pair, with bases 
 turned outwards ; and which, therefore, deflect the rays out- 
 wards, and, apparently, displace the object inwards. At the 
 extreme right of the series is the strongest positive pair, with 
 bases turned inwards ; and which, therefore deflect the rays 
 inwards, and apparently displace the object outwards. The 
 left hand (strongest negative) spectacles, we call the weakest 
 of the whole series ; and the right hand (strongest positive) 
 the strongest. We define each prism, not directly by its 
 
392 THEORY OF OCULAR DEFECTS. 
 
 angle of total deviation ii>, but by it's total proportion of de- 
 viation hj which is the goniometric tangent of this angle, or 
 = tan $ ; and which, when d is the distance between the 
 central points of the eyes, and g is tne convergence distance, 
 that is, the distance of the point of convergence of two parallel 
 rays when refracted by the prisms, has the value h = tan ^ = 
 
 : g being positive for positive, and negative for negative 
 
 prisms. We also often use the convergence distance g to 
 denominate the corresponding prismatic spectacles. In the 
 supposed connected series, the values of g form, therefore, a 
 series of numbers, as 10, 50, 100, + 00, 100, 50, 10; 
 
 and the values of - , which varies as the proportion of devia- 
 
 9 
 tion h, form a steadily increasing series of numbers, as 
 
 ~T\F> ""lAfJ ~~TW> 0> TOTT> Al T"V 
 
 For every degree of accommodation the eye is permitted a 
 certain play of the convergence, and, therefore, a certain 
 relative minimum and maximum of convergence, correspond- 
 ing to the relative range of convergence. If g' indicate the 
 strongest, and g" the weakest prismatic spectacles through 
 which a pair of eyes can see a given object clearly and singly ; 
 so that the eyes, in using the strongest prisms, exert their 
 relative minimum, and in using the weakest exert their rela- 
 tive maximum of convergence, it follows, from considerations 
 like those already stated with regard to accommodation, Jhat 
 
 111 
 
 = _ -- will be the relative range of convergence for 
 m g' g" 
 
 the given degree of accommodation. 
 
 Ifj indicate that portion of the relative range of convergence 
 with which the pair of eyes exerts the given degree of ac- 
 commodation most pleasantly, then the formula 
 
 will give the prismatic spectacles required ; and this, by 
 making^;' = ^, will be reduced to the arithmetic mean 
 
PRACTICAL RULES FOR SPECTACLES. 193 
 
 9 
 
 If, therefore, it be desired to determine the prismatic spec- 
 tacles that should be combined with given spherical spectacles, 
 the most rational method is to furnish the eyes with the 
 spherical spectacles, and then, while an object placed at the 
 distance for which the spherical spectacles are adapted is being 
 regarded, to determine the strongest and the weakest prismatic 
 spectacles, g' and g" , with the addition of which the object can 
 still be* seen clearly and singly ; in order to determine by 
 equation (18) the prisms necessary to be combined with the 
 spherical glasses : prisms of which the proportion of deviation 
 would be 
 
 When the necessary spherical glasses have not been fixed 
 upon, or when it is intended afterwards to prescribe these for 
 different conditions ; so that it is not practicable to work at 
 once with these glasses, the determination of the prismatic 
 spectacles is of a more general kind ; since it then furnishes 
 an index of the condition of the faculty of convergence of the 
 /pair of eyes. Under such circumstances the following method 
 may be adopted. 
 
 The eyes being first of all furnished with their manifest 
 emmetropic spectacles /,, an infinitely distant object is used as 
 a point of fixation. The eye is thus placed in the condition 
 of minimal accommodation. In this condition must be deter- 
 mined the strongest and the weakest prismatic spectacles, g' 
 and g" 9 with which the object can be seen clearly and ^ingly. 
 Then g^ shows the absolute manifest convergence distance, 
 
 or - the minimum of the ratio of convergence (the tan- 
 
 t/ 
 
 gent of half the angle of convergence of the two ocular axes 
 in the minimal convergence effort), and characterizes the pair 
 of eyes as hypogonic or as hypergonic, according as gj is 
 
194 THEORY OF OCULAR DEFECTS. 
 
 positive or negative. The prismatic spectacles adapted for 
 the state of minimal accommodation will then be 
 
 or, if it be desired to express them by the proportion of 
 deviation, 
 
 The pair of eyes must next be furnished with the weakest 
 spherical spectacles, Z 2 , with which each single eye in its 
 maximal accommodation can clearly see the infinitely distant 
 object. In this condition must again be determined the 
 strongest and the weakest prismatic spectacles, # 2 ' and g 2 ff , 
 through which the object can still be seen clearly and singly. 
 The latter, g%', corresponds to the maximum of convergence 
 action, so far as this can be exerted with the given accommo- 
 dation. We may therefore say, 
 
 or, expressed by the proportion of deviation, 
 
 i==*''-*" 
 
 as the absolute manifest range of convergence, according to 
 the extent of which the degree of stenogonia is to be estimated. 
 The prismatic spectacles needed for the maximal accommoda- 
 tion would be, 
 
 1 I-,/ , 3 /22V 
 
 T*=~5r + W 
 
 or, by proportion of deviation, 
 
 We have expressed by the letter i that part of the range of 
 accommodation which is employed in a visual act. According 
 to the formulae (20) and (22), in the accommodation i = the 
 prism g, and in the accommodation f = 1 the prism # 2 , is to 
 
PRACTICAL RULES FOR SPECTACLES. 195 
 
 be applied. Hence we may deduce the result that it is reason- 
 able, in the accommodation i, to use the prism determined by 
 
 -=+- (23); 
 
 9 9i 9* 
 
 of which the proportion of accommodation is 
 fc = (l 2) h l + M 2 . 
 
 Hence, therefore, with the spherical spectacles / (which, by 
 equation (5) or (13), confer upon the pair of eyes the power 
 to see at the distance s, and with the accommodation effort t), 
 we must combine the prismatic spectacles g, found by equation 
 (23), and having the proportion of deviation h. 
 
 With the spherical glasses (9) or (14), which are to be used 
 at the maximal distance s t , must be combined the prismatic 
 glasses #, which are found by equation (23) for i = 0, and 
 
 therefore with = , or g = g l ; that is, the prismatic glasses 
 
 9 9\ 
 found by equation (20). 
 
 With the spherical glasses (10) or (15), which are to be 
 used at the minimal distance s 2 ? must be combined the 
 prismatic glasses g lt given by equation (23) when i = 1, and 
 
 therefore = , or g = g% ; that is, the prismatic glasses 
 
 9 9i 
 found by equation (22). 
 
 We may yet observe that, following these conclusions, the 
 strongest and the weakest prisms, g' and g" , which can be over- 
 come in the accommodation i, will correspond respectively to 
 the formulae 
 
 11** / 0/1 N 
 
 -+-' (24) ' 
 
 and that the most suitable prism #, which is determined by 
 equation (23), will correspond equally to the formula given in 
 equations (20) and (22), and contained in equation (18), 
 
 o 2 
 
196 THEORY OF OCULAR DEFECTS. 
 
 when we substitute for g' and g" the values taken from equa- 
 tions (24) and (25) ; since, then, the right sides of the two 
 formulae (23) and (26) become identical. Furthermore, it 
 follows, from equation (24) and (25) that the relative range of 
 convergence for the accommodation f has the value, 
 
 In the preceding formulas, which serve for the determination 
 of the prismatic spectacles g, necessary to be combined with 
 the spherical spectacles /, we have introduced a quantity j, 
 which shows the proportion in which it is desirable to employ 
 the relative range of convergence for a given accommodation. 
 The average value ofj can only be deduced from the function 
 of normal eyes in actual use ; that is, by observing in what 
 degree a pair of normal eyes, in constant accommodation, can 
 exert or relax convergence, short of the production of denned 
 double images. In the absence of certain data, we may 
 assume j = -|, which would lead us to take the arithmetic 
 mean between the strongest and the weakest prism which the 
 eye in a given accommodation, was able to bear. We have no 
 observations upon the play of the convergence in constant 
 accommodation, or upon the relative range of convergence ; 
 but only upon the play of accommodation in constant con- 
 vergence, or upon the relative range of accommodation. The 
 latter may be easily instituted by the help of common spherical 
 spectacles, since the change of such spectacles during vision of 
 one and the same object changes the accommodation only, and 
 leaves the convergence unaltered. We may assume that, as a 
 general rule, applicable to all near and remote objects, the 
 effort of accommodation which a pair of eyes is able to exert 
 continuously, without strain, in a given convergence, is about 
 the mean between the greatest and least effort possible in this 
 convergence. Proceeding from this assumption, we will en- 
 
PRACTICAL EULES FOR SPECTACLES. 197 
 
 deavour to arrive at a conclusion upon the value of j, or upon 
 the greatest and least convergence effort which the pair of 
 eyes is capable of making under unchanged accommodation. 
 
 According to the foregoing formula we have the pair of eyes 
 with the accommodation effort t, bearing a strongest prism g', 
 and a weakest g". If we suppose these eyes to assume a 
 greater accommodation / 2 , then the strongest prism applicable 
 would be, by equation (24), 
 
 1 , _*2_ 
 
 ' 
 
 ffi' 9J 
 
 If we further suppose the pair of eyes to assume the smaller 
 accommodation z l7 then the weakest prism applicable would 
 be, by equation (25), 
 
 ffi" " 9*"' 
 
 Assuming also, that the above-named two prisms are together 
 equal to the prism #, with which the eye sees without strain 
 in the accommodation i lt it will be obvious thatt'j and 2 indi- 
 cate the weakest and the strongest accommodation that the pair 
 of eyes are capable of assuming in the constant convergence 
 brought about by the prismatic spectacles g. According to the 
 above-mentioned experiments, therefore, we should obtain as 
 a general rule, nearly, 
 
 i 2 -i = i-i l or z = i(z' 1 +* 2 ); 
 
 and, in order, for the present, to keep the calculation free 
 from the hypothesis that 2 i = i i l we may assume, 
 generally, that the ratio k exists between 2 i and i j\, and 
 that we have 
 
 The first two conditional equations are now 
 
 Li *? , _* 2 _I 
 ffi' ' 9f = ~~ i 
 
198 THEORY OF OCULAR DEFECTS. 
 
 1 1^ _Zj J_. 
 
 n " ~^~ a " - ~n'' t 
 
 9i 9^ 9 
 
 and from these it follows that 
 
 1 1 
 
 ffi ~ i* f 
 
 If we insert these values of i l and t' 2 in the third conditional 
 equation i* 2 * = Tc (i z^), and substitute for its value in 
 
 equation (26), substituting also for and their values in 
 
 / / 
 
 equations (24) and (25) we obtain the highly interesting result 
 that the quantity z, which exhibits the assumed accommodation 
 effort, wholly disappears : since the equation may be divided 
 by a factor which contains only the quantity i. The remain- 
 ing equation then affords forj the solution 
 
 Tc 
 j = ~ -J- -j- 
 
 + #L! _ 2" . . . (28). 
 
 This formula teaches, that the most suitable portion of the 
 range of convergence is wholly independent of the special value 
 i of the accommodation ; and that j indicates a proportion that 
 is of one and the same constant value for every degree of 
 accommodation. This value of j is found by the four prisms 
 ff\i $21 9\'i #2" ; anc * especially by the range of convergence 
 
 ~ ; , which corresponds to the two strongest prisms for 
 
 ffl 92 
 
 the least and greatest accommodation, and by the range of con- 
 
PRACTICAL RULES FOR SPECTACLES. 199 
 
 vergence / , which corresponds to the two weakest 
 
 prisms for the least and greatest accommodation. The ratio 
 of these two ranges does not, on an average, deviate much 
 from the value 1 ; and, if we call it approximatively = 1, 
 we obtain 
 
 and, when we substitute for k its average value 1, we obtain 
 
 /=* 
 
 We are thus led to the very noteworthy result that, while 
 it appears reasonable, in a given convergence, to exert half of 
 the manifest relative range of accommodation, it is also reason- 
 able, as a consequence of this disposition, in a given accom- 
 modation to exert half of the manifest relative range of 
 convergence. 
 
 By the value j = -J, all the foregoing formulae are simplified ; 
 and the process for determining the prismatic spectacles g will 
 be the following : 
 
 Determine the strongest and the weakest prismatic spectacles, 
 gi and g^', with which the eyes, furnished with the strongest 
 spherical spectacles / 1? can yet see an infinitely distant object 
 clearly and singly. Determine also the strongest and weakest 
 prismatic spectacles, g 2 ' and g 2 ", with which the eyes, fur- 
 nished with the weakest spherical spectacles 1 2 , can yet see the 
 same object clearly and singly. The prismatic spectacles g, 
 proper to be combined with the spherical spectacles/ which 
 are adapted for use at the distance s, and with the accom- 
 modation effort i, and are found by equation (5) or (13), may 
 be found by equation (23) or (26) by the formula 
 
 1 I -i I 
 
 If the prismatic spectacles are to be combined with spherical 
 spectacles adapted for use at the maximal distance s lt as by 
 equation (9) or (14), when i is = 0, therefore 
 
203 THEOKY OF OCULAR DEFECTS. 
 
 If they are to be combined with spherical spectacles adapted 
 for the minimal distance S 2 > by equation (10) or (15), when 
 2 = 1 then 
 
 If the prisms be designated by their proportions of devia- 
 tion A/, h^, A 2 ', A 2 ", we have, instead of equation (30), 
 
 instead of equation (31), 
 
 *=!(*,' + *,") - (34); 
 
 and, instead of equation (32), 
 
 = !(*,' + *,") . . . (35). 
 
 It is also manifest that, in these formulae, either the angle 
 of deviation, or the refracting angle of the glass, may be 
 substituted for h at pleasure. 
 
 If, for example, it were found in a given case that the 
 strongest and weakest prismatic spectacles that could be com- 
 bined with the strongest spherical spectacles had proportions of 
 deviation /*/ = 0'015, /*/' = 0*012 ; and that the strongest 
 and weakest prismatic spectacles that could be combined with 
 the weakest spherical spectacles had proportions of deviation 
 A 2 ' = 0-045, A 2 " = ' 051 tnen > according to equation (23), 
 we should combine with the spherical spectacles /, adapted for 
 use with the accommodation effort z, prisms having the pro- 
 
 portion of deviation h = 1^-1 (0-015 - 0'012) + 1 (0-045 
 
 "2i 2i 
 
 - 0-051) = 0-0015(1 - - 0-OOSi. For i == 1 h = 
 0' 0007 5 ; that is, there would be required very weak negative 
 
PRACTICAL RULES FOR SPECTACLES. 201 
 
 prisms (with bases outwards), having a proportion of deviation 
 0-00075, or, by page 77, an angle of deviation 0-00075 . 57'3 
 = 0'043, in order to produce the most comfortable vision. 
 If i = J, then h = ; that is, no prisms would be required at 
 all. 
 
 3. Anomalies of Position and Nutation. 
 
 While the convergence has for its sole purpose fixation, or 
 direct vision with the poles of the eyeballs, the application 
 produces single vision, or vision of the object in similar direc- 
 tions. The two functions are thoroughly independent and 
 essentially distinct. Defects of convergence produce, there- 
 fore, phenomena wholly different from those of defects of 
 application. 
 
 In a case of failure of convergence, direct vision with both 
 eyes is impossible, and squinting is necessarily produced. In 
 defective fixation the object seen no longer appears the chief 
 object, or the concentration point of the attention and con- 
 sciousness ; but as something accessory or collateral which 
 only engages the attention secondarily, or to which it cannot 
 be completely devoted : the incomplete fixation is therefore 
 united with a feeling of imperfect satisfaction. This feeling 
 of imperfect satisfaction of the visual organs, from deficient 
 attractive power of the object, is at its greatest, and hence the 
 value of the object as such is at its least, when neither eye 
 fixes. Generally, in a defective convergence, the conflict 
 between the two eyes terminates by one of them fixing, while 
 the other deviates so much the more, and sees only indirectly, 
 that is, not with the polar bacillus, and consequently squints. 
 As a rule, the angle of convergence of the two ocular axes 
 does not correspond to the angle of convergence of the visual 
 lines, and rays of light impinge upon different parts of the two 
 retina?. With this squint, arising from a defect of convergence, 
 there is on principle absolutely no double vision united. If 
 the power of application be sufficient, the affected bacilli 
 
202 THEORY OF OCULAR DEFECTS. 
 
 assume, notwithstanding the squinting, the indirect vision, and 
 the difference of the nerve fibres, parallel positions ; so that the 
 object is seen in like directions, and therefore singly. Double 
 vision first occurs when the application is insufficient. 
 
 If there be a defect of application, the affected bacilli do not 
 assume a parallel position ; but exhibit the object in different 
 directions, and produce double vision. This double vision, 
 produced by insufficient application, is not in principle com- 
 bined with squinting or indirect vision : the eyes may per- 
 fectly fix the object, and therefore see with their polar bacilli ; 
 although, nevertheless, these bacilli, from not assuming a 
 parallel direction, may produce double images ; that is, two 
 fixed double images. The striving after single vision is, how- 
 ever, generally greater than that after fixation. Consequently, 
 as a rule, the former preponderates over the latter, and compels 
 the eyes to abandon their fixation, and to squint just so much 
 that the different bacilli (one polar and one eccentric) which 
 are impressed, have the power to become parallel, and thus 
 to afford a single image ; in so far as the faculty of con- 
 vergence may be able to accomplish the required degree of 
 squint. It is only when the convergence is insufficient for 
 this purpose, that there remains a permanent diplopia caused 
 by the defect of application, and only moderated by the con- 
 vergence effort. 
 
 We learn from this that we have, as a consequence of defects 
 of convergence, squinting with single vision ; and, as a conse- 
 quence of defects of application, double vision without squinting, 
 which may give way to squinting with single vision ; and also 
 that, from eithef defect, squinting with double vision may be 
 produced ; in so far as in defects of convergence the application, 
 and in defects of application the convergence, may be insuf- 
 ficient to render single vision possible by means of increased 
 squinting. 
 
 Squinting with double vision is, as a rule, the result of the 
 complication of a defect of convergence with a defect of appli- 
 cation, and results from the tendency to fixation and to single 
 
PRACTICAL RULES FOR SPECTACLES. 203 
 
 vision. Generally speaking, that is, when the attainment of 
 single vision is not too difficult, or when the images are not 
 too far apart, the struggle between fixation and single vision 
 will end in the victory of the latter tendency ; that is, the 
 eyes will so place themselves, that the affected bacilli, by the 
 employment of a corresponding degree of application, will be- 
 come parallel, and furnish a single visual impression ; although 
 by this the fixation may be sacrificed, or, more frequently, 
 limited to a single eye. In general the direction of the parallel 
 bacilli will deviate from the direction of the middle visual line, 
 drawn to the centre of the forehead from the object ; and the 
 object will therefore appear, although single, in a false posi- 
 tion, or will be falsely projected. This false projection will 
 vary, accordingly as one eye or the other takes up the fixation. 
 
 Since either of the two faculties will the more easily be in- 
 sufficient, the more it is already overtaxed by its use in normal 
 vision, and as this strain will vary with the distance of the 
 object, or with the accommodation, it may very well happen 
 that a pair of eyes may behave very differently in near and in 
 distant vision, and may in the one or the other case squint or 
 not squint, see singly or double, project falsely or truly. 
 
 When it is desired to obtain a clear and complete know- 
 ledge of the peculiarities of the visual organs in all their func- 
 tions, which depend partly upon formation and partly upon 
 innervation, a variety of suitable experiments must be insti- 
 tuted. Of these some will require optical aids, some, instru- 
 ments for geometrical measurement, some, mechanical appa- 
 ratus, such as an ocularium, and graphic delineations and 
 diagrams. Moreover, these experiments must be partly during 
 monocular, partly during binocular vision. The theory of 
 ocular defects already given in this volume, as well as the 
 theory of ocular movements contained in my earlier writings, 
 especially in the Gesetze des raumlichen Sehens, and in the 
 Physiologische Optik, furnish the necessary data for the con- 
 duct of these experiments. Whenever practical ophthalmology 
 succeeds in discovering two anaesthetic agents, which, as atropia 
 
204 THEORY OP OCULAR DEFECTS. 
 
 paralyses tjae accommodation, shall paralyse respectively the 
 convergence (the motility or muscular apparatus), and the 
 application (the nutation of the bacilli), the use of such agents 
 would be of great value in establishing the defects of ocular 
 direction during rest of the convergence, and the defects of 
 position during rest of the application. In the absence of such 
 means, the accurate determination of all sources of defect would 
 be too laborious ; and, even when they are eventually disco- 
 vered, would be a proceeding by no means without difficulty. 
 
 But when the sole object of the examination is to fix upon 
 the most appropriate spectacles, and since defects both of con- 
 vergence and of application are to be combated by the same 
 optical means the prismatic spectacles we may be content 
 to forego the accurate determination of every defect. The 
 method already described will be fully sufficient, when the 
 strongest and weakest prisms are selected with reference to 
 the obtaining of an adequate visual effect : that is, an effect 
 by which the requirements of single vision and of fixation are 
 both sufficiently provided for. 
 
 It should also be mentioned that the combination spectacles 
 f, g, selected in the manner described, since they are adapted 
 only to the manifest, and not to the total, anomaly of refrac- 
 tion, direction, and position, are specially fitted only for the 
 actually present time. It is possible that they may admit of im- 
 provement after a while ; because, under the influence of their 
 use, latent defect may possibly become gradually manifest. 
 
 It is even possible that the addition of the prisms, deter- 
 mined in the manner described, may render manifest a part of 
 the defect that had remained latent under the application of 
 centric lenses. This would show itself when the eyes were fur- 
 nished with the piano-prismatic spectacles g, and if then, in 
 binocular vision, the strongest spherical spectacles Z 1? with 
 which an infinitely distant object could still be clearly seen, 
 were ascertained by experiment. 
 
PRACTICAL RULES FOR SPECTACLES. 205 
 
 4. Spectacles for dissimilar eyes. 
 
 Let us first suppose that the two eyes of a pair are alike in 
 respect of their refraction and accommodation, although not 
 normal, and that they have dissimilar defects of convergence 
 or of application. However considerable the dissimilarity may 
 be, it will always be practicable, by the methods described 
 under section No. 2, to find prismatic spectacles, with two 
 similar glasses, which shall produce at pleasure one of two 
 effects : either that both eyes shall see with their polar bacilli, 
 and therefore fix their apparent objects, although they may 
 not blend them into a single one ; or that the two eyes shall 
 see only one object, although in a false direction, and not 
 with both alike by the polar bacilli. 
 
 Direct vision with both eyes at the expense of double images 
 can scarcely ever, or at least very rarely, be the goal of a ra- 
 tional selection of spectacles ; and in almost every case the 
 second result would be the one aimed at ; namely, so to select 
 the prismatic glasses that single vision should be the result. 
 The fulfilment of the first condition direct vision with both 
 eyes is not thus unconditionally excluded, and can in many 
 cases be accomplished together with the other, when the state 
 of the convergence does not oppose too great difficulties. If 
 these difficulties should be insurmountable and their being so 
 would soon become evident in the course of the above-de- 
 scribed experiments we must aim at the attainment of single 
 vision, which will, however, be subject to the two following 
 imperfections. In the first place, one eye will see directly 
 with the polar bacillus, and the other indirectly with a lateral 
 bacillus ; and both eyes will therefore periodically lose their 
 direct vision. Secondly, the object will appear in a wrong 
 direction, or falsely projected. The first of these two deficien- 
 cies, namely, the unilateral fixation, cannot be remedied by 
 any optical means, but only by an operation on the muscle. 
 The second, however, the false projection, may be corrected 
 
206 THEORY OF OCULAR DEFECTS. 
 
 by the use of dissimilar prisms. The determination of these 
 is very simple. 
 
 Assuming that the combination spectacles, selected accord- 
 ing to the rules already laid down, have the effect of apparently 
 displacing an obj ect, at a distance of a inches, b inches too far 
 to the left ; then for the restoration of the right projection, it 
 would be necessary to place before each eye a prism with its 
 base turned to the left, and having a proportion of deviation 
 
 q = This prism, according to the foregoing definition, 
 
 would be negative for the left eye, since it would displace the 
 object inwards, and positive for the right eye, since it would 
 displace the object outwards. Hence, the two equal prisms 
 /*, determined as in the last section, must be exchanged for 
 two unequal prisms, which must have, for the left eye and 
 the right eye, the proportion of deviation of h q, and h + q, 
 respectively. 
 
 If the object were projected towards the right, then the two 
 prisms for the left and for the right eye would be h + , and 
 h-q. 
 
 We see, therefore, that a pair of eyes, alike in their refrac- 
 tion and accommodation, may always be furnished with 
 prismatic spectacles, that will afford them a single and rightly 
 projected apparent object, although it be not fixed with both 
 eyes. It is important to observe that the apparent size of the 
 object will be the same, as a rule, for both eyes; and that, 
 therefore, both eyes can participate, without strain, in binocular 
 vision. The proof of this assertion rests upon my theory of 
 apparent size and distance. According to this theory, the 
 
 apparent size and distance is determined by the quotient - 
 
 (p. 127) in which (3 represents, under normal conditions, the 
 tangent of the angle of accommodation, and under abnormal 
 conditions, or generally, the exerted accommodation effort ; 
 and y, under normal conditions, the tangent of the angle of 
 application, and under abnormal conditions, or generally, the 
 
PRACTICAL RULES FOR SPECTACLES. 207 
 
 exerted application effort. If we consider that, when the ob- 
 ject appears single, that is, to both eyes in the same direction, 
 the two affected bacilli are parallel ; and that they, therefore, 
 make nearly equal angles with the visual lines, or with the 
 lines drawn from them through the central points of the eyes ; 
 therefore, the application effort, y, is also nearly the same for 
 both eyes, unless they suffer from anomalies of position of 
 different degrees. Moreover, it has been supposed that there 
 is similarity of refraction and accommodation, and hence the 
 accommodation effort, (3, will also be alike for both eyes. 
 
 Since, therefore, the quotient has for each eye an equal 
 
 value, and since the distance of the actual or apparent object 
 is the same for each, so it must appear to each of equal magni- 
 tude. 
 
 Only in those cases in which the state of the position in the 
 two eyes, and therefore the angle of nutation of the bacilli in 
 the state of rest of the application is different, would the 
 quantity y be different for the two, since it indicates the effort 
 of application, not the angle of application or nutation. The 
 object would then appear to one eye larger, in the same degree 
 as the effort of application, y, of that eye, was smaller than 
 the same effort of the other. 
 
 Equality of apparent size is an important, or even an indis- 
 pensable condition of binocular joint vision; so that to it correct 
 projection must undoubtedly be sacrificed. If it be found, 
 therefore, that the tests prescribed in the second section of this 
 chapter, which have for their object to discover prisms adapted 
 for single vision, and, whenever possible, for direct vision also, 
 have the effect of showing the object, single, indeed, but of un- 
 equal magnitude for the two eyes, then we must conclude that 
 there exists a defect of application, either unilateral or unequal 
 in the two eyes. In such a case we must no longer endeavour to 
 remove the defect of projection produced by the given combi- 
 nation spectacles, but the inequality of the apparent size of the 
 two visual impressions received through them. The necessary 
 
208 THEORY OF OCULAR DEFECTS. 
 
 method of procedure is based upon the following considera- 
 tions. 
 
 When, as in Fig. 58, the two polar or eccentric bacilli, b and 
 &', are impressed, it is necessary for single vision that b e and 
 b' e r , the directions of these bacilli, shall be parallel. If there 
 were an equal defect of position in both eyes, the equality of 
 the application effort, y, would determine the equality of the 
 angles of nutation, e b c and e' b' c' . If, however, either one 
 eye only be aklitic, or each eye in a different degree ; that is, 
 if the bacilli of one eye, in the state of rest of the application, 
 form, with its radii, different angles from the bacilli of the 
 other, then equality of application effort, y, manifestly implies 
 inequality of the two angles e b c and e f b f c'. This inequality 
 may be attained, in a pair of eyes that, on account of an uni- 
 lateral defect of application, strive after it that is, after the 
 exercise of an equal application effort in both eyes by a 
 unilateral prism, or by two prisms of half the strength, and 
 with similar directions of deviation. Experiment with a 
 single prism upon one eye only leads directly to the result. 
 
 If, for example, the object appears too large to the left eye, 
 and therefore requires to be diminished to this eye, then the 
 application effort y, and hence the angle of nutation, e b c, must 
 be increased, and consequently the object displaced, apparently, 
 inwards, or to the right. For this purpose, it is necessary to 
 furnish the left eye with a negative (base towards the left) 
 prism, q. This prism will show, experimentally, when the 
 eyes are first furnished with the combination spectacles de- 
 termined as already shown, and then the negative prism 
 q is placed before the left eye, or a positive (base also 
 towards the left) prism q is placed before the right eye, or a 
 
 negative i before the left, and a positive ? before the right, 
 2 2i 
 
 that the visual impressions of the two eyes will assume equal 
 magnitudes. If the impression upon the left eye required to 
 be enlarged, then prisms with their bases to the left would be 
 necessary. Generally, therefore, prisms with their bases to 
 
PRACTICAL RULES FOR SPECTACLES. 209 
 
 the left produce diminution of the image of the left eye, or en- 
 largement of the image of the right ; and prisms with their 
 bases to the right produce diminution of the image of the right 
 eye, or enlargement of the image of the left. We must observe, 
 however, that this effect is only to be produced with a pair of 
 eyes having the supposed defect of position : since it is only 
 such eyes that strive after the inequality of the nutation angle 
 in question. Normal eyes, in which equality of the angles of 
 nutation is a necessary condition, only undergo, by the use of 
 unilateral prisms, a common deviation of the position of both, 
 with undisturbed equality of the angles e b c and e' b' c' . 
 The final result would be, that with the lenses / must be 
 
 combined prisms, of the proportions of deviation h |, and 
 
 We now proceed to the case of two eyes of unequal anoma- 
 lies of refraction or accommodation, a case of especial practical 
 importance, on account of its frequent occurrence, and also 
 because the want of clearness of visual impression which it 
 produces is, of all defects, that to which the eye is most 
 sensitive. 
 
 If the total state of refraction of both eyes were alike, and 
 only their accommodation different, if, therefore, both eyes 
 were either emmetropic, or in equal degree hypometropic or 
 hypermetropic, but in different degrees stenopic, and, therefore, 
 of larger and smaller, higher and deeper lying, ranges of ac- 
 commodation, it is manifest that both eyes would be made 
 emmetropic by the same lenses. If, then, a part of the range 
 of accommodation of one eye was coincident with that of the 
 other, so that, for example, with the emmetropic lenses for 
 
 both eyes, for the far point were the greatest of the two 
 
 common values, and for the near point were the greatest of 
 
 the two common values, then complete binocular vision would 
 
 p 
 
210 THEORY OF OCULAR DEFECTS. 
 
 require spectacles with equal lenses which should place the 
 
 accommodation effort between and , ; since the portion of 
 
 w' n' 
 
 the range of accommodation , is common to both eyes. 
 
 n w 
 
 If it be desired, also, that this portion should be divided in a 
 proportion z, the accommodation must correspond to the ordi- 
 
 nate - , + f (- , *} = . + ^-*. Instead of equation 
 w' \n f w'J n' w' 
 
 (5), we have, therefore, to determine the spherical spectacles, 
 by which the pair of eyes will see at the distance s, and with 
 the accommodation effort i, by 
 
 , + LrJ) . (36). 
 
 If w' were equal for both eyes, then, instead of the total 
 emmetropic lens, the manifest emmetropic lens l } should be 
 
 employed, for which w' = oo , or , = 0. In the formula 
 
 1U 
 
 JL _ JL _|_ _* L . . . (37), 
 
 n' represents the distance of the most remote of the two near 
 points. Generally, the determination must be guided by the 
 weakest of the two eyes. 
 
 Since, in the determination of the glasses immediately 
 necessary in any case, we are not interested in the total, but 
 
 only in the manifest defect, the larger corresponds to the 
 
 smaller w', or to the eye with the smallest accommodation 
 distance. If the emmetropic spectacles, ^ are determined by 
 
 this eye, we may retain formula (36) ; since we make f = 0, 
 
 and use for . the above mentioned smallest value. 
 w 
 
 If it should happen that the minimum of accommodation 
 effort of one eye were greater than the maximum accommoda- 
 
PRACTICAL RULES FOR SPECTACLES. 211 
 
 tion effort of the other, then binocular vision, with similar 
 lenses, would be manifestly impossible. This condition is 
 characterized by the weaker of the two strongest lenses, / t 
 through which the left and the right eye can see in the distance, 
 being weaker than the stronger of the two weakest lenses / 2 . 
 
 Without further reference to special cases, we may proceed 
 to the most general kinds of defect of one single eye. In all, 
 the problem is, how to furnish each eye with an image that is 
 clear, single, fixed, and of equal size with that of the other ; 
 and, when all four conditions cannot be fulfilled, how to com- 
 bine the first, second, and fourth, with direct vision of one, 
 and indirect vision of the other eye. In the practical solution 
 of this problem it is necessary to realize, practically, the follow- 
 ing fundamental principles of the law of the scheme of vision. 
 
 In every individual eye, the three chief functions of ac- 
 commodation, convergence (as regards the frontal axis), and 
 application, so influence one another that a given exertion of 
 one of them is alone sufficient to produce a given exertion of 
 the others, when they are free to act. This relation between 
 the three efforts, which varies with the absolute degree of one 
 of them, is the most pleasant to the eye. Thus, for example, 
 a single eye (the other being covered) does not accommodate 
 for a very near object perfectly, if this object be placed straight 
 before the eye, so that convergence and application are not 
 called into play, but only if the object be in the frontal 
 axis so that convergence and application are exerted in 
 the same degree as accommodation. Moreover, this rela- 
 tion, on account of the primitive independence of the three 
 faculties, permits to each of them a certain range, between 
 definite minimal and maximal values. Any departure from 
 this favourable relation produces visual strain, which interferes 
 with the duration of visual effort. And just as the three chief 
 functions of the same eye influence one another in monocular 
 vision, so the functions of one eye influence those of the other, 
 when the two are acting together in binocular vision : in so 
 much that the visual apparatus is not comfortable unless the 
 
212 THEORY OF OCULAR DEFECTS. 
 
 two eyes are exerting their functions in a certain definite pro- 
 portion to each other (in similar eyes in an equal degree) . The 
 functional relation between the two eyes also permits of a 
 certain range, short of the production of binocular visual strain. 
 In general we may say that the minimal or maximal exertion 
 respectively of one function, in monocular or in binocular 
 vision, requires the minimal or maximal exertion of the others, 
 as the condition of the most perfect vision with the smallest 
 degree of strain. But as regards the possible play of each 
 single function, and the visual strain produced by a given de- 
 viation of one or the other of them from the normal relation, 
 there exist many different conditions, due to the complication 
 of the visual organs. These again may, in defective eyes, be 
 in many respects modified. Generally there is, between the 
 accommodation and the application of the same eye, between 
 the accommodation of the two eyes, and also between the ac- 
 commodation of the pair and their convergence, the strongest 
 apparent union. Moreover, in binocular vision, each function 
 modifies the others very powerfully, in such a way as to con- 
 duce to single vision. It is especially difficult to exert different 
 degrees of accommodation with the two eyes; and rest of 
 accommodation of one eye very powerfully maintains rest of 
 the other. On the other hand, convergence of one eye towards 
 the frontal axis has a very great range ; that is, it is variable 
 within wide limits without the production of much strain ; but 
 it guides very energetically the direction of the other eye. In 
 other words, the ocular direction plays only a subordinate part 
 in monocular, but a very important part in binocular vision. 
 
 Under the guidance of these general principles, we are led 
 to the following very simple and practical method for deter- 
 mining by experiment the spectacles necessary for a pair of 
 eyes, the individuals of which suffer from any given different, 
 but symmetrical, defects of the three chief functions. 
 
 We first select for each eye its manifest emmetropic lens /,, 
 that is, the strongest lens through which each eye alone can see 
 at infinite distance. Each of these lenses, in monocular vision, 
 
PRACTICAL RULES FOR SPECTACLES. 213 
 
 produces the minimal accommodation of the eye to which it 
 belongs. We may assume, therefore, that both eyes together, 
 with these two lenses, exert, in binocular vision, the same 
 minimal effort that they each exert singly ; or that both eyes, 
 through a pair of spectacles composed of these separate mani- 
 fest emmetropic lenses, receive together a visual impression 
 that is clear, although it may not be single, or composed of 
 images of equal size. This assumption is supported by expe- 
 rience in eyes with similar defects ; and must be general, 
 applying to dissimilar defects also. If, however, the minimal 
 accommodations in binocular vision should be unexpectedly 
 found to have different values from the monocular, this must 
 be established by the simultaneous application of two glasses. 
 
 We must next ascertain for each eye singly the weakest lens 
 Z 2 , with which it can still see clearly in the distance, exerting 
 its maximal accommodation. 
 
 The supposition that both the eyes in binocular vision exert, 
 not only their minimum, but also their maximum of accom- 
 modation together, and also that degree of accommodation in 
 the interval between the minimum and the maximum, which is 
 proportionately expressed by , leads to the conclusion that, for 
 binocular vision at the distance s, with the accommodation ?', 
 we may calculate a lens for the left eye, as well as for the 
 right, by formula (13) 
 
 7=T +1 ir+i- (38); 
 
 since we substitute, for ^ and 1 2 , their respective values for each 
 eye singly. 
 
 Eyes with dissimilar refraction must therefore be furnished 
 with spectacles containing dissimilar lenses. It has, however, 
 already been remarked that, in binocular vision, the propor- 
 tion between the lenses calculated by the foregoing formula 
 may possibly require to be somewhat varied, in order to arrive 
 at the most beneficial effect. To this end it is useful to know, 
 for each eye, the range of the lens /, which renders vision pos- 
 
214 
 
 THEORY OF OCULAR DEFECTS. 
 
 sible at the distance s. This range is given by the variations 
 of the magnitude z,from to 1, and therefore by the two formulas, 
 
 4r = ^+i '.'.' . ..'.-. . (39), 
 
 and-_JL 
 
 (40). 
 
 If these two formula3 are calculated for each eye, the strongest 
 and the weakest lens for each eye will indicate the range within 
 which it possibly may be used. If the weaker of the two 
 strongest lenses is stronger than the stronger of the two weakest 
 lenses, it will be possible to furnish the two eyes with similar 
 lenses for use within this range, and naturally with different 
 values of the coefficient t. 
 
 "When the two eyes are furnished with the unequal lenses f r 
 above indicated, it must not be expected that these eyes will 
 be able forthwith to see the object clearly, with binocular 
 vision, at the distance s. It is possible that the convergence 
 and the application may yet offer impediments, requiring to be 
 set aside by prisms. If, for example, we suppose that each eye 
 is furnished with its ernmetropic lens, and fixes an infinitely dis- 
 tant object, the lenses 
 have the effect of 
 placing the apparent 
 object at e and e' 
 (Fig. 69). In order 
 that the eyes may 
 fix the objects e and 
 e', they must be able, 
 in their minimal ac- 
 commodation, to as- 
 sume a parallel di- 
 rection, making the angle of convergence nil. In order that 
 they may see the objects e and e' in the same direction, and 
 therefore singly, they must be able to make the angle of 
 
 Fig. 69. 
 
PRACTICAL RULES FOR SPECTACLES. 215 
 
 application nil, so that the polar bacilli b and b' assume at 
 the same time a parallel position, and coincide in direction 
 with the radii of the eyeballs. Generally it would be required 
 that absolute rest of the convergence and the application should 
 be united with the minimal accommodation. This is usually 
 impossible, the minimal accommodation generally requiring 
 the minimal convergence and the minimal application. Hence 
 it follows that the eyes cannot assume the above-supposed 
 condition. They must, according to their organization, either 
 fix in the parallel direction, and then see double, on account 
 of the divergence of the bacilli b and b' ; or they must so turn 
 themselves, for the sake of single vision, that the affected ba- 
 cilli, being parallel, form the application angles e b c and e'b'c', 
 corresponding to the minimal application, so that there will be 
 single vision with squint; or they must stop short both of 
 complete fixation and of complete single vision, so that they see 
 double and squint also ; or they must limit themselves to 
 unilateral accommodation, so as to surrender binocular vision. 
 If now we suppose, by means of a prism, the object e displaced 
 to d, and the e' to d f , it is manifest that the angle of nutation 
 e b c will correspond to the minimal application of the left eye, 
 and that the angle of nutation e f b f c' will correspond to the 
 minimal nutation of the right eye. Accommodation and appli- 
 cation will thus be brought into harmony ; so that, by the aid 
 of this prism, a clear and single impression becomes possible 
 in binocular vision. It only remains to ascertain whether the 
 convergence of the two eyes, corresponding to the directions 
 b d and b' d', may yet present a difficulty. If this were so, 
 the prisms would certainly be insufficient to regulate at the 
 same time the application and the convergence ; and the result 
 would generally be as follows : 
 
 If the discordance were of so slight a degree that it could 
 be overcome by visual strain, then the arrangement shown in 
 Fig. 69, with its complete optical effects, but with the visual 
 strain produced by a forced convergence, would be brought 
 
216 THEORY OF OCULAR DEFECTS. 
 
 about. If the discordance were so great as to overpass the 
 relative range of each single function, then, when the impulse 
 after direct vision exceeded the impulse after single vision, 
 direct vision in both eyes, after the position in Fig. 69, would 
 be brought about, but with false application ; that is, with a 
 deviation from parallelism of the polar bacilli b and b', and 
 with double vision of fixed images. In case, however, as 
 would most commonly happen, that the impulse after single 
 vision preponderated, and that binocular fixation was sur- 
 rendered, only one eye, for example the left eye c, would 
 remain in the position delineated, and would see directly. 
 The right eye c', on the contrary would turn itself and see 
 indirectly, so that its affected bacillus would continually be 
 parallel to the bacillus b of the left eye, and single vision with 
 indirect vision would be the result. 
 
 What has been said of the action of the emmetropic lenses 
 applies also, in principle, to every other pair of lenses ; a 
 certain degree of accommodation of any eye always requires 
 a corresponding exertion of its application and of its con- 
 vergence. The convergence of the single eye towards the 
 frontal axis has always a large range ; and the other eye is 
 turned with more energy to a corresponding position when 
 the convergence of the two axes towards each other is induced 
 by a common effort of accommodation. 
 
 We have yet to remark that, when a pair of eyes of like or 
 of unlike accommodative faculty, either without glasses, or 
 furnished with spherical centric glasses, are called upon to 
 exert different degrees of accommodation effort, they are 
 involuntarily made to squint; and this squint presents the 
 following peculiarities. If the left eye fixes, and exerts more 
 accommodation than the right, there is a stronger application 
 induced in the former than in the latter. The homonymous 
 double images thus occasioned can only be fused by the right 
 eye assuming a convergent squint. If the right eye take up 
 the fixation it will induce, since it accommodates less strongly 
 than the left, in the left a weaker application. The resulting 
 
PRACTICAL RULES FOR SPECTACLES. 217 
 
 crossed double images can only be fused by the left eye 
 assuming a divergent squint. 
 
 Hence is explained the singular fact that, when two eyes, 
 either in the unaided or the spectacled condition, work at 
 certain distances with unequal accommodation, one of them 
 will fix, and the other will squint; and that, according as one 
 or the other eye fixes, the squint of its fellow will be conver- 
 gent or divergent ; so that sometimes by changing the visual 
 distance from a near to a far point, the convergent squint will 
 change to a divergent one, or vice versa. The explanation 
 of these phenomena rests upon my theory of application. 
 
 Furthermore, this theory affords us the means of bringing 
 two unequal eyes to a common visual action. If with the two 
 lenses /, determined by equation (38) for each eye, two 
 prisms are united, which act in such a manner that the 
 apparent objects are placed in the points d and d' (Fig. 69) 
 from whence they require an application represented by the 
 angles ebd and e'b'd', and harmonizing with the accommoda- 
 tion for the distances bd and b'd'. the eyes will be placed in a 
 position to see clearly and singly without strain. Whether 
 both eyes would be enabled by these combination spectacles 
 to see directly, that is, with their polar bacilli, would be 
 dependent upon the state of the apparatus of convergence, and 
 could not be determined by anticipation. If fixation with 
 both eyes were not attainable, one eye alone would fix, and 
 the other see indirectly. We have said already that by 
 fixation is always intended direct vision with the polar 
 bacillus, and, therefore, the direction of the axis of the eye 
 b c, to the apparent object d, not to the actual object. Hence 
 direct vision may very well require a squinting position. 
 
 As regards the determination of the prisms #, to be com- 
 bined with the lenses /, this is accomplished in the most 
 simple manner by direct experiment with two equal prisms. 
 We seek, when the eyes are already furnished with the two 
 lenses f, the strongest and the weakest prismatic spectacles, 
 g' and g", with which the eyes can see the object at the 
 
218 THEORY OF OCULAR DEFECTS. 
 
 distance s clearly and singly, and then take for practical 
 application, prisms of the mean value between them : 
 
 1 1/1 l>v 
 
 g "2 V7 + 7V' 
 
 Instead of by g, we may also define these prisms by their pro- 
 portion of deviation A, and may say 
 
 using h also to express either the angle of total deviation or 
 the refracting angle. 
 
 In this determination of the prisms A, we have considered 
 only the single vision. For the binocular functions it is also 
 of great importance that the visual impressions of the two eyes 
 should be of equal magnitude. It has already been explained 
 that equality of apparent magnitude can be attained by varia- 
 tion of the two prisms before the left and the right eye, since 
 this equality depends upon the ratio of the accommodation 
 to the application effort in each single eye. If, therefore, 
 on the addition of two equal prisms h to the two dissimilar 
 lenses/, the object at the distance s should appear larger to 
 the left eye than to the right, it would be necessary to ascertain 
 
 what single prism q, or what two prisms -|-, with their bases 
 
 to the left, would produce equality of apparent magnitude. 
 When the value of the proportion of deviation q is found by 
 actual experiment, we can either give to one eye the prism 
 h + q, and to the other the prism /*, or to one the prism 
 
 h + -a & an( * to ^ ie ot ^ er ^ "~ Q- ? Generally we may ap- 
 ply the two unequal prisms h + mq and h (1 m) q- and 
 in many cases it will be necessary to use dissimilar prisms, 
 in order to obtain equality of the visual impressions. 
 
 For example, if it has been found that positive prismatic 
 spectacles (with bases inwards), of equal proportion of devia- 
 
PRACTICAL RULES FOR SPECTACLES. 219 
 
 tion h = 0*015, in combination with the lenses /, produce 
 single vision, but that the object appears too large to the left 
 eye or too small to the right, and that to equalize this dif- 
 ference of apparent magnitude a prism with its base to the 
 left is required, of the proportion of deviation q = 0*02, we 
 may, since this prism is negative for the left eye and positive 
 for the right, employ at pleasure, or according to circum- 
 stances, any of the following pairs : 
 
 0-015 0-02 =-0-005 and 0-015 = 0*015 
 
 or 0*015 - l . 0*02 = 0*005 and 0*015 + . 0*02 = 0*025 
 0*015 - j- . 0*02 = 0*008 0*015 + f . 0*02 = 0*028 
 0*015 = 0*015 0*015 + 0-02 = 0*035 
 
 Herewith it appears that the practical rules for the selection 
 of lenses and prisms for any abnormal pair of eyes, each eye 
 of which labours under a symmetrical defect, are exhausted 
 in all essentials. I think that, upon the basis of these rules, 
 the contemplated effect of obtaining binocular vision, with 
 common action of both eyes, clear, single, as much as possible 
 direct, and with the smallest possible amount of strain, is to 
 be attained by very simple experiments. 
 
 With regard to the technical manufacture of a combination 
 glass, when the necessary strength of the lens and the prismatic 
 deviation are determined, some remarks will be made at the 
 close of the volume, 
 
 5. Periscopic Combination Spectacles. 
 
 Whether any combination glass A or B is regarded as an 
 eccentric portion of a common flat lens AB (Fig. 70 or 71), 
 
 Fig. 70. Fig. 71. 
 
 O O 
 
 that is, of a lens ^ith a straight chord in section, or as an 
 eccentric portion of a periscopic lens (Fig. 72 or 73), that is, 
 
220 
 
 THEORY OF OCULAR DEFECTS. 
 
 of a lens with a curved chord in section, is, as far as concerns 
 
 its essential nature, a matter of indifference. Since the peri- 
 
 Fig. 72. Fig. 73. 
 
 O 
 
 scopic glasses are not distinguished from flat glasses by any 
 fundamental considerations, I have not devoted any special 
 attention to the former in the Theory of Ocular Defects. By 
 later observations, however, I have found that the aberration 
 in the flat combination glasses produces inconvenient effects, 
 which may be either remedied or very much diminished 
 by periscopic combination glasses. Upon this ground, and 
 because the disturbing phenomena produced by flat combina- 
 tion glasses are of the highest physiological interest, I seek 
 to draw the attention of ophthalmologists to the periscopic 
 combination spectacles* 
 
 The optical illusions produced by aberration in monocular 
 vision through a centric lens are well known. In order to 
 explain them it is generally necessary to depart from the 
 purely physical view, which, in the passage of rays from a 
 
 luminous point D, through a lens 
 A B (fig. 74) considers the lens 
 to be acting in its entire aperture 
 A B J an d takes into account the 
 entire cone of rays A D B. On 
 this supposition, the rays of such 
 a cone, by their refraction in the 
 lens, converge towards the re- 
 lative focal point E, which lies 
 nearly in the direction of the 
 principal ray c D ; and, by the 
 aberration, are spread over a 
 
 focal surface. This conception has, for an eye looking through 
 the lens, with a pupil considerably smaller than the lens, only 
 
 - 74> 
 
PRACTICAL RULES FOR SPECTACLES. 221 
 
 a very limited value : and scarcely any value at all for the ex- 
 planation of the physiological phenomena of aberration. An eye 
 so placed is acted upon only by rays actually passing through 
 its small pupil a &, and all others are non-existent for it. 
 These efficient rays form only a small pencil on the base of 
 the pupillary opening a b, and therefore an almost elementary 
 part of the whole cone passing through the lens A B. We 
 will call the former briefly the pupillary pencil. To such a 
 pupillary pencil, D c a, the three following essential laws 
 apply : first, that its rays a c, b d, converge nearly in a point 
 e, of the focal surface of the whole cone ; secondly, that 
 this point of convergence e is for a convex lens nearer to the 
 lens, and for a concave lens farther from the lens, than the 
 point of convergence E of the principal rays of the whole 
 cone, passing through the centre of the lens ; thirdly, that 
 the central ray of the pupillary pencil, and therefore the 
 whole elementary pencil, is caused to deviate by the refrac- 
 tion in the lens, and is turned outwards by a convex lens, and 
 inwards by a concave lens, since the ray a c is not directed 
 towards D, but towards the laterally placed point e. 
 
 From these three laws, in monocular vision through a 
 lens over an extended field, and especially over a vertical 
 plane, arise the following phenomena. Aberration of the rays 
 of the pupillary pencil is inconsiderable ; so that an extended 
 object, notwithstanding its boundaries, is seen only by means 
 of marginal rays, yet sharply, in so far as the eye can accom- 
 modate for the apparent distance of the marginal points. By 
 the second law, the points of a vertical plane lying laterally to 
 the ocular axis, in vision through a convex lens appear 
 nearer to, and in vision through a concave lens farther from, 
 the spectator, than the point of the same plane lying in the 
 axis ; that is, the vertical plane appears to be curved, and 
 through a convex lens to be sunken or concave, through 
 a concave lens to be convex. By the third law the field of 
 vision, in looking through a convex lens appears externally 
 enlarged or expanded, and a rectilinear square assumes the 
 
THEORY OF OCULAR DEFECTS. 
 
 form shown in Fig. 76, of a quadrangle with its sides concave 
 in a direction outwards. On the contrary, on looking through 
 a concave lens the field of vision appears externally diminished 
 
 Fig. 75. 
 
 Fig. 76. 
 
 or compressed, so that a rectilinear square assumes the form 
 shown in Fig. 76, of a quadrangle with its sides convex in a 
 direction outwards. 
 
 If the lens be small, about the size of a spectacle glass, so 
 that it does not permit a view over a large field of vision, all 
 the above deformities or distortions occur only in a pro- 
 portionately slight degree, and are hence but little noticed. 
 
 But if the lens is eccentric, or a combination glass, these 
 distortions, even if the lens be small, are observable in a 
 much higher degree ; since then the centric middle portion is 
 
 either absent or much 
 curtailed, and hence mar- 
 ginal rays come chiefly 
 into action. Moreover, 
 since in consequence of 
 the removal of the middle 
 portion, these marginal 
 rays are all from one side, 
 a square vertical surface 
 is caused to lose its sym- 
 metric form, and appears 
 
 through a convex lens, something like fig. 77, and, through 
 a concave lens, something like fig. 78. 
 
PRACTICAL RULES FOR SPECTACLES. 223 
 
 This considerable and un symmetrical distortion of the field 
 of vision is manifestly an objection to the eccentric glasses. 
 In order to estimate it rightly, however, we must not be guided 
 by the effect on monocular vision through a single glass, but by 
 the effect on binocular vision through a pair of spectacles. For 
 centric spectacles this effect does not much deviate from that 
 on monocular vision ; but it does so deviate for eccentric, and 
 therefore also for orthoscopic spectacles. In looking through 
 the latter the field of vision, as in monocular use, appears 
 extended or condensed at its boundaries, and a vertical square 
 exhibits curved sides, as in Figs. 75 or 76. But the apparent 
 curvature of a vertical plane is reversed, that is, through 
 orthoscopic convex spectacles it appears no longer concave, 
 but convex, and through orthoscopic concave spectacles no 
 longer convex, but concave. In vertical lines this curving 
 becomes especially manifest. 
 
 These appearances in monocular vision, and the change pro- 
 duced by changing from monocular to binocular vision 
 through combination glasses, are highly worthy of note, and 
 of great importance in the physiology of the eye. The apparent 
 arching of a vertical plane in monocular vision affords a re- 
 markable confirmation of my theory of the recognition of 
 distance : since it shows, first, that the faculty of recognizing 
 distance appertains to each single eye ; and, secondly, that 
 the single eye can simultaneously recognize different distances, 
 so that the recognition does not depend upon some general 
 function of the eye as a whole organ, but upon an elementary 
 function of the particular nerve-fibre or bacillus that is im- 
 pressed. The inversion of the arching in the transition from 
 monocular to binocular vision, and therefore, the seeming 
 approximation of the more distant images, and the seeming 
 recession of the nearer, teaches further that the judgment of dis- 
 tance does not depend exclusively upon the actual distance of 
 the apparent object or of its optical image, and therefore not 
 exclusively upon the accommodation for this image, nor ex- 
 clusively upon the position of convergence of the two eyes : 
 since first, in this transition from monocular to binocular 
 
224 THEORY OF OCULAR DEFECTS. 
 
 vision, the actual distance of the optical images of all points of 
 the vertical plane remains unchanged, and since this distance 
 is solely determined by the physical refraction of the rays ; 
 since, secondly, the accommodation remains unchanged, the 
 obj ect being as sharply defined afterwards as before ; and since, 
 thirdly, the position of convergence of the two eyes for all 
 points of the visual field must be the same. The appearances 
 on binocular vision can therefore certainly not be explained by 
 the doctrines that have hitherto prevailed in physiology ; but 
 readily by my theory of the recognition of distance. The 
 phenomena consequently afford to that theory an important 
 support, which I will now proceed to explain. 
 
 In the Gesetze des raumlichen Sehens, s. 117, and in the 
 earlier part of the present work (p. 127), I have shown that, 
 when a is the actual distance of an optical image, b the 
 accommodation distance, that is, the distance for which the 
 retinal bacillus that is acted upon accommodates, and c the 
 application distance, that is, the distance for which this 
 bacillus is applied, or which represents the angle of con- 
 vergence of the corresponding bacilli of the two eyes, the 
 
 apparent distance of the object has the value a' = a . 
 
 From this it follows that, when the actual distance a of the 
 optical image, and the accommodation distance b of the im- 
 pressed bacillus, both remain unchanged, but the application 
 distance c, or the distance of the point towards which the two 
 corresponding bacilli converge, changes, then the apparent 
 distance a' will become greater or smaller, in proportion as c 
 increases or diminishes. We have therefore to inquire what 
 changes the convergence distance of the optical images of the 
 different points of the vertical plane undergo in binocular vision 
 through combination spectacles; and whether these changes 
 are in unison with the alteration in the observed appearances. 
 Let us take a pair of normal orthoscopic spectacles A, B, 
 shown in diagram in Fig. 79. If c is an object in the horizon- 
 tal frontal axis, its optical image will be formed at D, since the 
 
PRACTICAL RULES FOR SPECTACLES. 
 
 225 
 
 rays c A and c B are refracted outwards through the glasses in 
 the directions A D and ED. If E be a point in a horizontal 
 drawn through c, the rays E A and E B, coming to the eyes, 
 
 Fig. 79. 
 
 would likewise be refracted outwards ; so that they, when D F 
 is parallel to c E, would converge towards F, in so far as no 
 aberration interfered. Aberration would displace the optical 
 image of E somewhat in front of the line D F, somewhat towards 
 F' for the left eye, and somewhat towards F" for the right. 
 Hence the horizontal line c E, in monocular vision through 
 either the left or the right eye, would appear concave like the 
 curve D F'. 
 
 If the two glasses A and B were centric, in vision with the 
 left eye the angular deviation F' A E would be equal to DAG, 
 and in vision with the right eye the angular deviation 
 F" B E=D B c ; and, therefore, the angle of convergence A F B= 
 A D B. But, since each glass is eccentric, or prismatic, this 
 equality is not produced ; the deviation of the ray going from 
 E to the right eye is somewhat greater than the deviation of 
 the ray going to the left eye, since the former passes through 
 
 Q 
 
226 THEORY OF OCULAR DEFECTS. 
 
 a part of the glass B, which, on account of its nearer approxi- 
 mation to the margin of the large lens, A B, increases the deflec- 
 tion in greater measure than the part of the glass A, through 
 which passes the ray for the left eye : since this part, by 
 reason of its being less distant from the centre of the large 
 lens, AB, produces a less degree of deflection. Consequently, 
 the corresponding rays for the two eyes converge towards a 
 point G, which is situated behind the line D F ; the angle of 
 convergence A G B is less than A D B ; or the distance of bacillary 
 convergence or application c, which is determined by the 
 point G, is increased. The actual distance a of the optical 
 images F' and F //; for the two eyes, and also the corresponding 
 accommodation distance 5, which, on account of the sharpness 
 of the visual impression, may be taken to be equal to a, cer- 
 tainly remain somewhat less than the distance of the point F ; 
 but, since the apparent distance of this point F is determined 
 
 by a'= a, then, because a=5, a'=c, that is, the point E, or 
 
 its optical image, which is F' for the left eye and F'" for the 
 right, appears to be situated at G. Hence it is established 
 that a horizontal line, c E, in binocular vision, through ortho- 
 scopic convex spectacles, would appear not concave, but 
 convex. 
 
 For the orthoscopic concave spectacles the foregoing condi- 
 tions are so reversed, that the horizontal CE will appear 
 concave. 
 
 If we now investigate the appearance of a vertical line, 
 erected upon the point c, let H be any point in this vertical 
 line. The rays which proceed from the point H to enter the two 
 eyes, incline symmetrically towards the glasses and towards 
 the eyes. If we follow the course of such rays, falling 
 obliquely on the refracting surfaces, and consider that the 
 refraction of such surfaces occurs in a plane which passes 
 through the incident ray and through the radius leading to 
 the centre of curvature of the refracting surface, we shall per- 
 ceive that the rays, after refraction in the glasses, and, therefore 
 
PEACTICAL RULES FOR SPECTACLES. 227 
 
 as they reach the eyes, when we place vertical planes through 
 them, are projected not upon the lines AD and BD, but upon 
 two lines A I and B i, which enclose an angle A i B, that is less 
 than the angle A c B ; or that the corresponding vertical 
 meridians of the two eyes, in which the higher lying point H 
 is seen, correspond to a greater convergence distance c, than 
 the meridians for the point c. And, although the actual dis- 
 tance a of the optical image of H, and also the accommodation 
 b for this image, are less, on account of the aberration, than 
 the distance of the image D, so that, therefore, in monocular 
 vision the vertical line c H will appear concave, yet in binocular 
 
 vision, because a! = a = c, and because c is greater than , 
 
 it will appear convex. 
 
 This appearance also will be reversed by the use of ortho- 
 scopic concave spectacles, in such a manner that the vertical 
 line, c H, will then appear concave. 
 
 The foregoing results afford a satisfactory explanation of the 
 peculiar appearances observed in binocular vision through 
 combination spectacles. They teach that these appearances 
 essentially depend upon the application of the bacilli, and, 
 therefore, furnish a further confirmation of the existence and 
 of the physiological effects of this function of the visual organs. 
 
 When we inquire into the geometrical ,.,. 
 
 conditions which impart to the eccentric 
 form of lens the peculiarity of producing such 
 an aberration that apparent curvature of the 
 field of vision necessarily ensues, we find on 
 more exact investigation that they are the 
 following. If we follow, as in Fig. 80, the 
 part E F of the ray that lies within the lens 
 A B, and which, since it becomes constantly 
 more convergent towards the point D, pene- 
 trates the lens more and more near to its mar- 
 gin, we find that for such a ray, e.g. for that which belongs to 
 the portion E F, the lens has only the physical value of a prism 
 
 Q2 
 
228 THEORY OF OCULAR DEFECTS. 
 
 E G F, of which the plane containing surfaces E G and F G are 
 the tangential planes of the lens at the points of entrance and 
 exit E and F. With the deviation of the ray from the axial posi- 
 tion D c towards the outer marginal ray D B, the angle E G F 
 of the supposed prism varies, and in the same degree the 
 deviation of the emerging portion of the ray from the direction 
 of the entering portion. The essential character of this 
 variation is that it is not uniform with the turning of the ray 
 towards the point D ; but that the angle E G F, and hence 
 the deviation of the ray by its refraction in the lens, is at first 
 small, and at last undergoes a great increase, for one and the 
 same angular bending of the ray. 
 
 The knowledge of the external cause of the apparent curva- 
 ture of the field of vision, gives at once the means of removing, 
 or of greatly diminishing, this inconvenience; and, since it 
 teaches for what forms of lens it occurs in the greatest and in the 
 least degree, affords a rational basis for judgment about the dif- 
 ferent forms of spectacle glasses. It is manifest that every form 
 of lens which increases the above-described want of uniformity 
 of the variation of the deviation of the ray of light towards its 
 margin, will increase the apparent curvature of the field; 
 while every form which diminishes the want of uniformity 
 will also diminish the apparent curvature ; so that the form 
 which produced a uniform variation of the deviation, would 
 set aside the apparent curvature entirely. 
 
 If we name a lens with regard to its sphericity by the forms 
 of its anterior and posterior spherical surfaces, calling the 
 anterior surface that which is external, or away from the eye, 
 and the posterior surface that which is internal, or towards the 
 eye, we find the following gradation from the most prejudicial 
 to the most beneficial form. 
 
 a. For the centric or eccentric convergence-lens (Fig. 81) the 
 concavo-convex, A, is the most prejudicial. The plano-convex, 
 B, is better, the biconvex, c, is better still ; and then follows 
 the convexo- plane lens D. The best of all is the convexo- 
 concave E. 
 
PRACTICAL RULES FOR SPECTACLES. 
 
 229 
 
 8. For the centric or eccentric divergence-lens (Fig. 82) the 
 concavo-convex lens A is the most prejudicial. The concavo- 
 
 Fig. 81. 
 B C 
 
 o 
 
 Fig. 82. 
 
 O 
 
 plane lens, B, is better, the biconcave, c, is better still ; and 
 then follows the plano-concave lens D. The best of all is the 
 convexo-concave E. 
 
 The most beneficial lens, in both cases, has the convexo- 
 concave form E, that is, it is curved towards the eye. This 
 form, when it is used for monocular vision, and therefore as a 
 centric lens, is more advantageous, the more its central curva- 
 ture approaches a circle described about the central point of 
 the eye. For binocular vision, or for two eccentric glasses, 
 the proper curvature must be determined by the considerations 
 stated below. A convexo-concave convergence- or divergence- 
 lens is periscopic. This name in no way distinguishes the one 
 lens from the other, by the fundamental peculiarities of each ; 
 it is only another expression for a lens with a convex anterior, 
 and a concave posterior surface. We are therefore led to the 
 result that periscopic spectacles are adapted to reduce the 
 apparent curvature of the visual field to a minimum. 
 
 As regards the distortion of the field of vision, namely, the 
 compression or extension of its outer portions, and the apparent 
 bending of straight lines, these defects, since they arise from 
 
230 THEORY OF OCULAR DEFECTS. 
 
 the inevitable, and, on account of the accommodation, directly 
 designed refraction of the rays of light, cannot be set aside by 
 any form of lens, and cannot even be essentially diminished. 
 
 By all this we are induced, as well for centric as for ec- 
 centric glasses, to turn the attention to the convexo-concave 
 form. 
 
 When r is the radius of the convex anterior surface, r 1 the 
 radius of the concave posterior surface, and n the index of 
 refraction (1'55) of glass, the focal distance of a periscopic 
 lens, measured from the apex of the convex surface of the 
 complete centric lens, with a supposed small thickness of glass, 
 
 r r* 
 and for parallel incident rays, will be / = 
 
 = 1*82 -/^ ; and the crossing- point of the chief rays, that 
 
 is, of the straight lines which unite the points of the object 
 with the corresponding points of its optical image, lies nearly 
 in the apex of the centric lens. Hence the optical effects of 
 such a lens are easy of demonstration. 
 
 In order to determine the most proper curvature for peri- 
 scopic combination lenses we will first consider the case of 
 normal orthoscopic spectacles. From these spectacles is required 
 their characteristic property, that the rays from any luminous 
 point shall be so refracted through both lenses as to appear to 
 both eyes to proceed from one and the same point. This is 
 only fulfilled when the apices of the complete lenses, of which 
 the two eccentric lenses, A and B, form parts (Fig. 83), coin- 
 cide. This occurs when these apices lie in the frontal axis F D. 
 The simplest solution of the problem is to cut the eccentric 
 glasses, A and B, out of the same large centric lens, A c B, at 
 the distance A B of the ocular centres. This arrangement sup- 
 poses, strictly taken, that the two eccentric glasses shall be put 
 before the eyes in the same positions that they occupied in the 
 lens A c B. But in this oblique position they would not be 
 practically applicable, and they must be so turned as to assume 
 a straight position in front of the eyes. Such turning, when 
 
PRACTICAL RULES FOR SPECTACLES. 
 
 231 
 
 it is not too considerable, has scarcely any appreciable influence 
 upon the refraction and deviation of the rays of light ; and 
 
 Fig. 83. 
 
 even this influence can be essentially diminished by the fol- 
 lowing disposition. 
 
 It is obvious that the fundamental property of the ortho- 
 scopic spectacles is also realized when the two glasses A and B 
 belong to the two centric lenses, A A' and B B', which are equal 
 to the lens A c B, but so placed, that their apices may coincide 
 in any desired point F of the frontal axis. Hence we may give 
 to the two glasses A and B any desired position before the eyes ; 
 and we can also still cut them from one and the same large 
 centric lens A c B, since, in general, the chord A F is not equal 
 to the chord A c, so that the glasses for this position may be 
 cut in distances A A' or B B', other than those already considered, 
 from the large lens. These changed distances may be deduced 
 from Fig. 83 by an easy drawing. 
 
 The case is of chief interest in which the two spectacle 
 lenses are to be placed perfectly straight ; that is, so before 
 the eyes that the central points G and H of their middle curva- 
 tures fall into the lines A G and B G, drawn through the eyes A 
 and B, parallel to the frontal axis. This, indeed, is the position 
 in which the use of glasses is most advantageous, and the 
 
232 THEORY OF OCULAR DEFECTS. 
 
 external view is the least embarrassed. For this position the 
 chord A F is in a remarkable way exactly equal to the chord 
 A c ; so that the eccentric glasses to be placed straight before 
 the eyes are precisely the same which are cut out of the large 
 lens A c B at the distance A B. 
 
 The large lens A C B, to which the lenses of periscopic spec- 
 tacles belong as eccentric portions, is conditioned by the focal 
 
 7* /"' 
 
 distance f = ... . . -, required by the accommoda- 
 
 (n - 1) (r f - r) 
 
 tion. Since the value of/ depends upon two quantities, r and 
 r', lenses of very different curvature satisfy every requirement. 
 Jn order to fulfil, as completely as possible, the purpose of 
 removing the apparent curvature of the field of vision, the 
 middle curvature of the periscopic lens must, when practicable, 
 be so great, that the central point of each eye corresponds to 
 the central point, G or H, of the middle curvature of each 
 glass. But it is manifestly impossible to give to the large lens 
 A c B, from which the spectacle lenses are to be cut, a smaller 
 radius than F A, which is the half of the distance A B = d, of 
 the centres of the eyes from one another. Since d about 
 equals 66 millimetres, or 2'44 Paris inches, the radius R of the 
 middle curvature of the periscopic lens, or the value of R = 
 1 (r f + r), may be assumed to be 33 millimetres, or 1-22 
 Paris inches. In practice we shall seldom obtain this outside 
 value, and R will generally equal 1-5 or 2 Paris inches. 
 
 It is very noteworthy that a spectacle lens, the middle 
 curvature of which has a radius of from 1*25 to 1*5 inch, 
 when it is placed straight before an eye, has the central point 
 of its middle curvature nearly coincident with the central 
 point of the eye ; so that periscopic spectacles of this kind 
 fulfil, as much as possible, the requirements stated. 
 
 When the middle curvature and the focal length of the 
 periscopic lens A c B are given, we have the two equations of 
 quality, 
 
PRACTICAL RULES FOR SPECTACLES. 233 
 
 Hence the two radii r and r 1 are completely determined, 
 since we find 
 
 r>= R - (n - 
 
 or, placing for ra, 1/55 ; and for R, 1*5 inches, 
 
 r =1-5 + 0-55 / >v/2-25 + 0-3025 / 2 inches, 
 r'= 1-5 - 0-55 /+ V 2-25 + 0-3025 / 2 inches. 
 Again, making n = 1/55, and R = 2 inches, 
 
 r = 2 + 0-55 / - V 4 + 0-3025 / 2 inches, 
 r / =2-0-55/+ V 4 + 0-3025 / 2 inches. 
 
 When, according to these formulae, we calculate the semi- 
 diameters r and r' for different focal lengths /, we find only 
 a little difference between these semi- diameters. Hence it 
 follows that a strongly curved periscopic glass for a given focal 
 distance must be ground with far greater accuracy than a flat 
 glass; and, therefore, it is the more important that the two 
 portions required for eccentric periscopic spectacles should be 
 obtained from one and the same original lens. 
 
 6. The Manufacture of Combination Glasses. 
 
 The ophthalmic surgeon must make his experiments with 
 single lenses and with single prisms, in order to determine 
 the combination that is required. For the greatest possible 
 accuracy it would be desirable to use plano-convex and plano- 
 concave lenses, and to place the additional prisms, when 
 possible, in direct contact with the plane sides of the lenses, 
 taking care also that the prisms shall not be rotated. This 
 would probably be best secured by quadrangular lenses and 
 prisms, instead of round or oval ones ; and a set of such glasses, 
 adapted to all the experiments with binocular vision, would be 
 very easily manufactured. 
 
 The business of the optician will then be to provide the 
 
234 
 
 THEORY OF OCULAR DEFECTS. 
 
 desired combination lens, in the form of a prismatic convex or 
 concave glass, made out of a portion of a larger spherical lens. 
 The larger spherical lens, out of which the combination lens 
 is cut, must always have the same focal length as the lens/ that 
 is required for the combination glass. The eccentric place, 
 from which the piece must be cut, is shown most easily and 
 certainly by the following simple geometric diagrams. In 
 Fig. 84, A B c is the anterior surface, that is, the surface turned 
 to the object, of a large positive lens of the desired focal 
 distance /, and D is its actual focus for the rays i E coming from a 
 
 r. 84. 
 
 Fig. 85. 
 
 point in front of it. If the combination with a positive prism is 
 desired, of which the angle of deviation is ^, or the proportion of 
 deviation is tan. i// = ft, the line D E must be drawn through 
 the focal point D in such a way that it may form, with the 
 axis of the lens, an angle B D E = //, or we make the ordinate 
 B H = h X B D. If the focal length / is measured from the 
 anterior surface of the lens, so that B D = /, then the point H, 
 through which the line D H must be drawn, will be at the 
 height B H = h ./. The line D H, so drawn, cuts the large 
 lens in the point E, which must be the central point of the 
 spectacle glass Fg. 
 
 If the object be to make a negative lens AC, of the focal 
 
PRACTICAL KULES FOR SPECTACLES. 235 
 
 length fj in combination with a negative prism ;//, for 
 which tan. ( ^) = A, then Fig. 85 furnishes the necessary 
 diagram ; since D is the focal point, and therefore B D =/, 
 and the angle BDH = ip, orBH = A and B D = h . f. 
 
 The combination of a positive lens f with a negative prism 
 h is accurately shown by Fig. 84, with the change that the 
 glass F #, which has its prismatic base turned towards the right, 
 would then be applied to the right eye instead of to the left. 
 
 In the same way the combination of a positive prism h with 
 a negative lens /is shown by Fig. 85, with the change that 
 the glass F g, which has its prismatic base turned towards the 
 left, would then be applied to the right eye instead of to the left. 
 
 The length of the curve B E, or of the chord B E from the 
 axis B of the large lens to the central point E of the spectacle 
 glass, will be nearly equal to the line B H or = h . f. When, 
 therefore, the most extreme accuracy is not required, we may 
 say that the central point E of the spectacle glass F g is at a 
 distance = hf from the axis B of the large lens. By this direct 
 measurement, or by the foregoing more exact construction, the 
 place is determined at which the combination glass is to be 
 cut out of its parent lens. 
 
 If we indicate the conventional diameter F g of a spectacle 
 
 lens by c, then the distance B g =.h . f --, and the distance 
 
 2 
 
 B F = h . f + -, and therefore the diameter or the aperture 
 
 2 
 
 A c, which the larger lens must possess, in order that the com- 
 bination glass F g may be cut from it, is = 2 B F =2h .f -f c. 
 It is important in practice that the large lens, from which 
 the combination glasses are cut, should not itself be much 
 larger than is absolutely necessary. Not only does increase cl 
 size increase the quantity of glass and the work of the grinder, 
 thus increasing the cost, but further, the desired combination 
 glass should be made as thin as possible, so as to preserve an 
 appearance not unbecoming, and to diminish the disturbing 
 optical effect of any great thickness of glass. On the other 
 
236 THEORY OF OCULAR DEFECTS. 
 
 hand, the requirements of the manufacturer, and the prompt 
 supply of any desired combination, demand that he should be 
 placed in a position to keep in stock only a small number of 
 the large lenses, from which combination lenses may be cut 
 as needed. Lastly, it is of advantage as regards price, in the 
 fabrication of spectacles containing two symmetrical combi- 
 nation glasses, that both of them should be cut symmetrically 
 from one and the same lens ; and this is also beneficial from 
 an optical point of view, since two glasses thus made will 
 coincide more accurately than any two cut out of different 
 
 All these advantages may be secured by the following con- 
 siderations. When B E < g E, or when h f < -, there can be 
 
 A 
 
 cut from the large lens, whatever its size, only one combina- 
 tion glass. The diameter A c of the large lens need not, there- 
 fore, be larger than the double of F g. When, on the contrary, 
 
 BE > g E, but < . g E, then from the large lens can be cut 
 
 two symmetrical circular combination glasses of the diameter 
 
 4 6 
 
 F g. When BE > g E, but < g E, then four such glasses 
 
 7T ' 7T 
 
 can be cut, and so on. 
 
 If we thus assume that we are dealing with a convex or 
 concave lens of definite focal length f, the manufacturer would 
 require to keep only four large lenses, of the diameters 
 
 2c, 2-273 c, 2-909 c, 3-546 c. 
 
 From the first of these, 2 c in diameter, he would be able to cut 
 only one combination glass, and one for which h is < 0*5 x 
 From the second, of diameter 2'273 c, he might cut two com- 
 bination glasses, for which h > O5 x ~, but < O637 . From 
 
 J J 
 
 the third, of diameter 2 '909 c, he might cut four, for which 
 h > 0-637 x -1, but < 0-955 ~. From the fourth, of dia- 
 
PKACTICAL RULES FOR SPECTACLES. 237 
 
 meter 3'546 c, he might cut six, for which h > 0*955 , but 
 < 1-273-^. 
 
 If we take 1/3 inch as the diameter c of the spectacle-glass, 
 or, allowing for loss in working, 1/5 inch, then the diameters 
 of the large lenses required are respectively 3, 3*41, 4'36, and 
 5 '32 inches. From the first is obtained one glass, for which 
 
 0*75 0*75 
 
 h < ; from the second, two, for which h > - , but 
 
 J J 
 
 ; from the third, four, for which h > , but 
 
 J J 
 
 -* 3 - ; and from the fourth, six, for which h > 1-432 , but 
 
 If circular glasses are not desired, but only oval ones, 
 having a horizontal diameter of c = 1*3, or, with allowance 
 for loss in working, = 1/5, and a vertical diameter c' = 1, or, 
 with allowances, = 1/2 inches, then we may obtain two, four, 
 and six glasses from lenses of smaller aperture. The four 
 large lenses, from which could be cut respectively one, two, 
 four, and six combination glasses, with the above-stated values 
 of h, may then be reduced to three, with diameters of 3, 3'42, 
 and 4*36 inches, since from the first can be cut all single and 
 
 0'75 
 double glasses for which h < - 
 
 When the glass-cutter is skilful, and therefore does not re- 
 quire so much space for working between any two of the 
 portions that are to be cut out, then the diameters of the large 
 lenses may be still further reduced, which is always much to be 
 wished. 
 
 The advisability of keeping in stock a certain quantity of 
 large lenses of definite focal lengths/, from which any desired 
 combination glasses can be quickly cut, is associated with the 
 disadvantage that many glasses must then be cut from lenses 
 
238 THEORY OF OCULAR DEFECTS. 
 
 larger than are absolutely necessary to yield them, and that 
 these glasses are therefore thicker than they need be. If we 
 attach to the greatest possible thinness of the spectacle-glasses 
 the great importance which, from the point of view of the 
 most complete fulfilment of their uses, would be justifiable, 
 then the large lens, from which the glasses are cut, should 
 itself only be made when these glasses have been determined. 
 It would then be made no larger than would be absolutely 
 necessary; and it would therefore have only the diameter 
 2 h f + c = 2 h f -f- 1*3, or, on account of the necessary loss 
 in rounding the glasses, the diameter 2 hf + 1'5 inch, so that 
 every combination glass would be cut close to the margin of 
 the large lens yielding it. 
 
 In photographic and other optical instruments, which consist 
 of several lenses, convex and concave glasses are often united 
 by means of a transparent cement. In the same way it is pos- 
 sible that a plano-convex or plano-concave lens might be united 
 to a plane prism. If to such a proceeding there are no prac- 
 tical impediments, the circumstance that it would involve 
 a less sacrifice of glass would plead in its favour. The opti- 
 cian would then require to keep only common lenses and com- 
 mon prisms, of definite numbers, and to unite them according 
 to hj,s needs. 
 
 This procedure would, however, involve the abandonment 
 of the periscopic form of the spectacle-glasses. Since this form 
 is essential to the completeness of the spectacles, the method 
 by cementing could not give the best instruments. 
 
 When it is the object to obtain periscopic glasses of so 
 strong a curvature that the central point of the eye is coinci- 
 dent with the centre of the sickle-shaped curvature of the 
 glass, we may wholly lay aside the experiments with prisms 
 to determine the peculiarities of the eye, and the calculation 
 for the necessary combination-glasses, and may proceed in the 
 manner following. 
 
 We take an apparatus which, as in Fig. 86, has the form 
 of a spectacle -frame, in which on both sides can be placed 
 
PRACTICAL RULES FOR SPECTACLES. 239 
 
 periscopic lenses of the already-mentioned size, with their 
 middle curvature c A = R, and which can be rotated at pleasure 
 round their central points c and c'. After 
 these lenses have been placed by rotation in 
 
 the position in which they best content the ,-'" x 
 
 eyes, we obtain immediately from the angle / 
 of rotation the necessary eccentricity of the j 
 spectacle-glass. If the judgment of the pa- \ 
 tient upon the minimum of visual strain is 
 uncertain, we may discover, by turning the 
 glasses into the extreme position inwards 
 and outwards, the outside limits of the ec- 
 centricity, corresponding to the strongest 
 and the weakest prisms, with which clear and / 
 single vision is still possible. The mean > 
 between these limits will then be the proper \ 
 eccentricity. s ^. ..<' 
 
 Lastly, we cannot forbear to mention, as 
 a disadvantage of all eccentric glasses, arising from their pris- 
 matic form, the unilateral chromatic aberration which fringes 
 the object with the colours of the spectrum. Although this 
 disadvantage is not considerable, it is greater than in centric 
 glasses. It can only be obviated by lenses composed of two 
 different kinds of glass. flint glass and crown glass united 
 together. 
 
 With regard to the care that should be exercised in the 
 manufacture of spectacles, it must be observed that the eccen- 
 tric require far more than the centric, and that the chief matter 
 of attention would be to secure that the horizontal diameters 
 of both glasses, which would point towards the common apex 
 of their parent lens, should be placed as accurately as possible 
 in a horizontal plane, or that any rotation of the glasses about 
 the ocular axes should be prevented. Such rotation, which 
 for centric glasses would be immaterial, in eccentric glasses 
 would cause the rays from the same point of the object to fall 
 upon different points of the two retinae. So long as the rota- 
 
240 THEORY OF OCULAR DEFECTS. 
 
 tion does not exceed a certain degree, we find, and this is of 
 much physiological interest, that single vision is preserved ; 
 but only by an upward and downward movement of the cor- 
 responding bacilli, and therefore with an unusual dragging of 
 the retina, which is at once recognisable by a feeling of strain. 
 Hence it is advisable that on every spectacle-glass the termi- 
 nations of the horizontal diameter should be marked by a 
 notch, so that the wearer would immediately be able to rectify- 
 any displacement that might be produced by accidental bend- 
 ing of the frame. 
 
 LONDON : PRINTED BY 
 
 SPOTTISWOODE AND CO., NEW-STREET SQUARE 
 AND PARLIAMENT STREET 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 JAN 2 1963 
 
 DEC 2 - 196! 
 DECS 138 H 
 
 LD 21-100m-7,'52(A2528sl6)476