THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA CTOMETRY GIFT OF Dr. Hugh ?. Brown iy z g I- u < ct. UJ < 'J Qi Q z >- U Ophthalmic Lenses DIOPTRIC FORMULA FOR COMBINED CYLINDRICAL LENSES THE PRiSM-DIOPTRY AND OTHER OPTICAL PAPERS WITH ONE HUNDRED AND TEN ORIGINAL DIAGRAMS Charles F. Prentice, M. E. "OPTICIST" (Second Edition) PUBLISHED BY THE KEYSTONE PUBLISHING CO. 809-81 1-813 North 19TH Street, Philadelphia, U.S./ 1907 All rights reserved Copyright, 1900, by B. Thorpe Publishes, op The Keystone Add'l GIFT iqoy PREFACE In the Treatise on Ophthalmic Lenses an endeavor is made, by- means of graphic and analytic methods, to guide the novice upon a path by which he may easily acquire a knowledge of lenticular refraction with- out recourse to mathematics. The text is, therefore, confined to the description of a series of diagrams, by aid of which the principles involved are presented in their natural order of succession. Shortly after publication of the above treatise, Dr. Burnett, of Washington, D. C. , kindly suggested the production of plastic models of combined cylindrical lenses, by placing an incomplete set of models of his own conception with the author for further elaboration. With it came the request, if possible, also to produce two combinations in which the cylinders were to be united at angles other than right angles. As a result of the author's research, during the time devoted to the construc- tion of the latter more especially, and with a view to establish confi- dence in the precision of these models, a complete mathematical demon- stration of the refraction by combined cylindrical lenses was first presented . in 1888. The purpose of republishing the Dioptric Formula?, in Section II, is to present the important results then obtained, with as much abridgment of the calculations as permissible. While the diagrams have been prepared with great care, yet they are somewhat at variance with the laws of true perspective, ov*-ing to the author's desire to strictly preserve therein all important circles and right angles referred to in the text. 293 Among the many attempts to solve this problem, this is the only one whose formuliE contain exclusively the known quantities, namely, the angle between the cylinders and their foci or powers. Furthermore, this is the only solution which has ever disclosed the sixteen laws inherent in a pair of superposed cylindrical lenses — a fact which may be easily veri- fied by comparing this solution with those of Bonders, Reusch, Heath, Jackson, Hay, Weiland, Suter and Thompson, whose formulae are also iar less simple. The main object in republishing the Prism-Dioptry and other original papers, in Section HI, is to present them collectively to the student of optometry, who in most instances will now find it very difficult to gain access to the numbers of the journals in which the various original articles have appeared during the past ten years. This action seems to be further justified by the frequent inquiries made for reprints, which have long since been exhausted, and also for the reason that American manufacturers have, since 1894, universally adopted the prism-dioptral system for their entire marketable product of prisms. This should also prove an incentive for all who make use of such prisms to become thoroughly familiar with the character and capabilities of the prism-dioptry. The manuscripts on prisms were generally approved, before original publication, by Dr. Burnett, to whom the author is indebted for having called attention to the necessity for a new system of numbering prisms, and, indeed, also for having suggested the name of its unit, the prism- dioptry. Therefore its original form of spelling is herein retained. For the sake of greater clearness it has also been deemed advisable to make several important additions to the original text, which, in its present form, may at least be considered authentic with reference to modern ophthalmic prisms. The appended revised papers will, it is hoped, prove of permanent interest, as they contain features of scientific value not to be found in any ]-.r.nd-book or treatise on optics. ^ -.^ ^ Charles F. Prentice GENERAL CONTENTS SECTION I OPHTHALMIC LENSES ii SECTION II DIOPTRIC FORMUL/E FOR COMBINED CYLINDRICAL LENSES . . 49 SECTION III THE PRISM-DIOPTRY AND OTHER OPTICAL PAPERS 99 CONTENTS OPHTHALMIC LENSES SECTION I PAGE Refraction 13 Prisms 16 Simple Lenses 20 Compound Lenses. 1. Congeneric Meridians (convex) 29 2. " " (concave) 37 3. * Contra-Generic " (convex and concave) 39 ToRic Lenses. Congeneric and Contra-Generic Meridians 44 Table of Dioitral Numeral^ 47 Tables of Crossed Cylinders. 1. Congeneric Meridians ^ -> 2. Contra-Generic Meridians S * Sec Diopiric Formula; for Combined Cylindrical Lenses. CONTENTS DIOPTRIC FORMUL/E FOR COMBINED CYLINDRICAL LENSES SECTION II PAOK I. Dioptric Formula for Combined Congeneric Cylindrical Lenses. 1. Relative Positions of the Primary and Secondary Planes of Refraction ■ • • 53 2. Positions of the Primary and Secondary Focal Planes 59 3. Relations between the Primary and Secondary Focal Planes 62 11. Dioptric Formul,« for Combined Contra-Generic Cylindrical Lenses. 1. Relative Positions of the Principal Positive and Negative Planes of Refraction 69 2. Positions of the Positive and Negative Focal Planes 72 3. Relations between the Positive and Negative Focal Planes 75 in. Dioptral FoRMUL/ii: FOR Combined Cylindrical Lenses. I. Relation between the Principal Planes of Refraction and the Refractive Powers of the Cylinders 81 IV. Sphero-Cyltndrical Equivalence 87 V. Verification of the Formulae 95 1. Tables in Verification of the Dioptric Formulse 97 2. Tables in Verification of the Dioptral Formulae 97 LIST OF PLATES FRONTISPIECE Half-Tone Plate of Dr. Swan M. Burnett's Models, Demonstrative of Cylindrical Refraction, constructed by the author in accordance with the Formulas for Congeneric Cylindrical Lenses. PLATES I AND II The Refraction by Combined Congeneric Cylindrical Lenses, demonstrated in three diagrams. Plate I on page 52. Plate II on page So. PLATES III AND IV The Refraction by Combined Contra generic Cylindrical Lenses, demonstrated in three diagrams. Plate III on page 68. Plate IV on page 86. CONTENTS THE PRISM-DIOPTRY AND OTHER OPTICAL PAPERS SECTION III PAGB The PRisM-Droi»TKV 99 A Metric System of Numbering and Measuring Prisms 105 1. The Relation of the Prism-Dioptry to the Meter Angle Ill 2. The Relation of the Prism-Dioptry to the Lens-Dioptry II5 The Perfected Prismometer 125 The Prismometric Scale 139 On THE Practical Execution OF Ophthalmic Prescriptions Involving Prisms • • 151 A Problem in Cemented Bi-Focal Lenses, Solved by the Prism Dioptry 157 Why Strong Contra-Generic Lenses of Equal Power Fail to Neutralize Each Other. i6i The Advantages of the Sphero-Toric Lens 167 The Iris, as Diaphragm and Photostat 173 The Typoscope 183 The Correction of Depli-tted Dynamic Refraction (PRtiiKYOPiA) 185 SECTION I OPHTHALMIC LENSES THEIR REFRACTION AND DIOPTRAL FORMUL/C WITH THIRTY-SIX ORIGINAL DIAGRAMS REFRACTION. § 1. The change wrought in the direction of oblique rays of light, on their passage from one transparent medium to another of different density, is called Refraction. As our proposed treatment of its manifestation by lenses is to be strictly elementary, we must first define the law of refraction, at least so far as applied to parallel rays, in air, impinging upon and pass- ing through transparent optical glass. In the accompanying diagram, Fig. 1, a piece of glass of considerable thickness, having parallel surfaces a-& and c-d, is presented as an isolated vertical section a-h-c-d, which is ex- posed to the oblique ray i in the same plane of the surrounding air. S?| p"! A For convenience we shall term the ray prior to its contact with the glass, the incident ray, i; the ray during transit within the glass, the refracted ray, e* e^ ;* and the refracted ray after exit, the final ray, f. * The use of superior indices wiU not prove conflicting j>s algebraic Taluee are excluded. 13 14 REFRACTION. § 2. Refraction manifests itself by an acute bend in the direction of an oblique ray of light, i, at the point of incidence, e^, in passing from one conducting mediimi to another, a-b-c-d, of different density. Hence, a ray passing from one into and through another medium is bent both at the point of entrance e^ and of exit c". /''^^y- Fig. 1. By virtue of the deflection or bend alluded to, the incident ray, (', must include a different angle, a, with the perpendicular p^ from that, Z^, of the refracted ray e^ e" ; and it is by the trigonometrical values s and s^ of these angles, which have been found to bear a constant proportion to each other, that we are enabled to give expression to the amount of deflection sustained by a ray in passing from one medium to another. § 3. Experiment has shoMoi that the proportion *r remains a constant value for any obliquity of a ray incident to the same medium, and yet, that it possesses a different value by substituting one medium for another. It has therefore been considered expedient to establish the value of -^ for all transparent media, in the specific case of a ray passing from air into them ; such values being known as the refractive indices of the substances. To illustrate the graphical method by whicli we may arrive at the di- rection of the refracted ray, when tlio index of refraction and the direc- tion of the iucident ray are known, we shall select the index for crown KKFRACTTON-. 15 glass = l.o, bv introtliicin*; the proportion ^, =^ ^ = 1.5 in the construc- tion as follows : After erecting the perpendicular p^, take from a scale of equal parts the value for s = ."), and transfer it between c' and h^ beneath the ray i, upon the line c^ b. In the same manner transfer the value for s^ — 2 from e^ to a^, upon the line e^ a, and in both of these points, &^ and a^, erect perpendiculars. The perpendicular at ¥ will intersect the ray, i, at a point, v, which limits the radius of a circle drawn from c' as a center ; and by the circle's intersection with the perpendicular at a} the point, x, defining the direction of the ray, e^ e-, is fixed. § 4. As a ray of light is propagated backwards or forwards on the same path, the index of refraction from a denser medium into air is the inverse proportion from that of air into the medium, hence ~ is the proportion by which the direction of the final ray, f, is to be determined when the direc- tion of the ray, e^ e", is known. We therefore erect at e- the perpendicular p^ and transfer the value of s^ = 2 beneath the ray e^ e- from e- upon the line e~ d; likewise the value for s = 3 from e- upon the line e- c, and erect, as before, in the points d^ and c^ the perpendiculars. The perpendicular at d'^ will intersect the ray, e^ e-, at a point, y, limit- ing the radius of a circle from the point e^; the point, z, at the circle's intersection with the perpendicular in r'^ establishing the direction of the final ray, /. As p^ and p- are parallel from the construction it follows that the ray / is parallel to i, and therefore of the same direction. PRISMS. § 5. Pursuant to the spirit of our intention to avoid mathematical for- mulaB, we shall seek to arrive at a conclusion respecting the deflection in- curred by a ray in passing through a medium with oblique plane surfaces, confining ourselves as before to isolated vertical sections. Fig. 2. Fig. 3. Specifically, we shall select two right-angled prisms of var3'ing angles, «* and a', with the rays t* and r incident perpendiculary to the vertical sides a'h^ and a^b^, so as to avoid refraction on the incident sides, as shown in the vertical sections, Fig. 2 and Fig. 3, respectively. At e- and e^ the rays i' and i^ suffer refraction in the proportion - • =-|, according to §4, and which, if carried out in the construction, as before indicated, deter- mines the directions of /- and /"', respectively, as shown. In the future we shall have occasion to refer to the line dv, which is the perpendicular from v upon a line coincident with the ray i^ when the latter is parallel to the base b^c^ of the prism. Fig. 3. Under such circumstances the displacement dv of the final ray P is associated with a mathematical dependency upon the angle, a^, of the prism, and the index of refraction -JL. See page 108. 16 PRISMS. 17 From tliG construction it follows that the ilnal ray f^ (Fig. 3) intersects the horizontal line /<° at n^; and /- (Fig. 2) at a more distant point, n-, not shown. By a comparison of the prismatic section Fig. 2 with Fig. 3, we observe that by a decrease of the angle from a^ to a- the perpendicular p^ has a greater tendency to parallelism with the horizontal line /i° than />*. 8uch parallelism being realized — when a- c^ is parallel to (r h- or a^ = 0° — would result in the value s^ vanishing in the incident ray i~, and s in the final ray f-, by virtue of the decrease of the angles of incidence and refrac- tion in the proportion 3 to 3, thus establishing the coincidence of the inci- dent and final rays, and placing the point (n-) of intersection at infinity respecting the horizontal line Zi". § 6. In general we may therefore be permitted to assume that the greater the angle, a, of obliquity of the surfaces (Fig. 4) the greater will be the de- flection of the final ray f, and the closer to the base he of the prism will be its intersection n with the horizontal line h^. In looking through a prism the refraction manifests itself by an apparent change from the true position of an object 0, to that of its image OS when viewed from the point n. The author's suggestion that this phenomenon should form the basis of compari- son in measuring prisms was adopted by American manufacturers in 1895. The unit of prismatic refraction is equal to a deflection (dv. Fig. 3) of one centimeter at one meter's distance, and is called the prism-dioptry. § 7. By confining our observations to the relative directions of the inci- dent and final rays, we may easily memorize the law of refraction, for a medium included within plane surfaces, in the following manner: 18 PKISMS. 1, a. The direction of a ray remains unchanged in passing through opposite parallel surfaces of a transparent medium, or b. The incident ray i and the final ray / are parallel when the former is projected obliquely upon a transparent medium included within parallel surfaces. 2, a. The direction of a ray is changed in passing through opposite oblique surfaces, by a deflection of the final ray / toward the region of their gi'eatest distance apart, or b. The incident ray i and the final ray / are obliqvie when the former impinges upon a transparent medium included within oblique surfaces, or c. The apex of the angle formed by an obliquity of the incident and final rays is always directed toward the apex of the angle of obliquity of the surfaces. The law of refraction (2) finds its graphical demonstration in the follow- ing figures, wherein we have introduced the medium glass as being inter- sected by imaginary vertical and horizontal planes, V and H, coordinate at ihe point of exit e- for the final ray /. Fig. 5. Fig. 6. Prism, Base vertical; Refraction horizontal. Prism, Base horizontal; Refraction vertical. § 8. The Figures 5 and 6 are of particular interest to us, as they illustrate a very vital element in our future consideration of the refraction by cylin- drical lenses, namely, that the refraction is strictly confined to the plane whose intersection with the mediimi defines the obliquity of its surfaces. Thus, for an obliquity of the surfaces in the horizontal plane H (Fig. 5), PRISMS. 19 ■we find tho refraction active in the horizontal plane (i" to p), and for an obliquity of the surfaces in the vertical plane 1' (Fig. (>)? the refraction is active in the vertical plane (i^ to f^). Here, in the sense that the final rays are confined to the plane of inci- dence, we may term the refraction passive in respect to its right-angled coordinate plane. Thus in Fig. 5 the refraction is passive with regard to the vertical plane, and in Fig. witlr regard to the horizontal plane. Fig. 7. Prism, Base oblique; Refraction diametrically opposed. § 9. It is evident that the refraction is active in one and passive in the other plane for a medium whose surfaces are oblique in but one plane, so that to obtain the refraction active in both fixed planes an obliquity of the surfaces relative to each plane would he necessary. In such a medium (Fig. 7), if we consider the refraction merely with regard to the horizontal obliquity of the surfaces, the final ray would take the direction P-Ji, and, if independently for the vertical obliquity, the final ray would assume the direction f^-v. Therefore, with due consideration to the obliquity in both planes, the refraction must include both properties of deflection and result in a final ray, /, which is directed to a point, m, defined by projection of the apportioned horizontal and vertical displacements, dh and dv. As this is a prism whose base is really set diagonally to the fixed right-angled coordinate system, the ray / must naturally be refracted in the direction of the greatest distance apart of the surfaces, through the point m, within tht; diagonally bisecting or oblique plane P. SIMPLE LENSES. § 10. Directing our attention to the effect produced by substituting a segment of a circle for the line or c- of the original prismatic section (Fig. 2), each succeeding point e~, e^, e* (Fig. 8) may be considered as one of a prism varying in its angle a-, a^ a* with that of its predecessor ; and if the construction be carried out for each incident ray i', v', i* the corresponding radial lines at the points, e-, e^, e* in this ease substituting the perpen- dicular jr heretofore mentioned, each final ray /-, /\ /* will be found to intersect an arbitrarily selected base line, /i°^ at the respective points nr, 71^, n*, to infinity. Fig. 8. Plano-convex section. In the so-called plano-convex lens. Fig. 8, the converging final rays f, f, p, corresponding to the more central incident parallel rays i^, i", i''* estab- lish points n^, n^, n' to infinity, and possess the remarkable feature of inter- secting each other at a common point F, termed the focal point, which is situated upon the central and direct ray i-f. According to § 4, rays ema- nating from the focal point F, will be emitted as parallel rays r', t®, %' and i. *AU future deductions refer exclusively to such rays. 20 SIMPLK LENSES, 21 The points n*, n^, rr, toward the leus, correspond to the more eccentric in- cident rays, and, in the sense that these fail to assist in tlic harmony of a union of the final rays at the focal point, are to be considered a disturbing element, giving issue to what is termed aberration. In tlie plano-concave lens, Fig. 9, the final rays p, /", f are emitted as diverging rays, which may be considered as emanating from the so-called virtual focal point F, situated on that side of tlie section in which the rays arc incident. Fig. 9. Plano-concave section. § 11. For either of the above lenses it is also obvious that the more acute the curvature of the circle, the greater proportionately will be the angles a^ a^, a*, limiting the obliquity of the surfaces, and the closer to the lens will be the focal point F. Further, as the curvature of the circle is de- pendent upon the dimensions of the radius, the latter must prescribe the dis- tance, D, of the focal point from the medium or lens whose index of re- fraction is known. This relationship involves mathematical formulae for which we refer the reader to special treatises* on the subject. The greater the deflection of the final rays p, f', f, the shorter will be the distance D, or, for an increase in the refraction we have a corresponding decrease of the focal distance. Hence we say that the refractive power of a lens is in inverse proportion to its focal distance. * Elementary Geometrical Optics, W. Steadman Aldis, M.A., Cambridge, 1886. Hand Book of Optics for Students of Ophthalmology, W. N. Suter, B.A., M.D., New York, 1899. 22 SIMPLE LENSES. § 12. If we express the unit of refraction by the numeral 1, for a lens whose focal distance D is equal to one meter or 100 centimeters, lenses of two. three, or four times the refraction would find the expression of their focal distances in ^, -J, \, the focal distance of the unit, or 50, 33J and 25 centimeters, respectively. Tlie unit above mentioned has been termed the Dioptry, and is now the standard of refraction in optometrical practice. Values beneath the unit are designated as 0.25D.,* 0.50D., and 0.75D,, their respective focal dis- tances being four meters or 400 centimeters, two meters or 200 centimeters, and one and one-third meters or 133-J centimeters. Those values which are higher than the unit are expressed in whole numbers, including their intervals as above. See page 47. § 13. Assuming the medium to divide the aerial space into negative and positive regions (Figs. 8, 9) as indicated by the sign — (minus) on the in- cident side of the mediimi, and the sign -\- (plus) behind the medium, we shall find the focal point on the positive side for all convex, and on the negative side for all concave lenses. In this sense the refraction for convex lenses is considered positive, and for concave lenses negative. Hence Fig. 8 is + ID., and Fig. 9 is — ID. ^M..........± t Fig. 10. § 14. By substituting, in Fig. 8, for the plane side, a curvature c^ con- centric with C-, the refractive effects of the sections. Fig. 8 and Fig. 9, are virtually united, as shown in Fig. 10. Owing to the concave curvature c\ the incident ray i will assume the direction e^-e", being coincident with the focal point F, which may also be practically accepted as the focal point for ''D here being the abbreviation for Dioptry. SIMPLE LENSES. 23 the convex cnrvahire r-, provided tlie thickness, t, of the medium is created infinitely small in proportion to the radii r^ and r". Eays emanating from the focal point F^ for a convex curvature c', being emitted as parallel rays, § 10, it conditionally follows that the ray / will be parallel to the ray i. The neutralization is the more complete when the curvatures c^ and c" are identical, and are brought in contact as shown in Fig. 11, which, however, is a special case. § 15. Hence, in a pair of united convex and concave sections of identical curvature, it follows that the effect of the one is neutralized by the other, respecting the existence of a focal point on either side of the medium. This, however, is strictly only true for lenses weaker than 9D. § IG. If the opposite curvatures be unequal, the final rays will unite at a focal point on that side of the medium which corresponds to the focal point of the more acute curvature. Fig, Periscopic convex section. Positive meniscus. Periscopic concave section. Negative meniscus. Eeferring to the periscopic convex and concave sections. Fig. 12 and Fig. 13, respectively, if we consider the refraction merely with respect to the front curvature c^, disregarding the existence of a terminating back sur- face, the incident ray i will assume the direction of the ray e^ /S toward the focal point F^, then within the medium. Accepting the plane e--p to be the limit of the medium, the ray e^ e^ 24 SIMPLE LEJSfSES. would suffer a second refraction, and result in the ray e--f-, directed to the focal point at F-. To eliminate this second or augmented refraction, it would be necessary for the ray e^ e^ to impinge upon the back surface c- perpendicularly at e*. A surface effecting this is obtained by giving it a curvature c- prescribed from the point F'^ as a center, in which specific event the ray e} and f^ with refraction in the vertical plane. In a practical experiment in which the lens is held at some distance from the eye, convex cylindrical refraction manifests itself by an apparent increase, and concave cylindrical refraction by an apparent decrease in the dimensions of an observed object in that plane which is at right angles to the axis. In the axial plane, the refraction being passive, corresponding dimensions remain unchanged. Fig. 20. Axis vertical; Refraction horizontal. Axis horizontal; Refraction vertical. Plano-concave Cylindrical Lenses. § 21. To obtain cylindrical refraction of equal amount in both planes, thereby reducing the focal line to a focal point, it would be necessary to combine two identical cylinders, or, to create a single lens whose opposite surfaces are right-angled coordinate cylindrical elements as shown in Fig. 21. Under such circumstances, however, the focal line PFH^ for the front surface c^ is slightly closer to the face of the lens than the focal line l-F^2' for the back surface c". Aside from this, in making a bi-cyliudrical lens it is difficult to insure the chief planes of refraction being strictly at right- angles to each other, so that failure in this is certain to increase the aber- ration. SIMPLE LENSES. 27 § 22. The greater the distance apart of the surfaces, c^ and c-, the greater will be the al^orrative distance, F^ to F-. Yet, as the tliickncss of the Fig. 21. Double or Bi-cylindrical Lens. lens may generally be accepted as a vanishing quantity in proportion to the focal distance, we may consider a common focal point to exist for both refracting surfaces. Fig. 22. Plano-convex Spherical Lens. § 23, Practicall}-, however, it would be better to create a single surface capable of producing this amount of refraction in the vertical as well as in the horizontal plane. With this object in view, we shall select the isolated vertical section described in § 10, and cause it to be rotated upon the central incident and direct ray, i-f, as its so-called optical axis, whereby a plano-convex spherical lens is obtained. See Fig. 22. Similar rotation of the sections Figs. 9, 12, 13, 14, and 15, would result in the so-created spherical lenses being charact'^rized by the pectioTis employed. 28 SIMPLE LENSES. It is evident that the incident and final rays will retain their relative obliquity during the rotation, so that all incident parallel rays have their corresponding final rays in the resulting cone whose apex is at the focal point F. To further illustrate, we may take advantage of § 9 in its application to a medium having only one surface which is spherically curved, and con- sequently obli Fig. 27. Concave Cylindro-cylindrical Lens ( — c* axis 180° c^ axis 90°). d> d2 ^,Ar:^^ «• el \ \ .L_. e" ft 'Z^>^^i i ! .G Fig. 28. Concave Sphero-cylindrical Lens {— s^ ^ — d^ axis 90°) — Double Form. Fig. 29. Ooncavs Sphero-cylindrical Lens ( — s^ O + c^ axis 180°) — Periacopic Form. In the above figures, i", e", and/" are associated with horizontal, and i», c*, and/* with vertical refraction. COMPOUND LENSES. 3. CONTRA-GBNERIC MERIDIANS (CONVEX AND CONCAVE). § 37. Hitherto we have considered different amounts of refraction, re- stricted to the same type, convex or concave, for the principal right-angled Fig. 30. Concavo-convex Cylindro-cylindrical Lens { — c' axis 90° ':i^ -[- c^ axis 180°). sections. In contradistinction thereto, and as a final complication, we may combine in a lens different or even like degrees of refraction, though of opposite type; namely, convex in one and concave in the other diamet- rically-opposed coordinate meridian. As an instance, we may select the compound lens Fig. 30, represented as consisting of a plano-concave, c% and a plano-convex cylinder, c", so combined as to place their active me- ridians at right angles to each other. 39 40 COMPOUND LENSES. Independently considered, each cylinder c^ and c- would have its focal line Z^ F^ l^, and V^ F- P, of original magnitude in the region of its sign — and + respectively, and consequently on opposite sides of the lens. k^- Fig. 30. Concavo-convex Cylindro-cylindrical Lens ( — c' axis 90° C: + c^ axis 180°). When the cylinders are associated, however, the final rays, which would have been restricted to the limits of the focal line F F~ P for the cylinder c^ will, by virtue of the dispersive effect of the cylinder d in the horizontal plane, be confined to an augmented focal line d- F~ d~, within the limits d--d~, for the outermost rays emanating from the point F^ of the virtual focal line, P F' I\ By a similar method of reasoning to § 2G, all final rays within the limits of the circle C will be accorded associated vertical and horizontal refraction culminating in their united intersection of a line d~ F~ d-, of the hori- zontal plane in the positive region behind the lens. Interception of these rays, by successive transverse vertical planes, will make manifest a dem- onstration of similarly arranged ellipses, respecting their greatest and least diameters, before and behind the focal line d"^ F- d-. By projecting the final rays into the region of their apparent emanation from before the lens, we would obtain a similar increase of the virtual focal line V- F^ V- to the magnitude d^ F^ d^, and to a reversal of the so-defined ellipses, respecting their greatest and least diameters, as shown by the dotted lines in the nega- tive region (Fig. 30). CONTKA-GENEKIC MEK1DIAN8. 41 § 38. Identical refraction is also preferably obtained in this instance bj eombinationp of spherical and cylindrical surfaces. Fig. 31. Concavo-convex Sphero-cylindrical Lena (+ s^ r' axis 90°) The combination of a convex spherical surface with the active meridian of a stronger concave cylinder creates a periscopic section which is concave ; whereas the combination of a concave spherical surface with the active meridian of a stronger convex cylinder results in a periscopic section which is convex. The identity of the refraction for these combinations becomes apparent by reference to the concavo-convex sphero-cylindrical lenses Figs. 31 and 32, in which, by a judicious selection of the respective spherical ipi ..i--"" « U-'- Fis. 32. Concavo-convex Sphero-cylindrical Lens ( — s^ O + ^* axis 180°). and cylindrical curvatures, according to § 32, the demanded positive and negative elements of refraction for the principal meridians of the crossed cylindrical lens, Fig. 30, are fulfilled. 42 COMPOUND LENSES. To illustrate the equality of formulae characterizing these equivalents, we refer to their correlative sectional diagrams Figs. 30c, 31c, 32c, in the order following : Fig. 30c. Fig. 31c. Fig. 32c. Formula Ic. — 1.5 cyl. axis 90° C + 3.5 cyl. axis 180°. (Fig. 30.) Refraction : ) , . . _^ Fio' 30c r — horizontal .^ -\- 3.5 vertical = — 1.5H 3 + 3.5V. Formula lie -f 3.5 spherical C — 5 cyl. axis 90°. (Fig. 31.) FirtlT * 1 — 5 + 3.5 horizontal C + 3.5 vertical = — 1.5H C + 3.5V. Formula IIIc. — 1.5 spherical C + 5 cyl. axis 180°. (Fig. 32.) Fig. 32c. \ Eefraction : j 1.5 horizontal 1.5 + 5 vertical = — 1.5H C + 3.5V. §39. These lenses being equivalents (see § 32), we here give the only rule required, for reasons later given (§40), for converting the cylindro- cylindrical lens (Formula Ic) into the concavo-convex sphero-cyHndrical lenses (Formulae lie and IIIc). Rule 3. Place the sum of both numerals of refraction as the numeral of the newly-created cylindrical element, giving to it both the sign and axis of either cylinder, and combine with the neglected cylindrical numeral and its associated sig^ as spherical. CONTKA-GENEKIC MEKIDIAIS'S. 43 Comparison of the iieriseopic lenses Figs. 26 and 29 with the lenses Figs, 31 and 32, respectively, exhibits a striking similarity in construction. The characteristic difference between them is that in the latter the cylindrical exceeds the spherical refraction, whereas in the former the reverse is the case. § 40. In a case of mixed (contra-generic) astigmatism, demanding the foregoing correction, it becomes necessary to determine the chief meridians — 1.5 and -\- 3.5 independently of each other, thereby obtaining the com- bination expressed by Formula Ic, as by an endeavor to correct through introducing a spherical element in any proportion or wholly of either equivalent (Formula lie or IIIc), an improvement in one meridian would always be attended by a proportionate derangement in the other, with a probability^ of the patient failing to appreciate the benefits of its applica- tion. It is only in consequence of this fact that the lenses of the Formulae lie and IIIc are rarely the direct result in subjective optometry, whereas, in cases of regular astigmatism with congeneric meridians, the lenses Ila, Illa and lib, 1 1 lb arc most apt to be. § 41. Astigmatism has, in the main, been attributed to asymmetry of the cornea, though the crystalline lens is often found to be implicated; yet, specifically, in a case of mixed astigmatism, in which the crystalline lens does not assist, it is improbable that the corneal surface can ever be of the form requisite to include reversed curvatures. Fig. 3G. In such instance the ametropia is rather more apt to be one in which an opposite type of astigmatism is in excess of an existing hypermetropia or myopia, respect- ively. Accepting tliis to be the case, such an eye would fall heir to the features accredited to hypermetropia or myopia respecting the "nodal points" and "amplitude of accommodation;" wherefore, in prescribing either of the aforesaid sphero-cylindrical equivalents, a preference might be given to that form wliich would be commensurate with the inherent physical and physiological developments above alluded to. TORIC LENSES. CONGENERIC AND CONTRA-GENERIC MERIDIANS. § 42. The properties of astigmatic refraction are also fulfilled in a lens by creating for it, opposite to its plane side, a single surface whose diamet- rically opposed principal meridians are of unequal refraction. Fig. 33. Fig. 34. Such a surface, called a torus, is shown in Fig, 33, wherein the curva- ture c^ of the radius r^ and refraction 3D. is rotated upon a vertical axis R so as to create the curvature c^ whose radius r- is chosen to produce 2D. In Fig. 34 two lenses are shown to be included w^ithin the surface so de- veloped and an opposite plane side, the one being a plano-convex toric lens L^ — the other a i)lano-concave toric lens L-. From the construction it follows that these lenses are each possessed of 3D. of refraction in the vertical, and 2D. of refraction in the horizontal meridian, so that the formulae for the same may be expressed by 44 TOEIC LE^'SES. 45- (AJ [+ r>D. Ref. 90° Z + 3D. Ref. 180°] Tor {L') (Bj) [— HD. Rof. 90° '2 — 2D. Ref. 1S0°] Tor (/>=) as a distinction to the correlative fomuila} Ay and B^ for a pair of crossed cylinders of identical refraction, (A.) + ;5 cyl. axis 180° C + 2 cyl. axis 90° (Ba) — 3 cyl. axis 180° C — 2 cyl. axis 90° and their sphero-cylindrical equivalents, respectively : ( + 2 sph. C + 1 cyl. axis 180° (Double Form). ^ ' 1 + 3 sph. C — 1 cyl. axis 90° (Periscopic Form). \ — 2 sph. C — 1 cyl. axis 180° (Double Form) . ^ ''^ i — 3 sph. C + 1 cyl. axis 90° (Periscopic Form). § 43. The rotary body shown in Fig. 33 may also be considered to have "been created by bending a simple cylindrical lens c^ to the radius r-. In such an attempt, the lens of the Formula A^ might 1)e obtained by bending a 3D. cylindrical lens to that radius, which effects a refraction of 2D., or a 2D. cylindrical lens to a radius producing a refraction of 3D. In the latter case the lens would merely require to be turned 90° so as to correspond with the rest of the formulae The inner or back surface would naturally also require to be restored to a plane, as indicated by the dotted parallelogram in Fig. 33. The suggested method being impracticable, the process of grinding must be resorted to; although this at present involves more complicated ap- paratus. A lens having one surface spherical and the other toric is called ■& sphero-toric lens. Its advantages are explained in a subsequent paper. § 44. A toric surface with contra-generic meridians is shown in Fig. 35, wherein the concave section c^, of the refraction — 3D., is rotated upon the vertical axis R, so as to create the convex curvature c-, whose refrac- tion is + 2D. 46 TOEIC LENSES. In Fig. 36 two plano-toric lenses L^ and L~ are shown to have the same toric surface as in Fig. 35, As the principal meridians in each of these lenses are convex and concave, we may write their formulas as follows: (Ci) [— 3D. Eel 90° C + 2D. Kef. 180°] Tor {U) (Di) [+ 3D. Kef. 90° C — 3D. Kef. 180°] Tor (L=) so as to distinguish them from the correlative formulae for crossed cylin- ders of identical refraction : (Co) — 3 cyl. axis 180° C + 2 cyl. axis 90° (Do) 4- 3 cyl. axis 180° C — 2 cyl. axis 90° and their sphero-cylindrical equivalents, which are respectively: j + 2 sph. C — 5 eyl. axis 180°. (^a) I _ 3 gph. C 4- 5 cyl. axis 90°. j — 2 sph. C + 5 cyl. axis 180°. (^3) I -I- 3 sph. C — 5 cyl. axis 90°. The student should practise transforming optionally chosen formulae by applying the rules given in § 34 and § 39, when he may refer to the appended tables to verify the correctness of his own work. NUMERALS OF Metric System REFRACTION Inch System Focal Distances Focal Distances Centimeters DiOPTRIES Approximates U.S. Standard Inches 400. 0.25 1:160 157.V 200. 0.50 1:80 78| 133.3 0.75 1:53 52i 100. I. 1:40 39| 80. 1.25 1:32 31 1 66.7 1.50 1:26 264- 57.1 1.75 1:22 22JI 50. 2. 1:20 ISri 44.4 2.25 I:I8 17.- 40. 2.50 1:16 15i 36.4 2.75 1:14 14t\ 33.3 3. 1:13 13i 30.8 3.25 1:12 12| 28.6 3.50 I:II 4 25. 4. 1:10 23.2 4.50 1:9 4 20. 5. 1:8 7| 18.2 5.50 1:7 n 16.7 6. l:6i 6A 15.4 6.50 1:6 6 14.3 7. l:5| 54 12.5 8. 1:5 4B 11.1 9. 1:41 4| 10. 10. 1:4 311 9.1 II. 1:31 3^^ 8.3 12. l:3i 3A 7.7 13. 1:3 3^^^ 7.1 14. I:2| 8|| 6-7 15. 2f 6.3 16. l:2| 2iV 5.5 18. l:2i 2t\ 5. 20. 1:2 IH 2.5 40. 1:1 If The above table has been arranged for comparison of the metric with the old system of numbering, in which 1 inch was adopted as the unit. A lens of 10, 20 or 40 inches focus is therefore represented as being ■^^, -^j^, or ^^ of the refraction of the old standard. The focal distances have been calculated upon the basis that 1 meter =^ 100 centimeters = 39.87 U. S. standard inches, through dividing each of these equivalents by the dioptral numerals. To render a harmony of the numerals of the two systems possible, it is found necessary to neglect slight fractional variations, as shown in the differences between the divisors in the 3rd with the figures of the 4th column. 1 dioptry being placed as equivalent to 5'^, lenses of 2, 3, or 4 dioptries may be calculated as ^*g = ^, ^^ --^ -^j, or ^*g =^ ^g, respec- tively, without m.iterially conflicting with the practical demands upon accuracy in a sub- stitution of one system of numerals for the other. 47 DIOPTRIES +025C.90' -H).50C90 +0.75C.90' + l.00t.9 0" + I.85CBTD' + l.5Da90' + f.75C90' -f£.OOC.9 0' +2.25C.90* +2.50C.9 +27 5(190'' +3.00 ceo' +3.2 5C.0O° +3.500.90" In the a del With the cy I. TABLE OF CROSSED CYLINDERS AND THEIR SPHERO-CYLINDRICAL EQUIVALENTS CONGENERIC MERIDIANS (CONVEX) OlOPTRIES +0.250 180° +0.500.160° +O75C.I80° + 1.00C.I BO* + I.25C.I BO" + IB0C.ia0° + 1.760.180" +2000180° +225tl60° +2S0C.1 80° +275C.1B0' +3.000.180° +325C.180° +3.S0C.IB0° +0250,90" B *'+0.2 50+02 5e +0.500-0250) *+0250+o.5 0e +0 7 30-0.5001 'Va25O+07 5e + I.0DO-075O) +0.2 50+1.006 + 1 250-1 OOO) +025O+I256 + 1.5 DO- 1250 +0250+1.606 + 1750-1. 500 +0250+1756 +2.OOO-1750 +02 50+2.006 +225O-aOO0 +02 50+2256 +25 00-2250 +0250+2.506 +2750-2500 +0250+27 56 +3.OOO-2750 +0250+3,006 +3250-3,000 +02 50+32 56 +3500-3250 +<150C90° ^+0.253+0 250) +050C-0.2 5e MM ■ "+0.600+0256 +0750-0250) "+0500+0506 + 1 000-0500) "+0500+0756 + I25O-075O) +0.500+1.006 + 1.5DO-I.OO0 +0500+1256 + 17 50-1.250 +050O+I.506 +2.OOO-I.5O0 + 05 004-17 56 +2250-1750 +0S0O+2.006 +2SOO-2iJO0 +0500+2256 +2750-2250 +0500+2506 +3000-2500 +OSOO+2.750 +1250-2750 +05 00+3.006 +3 500-3000 +075C9D° *'+0.2 52+0.5 OH ^075C-050e '+05 00+0.2 50) +0750-02 56 +07 50+02 56 + 1.00O-0.2 5O) +0750+0506 + 1.250-0.500 +07 50+0756 + 1,500-0750 + 0750+1.006 + I7 5O-I.OO0 +0750+1.2 56 +2.ODO-1.250 + 07 50+1506 + 2250-1.500 +07 50+ 17 56 +25 DO- 17 50 +C750+2.009 +27 5O-2.OO0 +0760+2256 +3000-2250 +0750+2506 +3250-2500 +07 50+2756 +350O-275(I> + I,00t9 0* *'+02 5C+075 + l75C-l.50e + 05 00+ 12 50) + 1750-1256 +D75O+I.0 0O1 + 1.750-1.006 + 1 000+0750) + 17 50-07 56 + I.25O+O.SO0 +1750-0506 + I.6OO+O2S0 + I75O-0.256 (,.) + 1. 750+0256 +2,000-0250 +1750+0506 +22 50-0500 + 1750+07 56 +26 00-07 50 + 1 750+l.DOe +2760-1.000 +1750+1256 +300O-I 250 +175O+I506 +325O-15O0 +1750+1756 +3500-1760 ■tE.OOC.OD" +0.250+175(1 +2.000-17 56 + O.5OO+I.5O0 +2000-1.509 +07 50+ 1250) +2.000-1.256 + 1.00O+I00O) +2000-1.006 +I25O+O750 +200O-D756 + 1.SDO+D5O0 + 2000-0.506 + I75O+O.250 +21)00-0256 +2.00O+0.256 +225O-Q2S0 +2.0 00+ 05 06 +25OO-O5D0 +2000+07 5© +2750-0 7 50 +2,0 00+1,006 +3,000-1 000 +2.000+1256 +3250-1250 +Z00O+ 1.506 +3.5 DO- 1500 +2-25 C.BO* + 02 5O+200(D +2.2 50-2.006 + 0500+1750) +2250-1756 +075O+1 500) +2250-1.506 + 1.0 00+ 1.250) + 22 50-1,256 + I25O+1.OO0 + 2250-1 006 + 1.5OO+O7 50 +2250-07 56 + 1.7 5O+O.5O0 + 2250-0506 +2.000+0^50 + 2250-0.256 B +225O+02S6 +25OO-O2S0 +2250+0506 +2750-0 6 00 +2250+0756 +3 00-07 50 +2250+1.006 +3250-1000 +2250+1.256 +35 00-1250 +2 50 0.9 0° +02 50+22 Sir +2.500-22 56 +0S0O+200O) +2500-2006 +0750+1750) + 2.50O-I756 + 1.00O+150O) +2.5 00-1506 +1250+1 250 + 2500-1256 + ISDO+1.OO0 + 2500-1.006 + 1.75O+O750 + 25 00-07 56 +2000+0500 +2500-0506 +22 50+012 50 +2500-0250 B +250:+0256 +2 750-0250 +2500+0506 +3000-0500 +2500+0756 +3250-07.50 +25 00+1.006 +J5OO-1.DO0 +2750.90° + 02 50+2.5 00) +2 7 50-2 506 + 0500+2250) +27 50-22 56 +07 50+21)00) +2 750-2006 +1OOO+I750 + 2750-1.756 + 1.250+1500 +2760-1 506 + 15OO+I.250 +2750-1 256 + 1.75O+I.OO0 +2750-1006 +200 0+075 +2750-0756 +Z2 5 0+05 00 +2750-0506 +25 00+02 50 +275O-D2 50 -B +2750+0256 +3OOO-O250 +2750+0500 +32 50-0500 +27 50+0756 +1500-0750 +5OOC.9 0' + 02 50+27501 +3.0 00-27 56 +0500+2500) +3000-2 506 + 07 50+2.250) +3000-2256 + I00O+200O) +J00O-2.006 +1250+1750 + 3.00O-I756 +1 500+1 500 + 3,000-1506 + 1750+1 250 + 3000-1.256 +2.000+1.000 +300O-I.006 +2250+0760 +3.000-0 756 +2500+0500 +3000-0500 +275O+D2S0 +3D0O-0256 , H-iom +3000+0256 +3250-0250 +3.0 00+0506 +3500-0500 +3.E5CiO° + 0.25O+3.00OI +3i5O-3D0e +0 500+27 50) +3250-27 56 +07 5O+2.50O) +3 2 50-26 06 + 1.O0O+ 2.250) +32 50-2256 + I25O+2.OO0 +3250-2006 +1 50 0+17 50 +3,250-1 7 56 + 175O + 1.5O0 + 3.250-1506 +2OOO+I.250 +3.250-1256 + 2250+1.000) +3250-1 006 +2500+0750 +3250-07 56 +2750+0500 +3 25O-D506 +300O+O25O' +12 50-0 256 B +3250+0256 +3.5OO-O.2S0 +3.50C.9ef + 0J 60+3.250) +15 00-3256 + 0.500+3000) +3500-31)06 +0750+2750) +J50O-2759 + I00O+250O. +3500-2506 + 1.250+1250 +3500-2.256 + l,5OO+aOO0 +3.50O-2DD6 + 1750+1.750 +3.500-1 756 +2.OOO+1.5O0 +350O-I506 +2250+1250 +3iD0- 1256 +2500+1000 +J5OO-1.OO0 +2750+0750 +J5DO-0756 +300O+060O1 +3500-0506 +325O+O250 +35 00-02 56 In the above formulae the first numerals apply to spherical, and the second to cylindrical refraction. In the appended signs, the upright and horizontal diameters ( | and — ) of the circles denote the axes 90° and 180°, respectively. With the exception of the diagonal column of spherical equivalents, each field contains both the double and periscopic form of convex sphero-cylindrical equivalent. For crossed concave cylinders it is merely necessary to reverse the signs + and — wherever they occur. DIOPTRIES -0.25 C.9 0° -050C.90' -0.7 5 C.9 0' -i.oocao -I.25C.90* ■1.50C.90° •l.75C.gQ* -2.00 C.9 0* -2.2509 0° -2 5 C.9 0° -2.7 6 C.9 0° -3.00 C.9 0* -32 5 C.9 0* -3.50C90° In the abo the a Practical eqi For crossed DIOPTRIES -0.25C.90° -0.500.90° -0.75C.90° -i.oocao" -I.25C.90' 1.5 0.90° -I.75C.90* -2.00C.90* -2.2509 0° -25 0.90° -2.7 5C.90* -3.00C.90* -32 5C.90* -3.50C90° In the abo the a Practical eqi For crossed II. TABLE OF CROSSED CYLINDERS AND THEIR SPHERO-CYLINDRICAL EQUIVALENTS CONTRA-GENERIC MERIDIANS (CONVEX AND CONCAVE) HOPTRIES +025C.I80' + O50C.(80- +07 5 CI eo' + 1000.180° + 1 25a; 80" + 1 500180° 4 1,75 CI 80° +2.00 01 60° +2260180° 41500,180" +2750180° +300C.I 80° +1260,180° +J60ai80° -025M0* +a25C-OiO® -tt25.-+0.50e +a.B0c-a7M) -0.25C+076e +075C-I.00O) -o25:+i.ooe + 1 000-1.250) -0.250+1.259 41 250-1 500) -0.250+1509 + I.50O-175® -0,250+1, 7Se + 1 750-2,00® -0250+2 09 +2D0O-225® -0250+2269 42250-250® -025042509 42500-275® -0,25042769 +2750-300® -0250+3009 43000-325® -0250+3259 +1260-350® -0250+3509 +1500-375® -0250+37 5© -OSOC.SO" + 0.25C-0.75 -2.00C+2.259 +0.50:-250(I) -2DO:+2.50e +0750-2750) -2.000+2 7 59 41.000-3000) -2.0 0043.006 + 1250-325® -2,000+3259 4I50O-J50® -200043609 +1750-375® -20DO+3759 42DOO-400® -200044009 42250-425® -2,000+4259 42500-450® -2,00044509 +27 60-476® -2000+4759 +3,000-5,0 0® -2,000+5.000 +3250-526® -2flOO+5.259 +1600-5,50® -2000+5.50O -2.25050° +0.251-2500) -225:+2.50e +050:-275(D -226C+275e +0 750-3.000) -225O+3.009 41.0 00-3250) -225043259 + 12SO-3.50® -22 5 0+3509 41.500-375® -225043759 + 1750-4,00® -2250+4.009 42000-425® -225044259 42250-450® -2250+4509 42500-475® -2250+4769 +27 5O-5.00® -226O+5.0 09 +3X)0O-525® -2250+5259 +3260-5,50® -2250+5509 +15 00-57 5® -2250+575© -25 0C.90' +D.25C-2.76(I) -250C+2759 +O5O:-3flO0 -250;+3.0D9 +0 750-3.2 50) -^50o+3.^5e 4I.OOO-350ffl -250043.509 + 1.250-375® -2,500+3759 41,500-400® -2,50044009 + 1750-425® -2500+4259 +2000-450® -£500+4609 +2250-476® -25 00+4759 42500-500® -25DO+5.009 +27 50-52 5® -2500+5259 +3.00O-550® -2 500+5.50O +1260-575® -2500+5759 +1500-600® -2500+6009 -2.75C.30- + 025C-M0a) -2.7SC+3.009 +0.50C-3.25(Il -275C+325e +0760-3.5 0O -27 50+35 09 41,000-375® -27 5043759 + I25O-4.00® -276O+4D0e 41,500-425® -275044259 + 17 50-450® -2750+4509 +200O-475® -2750+4759 +2250-6,00® -2760+5,009 42 600-525® -2750+5259 +27 50-650® -27 50+5509 +3 0DO-57 6® -27 50+5.759 +1250-6,00® -2750+6,009 +360C-B25® -2750+6259 -3.O0C.9O' + 0.25C-3.25(I) -3.000+3.259 +0.50C-3.50(Il -3.oo:+35oe +0760-3750) -3.0 00+37 59 41.000-41)00) -3.00044J)09 +1260-425® -3.0 00+42 69 41,600-450® -3.00O44.509 + 1750-475® -3000+4759 +20 00-5,00® -3,000+5,009 +2250-525® -300045259 42500-5500 -3.0DO45 5 0G +27 50-57 5® -3.000+6769 +I0 0O-6.0 0® -3.00O+6.0O9 +1250-625® -3DU0+S25© +3500-650® -3.00O+6.6D9 -3^5 too" +o.2B:-3,5Da) -3.25C+350e +050C-375(D -3.25C+375e +0750-41)00) -3250+4009 4I.OOO-4260) -326044259 + 12 50-4.5 0® -325O+4S09 41.600-4,75® -325 0447 59 +17 50-5BO® -32 6 0+5.0 09 +2000-625® -3,25046259 4225O-550® -32 5 045509 +2600-57 5® -2250+5759 +275O-G.0 0® -325 0+6000 +3 0O-6.2 5® -3250+G.259 +3250-650® -325046509 +350O-a75® -1250+6759 -3.5DC.90" + 0.25C-375(B -3.50C+375e +0.500-4 00) -3.50C+4JJ09 +0 7 50-4.250) -3 50 0+4.2 59 41.000-4.600) -3.50C44609 + 1.260-475® -3.500+4759 + 1 500-5,00® -3,50045009 + 1750-525® -3500+5258 +2000-550® -3500+55D9 42250-576® -15 00+57 59 4250O-GJ)0® -160046009 +2750-626® -3500462SO + 3.00O-650® -350O+B50O +3250-6,75® -1500+6769 +1500-7.00® -1500+700© In the above formulas the first numerals apply to spherical, and the second to cylindrical refraction. In the appended signs, the upright and horizontal diameters ( | and — ) of the circles denote the axes go° and 180", respectively. Practical equivalents can only be produced when the numerals of the formula coincide with those included in the adopted dioptral series, page 47. For crossed cylinders having the axes of the concave cylinder at 180° and of the convex cylinder at 90° it is merely necessary io reverse the axes throughout. SECTION II DIOPTRIC FORMULA FOR COMBINED CYLINDRICAL LENSES APPLICABLE FOR ALL ANGULAR DEVIATIONS OF THEIR AXES WITH SIX ORIGINAL DIAGRAMS AND ONE HALF-TONE PLATE SECTION 11 DIOPTRIC FORMUL/E FOR COMBINED CYLINDRICAL LENSES APPLICABLE FOR ALL ANGULAR DEVIATIONS OF THEIR AXES WITH SIX ORIGINAL DIAGRAMS AND ONE HALF-TONE PLATE DIOPTRIC FORMUL/E FOR COMBINED CONGENERIC CYLINDERS I. DIOPTRIC FORMULA FOR COMBINED CONGENERIC CYLINDRICAL LENSES. 1. RELATIVE POSITIONS OF THE PRIMARY AND SECONDARY PLANES OF REFRACTION. In the following theorems, a prior knowledge of the established mathe- matical deductions applied to lenses for parallel rays incident in the immediate vicinity of the optical axis, and in which the lens-thicknesses are considered vanishing quantities in proportion to the focal distances is taken for granted. The formulae here advanced are therefore dependent upon those which have not been carried beyond first approximations. Practically, in almost all cases that occur, the thicknesses of the combined lenses are very small quantities compared to the other dimensions involved, so that we shall consider the cylinders to be so thin that their centers actually coincide, and in which case the focal distances are to be counted from a plane perpendicular to the optical axis, in the optical center of the combined lenses. In Plate I, two combined convex cylindrical lenses are shown, which, though somewhat at variance with the prescribed conditions of thickness, will, however, better serve to make our subject clear. The dotted circle shown within the lenses, with its center at the optical center o^ shall represent the plane above alluded to. The passive or axial planes of the cylinders are shown by dotted paral- lelograms at A and a, bisecting each other under the angle Aoa = ^ in the optical axis at o ; and their active planes of refraction C and f, which are 53 54 DIOPTRIC FORMULAE. of necessity at right angles to their correlative axial planes, similarly bisect each other at the same point. Hence, <^ Coc = <^ Aoa = y. The compound lens thus presented consists of two congeneric cylin- drical elements, each of which, independently considered, will have its corresponding focal plane, which, for convenience, we may term an elementary focal plane of the combination. Thus, E^ and E.^, at the focal distances f^ and /.,, are the elementary focal planes for the cylinders C and c, respectively. The cylinder C will consequently have the proj^erty of deflecting a ray, incident at D, perpendicularly from Z>j, in the plane E^, to the point Zj of the axial plane A^ Z^ ; whereas, the cylinder c will have the property of deflecting a ray incident at the same point, perpendicularly from Z>2, in the plane E^^ to the point K^, of the axial plane a^ o^. The greatest amplitude of deflection for C will therefore be D^Z^ in the plane E^, and for c will be D^ V^ in the plane E^. It is further manifest that the refracted ray D V^ V^, contributed by c only, in attaining its greatest deflection D.^ F^, in the plane E.^, will penetrate the plane E^ at V^, and in it present a proportionate deflection Z>i V^. D^ Zj and D^ V^, being amplitudes of deflection reduced to the same plane ^j, will then bear the same relation to each other as their corresponding refractions. Thus or, Z>iZ, = y-, when D^V^ = -j-, J\ Jn and which may easily be shown to be the case when the deflections are measured in a plane one inch from the lens.* Provided, therefore, that the deflections are measured, within the same plane, from a point D^ of the same line of incidence DD^, we may deter- mine the resultant of two deflections D^Z^ and D^V^^ for any angular deviation existing between them at A, by the physical law governing similarly united forces. E>^M^, as the diagonal of the parallelogram A Fj M^ Z^, will consequently be the resultant deflection accruing from a combination of the cylinders C and c. ♦Refraction and Accommodation of the Eye, by E. Landolt, M.D., Paris, translated by C. M. Culver, M.A., M.D., Philadelphia, 1885 (see page 58). CONGENERIC CYLINDBttS. 55 As each cylinder contributes a plane of active and one of passive refraction, we shall evidently obtain two resultant principal planes for their combination, the one of greatest refraction, commonly called the primary plane, DD^o^o, intersecting the angle Coc = y between the active planes of refraction C and c, and one of least refraction, termed the secondary plane, dd./).^o, intersecting the angle Aoa = y between the passive or axial planes A and a. The primary plane, in penetrating the plane E^, will consequently divide the angle C^o^c^ = Coc = y into D^o^c^ = « and Z>i^iC, = /5. In the plane £^ we shall then find the angles a and /3 to be directly dependent upon the associated deflections D^Z^ and Z?, V^ for the point D^. In the plane £^ a similar division of the angle A.^o,a.^, by the secondary plane, will be rendered dependent upon d.,v.^ and d.^z.^ for the point d.^. As to this, the diagram is believed to be sufficiently clear, without further reference. Since the resultants D^Jlf^ and d^?^i., define the directions of the refracted rays DAf^ and dm.^, it is further evident that for D and d to be points of the primary and secondary planes, respectively, they will have to be so chosen that D^M^ and d.^m^ shall be directed towards the optical axis oo^o.,_ ; and, as we shall later learn, this is but one of the restrictions which renders a diagram somewhat difficult of construction. The resultant deflections Z>i M^ and d.pi., are therefore shown in the primary plane, coincidetit with Z?,^i, and in the secondary plane coincident with d.p.j^^ respectively. For all intermediate points of the circle, the resultant deflections deviate from the optical axis. This has been taken advantage of in constructing Dr. Burnett's models, and in determining the directions of twelve refracted rays in each of the figures 2, Plates II and IV. The position of the primary plane DD^o^o, shown as dividing the angle C^o^c^ = y bo that r = « + /5, (1) will then be determined by fixing the relations existing between « and /?. In the plane E^, from the triangle D^Z^M^, we have D,Z, : Z,i7/, = sin < Z,M,D, : sin < Z,Z>,/7/,, < Z,M,D, = < D,o,c, = a, 56 DIOPTRIC FORMULAE. by parallelism of Z^M^ and c^o^ ; and, for similar reasons, .*. D^Z^ : Z^M^ = sin a : sin /J, .-. Z?iZi : D, Fj = sin « : sin z? (2) In the oblique plane DD^ V.^ we find D, V, ■D,V.,= DD, : DD, ; ar, as DD^ and Z?/?^ are the focal distances f^ and f.^ of the cylinders C aad jZj sin a /i D,V^ BiiiiS f^ W Since D^o^ is the radius of the circle indicated, we may, for convenience, ascribe to it the value 1. We shall then have Z?jZj = sin <^ A^i'2'i, < A^i^i = C,o,Z, — < n,o,C,. . •. < n,o,Z, = 90° — fi. .-. D,Z^ = sin (90° — ;5) = cos ,5. ... (5) In the plane A ^^ similarly find AF, = sin <^n,o,v„ .-. <^D,o,V, = 90°— «. .-. Z?,F, = sin (90°— «) = cos a. . . (6) CONGENERIC CYLINDERS. 57 Substituting the values for Z?, Z, and D^ V.^ from (5) and (6) in the equa- tion (4), we obtain, cos /J sin a yj cos a sin /? f., or, by multiplying both members of equation by 2 and transposing, 2 cos iS sin /? = 2 cos « sin « ^^ . •. sin 2/3 = sin 2a 4- (7) The position of the secondary plane dd^op^ shown as dividing the angle A^o.fl^ = y into d^o^a.^ = a and d^o^A^ = ,5, provided d^o^ is perpen- dicular to D^o^, will be determined by similarlj'^ fixing the relations between a and /?. Here it can also be shown that d^2 : d^v.2 = cos « : cos ,3 (8) 2 d,2^ :flf,0, =/ :/, (9) d^z^ = sin /? (11) d.,v.^ = sin a (12) whereby, as before, sin 2/9 = sin 2a ^ . We therefore conclude that : 1. The primary and secondary planes of refraction are at right angles to each other for any angular deviation of the axes of two combined congeneric cylindrical lenses. In a further consideration of the relation (7), sin 2/3 = sin 2 a-=y^,weob- serve that the sines of the angles 2a and 2/3, which are each always less than 90°, merely differ by the co-efficient ^--. If, therefore, /^ = /^, which is the case when the cylinders are of equal refraction, the sin 2/3 will be equal to the sin 2«, and which can only be the case when « = ^3, or, as « + /3 = r, when « = ,? = -L-- hence ; 58 DIOPTRIC FORMULA. 2. For combined congeneric cylinders of equal refraction, the primary plane equally divides the angle between the active planes of the cylinders, and the secondary plane similarly di- vides the angle between the axial planes of the cylinders. In case, however, /j > /,, which is the case when the refraction of the cylinder C is greater than r, then sin 2« > sin 2/3, or, when « > /3, so that 3. For combined congeneric cylinders of unequal refraction, the primary plane, in dividing the angle between the active planes of the cylinders, will be nearer to the active plane of the stronger cylinder, and the secondary plane consequently nearer to the axial plane of the same cylinder. This is also demonstrated in the diagram. As, for a combination of two cylinders, C and r, under given angular deviation of their axes, the only known quantities will beyj,y^, and r^ it will be necessary to express « and /5 in terms ofy], /^, and y. This is accompHshed through the equations (1) and (7) : r = « + /5 sin 2/9 = sin 2a A^ when, after proper substitution and reduction, we obtain : /I , 1 /, +/, cos2r ,,. cos « = -v/ 6 + o" — ^ . . . (I) It will be unnecessary to seek ,9 in the same manner, since, through (1), we find ^ = Y — "• When reducing this formula, for any given value of ^, pursuant to rea- sons later given, it should be observed thaty, > /,, in which case a, within the angle y^ is to be counted from the axis of the weaker cyHnder. CONQENERIC CYLINDERS. 59 2. POSITIONS OF THE PRIMARY AND SECONDARY FOCAL PLANES. As the plane DD^o^o is the primary plane, it follows that all parallel rays incident in it between D and o will, after refraction, intersect the op- tical axis ooj at some point, which will be a point of the primary focal line. Therefore the resultant ray DM^M^, in attaining its greatest deflection Z),v?/j in the elementary plane E^, will establish the position of the primary focal line, through its previous intersection of the optical axis oo„ at the point Oy In the secondary plane dd^o^o, for similar reasons O.^ will be a point of the secondary focal line, though this point of intersection of the final ray dm^m.^ with the optical axis is more distant, in consequence of the inferior deflection d^m^ in the plane E,. Similar resultant deflections, at opposite cardinal points of the circle within the lens, define the directions of their corresponding refracted rays. These rays not only limit the major and minor axes of the ellipses shown in the planes E^ and E.^^ but also determine the lengths of the focal lines at 6>j and O.^. Thus O.^M^ represents one-half of the secondary focal line at O^. The primary focal line, in the secondary plane, perpendicular to YO^ at 0„ has been omitted, to avoid possible misinterpretation of more impor- tant points of reference in the diagram. All rays parallel to the optical axis, incident at intermediate points of the circle within the lens, will, upon refraction, intersect the planes E^ and E.^ at correlative points of the ellipses drawn thereon. The circle of least confusion, T, will lie between the planes E^ and E^. (See Plate II, Fig. 2. ) Its position may be determined through a simple formula given by Prof. W. Steadman Aldis, of the University College, Auckland, New Zealand, in his discussion of the focal interval resulting from rays obliquely incident upon a spherical lens. * Our object being to determine the distances of the primary and second- ary focal lines, or planes, from the principal plane within the combined cylinders, we shall proceed as follows : *Elementary Treatise on Geometrical Optics, W. S. Aldif?, M.A., Cambridge, 1886 (kcc papeSO). €0 DIOPTRIC FORMULA. In the primary plane DD^M^^ we have DY: DD, = YO, : D,M,. Substituting, DY = O^o = F^ as the primary focal distance; YO^ — D^o^ = radius = 1. . F = - -^^ (26) In the parallelogram D^ V^M^Z^, the angle between the forces, D^ V^ and Z>iZj, being equal to <^ C^OyC^ = y, we have, as the resultant deflection, D,M, = V {D,z,f + (A y,y + 2 (A K) (Ai^i) cos r, (27) in conformity with the statical formula, R= ^/ pt^ 0^^ 2.PQ cos r, for forces P and j2> acting at the same point, within the same plane, under the angle y. Substituting in (27) the value of D^Z^ = cos ii, from (5) ; and of A K f f = "— A ^r from (3), = —- cos a, from (6), we obtain, D^M^ = -i/cOS^ /3 + (f^) COS^ a + 2 -^y- cos a COS /S COS ^. Introducing this value for D^M^ in (26), F, = ^' • ■ . (28) I // \2 / -1/ COS^ /5 + ( "y- ) COS'^ a -\- 2 ~ cos a COS /3 COS 7- By substituting the proper values for a and i5, from equation (1), after adequate reduction we obtain 17 J\Ji , ^^\ [/. C/i +/. COS 2r) + (/. +/,) (/, + >//M^2/y: cos 2r +/*)] (30) CONOBNERIC CYLINDERS. 61 Transforming, and substituting 1 — 2 sin^ r for cos 2)', we may, for con- venience in calculating, preferably write F. /i/^ . (II) ^(/i±/Z-/j,^ sin=^ r + (/, +/,) ^l^-^^^'^-^-AA sin' r When the cylinders are of equal refraction, y"^ being equal to/j ==/, the above assumes the simple form, F,= —~^—~ (IV) 1 + cos y In the secondary plane dd.,XO,, we have dX : dd, = XO, : djn,. Substituting, dX = 0.,o = F, as the secondary focal distance; dd,=/,; XO.^ = radius = 1. .-. F,= -f- (31) In the parallelogram d^v^m^s.^, the angle between the forces, d^v^ and d^.,, being equal to <^ v./l.^z.^ = 180° — <^ A,o,a., = 180° — y, . • • djn, = t/ {d,zj' + (d,v,y + 2 id,v.,) (d,z,) cos (180° —r). Substituting the value for d.^z.^ = -y- d^z^, from (9), = ^ sin /?, from (11); and ior d.^)., = sin «, from (12), we obtain, d^m^ = -*/ ("y- ) sin- /? -f sin^ a — 2~ sin a sin /3 cos y] i^rhich, introduced in (31) and being multiplied in the numerator and de- nominator by ~ , gives 62 DIOPTRIC F0RMULi?5. /■ -* /sin' ,3 + ("t") sin^ ''- — 2 y sin 'x sin ^ cos /- Through substitution of the proper vahies for « and /J, from equation (1), after suitable reduction we find F^ /-/- ^ ^lU. (A +A cos 2j') + (/, +/.)(/.~i//,^+2/,/, cos 2r+//)] (33) Substituting, cos 2/' = 1 — 2 sin^ ;-, ^/(A+A)^ -/,/, sin^ r - (/, +/.)^^-^'"±^' -/,/. sin' , (HI) This formula, reduced for cylinders of equal refraction, /^ being equal to y^ = /, becomes F, = ^^ (V) ^ 1 — cos y It may be of interest to note that these formulas diSer from those given for F^ merely by the minus sign in the denominator. The preceding formulae being alike applicable for combinations of con- vex or concave cylinders, the fociyi andy"^ are to be introduced as positive values, merely with the restriction that/^ be greater than or equal to/, in either case. 3. RELATIONS BETWEEN THE PRIMARY AND SECONDARY FOCAL PLANES. Since F^ and F.^ have been shown to be dependent upon /j, /,, and /', it is evident that, for fixed values of f^ and f.,^ the resultant foci will be ren- dered dependent entirely upon whatever value may be given to the angle y. It is further obvious that the refraction of one cylinder will be affected CONGENERIC CYLINDERS. 63 most by the other when their axes coincide, or when ;' = 0°, and least when their axes are at right angles to each other, or when ;' = 90°. We shall, consequently, fix upon the limits of Fi and F.^ for these ex- tremes of y. Introducing ^ = 0°, and consequently cos 2^ = + 1, into the formulae (30) and (33;, we obtain, for/, > /„ rr f\fi J\Ji V 4 [y; (/,+/.) + (/■ + /.) a -/■ -/J] ^ For F, = f{\ ^ we shall have as the refraction -=: = — + -p? consequently, 4. When the axes of the congeneric cylinders coincide, the primary focal plane will correspond to that focal plane which is defined by the smn of the refractions of the cyhnders, whereas the secondary focal plane will be at infinity. This is shown in Plate II, Fig. 1 . Introducing y = 90°, and consequently cos 2r = cos 180° = — 1, into (30) and (33), we have, for/ >/, C J\Ji JlJ'i f p J I Ji fxJ'l f _ .-. F,:F, =./,:/, (35) * 00 The sign for infinity. 64 DIOPTRIC FORMULA. As/j and/2 correBpond to the positions of the elementary planes E^ and E^^ it follows that 5, The primary and secondary focal planes coincide with, their correlative elementary focal planes, when the axes of the congeneric cylinders of unequal refraction are at right angles to each other. This is demonstrated in Plate II, Fig. 2. In the same relation (35), if/ = f.^^ then F^ = F^, or 6. The primary, secondary, and elementary focal planes all merge into one plane, when the axes of the congeneric cylin- ders of equal refraction are at right angles to each other. As in this case we have but one focal plane, the refraction corresponds to that of a spherical lens. F^ being chosen to signify the primary focal distance, it will have to be less than F.^, yet if/ > /^, we should find, in consequence of the relation (35), that F^ > /^,. To retain the significances of F^ and F,, it will there- fore be necessary to substitute/ by the greater given value of cylindrical focus, and/ by the lesser, as stated under the formulae, page 62. Owing to the previous considerations; between the limits of 0° and 90° for Y, we are then to conclude that /\ will vary between /■ "j r and /, while F^ varies between 00 and /, as the nearest and most remote limits of focal distance for F^ and F.^^ respectively. As an illustration, let Fig. 1, Plate II, represent two combined convex cyhnders of unequal refraction, with their axes coincident, and so united as to permit of the rotation of one of the cylinders upon the true planes of their faces, about the optical center 0. CONGENERIC CYLINDERS. 66 In the position shown (f = 0°), the shortest possible focal distance /^j of the primary focal line will be > ' '•' , which corresponas to the combined refraction, ---\- . , of the cylinders in the active plane. In the secondary y 1 y 2 plane, F.,=(x>; consequently, ~e^ = — = 0, which corresponds to the re- fraction in the axial or passive plane of the cylinders. The slightest change in the position of one of the cylindrical axes will give rise to a definite value of the angle y in the Formula III, thereby bringing F.^ within the limits of finite distance, while decreasing the value of F^ in the Formula II. For each successive increase in the angle r, the primary focal plane cor- responding to F^, will recede farther and farther from the combined lenses towards F^, while the secondary focal plane, corresponding to F.^, ap- proaches nearer and nearer from oo to F.,, until ^ = 90°, when /^j will have reached F^ on the moment that F, merges into F.,, as shown in Plate II, Fig. 2. Rotation of one of the cylinders is thus associated with corresponding changes in the distances F^ and /^,, while the movements of their correla- tive focal planes will be in opposite directions to each other ; which proves that : 7. The primary and secondary focal planes are conjugate planes, subject to variations of the angle between the axes of the congeneric cylinders. In order to comply with this law, in constructing the Plate I, it has been necessary to select elementary foci in marked disproportion to the curva- tures of the cylinders ; otherwise the secondary focus F, could not be brought within the space allotted for the diagram. DIOPTRIC FORMUL/E FOR COMBINED CONTRA-GENERIC CYLINDERS / \ \ \ ,^ \ \Vv; V .A \ V %\ < \\ 1 l> 1— ( \\ \\\\'\ aK \\ \\ \ \ 1— I \\ ~ v. > '''' ' ' \ '' '\ \ \\ \ \ w v V- Y, ".^ "\ 'j V ' \ \\\\^^.^\ H \ \ V "">Vt'-'\ A ^■"v^/1 \ < \ \ \V ''\ \/ / k<^ v^^^ll \ \ ►J \ \\' ' ■> y X <*'s / \ ' // '"' '^ "S-U---- -~c«' •V'A -' '^'' / V#-c^--- -v5i=^ k\-->,V-' — ■«- "Vy \^ to l\l ;''''V/;iim ^ ^ CO ^^"^^^''V\ W'' r^ '-''^V'^v ■^^ 2 ^^'■'■'-<\, \ \- / ' ^^'''3'' ^-i^-r^ ■iy^ Ul \^'' \ /'■ ""^ '^^V''''''\^ '*^ ^T\ Q _I ', v\\ "\( 'Jv v^^i^^ V y \ s. \ ^^'^^..^^^^ z in 2 _l < ' W^^'%% X '' V ^1 O CC u ' '/ V '-/^""\'/\'' I '' V- - - - ~ ~ \ V i 2 t- w ' / /i A / c " ^ - ^ \ \ K / m -J 5 ir t-* '/ vi-T=> — --v=\-\ -A--- — \'\ 1 z >- s >- a. I O o ? " 9 / U''' \ ~ ~-V- \' \ ' \ - '; / ' U- o d E ° 1 / \\\ ''^^ A'A □^ UJ X '^ V '^' cc ul O ^^Y-^;- \ I 1- < a: \ \'"--. \ (- 1^ \ \ ^ o \ ' o V / II. DIOPTRIC FORMUL/E FOR COMBINED CONTRA-GENERIC CYLINDERS I. RELATIVE POSITIONS OF THE PRINCIPAL POSITIVE AND NEGATIVE PLANES OF REFRACTION. In a combination of convex and concave cylinders, we can no longer have the primary and secondary planes, which we have learned to consider as planes of greatest and least refraction, but, instead, we shall have a plane of greatest positive and one of greatest negative refraction, synonymouslj' with the generally-adopted distinction between convex and concave lenses, as designated by the signs + (plus) and — (minus), respectively. As the refractions of the convex and concave elements in the combination are op- posing forces, the plane of greatest positive refraction will evidently lie be- tween the active plane of the convex and the axial plane of the concave cylinder, whereas the plane of greatest negative refraction will be between the active plane of the concave and the axial plane of the convex cylinder. In Plate III, therefore, the plane DD^o^o of greatest positive refraction is shown between c and A^ and the plane dd^o^o of greatest negative refrac- tion between C and a. These planes, being at right angles to each other, divide each of the angles A^o^c^ and C^o^a^ into « and {i. To establish the formulae for combined contra-generic cylinders, we shall therefore have to ascribe another significance to the angles a and /3. 69 70 DIOPTRIC FORMULAE. The deviation of the axes Aoa is equal to angle A^o^a^ = y, and, oince c^o^ is perpendicular to a^o^, a + /? + ^ is equal to 90° ; consequently, a + /? = 90° — r (36) The elementary focal planes E^ and ifj, corresponding to the focal dis- tances f^ and/j, respectively, are exhibited on opposite sides of the com- bined cylinders ; since E^, for the concave cylinder, is virtual and in the negative region before the lens, whereas E^, for the convex cylinder, is in the positive region behind the lens. Consequently, for the point Z>, the convex cylinder c contributes as its greatest amplitude of deflection D^Z^, perpendicular to a^o^ in the plane Ey The greatest amplitude of deflection for the concave cylinder C is D^ V^, perpendicular to A^o^ in the virtual plane E,^. As the incident ray at Z) will be refracted by the concave cylin- der, as if emanating from a correlative point F^ of the virtual axial line V^Oq, it is evident that the direction of the ray refracted by it will be V^D V^ The proportionate deflection contributed by the concave cylinder, measured in the plane E^ will consequently be D^ Vy Provided the point D is properly chosen, it will be a point of the plane of greatest positive refraction, that is to say, when the resultant deflec- tion D^M^^ accruing from the associated deflections D^ V^ and D^Z^ in the parallelogram of forces D^ V^M^Z^, is directed towards the optical axis. To insure D^M^ being so directed, it is obvious that the associated de- flections, Z>jZj and D^ V^ must also be measured in the plane E^, in the positive region behind the lens. Similar reasoning will apply to the point d as being in the plane dd^o^o of greatest negative refraction. In this instance d^m^ being a force di- rected from the optical axis, in the plane E^^ is to be taken negative, synon- ymously with the plane of greatest negative refraction. The relations between a and /? are to be determined by an analogous method to the one given for congeneric cylinders, whereby we obtain sin 2a = sin 2^3^.^ (37) as defining the positions of the planes of greatest positive and negative re- fraction, which are again at right angles to each other. CONTRA-GENERIC CYLINDERS. 71 We here also find the sines of the angles, 2a and 2/?, to differ by the co- f 90° V efficient —-. Hence, wheny^=y], we shall have « = /J^: — -^ or, /o ^ 8. For combined contra-generic cylinders of equal refraction, the plane of greatest positive refraction equally divides the an- gle between the active plane of the convex and the axial plane of the concave cylinder ; and the plane of greatest negative re- fraction similarly divides the angle between the active plane of the concave and the axial plane of the convex cy Under. In case/o >/„ then ;5 > « ; or, 9. When the convex cylinder is stronger than the concave cyUnder, the plane of greatest positive refraction will be nearer to the active plane of the convex, while the plane of greatest negative refraction will be proportionately farther from the ac- tive plane of the concave cyhnder. In case/, > /, then « > /3 ; or, 10. When the concave cylinder is stronger than the convex cyhnder, the plane of greatest negative refraction will be nearer to the active plane of the concave, while the plane of greatest positive refraction will be proportionately farther from the active plane of the convex cylinder. This is manifest in the diagram. The values of a and /? may be expressed in terms of/,, /„ and ?' in a sim- ilar manner to that shown in the previous theorem, when it can be shown that. cos V2^2 /o — /, cos lr 2 Vf^-yj, cos 2r +// (VI) This and the transposed equation (36), /? = 90° — (^y -\- a)^ suffice to locate the positions of the principal planes of refraction ; the angle a being counted from the axis of the convex cylinder. 72 DIOPTRIC FORMULAE. 2. POSITIONS OF THE POSITIVE AND NEGATIVE FOCAL PLANES. The positions of the positive and negative focal planes will evidently here also be determined by the resultant rays, DM^ and dm^, and their correla- tive intersections with the optical axis at O^ and 0„. O, Wj will therefore represent one-half the focal line in the positive region behind the lenses, and O^M^ one-half the virtual focal line in the negative region before the same. The elHpses shown in the planes E^ and E^^ are of the same significance in this as in the preceding combination. In the plane of greatest positive refraction, Z>Z>, YO^, we have DY.DD,^ YO^ -.D.M,. Substituting, DY = O^o = F, as the positive focal distance ; YO^ = Do =- radius = 1. ••■^.= 7^ <''^' In the parallelogram D^ V^M^Z^, the angle between the forces, D^ F, and Z>jZj, is equal to 180° — y, since D^Zy : Z^o^, and D^ V^ A^Oy ••• A^i =V (,D,Z,f+{D,V,y+2 iD.ZJ (D,V,) cos (180''— r). In the oblique plane D^Vf^DV^Dj, we find, Z>o V^ = sin <^ D^o^Aq = sin <^ Dfi^A^ = sin /S. CONTRA-GENBRIC CYLINDERS. 73 .-. D,l\ =-^J. siny5. /^,Z, = sin (, yT/,, equation (47) be- comes, F, = I r -i 7 ' A / cos^ ""'"(/) ^^^^ '^ — ^/ ^^^ ^ c®^ " c®^ y and which, through equation (36), may be given the form : F =^ I'^i L/i (/. -/o cos 2r) + (/o -/)C/o + T//o"^-2/o/cos2r+/,0]' (48) Substituting, cos 2^ ^= 1 — 2 sin^ ;', -^.p_ ^ -^'-^° . . . (VII) ^iJ±-Ql +/o/8inV+ (/o-/x) ^^Az:^^ +/o/.sinV. This formula, when reduced for cylinders of equal positive and negative refraction, /o being equal to/, =/, assumes the simple form F, = -^ (IX) ^ sin /- ^ ^ In the plane of greatest negative refraction, d^m^dO^X, we obtain, dX : dd^ = XO^ : d^m^. Substituting, dX = 0,p = — /%, as the negative focal distance; XO^^ = do = radius 1. .-. _/; = — /-; (49) since d^m^ is to be taken negative. 74 DIOPTRIC FORMULiE. In the parallelogram d{L\m^z^^ the angle between the forces, d^v^ and fl?,^,, is again 180° — y\ hence, d^^^ = V {d^2j + {d,vy + 2 {d^z;) {d,V^) COS (180° —r)- In the oblique plane d^v^dv^d^, we find, d^v^ : dd^ -— d^^v^ : dd^. d,v^ = sin « Z>„<7o4 — < AMo) = sin (90° — < D^o^A,) = sin (90° — ;5) = cos fi. dd^ =/u. .'. flfjZ'j = "— cos /?. d^z^ = sin <^ ^i' = 0^ and Y = 90°, as in the previous theorem. When Y = 0°, or cos 2^' = 4- 1, from the equations (48) and (50) we find, for/„ > /;, ■p / 1 / ' JiJo ' ~ VK-A i/o-A) + (/o-/i) (/o +/o-/x)] ~/o -/ ~ Vii^A (Ao-A) + (fo-A) (/o-/o+y;)]^ o^=~ ^• .-. F,:-F, = /^:- ^. (51) Jo J \ For F^ = > x» we have as the refraction ->, = y y-; consequently, Jo /l ^ \ J \ Jo 11. When the convex cylinder is of greater refraction than the concave, and their axes are coincident, the positive focal plane will coincide with that focal plane which is defined by the difference of the refractions of the cylinders,* whereas the negative focal plane will be at infinity. *Or the sum of their refractions when taken as positive and negative elements. — F. 76 DIOPTRIC formul.t:. Placing Y = 0°, or cos 2^ = + 1, in the equations (48) and (50), we have, for/, >/„, -/i/o fi/o „ F,= -Fo = V\ [/ (/i -/o) - ( /, - /o) (/o + /, -/o)] /i/o yi/o !/*[/: (/. -/o) - (/ -/o) (/o -/ + /o)3 /. - /« /i /o For — /^) = — //' ^'® have as the refraction A /o ^ = — \^ -2- J consequently. 12. Wtien the concave cylinder is of greater refraction than the convex, and their axes are coincident, the negative focal plane will coincide with that focal plane which is defined by the difference of the refractions of the cylinders, ^'^ whereas the positive focal plane will be at infinity. This is shown in Plate IV, Fig. 1. Introducing y = 90°, or cos ly = cos 180° = — 1 in the equations (48) and (50), we have, for/ ^/j, "^"^ ~ l/K/ (/ +/o) + (/o -/) (f.-fo -/,)]^~ /x' "^ " ^" .-. F, ■.-]■,=./,: -I, (53) • Or the Bum of their refractions when taken as positive and negative elements. CONTRA-GENERIC CYLINDERS. From which we deduce : 13. The positive and negative focal planes coincide with their correlative elementary focal planes, when the axes of the contra- generic cyhnders are at right angles to each other. This is demonstrated in Plate IV, Fig. 2. Between the hmits of 0° and 90°, fory„ >/,, we have consequently found F^ to vary between the limits of r r ^.nd /j behind the combined Jo J I lenses, while F^ varies between the limits of ooand/^ on the incident side of the same. The convex being stronger than the concave cylinder, it is evident when their axes coincide that their combined refraction will be equal to that of a periscopic convex cylinder, since ~ =z ~ T ^"^ ^^^ active plane ; and ^\ J\ Jo -^ = - = in the passive plane. /; oo Between the same limits, when/j >y'o, F^ will vary between / 'S and J I Jo /„ on the incident side of the combined cylinders, while F^ varies between 00 and/i behind the same. (See Plate IV.) In this case, when the axes coincide, it is evident that the resulant refraction will be equal to that of a periscopic concave cylinder, since ^ = — (^ J- ) in the active plane ; and -j^ = — ^ = in the Fo V/o /i -^ F^ CO axial plane. Therefore, with an inequality in the refractive powers of the cylinders, rotation of one of them, from 0° to 90°, will be associated with correspond- ing changes in the position of the resultant focal planes, between the limits of infinity and the focus of the weaker cylinder on the one side, and between that focal plane which corresponds to the difference of their refractions and the focus of the stronger cylinder on the other. Since in this case the approach of one focal plane is accompanied by a corresponding 78 DIOPTRIC FORMULiE. recession of the other on the opposite side of the lenses, their movements are, as in the previous theorem, in opposite directions. When the cylinders are of equal refractive j^ower, f^ being equal to /„, it follows, from the relation (53), that F^ = F^, so that between the hmits of 0" and 90°, F^ will vary between infinity and/^ on the positive side, while Fq varies between infinity and /„ on the negative or incident side of the combined cylinders. Consequently, when the axes coincide, + Fi= -r oo and — -^o = — 00 . This is evident, since the refractions of equal convex and concave cyhnders, under such circumstances, neutralize each other throughout. By the previous considerations we therefore here also find : 14. Tlie positive and negative focal planes are conjugate planes, subject to variations of the angle between the axes of the contra-generic cylinders. The diagram, Plate III, has been constructed in accordance with the foregoing provisions. For practical purposes, it will be found more convenient to use the formula in the next chapter. DIOPTRAL FORMUL/E FOR COMBINED CYLINDRICAL LENSES PLATE ir III. DIOPTRAL* FORMUL/E FOR COMBINED CYLINDRICAL LENSES. I. RELATION BETWEEN THE PRINCIPAL PLANES OF REFRAC- TION AND THE REFRACTIVE POWERS OF THE CYLINDERS. As the task of reducing diophies to their focal distances would render calculation by the preceding formulae somewhat arduous, we may here give the formulae, expressed in refraction, which will be found especially convenient when applied to combinations of the metric system. Since original publication, these formulae have been given their simplest possible fomi. The new formula?, IIZ?, IIIZ?, VIIZ? and VIIIZ? are now- introduced as sequences to the original formulae, which are also given, and whose transformations have been accomplished through convenient substi- tutions from the equations 54a, 54 and 55. For the focal distance F. we have as the refraction ^ = /?„ and for/, and /, similarly, -^ = r and -j = n, which designate the dioptral powers of the cylinders. By these, and similar substitutions for other foci, we may give the pre- ceding formulae (I) to (^X), the following form : *The choice of this adjective would seem justifiable, since the unit "dloptry " has been chosen in distinction to "dioptric," which, though rclnti-d, hiis auother significance. «1 82 DIOPTRAL FORMUL.«. THE DIOPTRAL FORMULAE FOR COMBINED CONGENERIC CYLINDERS. cos a = A -f i r, + r^ cos 2r y^ 2 i/rj2_|-2r,r, cos2r+ r/ = i (''i + ^2 + I'V, + ''z)' — 4r,r, sin 2 J-). (IID) ^2 = V"! (r, + r,)' — r,r, sin V — {r, -\- r,) V^i^r, + r,f — r,r, sin'T- = i C'', + ''. — l^Vi + ^-2)' — 4^1 n sirT^. (IIIZ?) To retain the significances of R^ and J^.,, in calculating, rj should repre- sent the greater cylindrical refraction. When the cylinders are of equal power, then r, = r.^ = r, so that ^^ = r (1 + cos r) (IVZ^) /?, = r (1 — cos r) cvz;) THE DIOPTRAL FORMULA FOR COMBINED CONTRA- GENERIC CYLINDERS. _ /^ _L ^ r^ _ y-p COS 2 r y^r^ COS 2r -f ^' -^«= V2+2 7r^=^ir=^^=^'- • • • ^'^'''^ ^1 = V\irx—'^J' + ^%sinV + {r^ — r„) l/i(^i — O' + ^^o sin^* r- iC'-i — ^»+ l-^C'-i — ^oy + 4r,r„sinv) (VIIi9 — ^0 = — 1/* (r J — r„)' + rj r^ sinV + (^o — ^,) 1 ' i (^, — ^o)' + ''i ^0 sinV- = i(^i-^o--l/(^,— '-o)' + 4r,r„sinV) (VIIIZ;) CYLINDRICAL LENSES. 83 when tlio cylinders are of equal power, then r^ z= r^^. r^ henco R^ = rQmr (IXZ?) — R^ = — r sin y (XD) If, in (IID) and (IIIZ>), the convex element r, be replaced by the concave element — r^, we obtain (VIID) and (VIIIZ>). By the aid of the preceding formulse we may also arrive at the following signiticant facts. The formula) dlD) and (lllV) may be written : R;' = ^(r, -i r,y^r,r, sin'^ r + (r, + r,) l/j (r^ + rj' — r,r, sin' r), /?/ = ^ (r, + rj — r^r.^ sin' ^ — (r^ + r,) Vi(r^ -f- r J' — r,;% sin* r, which, by addition, result in the equation, /?, - + R, - ^ (r J + rj' — 2^1?-, sin V- . •. (R, + ^.)-^ - 2^, /?, = (r , + r,)^ _ 2r^r, sin -' y . • . (^j + R,r = (r^ + r,y — 2r^r , sin* r + 2/?^/?,. Multiplying (IIZ>) by (III/?), we find, 2^?,/?., = 2rjr, sin-'r (54a) .-. R^-^ R,^^r,-{-r., (54; From which we conclude : 15. The Slim of the primary and secondary refractions is a constant, being equal to the sum of the elementary refractions for any combination, and all deviations of the axes of two combined congeneric cylinders. In the same manner, we obtain from the formula) (VIIZ>) and (VIIIZ?), R^ — R^^r, — r„ (65) 84 DIOPTRAL FORMULAE. and therefore here also find, 16. The sum of the principal positive and negative refrac- tions is a constant, being equal to the sum of the positive and negative elementary refractions for any combination, and all deviations of the axes of two combined contra-generic cylinders. Ab the total inherent refraction always remains the same for any combi- nation, the angle y merely performs the function of allotting the proportions of refraction R^ and R^^ or R^ and R^^ in the resultant principal planes. By the equations (54) and (55), calculation may be greatly simplified. R^ being determined for a specific value of r, we may readily determine R^ and R^^ by transforming these equations, as follows : R. = r,^r,-R, — R^=r, — r^ — Ry This is demonstrated in the appended tables, although it has not been utilized in calculating ; on the contrary, a study of these led to the above deductions. SPHERO-CYLINDRICAL EQUIVALENCE PLATE IV THE REFRACTION BY COMBINED CONTRA-GENERIC CYLINDRICAL LENSES Fig. 1 Fig. 2 ^-^F.-^h :' : p = J "J 'i ' J .'1 Jo CHAS. F. PRENTICE COPYRIGHT, 1988 IV. SPHERO-CYLINDRICAL EQUIVALENCE. Since, for any combination of cylinders, the principal planes of refrac- tion are at right angles to each other for all values of y^ there can be no reasonable doubt, owing to the provisions made at the opening of this demonstration, as to the equivalence of a sphero-cylindrical lens to one composed of combined cylinders. However, as the use of such lenses is at present confined to the correction of errors of refraction in the human eye, it is evident, from the movements of the eye behind the fixed lens, that the visual axis cannot at all times coincide with the optical axis of the lens chosen ; therefore, in those instances where substitution of one form of lens for the other proves to be unsatisfactory, the cause might seemingly be explained by a possible difference becoming manifest for the more peripheral incident rays, though these be equally distant from the optical center of each lens. In other words, the available field in the one may be greater or less than in the other ; yet even this would probably only be appreciable in lenses of extreme curvature, and possibly in combinations where the cyhnders differ widely in power. Ht)wever, this would remain to be shown. To substitute a sphero-cylindrical lens for combined cylinders is a proposition which merely demands that the focal interval should be the same, at the same distance from the principal plane, at the optical center, for each of the compound lenses. The distances F ^ and F.^ being deter- mined for any angular deviation r of the axes, in a combination of congeneric cylinders, for instance, the substitution is accomplished by making a sphero-cylindrical lens in which the focus of the spherical F F element is equal to /% , and of the cylindrical element is equal to ' ' or, i^i — i"\ if expressed by refraction, = -=r sph. = -=: cyl. = — -. 87 88 SPHERO-CYLINDRICAL EQUIVALENCE. Should it be desired to place the primary and secondary planes of the sphero-cylindrical lens so as to coincide with those resulting from a combi- nation of two definitely placed congeneric cylinders, it will be necessary to refer to the formula (I) and to the laws 2 and 3. Comparing the sphero-cylindrical equivalent with its corresponding rotating cylinders, reference being had to Plate II, Figs. 2 and 3, we find a decrease in the angle y from 90° to zero to effect a corresponding de- crease in the spherical element F.^, from the focus/, to oo; this being asso- ciated with a cylindrical element of the focus Fc, which constantly increases from the focus /^ \ to ^^1 . In other words, a gradually decreasing potency of the spherical refraction —=: from — . to — =0, gives way to a proportionately increasing cylindrical refraction — , from -z. re J I — to --^ -f- — ^ . As an instance, if/. = / = / ^r will increase from A A Jl ' ^<^ 112 — -r- = to 7- , or twice the refraction of either cylinder. In this j\ Jl. J case, all successive values of cylindrical refraction will therefore be inhe- 2 rent between and -7- . Should a means be devised to suppress the spherical element for each successive value of ^, the remaining varying cylindrical element being thus rendered available for measuring corresponding degrees of astigma- tism in the eye, the formulae here advanced would prove of service in ob- taining the graduations upon the rotating scale of such an instrument. While there are few cases of astigmation which demand correction by combined cylinders, we may nevertheless be permitted to passingly allude to certain methods of procedure in such instances. We shall confine the subject to congeneric cylinders. In a case of astigmatism which has been found to be corrected by two cylinders combined under the angle /', the lenses should be withdrawn from the trial frame, when they are to be superposed with their plane surfaces in contact; and in such manner as to facilitate their being rigidly held in the required position for y. The positions of the principal planes of refraction may then be estimated for this fixed combination, the same as if it were a single lens, though without regard to the exact nature of the elements constituting it. The SPHERO-CYLINDRICAL EQUIVALENCE. 89 powers of the principal planes of refraction will be revealed by neutralizing with the lenses from the trial set. The spherical and cylindrical elements thus determined are then to be substituted in the trial frame, when rota- tion of the cylinder will lead to that position of it which produces the best acuteness of vision. The spherical and cylindrical elements will probably then also bear of further modification, in case any error may have been made at the outset. In lieu of this practical method, recourse must be had to the formula;. It having been shown that successive changes in the angle y are asso- ciated with corresponding changes in Fy and /v^, the above substitution would indeed seem advisable, since the present appliances for grinding bi- cylindrical lenses are not constructed with sufficient precision to enable opticians to fix the relative positions of the cylinders beyond mere approx- imation. As an illustration, let us select two congeneric cyUnders of equal foci, say 20 inches, combined under the angle y = 60°. Introducing these values in the formulae (IV) and (V), we find, F = 2Q = 20 _ 1 + cos 60° 1 -f 0.5 ~ ' F = ?-^_ = - J-^^^ = 40 1 — cos 60° 1 — 0.5 We then obtain the cylindrical refraction ~=r , for the desired sphero- cylindrical equivalent, from the equation, 1. _ Jl = 1 X F~F~ (56) Substituting herein the calculated values for /-', and F., gives. _1 J_ _ i _ _J_ _ i .nearly) 13.33 ^~~ Fc~ 19.99 ~ 20 '^'^^^^^y^- -=^=27^ being the spherical element, we therefore have the sphere- F, -40 cylindrical equivalent, 4^ 'p^- - ^ "y^- 90 SPHERO-CYLINDRICAL EQUIVALENCE. as an available substitute for the cylindro-cylindrical lens, ^r^ cyl. axis 0° Q ^^ cyl. axis 60° 20 -^ 20 *^ without regard to a definite position of these lenses before the eye. By way of comparison, allowing the optician to make an error of appa- rently 60 small an amount as 2°, in producing the same cylindro-cylindrical lens, we obtain, by introducing ^ = 62° in the same formulse, ^ 20 20 _ 20 ^ "" 1 + cos 62° ~ 1 + 0.469 ~ 1.47 ~" ' P _ 20 20_ _ _^ _ 37 70 ^^ ~ 1 — cos 62° ~ 1 — 0.47 "~ 0.53 ~ Substituting these values in the equation (56), we have, _1 1 ^1 1_ 13.61 37.73 ~ /^c~" 21.29 ' from which we obtain the sphero-cylindrical lens. 37.73 ^ ^ 21.29 Had the optician been required to make a sphero-cylindrical lens 27r sph. Q ^x cyl., his execution of it presenting such discrepancies as „ sph. Q cyl., would certainly be rejected as being unsatisfac- tory, on account of the notable difference of 2.27 inches in the focal distance of the spherical element. On the other hand, instances are likely to occur in which it will be im- possible, by the advanced method of neutralization to accurately arrive at the sphero-cylindrical equivalent. Since ^q cyl. axis 0° C 20 ^^^^ ^^^^ ^^° = 3773 ^P^^- ^ 21^ ^^^' we should evidently be unable to accurately neutralize such spherical and cylindrical elements by any of the lenses from the trial set. In those instances, therefore, where satisfactory neutralization of the principal planes of refraction in a pair of combined cylinders cannot be SPHERO-CYLINDRICAL EQUIVALENCE. 91 attained, the cylindro-cylindrical lens will have to be chosen, again under the proviso, however, of a faultless mechanical execution. However, as in most instances a sphero-cylindrical equivalent will be available, we are to suspect error in our estimate of the refraction of an eye which seems to demand cyUnders combined under acute or obtuse angles. The following is a case in point : A cylindro-cylindrical lens — ^Ti cyl. axis 0° C — j?) cyl. axis 70° had been prescribed for Mr. G. B. 0., of New York, by his oculist in Philadelphia, in 1880-1. With this lens the vision = -, for the left eye. In this instance the sphero-cylindrical equivalent was obtained as follows : The lenses being congeneric concave cylinders of equal refraction, by the formulaj (IV) and (V), for/= 40 and y = 70°, we have, it being admissible to neglect the fractions for such focal distances. By law 2, we find the position of the cylindrical axis eq\ial -^ = 35*, and consequently the sphero-cylindrical equivalent, — gQ sph. O— ^^ cyl. axis 35°. This lens was substituted with the knowledge and to the entire satisfacr tion of the patient. It is therefore obvious that the meridian (125°) of greatest refraction in the eye had not been disclosed by the oculist's diagnosis. The M'^eak spherical element in the substituted lens, while being an appreciable factor to the patient, might easily have been overlooked by the practitioner. In similar cases, the advanced formulae must prove of value in fixing upon the true state of the refraction. VERIFICATION OF THE DIOPTRIC AND DIOPTRAL FORMULA V. VERIFICATION OF THE FORMULAE. In the following tables, the Dioptric and Dioptral FommlgB have been separately applied to combinations of cylinders of the inch and metric sys- tems, respectively. It would be inadmissible to substitute the generally adopted inch-system equivalents for dioptrics, in calculating, on account of frequent repetitions of the former as factors in the dioptral formula?, which would naturally increase the neglected differences to an unwarrantable de- gree. For the purpose of obtaining reliable results, the calculations have been carried to the fifth decimal point under the radicals. The angles 30°, 45°, and 60° have been chosen so as to exhibit appreciable differences in the corresponding resultant refractions, which are thereby also brought within the lens-series of the inch and metric systems. The elementary foci and refractions have, in a measure, been arbitrarily selected, it being noticeable that the secondary refraction will generally be beyond the limits of neutralization for combinations of weaker cylinders whose axes deviate less than 30°. The Approximates given for refraction, in Table 1, will at times appear to conflict with the laws 15 and 16; this, however, is to be attributed to changes of proportion occasioned by the adopted substitutions. To substantiate the resultant refractions given in the tables, through the practical test of neutralization, the cylindrical axes should first be accu- rately determined, and their deviation effectively maintained while the plane surfaces of the cylinders are kept in absolute contact. In holding these lenses while neutralizing, great care should be exercised to prevent slipping, as the slightest variation in the position of the axes will prove misleading. In this practical experiment, the observer's eye will generally fail to appreciate the neglect of fractions made necessary through using lenses from the trial case. 95 96 VEEIFICATION OF THE FOBMTJL-SJ. In erplanation of the tables (1) on the following page, under the caption "Elementary Foci," are given, for instance, the foci (/j = 16 inches, and /„ = 24 inches) of the cylinders whose "Axial Deviation" is 30°. On the same horizontal ruling are given 10.2576 inches as the "Primary Focus," and 149.7422 inches as the "Secondary Focus." The nearest practical equivalents, expressed in refraction, are shown to be 1-10 and 1-160, rc- spectivety, which in practice will be found to be the lenses most closely ap- proximating neutralization of the principal meridians of the combined cylin- ders. In the second set of tables (2), under the heading "Elementary Eefrac- tion," the cylinders are expressed in dioptrics, and in the right hand ver- tical column the laws mentioned on pages 83 and 84 are forcibly exem- plified. The Dioptral Formulas on page 82 were applied in these tables, and will generally be found most convenient for use by the student who may desire to solve similar examples. In this event, great care should be exer- cised to retain the proper meaning and proportions of r^, r^ and r^, as in- dicated by the respective signs +, — , > and < in the left hand vertical column. VERIFICATION OF THE FORMUL-S. 97 I. TABLES IN VERIFICATION OF THE DIOPTRIC FORMUL/E. FOR COMBINED CONGENERIC CYLINDERS. Elementary Foci. Axial Deviation. Primary Focus. Primary Refraction. Secondary Focus. Secondary Refraction. /, < U 16 3 24 fv 7 + i^i {Approximafe. ) -n [Approximate. ) —1/32 —1/22 —1/16 — 14C+ 10 tl u U (( 30° 45° 60° 16.9799 13 2046 11.2537 + 1/16 + 1/13 + 1/11 32.9799 21.2040 16.5870 /or2 7 R, (Approx.) B, (Approx.) 2.5 C 1.5J9- i( K (I « 30° 45° 60° 3.75D. 3.46 3.09 3.75I>. 3.5 3. 0.25D. 054 0.91 0.25D. 0.5 1. 4D. 4 4 FOR COMBINED CONTRA-GENERIC CYLINDERS. Elementary Refractions. Axial Dev't'n. Positive Refraction. Negative Refraction. Ri — Ba= ri — ro ri<—ro 7 + i2i (Approx.) -B, (Approx.) + 4C— 2.75Z). « It 30° 45° 60° 2.397D. 3.062 3.564 + 2 5D. + 3. + 3.5 1.147D. 1.802 2.314 — 1.25D. —1.75 —2.25 + 1 25Z?. + 1.25 + 1.25 '•i < —^0 7 -\-E, (Approx.) -Bo (Approx.) 'i- ro + 2C— 2.75D. <( « 30° 45° 60° 0.8561). 1.325 1.690 + 0.75D. + 1.25 + 1.75 1606D. 2.075 2440 — 1.5D. — 2. — 2.5 — 0.75D. — 0.75 — 0.75 SECTION III THE PRISM-DIOPTRY AND OTHER OPTICAL PAPERS WITH SIXTY-FIVE ORIGINAL DIAGRAMS THE PRISM-DIOPTRY. In the year 1890 the author advocated a " Metric System of Numbering and Measuring Prisms,"* involving the principle that prisms should be numbered according to their refractive powers, instead of by their refract- ing angles, or angles of minimum deviation. As prisms notably possess the property of apparently changing the position of objects seen through them, it was proposed, in the new system, that the tangent-distance f between the object and its virtual image should form the basis of compari- son in measuring the relative strengths of prisms. The tangent-deflection of one centimeter, measured in a plane one meter from the prism, was, therefore, arbitrarily though befittingly chosen as the new unit of prismatic power, and was named the prism-dioptry. In measuring the refraction of prisms, however, the same as for lenses, it is necessary that the incident pencils of light should be composed of parallel rays, so that the theoretical distance of one meter must in practice be increased to at least six meters. The Prismometric Scale, | which is to be placed exactly six meters from the prism, therefore, represents the prism-dioptry as a six-centimeter dis- tance. Scales which are computed for a shorter distance than six meters have been placed upon the market, but, as demonstrated on page 148, are wholly unreliable. ♦Archives of Ophthalmology, Vol. XIX, Nos. 1 and 2, 1890 ; Vol. XX, No. 1, New York, 1891. Archly fiir Augenheilkunde, Band XXII, Berlin, 1890. The Ophthalmic Review, discus.sion by Dr. Swan M. Burnett, Vol. X, No. 3, London, 1891. tOfficinlly adopted by the section of Ophthalmology of the American Medical Association, Waahing- ton, D. C, 1891. "To Mr. Prentice alone belongs the credit of having proposed as a standard prism one which produces a deflection of one centimeter at one meter's distance, and no advocate of the centrad ever hinted at it until the appearance of his paper in the Archives of Oi)hthalmology. We owe the simplicity of that idea to Mr. Prentice ; let us not deprive him of whatever honor belongs to the conception."— J/edtcaZ News, Philadelphia, May 2, 1891. jThe American Journal of Ophthalmology, Vol. VIII, No. 10, St. Louis, 1891. Les Annales D'Oculisticuxe, Paris, Julv, 1892. 101 102 THE .PRISM-DIOPTRY. The author was the first to recommend that the figure of a prism /\, used as an exponent* to the prism numerals, should be the symbolic sign for the prism-dioptry, it also being the letter D of the old Greek alphabet. By this means, one prism-dioptry (1^) is readily distinguished from the prism of one degree (1°) refracting angle, and, in fact, from prisms of any other system. The Dioptral f System | of numbering prisms alone possesses the great desideratum of establishing a direct and simple relation between the prism- dioptry and the lens-dioptry, as demonstrated by the authors' law,|| that ' ' a lens decentered one centimeter will produce as many prism-dioptries as the lens has dioptrics of refraction." Thus a lens of 1 D. decentered 1 cm. will afford l'^ ; a lens of 2 D. decentered 1 cm. Mali produce 2^, etc. The Fig. 1. Fig. 2. prism-dioptral power is also in direct proportion to the amount of decentra- tion, so that a lens of 2 D. decentered J cm. gives 1^ ; whereas, if the same lens is decentered 2 cm. it produces 4^, and so on. It is, therefore, only the size of the lens which in practice will set a limit to its prismatic power. * Concerning this exponent (a) sec paper by Dr. Swan M. Burnett in Annals of Ophthalmology and otology, July, 1894, Transactions International Ophthalmological Congress, Edinburgh, 1894, and the Refraetionist, December, ] 894. The figure of a triangle, no matter how placed in respect to the position of its sides, refers exclusively to the prism-dioptry, being so recognized by American manufacturers. t First used in Prentice's Dioptric Formulse for Combined Cylindrical Lenses, monograph, New York, 1888. " The selection of this adjective would seem justifiable, since the unit ' Dioptry ' has been chosen in distinction to 'Diop'ric,' which, though related, has another significance." Thus, a 40-inoh tele- scope lens is a member of a dioptric system, whereas, a 1-dioptry lens is specifically a member of the dioptral system. In the English language we have an analogy to dioptry and dioptral in the spelling of ancestry and ancestral. X " Having, by elaborate practical test, fully convinced ourselves of the preeminent advantages of the Dioptral System in the art of manufacture, we have discarded the old degree system entirely, and are now manufacturing prisms which are more accurately ground than ever before." Circular issued to the optical trade by the American Optical Company, Southbridgc, Mass.; also catalogue, 1894. "Our prisms are now ground to conform to the metric system." Catalogue of the Bausch & Lomb- Optical Company Rochester, N. Y., 1895. I Text-book of Ophthalmology, page 141, Drs. Norris and Oliver, Pliiladclphia, 1893. TexMx)ok of Diseases of the Eye, page 201, Henry D Noyes, M.D., New York, 1895. THE PRISM-DIOPTRY. 103 In Fig. 1, abc represents a vertical section of a 1 D. plano-convex lens, with three parallel rays /j, z,, Zj, separated by one centimeter distances, which are incident upon its plane side. These rays, after refraction, are collectively directed to the focal jjoint v^ and therefore suffer perpendicu- lar deflections in the focal plane,* dv^ which are equal to the correlative decentrations of the rays z\, z^, i^ at their respective points of refraction. As the spherical surface may be considered as being built up of an un- broken succession of infinitely small prisms of gradually and slightly vary- ing angles, it is to be noted that the three chosen prisms, shown in their order of 1^, 2^ and 3^, correspond to the respective decentrations of 1, 2 and 3 centimeters, and, therefore, produce correlative deflections in the focal plane, dv^ exactly the same as the spherical surface at the same points of refraction. Some recent authors have failed to comprehend this unequivocal precision. In Fig. 2 three concentric curvatures are shown to represent, respec- tively, the spherical or cylindrical surfaces of 1, 2 and 4 dioptry lenses, in which the same prism of 1^ occupies a different position (decentration), relatively to the optical axis, on each of the lenses. Beneath each section is given the dioptral power of the lens, which, being multiplied by the decentration in centimeters, shows the same pris- matic power of 2^ to exist at a different though definite point on the sur- face of each lens. Thus it is seen that every lens, whatever its dioptral power, contains all possible values of the prism-dioptry, which means that the prism-dioptry itself must constitute a distinct part of every lens of the dioptral system. The prism-dioptry, therefore, stands unchallenged in its unique ability to harmonize all of the refracting elements in the optometrical lens-case by establishing a complete and inseparable relationship between prisms and lenses. We need only to remember the centimeter in connection with the prism-dioptry, as we do the meter in its relation to the lens-dioptry. •"The great and enduring work of Gauss on the elucidation and simplification of optical laws has among its cardinal elements four planes— the anterior and posterior focal planes and the two principal planes (Haupt-Ebenen); and the proportion of the size of image to object, as elucidated by the formula of Helmholtz, is calculated on the tangent plane. . . . This plane can, in the case of prism-deflec- tion, be regarded in the same light as the focal plane of the standard lens. . . . This method was first suggested and made practical by Jlr. C. F. Prentice, of New York, . . . who has gone very thoroughly into the mathematics of the subject in his paper."— 77i€ Ophtfialmic Keview, a monthly record of ophthalmic science, London, England, January, 1891. A METRIC SYSTEM OF NUMBERING AND MEASURING PRISMS. Revised reprint from the Archives of Ophthalmology, Vol. XIX, No. 1, 1890. Also translated by the author, and published in Arehiv f\ir Anjrenheilkunde, Band XXII, Berlin, 1890. Introductory ier7?ia /•/.>■ by Dr. Swan M. Burwlt. "The old method of numbering prisms simply by the angular deviation of their sides is, confessedly, inaccurate and unscientific. Any attempt to .supplant this by one more accurate, and to place the no- menclature of prisms on the same basis of scieutiflc exactness as the other optical appliances in the hands of the practical ophthalmologist is, therefore, deserving of consideration. The method proposed by Mr. Prentice, in the following paper, not only docs this, but does it in a manner and according t<> principles which are familiar to even the less sicentitic practitioners. To have the same unit (the meter) of measure and comparison for all refracting apparatus and imiform with the nomenclature em- ployed in the designation of anomalies of refraction and muscular equilibrium, gives a simplicity which is not only commenda,ble in itself, but tends to render the study of the practical use of prisms easier and more comprehensible to the student. This is particularly apparent in the connection the author estab- lishes between the prism-dioptry, the lens-dioptry and the meter-angle. Not the least important part of the contribution is the description of the instrument Mr. Prentice has devised for illustrating his idea and for testing the relraction of prisms generally." The present method of designating prisms by the angular deviation of their refracting surfaces, is open to the objection that we thereby define only an isolated feature of their construction, to the utter disregard of the varying powers of refraction, which must result from the use of refracting substances having different indices of refraction. With a view to securing greater accuracy and uniformity in our utili- zation of the refractive properties of prisms, the following system of num- bering, which the author believes to be feasible, as well as suited to the requirements of optometrical practice, is presented. IjQi abc, Fig. 1, represent a prism, with the ray z incident perpendicularly to ab, and we shall have dv as the deflection accruing from the refraction at e. Similarly, d V will represent the deflection arising from the refraction at the same point e^ for a prism, ABC, of greater angle. 105 106 A METRIC SYSTEM OF NUMBERING AND MBASDRING PEI6MS. We shall then have But dV : de = d{V^ ; c c ^ Fisf. 2. have <^ abc = /5, as the angle of the prism ; <^ iep^ = /?, the angle of inci- dence ; '^pev, the angle of refraction ; <:^ dev = r, the angle of deviation. Consequently the index of refraction sin <^pev sin (<^ ped + \ corresponding to the angle y, being equal to cie. tang ^ = 1„, tayig y = fang 31' 48" we have, in meters, '^ = 0.009250. A prism producing a deflection equal to the tangent of 31' 48", equal to 0.00925 at a distance of one meter, will therefore correspond to the accu- rately ground prism of one degree refracting angle, with an index of 1. 53. In case glass of another index were used, it would be necessary to vary the angle of the prism, so as to satisfy the conditions of refraction for pro- A METRIC SYSTEM OF NUMBERING AND MEASURING PRISMS. 109 ducing the aforesaid deflection, and it is therefore obvious that manufac- turers will be privileged to adopt any correlative proportion of angle to in- dex which will satisfy the demands for any tangent-deflection which it may be determined to adopt as a unit. Supposing the chosen unit of deflection, tang ^, to be slightly greater than the above, say exactly equal to 0.01, or one centimeter, our series of prisms would then be : 1 Prism-dioptry producing a tangent deBection = 1"» in the meter-plane. = 2«» " = 3«" " A system of numbering prisms in terms of prism-dioptries, could therefore be adopted which would satisfy all the conditions here set forth. Such prisms could be measured by noting the deflection they produce upon the index line of a coarse centimeter scale, placed at right angles to the line of sight, at the distance of one meter.* (See Fig. 3.) y~ Fig. 3 While a restriction of this character offers the advantage of a ready ocular means of verifying the correctness of the prisms, there are at present however, many difficulties to be overcome in manufacturing them. Calcu- lation would disclose the fact that such prisms would require to be ground to degrees, minutes and seconds, so that comparatively few prisms out of a lot, at the close of our effort to produce them, would be found to actually meet the aforesaid requirements. This would so heighten their cost as to render them impracticable, except as diagnostic instruments in the consultation room. Even these, however, could be substituted by the prism-mobile, which consists of two prisms rotating before each other in opposite directions, and * For reasons given in the paper on the Prismometric Scale, perfect accuracy will only be insured ■.vhcn this scale is enlarged to the dimensions required for its use at a distance of six meters. 110 A METRIC SYSTEM OF NUMBEKING AND MEASURING PRISMS. which will afford the most ready means of filling a demand for definite de- flections, inasmuch as the rotation of the prisms, from 0° to 180°, pro- duces all possihle deflections from one millimeter upward. The instrument could easily be graduated to read to centimeters, and tenths, or millimetei's of deflection. For the determination of muscular insufficiencies of comparatively low degree, and to render the instrument as light as possible, a special cell, to contain weak rotating prisms, could be devised, similar to that of Dr. Eisley, to fit the trial frame. We may venture to assert that the prism, although the simplest element in Dioptrics, is the most difficult to manufacture, when required to be exact; and we shall therefore be obliged, for the present at least, to use existing commercial prisms for spectacle glasses. We shall subsequently show that these may be profitably utilized, by assigning the unavoidable variations of deflection, consequent to the manu- facture of such prisms, to their proper places, as members of the new system. By actual experiment the author has found imported prisms, represented as being of one degree (1°), to produce defiections varying between 9 and 13 millimeters, and which, if reduced to the basis of our standard of one centimeter, are to be designated as 0.9 and 1.3 prism-dioptries, respectively. Similar discrepancies of deflection are found to exist throughout the entire series of imported prisms now in use, so that we shall have adequate variety, covering almost every required interval of the new system; where- as, by the earlier method, an optical prescription, although required to be exact, is constantly exposed to the danger of being reduced to little else than a ticket of chance in an optical lottery. Farther on an instrument will be described which the author has devised to determine the power of prisms in dioptries and fractions, thus making it possible for the optician to select from his stock the one that shall fill the requirements of deflection sought. Indeed, by its use, we may hope to have manufacturers ultimately furnish us prisms, in packages, assorted and marked with the number indicating their power in dioptrics.* *• This expectation lias been fulfilled by American manufacturers ever since 1S94, see page 102. A METRIC SYSTEM OF NUMBERINQ AND MEASURING PRISMS. Ill THE RELATION OF THE PRISM-DIOPTRY TO THE METER- ANGLE. In the accompanying Fig. 4, E^ and E.^ represent the centers of rotation of the eyes, and OM the median Hne, bisecting the base-line E^ E^ at M. For the point of fixation 6>, corresponding to the angle of convergence, y^ for the eye E^^ we have b = ti8 «.< sin Y^ 0E\ b AIM. or if C, = 1 meter, sin y^ = b = the deviation re- quired of the eye which is optically adapted for the point O. The base-line, b, having a constant value for each inter-pupillary distance, 23, we have the angle y^ solely dependent upon the varying metrical values of Cj, so that the unit-angle of convergence has been designated, by Nagel, the meter-angle, when Q = 1 meter. Hence 1 ma = arc siji -7^-= arcsm -t-= b ; arc sin arc S171 b Ci arc sin t7-=2 b; 7i 2 ma = arc sin -^ a. s. f. We may now proceed to find the value of the prism in prism-dioptries, which, being placed before the eye, with its base in, shall substitute an effort of convergence to the same point. The deviation produced by a prism of one prism- dioptry, at one meter, being equal to <5, Fig. 4, it is evident that the prism which shall be required to produce a deviation b =^ b^=^ ^^, will have to be i? "" b ' times greater than one prism-dioptry (l^),t conse- ^'^•^' quently = ^.l^ If the same deviation is to be produced at the dis- tance a, = — of one meter, when C^ = 1 meter, for the meter-angle, the prism will require to be Y times as great, consequently t See page 102. 112 A METRIC SYSTEM OP NUMBERING AND MEASURING PRISMS. 1 ma = Kry.l^ But V='^ and 7 = ~ Similarly, if the deviation d is to be produced at the distance a^ =z ~ of one meter, when Q = ^ meter, for two meter angles, the prism will re- quire to be Z times greater than tj.I^, or Z.-q.l^. Consequently 2ma = Z.-qA^ But Z = — and v = ~r .•.2ma = ^. -^ . 1^ o .-. ;^war= J- . 4. 1^ = '-5!^1A . . . (I) When convergence is confined to comparatively small angles, we may regard the sine, tangent, and angle as being equal to one another. In other words, we may consider a^ = C^ = 1, a., = Q = ^, etc., so that l».a=i.A.l^ = A.i. 2 raa = -!- . 4- . 1' = 2 '- . 1' "2 Under thes^ circumstances one prism-dioptry differs from the meter- angle by the co-efficient -y-. This is due to the selection of a compara- tively small unit-deflection 5. If we had chosen a greater unit-deflection, say S = d, then one meter-angle would correspond to one prism-dioptry exactly. A prism, producing so great a deflection as half the inter-pupil- lary distance, would, however, give too great an angle for the unit and lowest degree of prism, unless others, as fractions of the unit, were in- cluded in the series. For instance, if 5 =: ^ equal the deflection for one A METRIC SYSTEM OF NUMBERING AND MEASURING PRISMS. Ill prism-dioptry, prisms of lesser refraction might be designated as 0.25"^, O.S'^, 0.75^, when their respective deiiections are } b, ^ b, and f b. Such a, selection would, however, possess no particular advantages, since b will generally be a variable quantity for different individuals ; besides, it will not be admissible to approximate the factor — for considerable degrees of convergence. A strict consideration of the co-efficient „ — . , under such circumstances, will be imperative, without regard to any particular choice of the unit deflection, so that for considerable degrees of convergence the prism-diop- tries will have to be determined by the Formula I. The following table exhibits the errors committed in estimating the value of the angle of convergence, when Formula II is substituted for Formula I, in reducing meter-angles to prism-dioptries. METER-ANGLES REDUCED TO PRISM-DIOPTRIES, FOR AN INTER-PUPILLARY DIS- TANCE OF 64 MILLIMETERS (3 = 32 "•/„,). STANDARD UNIT DEFLECTION FOR 1 PRISM-DIOPTRY = '^ = 0.01 = !""■ AT THE METER-PLANE. Distance from the Object of Fixa- tion to the Sine of the Meter- Angle. Tangent of the Meter- Angle. Value of the Angle of Convergence. Value of the Angle of Convergence when sine, tangent and angle are accepted as equal. Eye. In Meter- Angles. In Prism- Diop- tries. In Degrees ex- actly = A De- viation. In Meter- Angles. In Degrees arc sin. In Prism- Diop- tries. In Meters. In Milli- meters. A Deviatitm arc tang. 1 i i \ 20 1,000 500 333.3 250 200 100 50 0.032 0.064 0.096 0.128 0.160 0.32 0.64 0.0320163 0.0641309 0.0964465 0.129061 0.162080 0.337765 0.832919 1 2 3 4 5 10 20 3.20163 6.41309 9.64465 12 9061 16.2080 33.7755 83.2919 1°50' 1"6 3° 40' 9" 9 5° 30' 32" 1 7°21'14"4 9° 12' 24'' 6 18° 39' 46" 7 39° 47' 30" 1 2 3 4 5 10 20 1°50' 1"5 3° 40' 3" 5° 30' 4"5 7° 20' 6" 9° 10' 7"5 18° 20'15" 36° 40'30" 3.2 6.4 9.6 12.8 16 32 64 1° 49'68"2 3° 39'43" 5° 29' 0"9 7° 17'38"9 90 5'24"8 17° 44'40" 32° 37' 8"5 It is apparent that such a substitution will be admissible, up to five meter-angles, where the difference between the deviation produced by the prism, and the value in degrees of five meter-angles amounts only to 4' 42" 7. When we consider that a muscular insuflQciency of five meter- angles is entirely beyond the limits of optical correction, the latter, in fact, being confined to deficiencies of about one meter- angle, or less, it is obvious that a substitution of the sine for the tangent will be justifiable. 114 A METRIC SYSTEM OP NUMBERING AND MEASURINa PRiSMS. Such being the case, the subject of prismatic corrections becomes wonder- fully simple. For instance, for an inter-pupillary distance of 60 millimeters = 6 "", the base line will be 3 ™, when, according to Formula II, 1 fna = j^ = 3 prism-dioptries. Similarly, for an inter-pupillary distance of 50 millimeters, the base-line being 25 millimeters, equal to 2.5 centimeters, the meter-angle will be equal to 2.5^. Thus, for each inter-pupillary distance, we find a different prism neces- sary to supplant the meter-angle. This is but natural, since greater demands for convergence will be necessary in wide than in narrow inter- pupillary distances. This leads us to the final and simple rule : Read the patient's inter-pupillary distance in centimeters, when half of it will indicate the prism-dioptries required to substitute one meter-angle for each eye. One could scarcely hope for a more convenient method than to find the prism-dioptries, corresponding to one meter-angle, expressed in the patient's features. There will, however, frequently be occasion to supply less than one meter-angle, as indicated in the following tabulated examples. Pupillary distance, 2Z» == 56 60 64 68 millimeters Base-line ... 6 = 2.8 3 3.2 3.4 centimeters 1 meter-angle . . . = 2.8 3 3.2 3.4 prism-dioptries } meter-angle . . . = 0.93 1 LOG 1.13 ^ meter-angle . . . ^ 1.4 1.5 1.6 1.7 " " According to our standard, a prism of 0.9'^ will produce a tangent deflection of 0.9 of a centimeter, or 9 millimeters, and a prism of 1.1'^ a deflection equal to 1.1 centimeters, or 11 millimeters. It will therefore be possible to select these, by aid of an adequate instrument, from a paper of I*' prisms of foreign manufacture, since the latter are found to produce deflections varying between the same limits. Prisms of 1.3'^ to 1.6^ will similarly be found among prisms of 1^°, and so on. A METRIC SYSTEM OP NUMBERING AND MEASURING PRISMS. 115 Later we shall describe the instrument, called a prismometer, to distin- guish it from optical theodolites and goniometers of the physical laboratory, which, we believe, offers an advantage over the present methods of meas- uring prisms, inasmuch as it makes it possible to measure prisms to the nicety of fractions. * The general principle evolved, therefore, affords a new means of verifying the correctness of prisms in a simple manner, and must assuredly serve its purpose, whichever standard unit-deflection, at one meter distance, it may be ofl&cially determined to adopt, although it is believed that the centimeter has been shown to possess such decided advantages as to be worthy of favorable consideration. THE RELATION OF THE PRISM-DIOPTRY TO THE LENS- DIOPTRY. Some of the advantages of the prism-dioptry, as the unit of measure for the refraction of simple prisms, having been shown, and whereas prisms are frequently combined with spherical lenses, it is here proposed to further consider the relations of the prism-dioptry to such combinations, as well as to the equivalents which are to be obtained by a mere decentra- tion of the spherical lenses themselves. In the accompanying figure 5, a lens is shown with its principal anterior and posterior foci /and F, equidistant from O, upon the optical axis/OF. Fig. 5. The ray, ze, which is parallel to the optical axis, and incident at an eccentric point e, of the lens, being refracted to the focal point F will sus- * In the original paper, the first, and consequently more or less cnide prismometer, was described as being capable of measuring prisms up to 20 a. As in subsequent papers this was shown to have been an error, the reader is referred to the descriptions of the Perfected Prismometer, and the Prismometric Scale. 116 A METRIC SYSTEM OF NUMBERING AND MEASURING PRISMS. tain a deflection dF, in the focal plane, which will always be equal to the decentration Oe. A ray, de^ which is parallel to the optical axis and incident from the opposite direction, will be refracted to the focal point/, and, if received by the eye at E^ will be projected by it in the prolongation oife to v. Consequently dv = /0> ophthalmologists as well as avoid conflict with the practical methods of manufacturing opticians. At the close of the author's investigations, he felt that this had not only been accomplished, but that also an instrument in support of that system had been offered as a valuable assistant to opticians. The author's familiarity with the routine of manufacture would not allow him to lose sight of the practical side, so that this, being a matter of primary importance to opticians, was kept well in view from the outset. In advocating the metric system and the use of the prismometer, we shall, therefore here only do so in so far as they relate to the interests of manu- facturing and dispensing opticians; the advantages of the system to ophthalmic practice having been previously set forth. The author's argument in favor of the metric system was, and is based upon the former unavoidable variability in the angles of our prisms, and which must result from the foreign process of manufacture. Although this has been indicated in the previous papers, we shall here take the liberty of quoting from a paper read in connection with the author's exhibit of 125 12() THE PERFECTED PRISMOMETER. the prismometer before the ?^ew York Academy of Medicine, October 30, 1890 : "It would, however, be exceedingly difficult and correspondingly expen- sive to manufacture prisms producing only fixed intervals of deflection. To render prisms sufficiently inexpensive as spectacle glasses it is necessarj' that they should be produced in large quantities at one grinding." "The process at present consists in fastening a number of slabs of glass, by means of pitch, or other resinous material, upon a metallic surface- tool. The friction in grinding generates more or less heat, which at times is sufficient to soften the pitch and cause it to yield beneath the slabs. Some slabs will shift more than others, so that the prism-angles will vary more or less throughout. Besides, the imderlpng layer of pitch can never be of a uniform thickness." Were it not for these facts, competition alone would undoubtedly long ere this have resulted in greater uniformity. By means of the prismometer the author has found prisms, more especially of low degree, to vary Ijetween ten per cent, and thirty per cent, of their indicated numbering. It is obvious that if manufacturers were obliged to discard all those prisms which varied from desired fixed intervals of prism-angles, minimum deviation, or any other designated deflection, the price would have to be increased on the perfect prisms sufficiently to compensate for the cost of those rejected, and which would havo consumed equally as much material, time, and labor to produce. Without being confined to the deflections which should, by calculation, correspond to the prism-angles and index, the author found, by means of the prismometer, among a series of prisms, of best Parisian manufacture, only the following number to produce deflections which were even alike : Three doz. prisma 1° 2° 3° 4" 5° Number alike 6 = 1.1 7 = 2 6 = 3.1 6 = 3.7 8 = 4.6 prism dioptriea Balance varying between 0.8& 1.6 1.8 & 2.5 2.6& 3.2 3.4 & 3.9 4.4& 4.8 " " These prisms were taken from original packages, and may be credited with having been made of the same material, at the same time, and upon the same tools. Greater precaution on the part of the manufacturer could not be expected. To the careful reader of the author's papers it must have been apparent THE PERFECTED I'RISMOMETER. 127 that stress had nowhere been laid upon the possibility of a variability in the index, but, on the contrary, that all his deductions were referred to the commonly accepted index of 1.53. The privileges, however, were mentioned* of which manufacturers might avail themselves, both in respect to prism-angle and index, in seeking to provide prisms of the desired properties. To any one familiar with the use of optical theodolitesf and spectrome- tersj it must further be apparent that an endeavor to measure tlic minimum deviation, with prisms of small angles especially, is very tedious and diffi- cult. The apparatus is expensive, requires a degree of accuracy in manipu- lation, and a knowledge in the reading of verniers, with which opticians cannot readily be made familiar. To mount such prisms accurately upon the table of the spectrometer, and to rectify the various adjustments of the instrument, are tiresome and slow operations which alone are sufficient to condemn its daily use by opticians whose work must necessarily be expe- ditious. In the physical laboratory, however, the instrument is undoubt- edly invaluable. If the use of an instrument is to be abandoned in meas- uring the minimum deviation suggested by Dr. Jackson, we shall find that manufacturers will simply divide the prism-angles by two (2), for the new nomenclature, and so give us the old culprit disguised under a new name. There would be great commercial convenience to be sure, in being able to dispose of the same prism under two names, but no reform in the inter- est of scientific exactness could be effected without measurement. Will it be policy under such circumstances to adhere to the minimum deviation merely for principle's sake? As the prismometer is intended to measure the refraction of prisms, in terms of the prism-dioptry, it may be well, for the benefit of those who may have found its simplicity obscured by the mathematical portion of the previous papers, again to explain its principles in more simple and somewhat different terms. We know that a lens-dioptry is the unit of refraction, and corresponds to a lens of one meter focus. Fig. 1. *Fage 109. t Lehrbuch der Pliysik., Prof. Joh. Miiiler, Braunschweig, 1878. % Practical Physics, Glazebrook & Shaw, London, 1889. Elements of Physical Manipnlation, Prof. Ed. C. Pickering, Boston, 1878. 128 THE PERPECTKD PRISMOMETBR. The prism-dioptry, since lenses are but a fusion of prisms of varying angle, may then also be said to be the linear deflection which the refracted ray sustains at the focus of a meter-lens, when the incident ray impinges 1 meter = 100 centimeters. A IP. D. ! 1 D. Fig.]. upon a peripheral portion of the lens one centimeter from the optical center (Fig. 1). The prism-dioptry therefore also represents the measure of the angle of deviation /'„ for an eccentricity or decentration of one centimeter (Fig. 1). 1 meter = lOO""- JCWI f 2 D. 3^ meter. Fig. 2. A ray impinging upon the same point of a 2-dioptry lens (Fig. 2) will sustain the same unit-deflection at its focus ^ meter, and will therefore find the measure of its angle of deviation y,, expressed by twice the deflection at the meter-plane, or 2 prism-dioptries. A lens being decentered twice or half as much will produce twice or half as many prism-dioptries as the lens possesses lenticular dioptrics of refraction.* The prism-dioptry is therefore l)ut a sequence to the lens-dioptry. Nothing can be more simple. Thus the prism-dioptry represents the proportion 1:100, which is expressive of a wrade of angular inclination in daily use by engineers and scientists the world over. To reduce prism-dioptries to degrees of angular deviation, it is only necessary to divide the prism-dioptries by 100, when they will * Page ns. THE I'KRFEOTEl) I'UISMOMETEU. lii!> represent the tangents to correlative angles in degrees, which are to be readily found in any table of goniometrical lines. Since different lenses, through varying decentrations, will produce different values of the angles of deviation 'Vj "^2 • • • ,. how will it be possible to determine the value of such angles in degrees, minutes, and seconds? The instrument is yet to be invented. The prism-dioptry and the prismometer* solve the problem, and in a manner simple and rapid enough to any one of ordinary intelli- gence. Since the therapeutic value of prisms is conceded, and their combination with lenses in practice frequent, the prismometer has been constructed with due regard to such combinations, making it possible by its aid to utilize to advantage the prismatic action due to decentration of the lens, for the purpose of offsetting the error which invariably exists in the associated prism, after the combination has been ground. Would it not then seem unwise and even arbitrary to hamper the dispensing optician in the practical fulfillment of his work by forcing him to a system of degrees merely because it harmonizes with the designation of a strabismus which is incorrigible by prisms, or with the graduations found upon perimeters, ophthalmometers, etc., which have no connection with prisms whatever ? The metric system certainly possesses the commendation of reducing all the glasses of the trial case to a uniform nomenclature in dioptrics. This alone should be considered a practical advantage, fully offsetting the merits of a theoretical minimum deviation which cannot at present be expeditiously verified by any known means of accurate measurement. With a view to convenience and simplicity, let us learn to comprehend the power of our prisms by their limits of refraction, shown by the solid triangles in the preceding figures, when it will become wonderfully easy to fit these into meter-angles, or for that matter to any other angles in space, without necessarily confounding prism-dioptries with meter-angles, or meter-angles with "deviations of the e3'es in height," as stated by Dr. Landolt. f The latter mistake could only be the result of a misconception of the definition of the prism-dioptry and its relations to the meter-angle. In recommending the metric system to the profession and practical * Now also the Prismometric Scale. t "On the Numbering of Prismatic Glasses," ArcTiives of Ophth., XIX, No. 4, 1890. 130 THE rKRFECTKD PRISMOMETER. opticians, the author in conclusion begs to call attention to its superior advantages, as follows : 1. From a mechanical point of view, by taking the unavoidable difficul- ties of manufacture into consideration. 2. From a commercial and pecuniary point of view, by avoiding un- necessary expense in the production of prisms. 3. By the prismometer, which enables opticians to accurately fdl the demands of the system.* Any system which neglects these important considerations cannot be considered progressive, nor can it effect a reform in the present necessarily haphazard endeavors of the dispensing optician, with whom so much of the blame and responsibilit}^ must rest. Dispensing opticians have always been on the alert to meet the requirements of the profession, and will no doubt gladly avail themselves of a system and an instrument which will enable them to sustain their repvitations as mechanicians. Taking all the facts into consideration, it suffices to say, that we have prisms of almost every imaginable deflection on hand in the market to-day, so that it merely requires an instrument of simple construction, which may be used in making the proper selection with accuracy and despatch, and this is precisely what is claimed for the prismometer, which it is the author's purpose here to describe. In the accompanying illustration. Fig. 3, the essential operative parts of the instrument are shown to be moimted upon a triangular truss which is pivoted by a suitable joint to a pedestal, so as to permit of convenient inclination of the whole. The graduated bar is rigidly supported near its extremities, upon the truss, by two short studs or pillars, the latter being slightly higher than the radius of the circular stage, which is supported at its back by a rod, fitted, sliding, and acted upon by a spring within the bar, so as to auto- matically effect contact of the face of the stage with the knife-edge, which is also mounted upon the truss, between the stage and the pinhole eye- piece. ♦Since this paper was written, American manufacturers have so perfected the art of grinding that they are now able to furnish prisms which are accurately numbered In prism-dioptries, thus causing the prismometric scale to practically supplant the prismo- meter as an instrument for verifying measurements. THE PERFECTED PRISMOMETER. i;u The divisions of the graduated bar, numbered 2, o, 4, up to 10, are placed ^^ h h i) "P to tV of ^^^6 meter* length, counted from tlie knife-edge, which represents the zero-end of the scale. A vertical plane, arranged to slide upon the graduated bar, termed the index-plate, is provided with the index-line, marked zero (0), and two graduations at the right-hand upper Fig. 3. edge, marked 1 and 2, which, being equal to correlative centimeter defec- tions at the meter-plane, correspond to their equivalents in prism-dioptries. To facilitate subdivision of these graduations the index-plate is provided with a transverse slide bearing its allotted part of the index-line, which ia rendered adjustable by a milled head and micrometer screw, the first com- plete rotation of which will cause this section of the index-line to travel from to 1, the second complete rotation taking it from 1 to 2. The milled head, being divided into 100 parts, enables us, by its graduations, to determine the position of the index-line of the transverse slide, relatively to the graduations upon the face of the index-plate, in lOths and lOOtha. * It has been found convenient to construct the instrument to half sc«de throughout. 132 THE PERFECTED PRISMOMIETER. Tims, in the accompanying figure (4) we read from the face of the index- plate "1" and from the milled head y^ths and y^-^ths, or 1.25 for the posi- tion of the index-line of the transverse slide. As all readings of deflection must be reduced to the meter-plane^ it will be necessary to note the position of the index-plate, which must at all times correspond to one of the graduations of the bar. Consequently, if the above reading is taken from the index-plate, when placed at the figure "2" of the bar, we shall have twice the number of prism-dioptries at the meter- plane, or 1.25 X 2 = 2.5^ For a reading of "2," from the index-plate, when placed at the gradu- ation upon the bar marked "10," we have 20"^, which is the maximum measuring capacity of the instrument. In other words, it is merely neces- sary to multiply the readings of the index-plate by that figure upon the bar which defines the position of the index-plate upon it. Before placing a prism in position for measurement, it is necessary to carefully determine its base-apex line. This is accomplished by such slight rotary adjustment of the prism before the eye, until a line, situated at a convenient distance, is sighted as an unbroken one, being precisely the same method which we employ in determining the axes of cylinders. For convenience of registration, ink dots, in collimation with said line, should be applied to the prism. The stage is provided with a series of horizontal lines, engraved upon it, to facilitate perfect adjustment of the base-apex line of the prism, which is to be introduced between the stage and the knife-edge, with its apex to the right, and gradually forced downward while the ink dots pass successively from one horizontal line of the stage to the other, until the upper edge of the prism exactly bisects the circular opening in the stage. In this position the prism will exactly cover the lower half of the opening, while its lateral upper edge will be in collima- THE PERPECTBD PRISMOMETER. 183 don with the lower edge of the transverse slide. On completion of this adjustment it is of the utmost importance that the ink dots should coincide with one of the parallel lines of the stage. The observer's eye being placed before the eye-piece, will now perceive the upper edge of the index-plate, and the index-line at zero of the transverse slide, in their true positions, whereas the lower portion of the index-plate, with its index-line, being seen through the prism below, will appear displaced to the right. The position of the observer's eye is now to be carefully maintained, while the graduated milled head is operated with the right hand, until the index- line of the transverse slide has been shifted sufficiently to the right to make contact with the lower index-line seen through the prism. Perfect coinci- dence of these lines is necessary for an accurate determination of the de- flecting power of the prism at any distance. It will consequently be well to previously remove any roughness of the upper base-apex edge of the prism by grinding it to a flat dull edge, and, to be very precise, to take the mean of several readings while the prism is in an undisturbed position. As an example, we shall suppose the prism to have been carefully adjusted in the manner described, and that our readings for three positions upon the bar from the index-plate are as follows : 2(i Graduation of the bar, index-reading ^ 1.57 X 2 = 3.14^ 3d " " " " " =1.05X3 = 3.15^ 4th " " " " " =0.78X4 = 3.12'^ 041 A Mean: -^ = 3.13^ + This precaution, in the interest of exactness, may appear to be unnec- essary to some, yet it is here introduced as an exhibit in favor of the capa- bilities of the instrument. The prismometer is particularly valuable when it is desired to measure the inherent prismatic action of decentered lenses, and their combinations with prisms. In such cases it will be necessarj'^ to remove a peripheral portion of the lens by grinding it to a dull flat edge, as shown in the accompanying Fig- ure 5. The lens is then to be placed upon the stage with the flattened edge up, so as to cover half the stage opening ; the index-line of the transverse slide 134 THE PERFECTED PRISMOMETER. having been previously adjusted to zero (0). If, in sighting through the eye-piece, the index-line appears disjoined, it will only be necessary to shift the lens slightly to the right or left to re-establish coincidence of the lines, when the lens is said to be centered. While in this position ink dots Fig. 5. should be placed upon the outer edges of the lens over a centrall}^ situated horizontal line of the stage, as shown. For this centered position of the lens, in sighting through the eye-piece, we shall find the index-line at zero (0) un- broken, while the lower half of the index-plate will be enlarged or dimin- ished according to the character of the lens employed. Supposing the lens be 3 D. convex, we shall find the index-plate to present this view (Fig. 6) when it is placed at the graduation marked " 3 " upon the bar. Normal Plate. Fig. 6. — Magnified Plate. The lower half of the index-plate is provided with a red line, indicated by a dotted line in the figure, corresponding to a deflection of l'^, and which appears proportionately magnified. As it will be inadmissible, in our readings, to place a magnified scale on a par with the normal scale of the prismometer, it will be necessay to register the magnified unit upon the upper portion of the index-plate, for reference and comparison during de- centration of the lens. To accomplish this we displace the index-line of TUB PERFECTED PRISMOMETER. 136 the transverse slide until it coincides with the red line (dotted line, Fig 7), which, as far as the lens is concerned, now represents and takes the place of 1^ on the index-plate. _t^_i =E^ Di Fig. 7. Now, by slowly shifting the lens to the left, we shall observe the lines of the lower index-plate to shift to right (Fig. 8). Fig. 8. When the 3 D. lens has been decentered one centimeter, experiment shows that the lower black index-line cuts the index-line of the transverse slide above. Bearing well in mind that the position of the upper index- line now represents "1," and that our reading has been taken for a position of the index-plate upon the bar at "3," we have 3^ as the result of decen- tering a 3 D. lens one centimeter.* In case, however, that the refraction of the lens as well as its decentra- tion have not been previously determined it will be necessary to note the following (see Fig. 6) : normal plate 1 It is evident that the magnified plate normal prism-dioptries sought = -^ = ~=, so that the 1 D the magnified readings, for convex lenses, which will be when Z> > 1, and the the diminished readings, for concave lenses, which will be when Z) < 1. * Page 117. 136 THE PERFECTED PRISMOMBTER. It is therefore only necessary to divide the magnified or diminished read- ings by D. The value of Z>, as we have seen, is determined by first centering the lens. It will have a different value for different lenses, and will depend upon the distance of the lens from the index-plate. In fact, D represents the mag- nifying or diminishing power of lenses for any position of an object, when viewed through them, and which may be placed within their respective focal distances. For the 3 D. convex lens, at the graduation upon the bar marked "3," measurement by the instrument shows Z> to be equal to 1.2, Fig. 7. Suppose we decenter a -f 3 D. lens until we obtain a reading say of 0.6^, which is of course a magnified reading, we then have — ^ = -^ (magnified reading) = 0.5 of the normal prism-dioptry at the distance ' '3," or 1.5 normal prism-dioptries at the meter plane. In measuring sphero-prismatic lenses we shall therefore find that the value of the constant prism can either be increased or diminished by a de- centration of the lenticular element of the combination, a decentration of 5°/ni in the above instance being sufficient to contribute 1.5'^ ad- or ab-duc- tive as occasion may demand. By such means it will be possible to counteract the inaccuracies which invariably exist in the associated prism after the combination has been ground. When the lens is combined with a prism, the flattened dull edge should be cut parallel with the true base-apex line, the latter being regis- tered with ink dots and adjusted upon the stage as usual. The most ready means of measuring such a combination — for example, -f- 3 D. spherical combined with 2^ (constant prism) — will be to place the index-plate at the distance upon the bar marked "3," when, as before, the lens magnification Z> = 1.2, and which may be more conveniently de- termined by previously centering a spherical lens of the same refraction. Now, by deductive reasoning, we know that 2 normal prism-dioptries will be equal to f'^ at \ the distance, and this would require to be 1.2 greater at the same distance to appear as the properly proportioned magnified de- flection seen through the lens, consequently f of 1.2 =0.8 magnified prism- dioptries. We therefore set the line of the transverse slide so as to read 0.8'^ at the distance marked "3," upon the bar, and proceed to decenter the lens until the lower index-line cuts it, when we shall have the desired THE PERFECTED PRISMOMETER. 137 2 normal prism-dioptries. We may utilize the rule to prove the result : 0.8 mag. prism-dioptries X 3 = -jc- ^o ^^ ^ n<7r/«a/ prism-dioptries. Since lenses are capable of providing as many prism-dioptries as they pos- sess lens-dioptries of refraction, it also follows that we shall occasionally be enabled to secure a considerable proportion of prismatic action by decen- tration alone, provided the spherical lens is of proportionately greater strength. For instance, the 3 D. lens will produce 3'^ for a decentration of 1""- so that an available decentration of 3^"y,„ could in itself be relied upon to furnish l'^ of the 2^ in the lens forming the subject of our example. To facilitate measurement of concave sphero-prismatic lenses, the stage is provided with a rotating disk, within, containing three prisms of varying power, with their bases down, and which may be successively carried before the lower half of the opening in the stage as occasion may demand. The object of these prisms is to counteract the prismatic action in the yertical plane, which would otherwise manifest itself by a confusion of the transverse slide in its contact with the loMcr portion of the index-plate (Fig. 9), as a result of sighting through the upper peripheral edge of a concave lens (acting as a prism with its base up) when placed in proper position on the stage. The extent of the confusion of the parts, as shown in the figure, will naturally depend upon the strength of the lens, so that rotation of the disk will reveal the prism best calculated to re-establish contact, as shown in Fig. 10. Our choice of the prism being made, the lens is to be removed from the stage so as to rectify the position of the disk-prism before the index-line at aero (0), which should naturally present a perfect vertical line to view. As an example, let us suppose the combination — 3D. sph. 3 2"^ (constant prism) to be presented for measurement. We should first select a concave 3 D. lens, centering it upon the stage as described, and discover a confusion of the index-plate, at " 8 " upon the bar, as shown in Fig. 9. 138 TUB PERFECTED PRISMOMETER. It will be found that the first prism of the disk proves sufficient to re-establish contact, as in Fig. 10, I' I" e I Fig. 10. Removing the lens, rectifying the disk-prism, and replacing the lens, we find the diminishing power of the lens Z?= 0.83. Our object being to secure 2^ at the meter-plane, it follows that \ of this will have to be th(^ = 0.67 in the 2^ reading from the index-plate at *' 3 " upon the bar, or „ absence of diminishing power, and consequently 0.67 X 0.83 = 0.55 as u result of diminution by the lens. The index-line of the transverse side is therefore to be set to 0.55. The spherical lens is now to be replaced by the sphero- prismatic lens, with it-; base- apex line marked and adjusted upon one of the horizontals of the stage, and shifted upon this to the right or left, until the lower index-line cuts the index-line of the transverse slide. While the sphero-prismatic lens is in this position, an ink dot is to be placed upon it at the knife-edge, as the dot is intended to ultimately occupy the center of the frame in which the lens is to be mounted. Such can be the accuracy of the optician's work, with the aid of the prismometer for the metric system, and of which oculists in America may readily avail themselves by a simple request to have their diagnostic prisms re-numbered by measurement upon the instrument. By these explanations the author hopes to have succeeded in conveying the fact, that his object has not only been to promulgate a theory^ but also to render it useful and fully subservient to practice^ and in the absence of which it should, like many another, only live in minds, and mould in books. THE PRISMOMETRIC SCALE Revised reprint from the "American Journal of Ophthalmology," October, 1891. During the past two years "The Metric System of Numbering and Measuring Prisms"* has been a subject of considerable discussion, although the exact nature of its unit, the prism-dioptry, does not seem to have been generally understood, while its practical advantages to opticians, "of whom accurate work is expected," have been wholly disregarded in some recent criticisms, in which it has been compared with Dr. Jackson's and Dr. Dennet's equally as scientific though less convenient systems. It is, there- fore, now proposed to call attention to a still more simple feature of the metric system, with further explanations, yet with the understanding that the reader is familiar with its general principle and applications aa originally explained. The prismometric scale, preferably drawn upon heavy paper or card board, consists of a series of numbered gradations, " 6 centimeters apart," O I 6 8 9 JO Index MM I M I i I I I Mil I I III Line Fig. 1. with an index-line at zero, longer than the rest, as shown in Fig. 1, which being just six times greater than the "coarse centimeter scale" le- asee page 105. 139 146 THE PKISMOMETEIC SCALE. ferred to in the author's first paper, is intended to be placed at a six times greater distance, or "6 meters" from the eye; when simple prisms may be measured by it according to the manner originally set forth. The scale is also subdivided to quarters, thus making possible the measurement of prisms from 0.85 to 10 prism-dioptries. The average deflections produced by our foreign commercial prisms, marked 1° to 5°, will be found to correspond closely to this scale up to the fifth division. In applying the scale to the measurement of sphero-prismatic lenses, it is evident that the index-line will be rendered more or less indistinct in viewing it through such a lens, so that the lenticular element of the sphero- prismatic lens will require to be fully neutralized by a contra-generic lens of the same power, when, by shifting the neutralizing lens from right to left, it will be possible to secure a position for it which will leave us the prismatic deflection which it is sought to attain by the inherent prism of the entire combination. The procedure is best explained by the following example : The optician "Ijeing requested to grind a sphero-prismatic lens of + 3 D. sph. 3 3^, selects from his stock a prism which is rough on one side, and which he consequently is obliged, from its marhing, to take for granted is a prism of 3°. He then grinds the rough side to + 3 D. spherical, when according to the old method, he naturally assumes that he has accomplished the full object of his purpose. It is now suggested that lie carefulty determine the optical center of a concave lens of 3 dioptrics, and mark this point with an ink dot, placing the opposite side of this neutralizing lens in contact with the spherical side of the sphero-prismatic lens which it is desired to meas- ure. He is next requested to hold the entire combination before his eye, at exactly 6 meters from the scale, the precaution being taken to have the base-apex line of the sphero-prismatic lens horizontal, with the base to the left, and in such a manner that the upper edge of the entire combination covers only the lower half of the pupil. The index-line observed through the lenses will then appear to be displaced toward the right, relatively to the graduations as seen through the uncovered upper portion of the pupil; In the event of the index-line appearing to be displaced more or less than the required graduation marked "2," tlio operator has only to shift the THE PRISMOMETRIC SCALE. 141 neutralizing lens carefully to the left or right, until the index-line exactly cuts the second graduation. Care should be exercised not to change the position of the sphero-prismatic lens during this act, and while in this position, an ink dot should be placed on the sphero-prismatic lens, precisely opposite to the dot on the neutralizing lens. The former then indicates the point which should form the center of the glass in the spectacle frame. The reasons for this will be obvious from a consideration of the following figures : A Fig. 2. Fig. 3. The concave lens ABC in Fig. 2, with its center at B, neutralizes the plano-convex lens abc, thus securing the effect of a prism acd, just at the opposite points Bb. By shifting the neutralizing lens, as shown in Fig. 3, the effect of a prism of greater angle is obtained. It is, consequently, possible, within reasonable limits, by this means to correct any inaccuracy which may have existed in the original rough prism. The same effect is obtained in sphero-cylindro-prismatic lenses, by neutralizing the cylindrical element with an additional and carefully adjusted coiitra-generic cylindrical lens, though this is naturally a little more difficult. Opticians who keep sphero-cylindrical lenses in stock will generally find it more convenient to use these in neutralizing compound lenses involving prismatic power. It is obvious that it will be much easier to hold and shift a neutralizing lens which consists of only one piece of glass. In shifting the neutralizing lens, great care must be exercised to keep both cylindrical axes parallel in case a change from their coincidence becomes necessary to secure the desired prismatic power. We shall preface a further discussion of this question with a few simple optical definitions, which the author holds to be indispensable to a thor- ough understanding of the subject, and which, much to the author's regret, and for reasons too obvious to mention, Avere not presented by him in the previous papers. 1. The optical center of a lens is a point situated upon a line called the 142 THE PRISMOMETRIC SCALE. optical axis, which must be perpendicular to both the anterior and posterior surfaus of the lens. 2. Direct Pencils. — Rays of light which are emitted from a luminous point upon the optical axis will be refracted and directed to a conjugate point upon the same axis, it being specifically noted that the axes of the incident and refracted pencils of light and the optical axis of the lens must coincide. 3. Oblique Pencils. — In any case where the axis of the incident cone of light does not coincide with the normals to the surfaces of the refracting medium, whether it be a lens, prism or plate, the refracted pencil will no longer be a circular cone of light ; but, it will be a pencil bounded by a surface which penetrates and defines the illuminated area of the medium and two focal lines, which are at right angles to each other and the axis of the refracted pencil (see Fig. 6), Fig. 6. The same laws apply to the reflection by spherical surfaces of direct and oblique incident pencils of light, and their mathematical elucidation is given by Profs. R. S. Heath and W. Steadman Aldis, in their recent exhaustive treatises on Geometrical Optics. In illustration of the above definitions, let the curved line in Fig. 4 represent the spherical surface of a medium with a greater density than air, THE PRISMOMETRIC SCALE. 143 when perpendicularly incident conical pencils of light, projected upon it from successive points A, B, C, will have their respective conjugate foci, /, upon the correlative radii with which the axes of the incident pencils coincide. If the refracted pencils, within the medium, are to have focal points outside of the medium, the axes of these pencils will have to be perpendiadarly intercepted by the second surfaces as shown by the heavy lines in Fig. 5 ; and in the event of the second surface occupying an oblique position, ab^ Fig. 6, with respect to the pencil A^ the medium must be considered as a lens, having its optical center upon the axis An of the incident pencil, with the prism abc added to it. The circular cone of light, within the medium, will then project an el)ipti(ja,l area of illuminatioti, Zf, Fig. 7, upon the second surface, as the Fig. 7. axis of the pencil is here oblique^ and the refracted pencil ceases to be a circular cone, projecting itself outside of the medium as an astigmatic pencil, of which/, and/, are the focal lines at right angles to the axis, the whole being deflected toward the base of the inherent prism P. 144 THE PRISMOMETRIC SCALE. While this optical phenomenon, which in this case we may term a sphcro-cylindro- prismatic action, may be new to many, it has been known to physical science since Kummer, in 1860, first called attention to the theory by which it was mathematically demonstrated. Its significance to optometrical practice may, perhaps, be treated of in the future.* The fact, however, may be experimentally, though crudely, demon- strated by placing a plano-convex lens of 8 D. directly between a light at 20 feet, and a screen receiving its image. On interposing a prism of 20^, for example, with its base down, and in a manner to insure contact of the plane faces of the glasses, the image will be observed to change both its form and position upon the screen. By drawing the screen slightly nearer to the lens, a horizontal though imperfectly defined line corresponding to/, will become manifest, and by increasing the distance between lens and screen a vertically elongated looped figure, closely resembling a line at f.^, will appear. When a circular cone of light, C, Fig. 8, from a short definite distance falls obliquely upon the face of a simple prism, we again have an elliptical area of illumination, and the refracted rays witlibi the medium will assume a direction as if emitted from the focal lines z;„ z\^ reaching the second surface of the prism, and being refracted by it to the eye at £", as if projected from the lines Fj , V.^ , on the opposite side of the prism. There is one exception to this result, and that is when the axis of the incident pencil assumes a direction which is subject to minimum deviation, Normal. Fig. 8. in which event the emergent pencil will appear to diverge from a pointy at the same distance from the anterior surface as the original source of light C. In the case of a plate, the emergent pencil will also be of astigmatic * Now mentioned in Hand Book of Optics, for Students of Ophthalmologj-, W. W. Suter, M.D., 1899. TilE PRISMOMETRIC SCALE. 145 form, with the difference that it will appear to })rocced from a pair of focal lines located upon an \xyj\s. parallel to the axis of the incident pencil. This sphero-cylindro-prismatic action, on the part of a simple prism may be experimentally demonstrated in the following manner. Construct the figure MO (to the left in Fig. 9, in which the width of the principal lines is, say, 2 inches, and the distance apart of the perpendiculars is Fig. 9. about 24 inches), and place it at right angles to the line of sight, at a dis- tance of about 6 feet from the eye, before which a prism of 10^ is given considerable inclination to the visual axis, with its base in or out, and as shown in the diagrams, to the right in Fig. 9. The eye in each instance is to be placed directly opposite to the figure M. In both cases the prism is shown not only to have changed the position of the solid cross O, but also to have altered the dimensions of its vertical and horizontal bars in com- parison with M. "With these facts in mind we may return to our subject of measurement. In Fig. 10 the relative positions of the object of fixation (9, the prism, and the eye are shown. It is evident that the perpendicularly incident axis OV oi the conical pencil of rays emitted by the object O coincides with the visual axis, and that the axis of the refracted pencil F/* does not enter the eye, although it does define the deflection 01 which it is desired to ascertain. The axis of the refracted pencil, d^Ey which does enter the eye, however, will result from that incident pencil whose axis is oblique relatively to the normal at d, and it will therefore be a ray approaching 146 THE PRISMOMETRIC SCALE. direction for minimum deviation and will consequently suffer less deflec- tion, Oif than the refracted pencil whose axis is VP. Now if, as is the case with the prismometer, the observer reads the deflection 0< at the definite distance marked, say, "10," upon the graduated n bar, it is evident that an error will be committed, since 10 times Oi will be less than 10 times 01; * yet this seeming weakness in the author's pre- vious papers has escaped detection by the critics of the prism- dioptral system, and for the consolation of whom let it now be said that there could have been no reasoning so clever or ingenious on their part as to have made this error any the less apparent, even in a prism of io°y by merely contrasting the differences between arcs, sines and tangents, in a choice for the unit of measurement. Besides, a mere consideration of the well-known relative goniometrical values of these has not hitherto been pertinent to the discussion, since the proposed unit, the prism- dioptry, is not a goniometrical unit, but an optical unit. The desire to multiply any unit in optics should be curbed by a knowledge of the fact that all the fundamental optical laws are based upon the assumption and acceptance of values of limited m.agnitude, and that there is therefore apt to be a point where unreasonable multiplication of an optical unit will contradict the actually existing optical phenomenon. A warning to this •This will be equally true for mcasuremants taken from an arc at short finite dJatance. THE PRISMOMBTRIG SCALE. 147 effect was given in speaking of the decentration of lenses (see page 116 of the author's second paper). Even thickness, a dimension which we are taught to neglect with respect to ophthalmic lenses, becomes an appreciable factor in prisms above 8^, when we attempt to measure their deflection at short finite distance. This will be apparent from the following considerations. It has been shown that the ray, which in the nearest limit reaches the eye, is the axis Od, Fig. 11, of an oblique pencil, being refracted within the prism ABC from d to ^„ and thence in air to the eye E, which projects it to 2, upon the scale 01. For a given thickness of prism, this is the only pencil which will be received by the eye, since, if we increase the thickness by allowing the plane A^B^ to represent the anterior surface of the prism, the original incident axis Od will be refracted at v instead of RRF< ■^Tcri \ \ 25. \ DISTANCE \ 3 \ 40 33-3 2 \ 50. >^ .... 571 ■ 66 So V, 4- ._ ■^ .^^^.... ■33 .1 "^ .j^OO ■ ""^ -— __ ^^^ "^ — I i ""'X^ 2 —AM =LITU DE-/! ccor IMOD ATior l:-f3 5.-2 1 5— 1.75-1. 25-075- o- 25—0 1 1 . 3IA • -"T.—o c; <^ 7^0. 75, _-: 3 — C. F. PRENTICE, DEL. , 1887 188 THE CORRECTION OF DEPLETED DYNAMIC REFRACTION. total accommodation reijuired at 25 cm. is used in reading at 33J cm. distance, because the corrections then more closely correspond to Donders' table, and also for the reason that the author has found it more satisfactory to thus calculate in practice. For instance, the emmetrope, whose ampli- tude of accommodation is 2.5 D.^ should have 3 D. of refraction to read at 33^ cm., but, as he must also have one-third of the accommodation required at this distance in reserve, he should have 4 D. of refraction to read comfortably. Since his dynamic refraction is 2.5 D. Ave therefore give him -f 1.5 D. glasses to supply the deficiency. If his range of accommodation is normal, we find, by the chart, that he should be 50 years of age. In this manner the author frequently estimates the ages of persons with surprising accuracy. The lenticular corrections, inserted in the field of the chart, between the ages of 45 and 80 years, added to the corresponding amplitude of accom- modation, less the acquired hyperopia noted beneath, are found in each instance to amount to 4 D. Consequently, in any case, the requisite 3 D. to read at %o\ cm. represents three-fourths of the total power supplied both dynamically and artificially, by lenses. Of course, the corrections in the table, are only applicable in such cases where the reading distance is no greater than 33 J cm. , and wherein the amplitude of accommodation is found to correspond to Donders' deter- minations. The chief value of the chart therefore exists only in the fact that it serves to show, by comparison with any given case under examination, to what extent its amplitude of accommodation differs from the accepted normal state as determined by Donders. This cannot be too highly estimated, however, for Landolt * says : ' ' Donders' diagram corresponds so perfectly to the natural condition of things, that, in every case where the amplitude of accommodation is less than is indicated thereon, we may safely diagnose a weakness of accommodation, and, in case of any considerable difference, we may admit a paresis of this function." A knowledge of the patient's condition of health and habits will of course assist greatly in arriving at a definite decision. The piindinn proximum is tliat point located at finite distance, at which *LandoU's work, rR§e ^^^- THE CORKECTIOM OK DEIVLETED DYMAiMIC REFKACTION. 189 the patient is still able to distinctly sec small printecl characters, such as diamond type, dots, or fine lines. The size of the test-objects should, of course, be in proportion to the visual acuteness of the eyes, since there are ])ersons, who, though possessing a good range of accommodation, cannot read fine print at any dis- tance, simply because their visual acuteness is in- sufficient. Therefore, in cases where the visual acute- ness is -y- , or less, it is preferable to use more heavily executed test-objects. For a normal visual acuteness, the author finds it convenient to use the characteis liere shown and which are engraved on a circular disk mounted in a handle, to be held by the patient. On flic obverse side of the disk, the same, though more heavily drawn figure, a, is used when the visual acuteness is subnormal. To determine the position of the pimctum proxi- muni, the patient is requested to binocularly fix the central dot of the test-object at about his usual read- ing distance, and to gradually draw it nearer to the eyes until it just begins to blur; the latter effect being made more noticeable to the patient through the tendency of the dot to fuse with the lines of the square which encloses it. The punctum proxi- mum is reached the instant before the blurring is observed. This experi- ment should be repeated several times, and it is generally also advisable to verify it by moving the dot slightly nearer than the punctum proxi- mum, having the patient fix the dot attentively while it is gradually withdrawn to the position where it again appears sharply defin^ By using a tape, Avhich is graduated to dioptrics of refraction, we may read directly from its graduations, the amplitudes of accommodation in emme- tropia, provided the distance is measured from the cornea to the test-object, on the median line. The same procedure will apply in any case of ametropia, when the distance glasses which correct it are worn at the time the above measurement is made. In the latter case, calculation will of course be greatly simplified. To those familiar with the art of fitting, glasses it is, in most instances, 190 THE COEEECTION OF DEPLETED DYNAMIC REFEACTION. comparatively easy to determine the proper distance glasses, as well as for them to predict, with reasonable certainty, that the lenses will be worn, with comfort. But when the case is complicated with a loss of dynamic refraction, and unless extreme caution is used, there is great danger in giving the patient an over-correction, thus ultimately making a change in the prescribed reading glasses necessary. It is therefore of great importance to the optical practitioner, particularly if the cost of a subsequent change in the reading glasses is to be borne by him, that he should be able to predict the correctness of the reading glasses with the same degree of certainty that he feels in respect to the glasses which he prescribes for distance. In prescribing for depleted accommoda- tion there are at present two methods in vogue : 1. The impirical method; that of prescribing the glasses which have been accorded to a given age by Bonders in his table. 2. The physical method; that of locating the punctum proximum in each individual case, and using this as the basis for calculating the loss of ac- commodation which is to be compensated for. The latter is the only accurate and reliable method, j^et even in apply- ing it we are frequently hampered by the patient's indecision as to the dis- tance at which he habitually performs his near work. In the practitioner's office the patient may indicate the distance as being thirteen inches, whereas at his own occupation he perhaps finds that it is twenty. To avoid this feature of uncertainty as much as possible, it is consequently prudent to have the patient state the nature of his occupation, and to have him assume his accustomed position of the head, arms and body when engaged in near work. Then measure the distance from the eye, during convergence to the median line, by the dioptral tape, and note it as the desired reading distance, which is after all the real and only distance to be considered in the calculation for reading glasses. As an example, let us take an emmetrope who has a rang^ of accommo- dation of 2 D., and who indicates, by the aforesaid measurement, that he desires to see at the distance which corresponds to a refraction of 2.35 D. by the tape. As this amount should represent only three-fourths of the total refraction, to allow him one-fourth in reserve, it follows that the total THE CORRECTION OP DEPLETED DYNAMIC REFRACTION. 191 refraction should be 3 D. As he is capable of contributing 2 D. , dynami- cally, we give him -\- ^ D. glasses to supply the deficiency for reading. Therefore, to ascertain the reading glasses for any given case of emme- tropia^ we have only to follow the simple rule : Increase the refraction corresponding to the desired reading distance by one-third, and subtract therefrom the patient's amphtude of accommodation. It happens occasionally, as in the above example, that the punctum proximum is too far distant to enable the patient to see the test-object dis- tinctly. In such cases it is convenient to assist the patient by a lens which will enable him to do so. Let us suppose that we have assisted the patient impirically by a 1 Z?. lens, when he will, by the use of his 2 D. of accom- modation, be able to see the test-object distinctly at the 3 D. distance. By deducting the 1 D. lens we then find his amplitude of accommodation, which is 2 Z>. , and proceed by the rule given. In ametropia^^ provided it is corrected by the distance glasses, which then virtually render the patient emmetropic in accommodation, we pro- ceed by the same rule. The full reading correction will in this case be equal to the amount found by the rule, plus the distance correction. Should the accommodation be insufl5cient to definitely locate the punctum proximum, the patient, here too, is to be assisted by a lens which will likewise have to be deducted to ascertain the amplitude of accommodation. An exception to an application of the rule is found in those cases of myopia where the punctum proximum is nearer than the desired reading distance. In such cases it is customary to deduct the refraction corre- sponding to the desired reading distance from the distance correction. In some instances, where the amplitude of accommodation is considerable. * When the ametropia is not corrected, that is to say, when the distance glasses are not worn during the measurement of the range of accommodation we must resort to the formula : a^p — r. In hyper- opia the punctum remotum is behind the eye, therefore the refraction is negative, so that a=p — ( — r)=p -\- r which means that the range of accommodation is equal to the refraction at the punctum proximum, plus the refraction of the lens which corrects the hyperopia. In myopia the amplitude of accommodation is equal to the difference between the refraction at the near point, and the refraction of the lens which corrects the myopia. For a more exhaustive discussion of this subject, the reader is referred to Dr. Landolt's work, in which the physical portion is treated at greater length and more lucidly than in any other medical publication which has come to the author'a notice. 102 THE COERECTION OF DEPLETED DYNAMIC BEFBACTIOlf. and more especially in young persons, the patient will prefer to use his distance correction for all purposes. From this discussion it must be evident to anyone proficient in the practice of optometry that, as a matter of fact, greater skill and knowledge is required to scientifically determine the proper glasses for reading, com- monly known as presbyopic corrections, than is needed to ascertain the lenticular requirements for distance. Nevertheless, some oculists have ex- pressed the opinion that opticians should only be permitted to adapt glasses for presbyopee. If the same medical gentlemen were better informed ia optics, they would imdoubtedly deny opticians all rights in the matter. CLINICS IN OPTOMETRY By C. H. Brown, M.D. Graduat<> Uniyersity of Pennsylvania ; Professor of Optics and ftefraction ; formerly Phypiciaa in Philadelphia Hospital ; Member of Philadelphia C-onnty, Pennsylvania Stat* and American Medical Societies "Clinics in Optometry" is a unique work in the field of practical refraction and fills a want that has ^^^^^^^^^ been seriously felt both by oculists l^lTn^^BHI '^'^^ optometrists. ^^^^^^^^^^^ The book is a compilation of ppTnTIi^LMBIj optometric clinics, each clinic being complete in itself. Together they cover all manner of refractive eye defects, from the simplest to the most complicated, giving in minutest detail the proper procedure to follow in the diagnosis, treatment and correction of all such defects. No case can come before you that you cannot find a similar case thoroughly explained in all its phases in this useful volume, making mistakes or oversights impossible and assuring correct and successful treatment. The author's experience in teaching the science of refraction to thousands of pupils peculiarly equipped him for compiling these clinics, all of which are actual cases of refractive error that came before him in his practice as an oculist. A copious index makes reference to any particular case, test or method, the work of a moment. Sent postpaid on receipt of $i.50 ies, 3d.) PUBLISHKD BY The Keystone Publishing Co. 809-81 i-Si.^ North 19x11 Street, Philadelphia, U.S.A. THE OPTICIAN'S MANUAL VOL. I. By C. H. Brown, M. D. Graduate University of Pennsylvania; Professor of Optics and Refraction; formerly Physician in Philadelphia Hospital; Member of Philadelphia County, Pennsylvania State and American Medical Societies. The *^' ; OPTICIANS >!' MANUAL The Optician's Manual, Vol. I, was the most popular and useful work on practical refraction ever written, and has been the entire optical education of many hundred successful refractionists. The knowledg"e it contains was more ef- fective in building- up the optical profes- sion than any other educational factor. It is, in fact, the foundation structure of all optical knowledge as the titles of its ten chapters show: -Introductory Remarks. -The Eye Anatomically. -The Eye Optically; or. The Physiology of Vision. -Optics. -Lenses. -Numbering of Lenses. -The Use and Value of Glasses. -Outfit Required. -Method of Examination. -Presbyopia. The Optician's Manual, Vol. I, was the first important treatise published on eye refraction and spectacle fitting. It is the recognized standard text-book on practical refraction, being used as such in all schools of Optics. A study of it is essential to an intelligent appreciation of its companion treatise, The Optician's Manual, Vol. II, described on the opposite page. A comprehensive index adds much to its usefulness to both student and practitioner. Bound in Cloth — 422 pages — colored plates and illustrations. Sent postpaid on receipt of $1.50 ^63, 3d.) Chapter L- Chapter H. Chapter III. Chapter IV. Chapter V.- Chapter VI. Chapter VII.- Chapter VIII.- Chapter IX.- Chapter X.- Published by T^HS Keystone Publishing Co. 809-811-813 North 19TH Street, Philadelphia, U. S. A. THE OPTICIAN'S MANUAL VOL. 11. By C. H. Brown, M. D. Oraduate University of Pennsylvania; Professor of Optics and Refraction; formerly Physician in Philadelphia Hospital; Member of Philadelphia County, Pennsylvania State and American Medical Societies. The Optician's Manual, Vol. II., is a direct continuation of The Optician's Manual, Vol. I., being- a much more advanced and comprehensive treatise. It covers in minutest detail the four great subdivisions of practical eye re- fraction, viz : Myopia. Hypermetropia. Astigmatism. Muscular Anomalies. It contains the most authoritative and complete re- searches up to date on these subjects, treated by the master hand of an eminent oculist and optical teacher. It is thor- oughly practical, explicit in statement and accurate as to fact. All refractive errors and complications are clearly explained, and the methods of correction thoroughly elucidated. This book fills the last great vi'ant in higher refractive optics, and the knowledge contained in it marks the standard of professionalism. Bound in Cloth — 408 pages — with illustrations. Sent postpaid on receipt of $1.50 (6s. 3d.) Published by Thk Keystone Publishing Co. S09-811-813 North 19TH Street, Philadelphia, U.S. A. THE PRINCIPLES of REFRACTION in the Human Eye, Based on the Laws of Conjugate Foci By Swan M. Burnett, M. D., Ph. D. Formerly Professor of Ophthalmology and Otology in the Georgetown University Medical School; Director of the Eye and Ear Clinic, Central Dispensary and Emergency Hospital; Ophthalmologist to the Children's Hos- pital and to Providence Hospital, etc., Washington, D. C. In this treatise the student is given a condensed but thor- ough grounding in the principles of refraction according to a method which is both easy and fundamental. The few laws governing the conjugate foci lie at the basis of whatever per- tains to the relations of the o1)ject and its image. To bring all the phenomena manifest in the refraction of the human eye consecutively under a common explanation by these simple laws is, we believe, here undertaken for the first time. The comprehension of much which has hitherto seemed difficult to the average student has thus been rendered much easier. This is especially true of the theory of Skiascopy, which is here elucidated in a manner much more simple and direct than by any method hitherto offered. The authorship is sufficient assurance of the thoroughness of the work. Dr. Burnett was recognized as one of the great- est authorities on eye refraction, and this treatise may be described as the crystallization of his life-work in this field. The text is elucidated by 24 original diagrams, which were executed by Qias. F. Prentice, M.E., whose pre-emi- nence in mathematical optics is recognized by all ophthalmol- ogists. Bound in Siik Cloth. Sent postpaid to any part of the world on receipt of price, $1.00 (4s. 2d.) Published by Thk Keystone Pubi^ishixg Co. 809-811-813 North 19TH Street, Philadelphia, U. S, PHYSIOLOGIC OPTICS Ocular Dioptrics — Functions of the Retina — Ocular Movements and Binocular Vision By Df. M. Tscheming IHrector of tlie Lalioratory of Ophthalmology at the Sorbonne, P&ris. AUTHORIZED TRANSLATION By Carl Wciland, M. D. Fomer Chief of Clinic in the Eye Department of the Jefferson College Uospitai, Philadelphia, Pa. This book is recognized in the scientific and medical world as the one complete and authoritative treatise on physiologic optics. Its distinguished author is admittedly the greatest authority on this subject, and his book embodies not only his own researches, but those of the several hundred investigators who, in the past hundred years, made the eye their specialty and life study. Tscherning has sifted the gold of all optical research from the dross, and his book, as now published in English, with many additions, is the most valuable mine of reliable optical knowledge within reach of ophthalmologists. It contains 380 pages and 212 illustrations, and its reference list comprises the entire galaxy of scientists who have made the century famous in the world of optics. The chapters on Ophthalmometry, Ophthalmoscopy, Ac- commodation, Astigmatism, Aberration and Entoptic Phenom- ena, etc. — in fact, the entire book contains so much that is new, practical and necessary that no refractionist can afford to be without it. Bound in Cloth. 380 Pages, 212 lilustratioas. Price $2.50 (IDs. 5d.) Published by The Keystone Publishing Co. 809-81 1-813 North 19TH Street, Philadelphia, U. S. A. SKIASCOPY AND THE USE OF THE RETINOSCOPE By Geo. A. Rogers rormci'iy I'rofessor in the Northern Illinois College of Ophthalmology and Otology, Cliii'afro; Principal of the Chicago Post-Graduate College of Optometry; Lecturer and Specialist on Scientific Eye Refraction. ■^B^ A Treatise on the Shadow Test in its Practical Application to the Work of Refraction, with an Ex= planation in Detail of the Optical Principles on which the Science is Based. This work far excels all previous treatises on the sub- ject in comprehensiveness and practical value to the refrac- tionist. It not only explains the test, but expounds fully and explicitly the principles underlying it — not only the phe- nomena revealed by the test, but the why and wherefore of such phenomena. It contains a full description of skiascopic apparatus, in- cluding the latest and most approved instruments. In depth of research, wealth of illustration and scientific completeness this work is unique. Bound in cloth; contains 231 pages and 73 illustrations and colored plates. Sent postpaid to any part of the world on receipt of $1.00 (4s. 2d.) Published by The Keystonk Publishing Co. -S11-813 North 19TH Street, Philadelphia, U. S. A. TESTS AND STUDIES OF THE OCULAR MUSCLES By Ernest E. Maddox, M. D., F. R. C. S., Ed. Ophthalmic Surgeon to the Royal Victoria Hospital, Bournemouth, England; formerly Sj-me Surgical Fellow, Edinburgh University. |j Tests and Studies j! of the ; OGular Muscle|, This book is universally recog- nized as the standard treatise on the muscles of the eye, their func- tions, anomalies, insufficiencies, tests and optical treatment. All opticians recognize that the subdivision of refractive work that is most troublesome is muscular anomalies. Even those who have mastered all the other intricacies ot visual correction will often find their skill frustrated and their efforts nullified if they have not thoroughly mastered the ocular muscles. The eye specialist can thoroughly equip himself in this fundamental essential by studying the work of Dr. Maddox who is known in the world of medicine as the greatest in- vestigator and authority on the subject of eye muscles. The present volume is the second edition of the work, specially revised and enlarged by the author. It is copiously illustrated and the comprehensive index greatly facilitates reference. Bound in silk cloth — 261 pages — no illustrations. Sent postpaid on receipt of price $1,50 (6s. 3d.) Published by The Keystone Publishing Co. 809-81 1-813 North 19TH Street, Philadelphia, U. S. Optometric Record Book A record-book, wherein to record optometric examina- tions, is an indispensable adjunct of an optician's outfit. The Keystone Optometric Record-book was specially pre- pared for this purpose. It excels all others in being- not only a record-book, but an invaluable guide in examination. The book contains two hundred record forms with printed headings, sug-gesting, in the proper order, the course of ex- amination that should be pursued to obtain most accurate re- sults. Each book has an index, which enables the optician to refer instantly to the case of any particular patient. The Keystone Record-book diminishes the time and labor required for examinations, obviates possible oversights from carelessness, and assures a systematic and thorough ex- amination of the eye, as well as furnishes a permanent record of all examinations. Sent postpaid on receipt of $1.00 (4s. 2d.) Published by The Keystone Publishing Co. 809-811-813 North 19TH Street, Philadelphia, U. S. A. 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