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| author | www-data <www-data@mail.pglaf.org> | 2026-07-10 19:06:16 -0700 |
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| committer | www-data <www-data@mail.pglaf.org> | 2026-07-10 19:06:16 -0700 |
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+ right: 84%; + text-align: right; + font-size: 0.9em; +} + +.label:hover { + background: aqua; +} + +.fnanchor { + vertical-align: baseline; + position: relative; + top: -0.5em; + margin-left: 0.05em; + font-size: 0.8em; + font-weight: normal; + font-style: normal; + white-space: nowrap; + text-decoration: none; +} + + +/* Transcriber's notes */ +.transnote { + background-color: #F2F2F2; + color: black; + font-size:smaller; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; +} + +.transnote > p {text-indent: 0em;} + + +/* Illustration classes */ +.illowe39_3750 {width: 39.3750em;} +.illowe36_2500 {width: 36.2500em;} + + </style> + </head> + +<body> +<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 79069 ***</div> + + +<div class="transnote"> +<p><b><a id="Transcribers_notes"></a>Transcriber’s notes</b>:</p> + +<p>This e-book comprises the text of a lecture delivered on November +27, 1800 and subsequently published in the <i>Philosophical Transactions +of the Royal Society</i>, Issue 91 (1801), pp. 23–88</p> + +<p>It retains the original page numbering and includes 57 Figures that +are grouped together in seven Plates near the end of the lecture text. +Cross references to the figures have been hyperlinked to the relevant +plate containing the figure but return links back to the x-refs are not +possible. Footnotes have been numbered consecutively and moved to the +end. They are hyperlinked in both directions. The text contains +sections of complex mathematics and some unfamiliar symbols.</p> + +<p>New original cover art included with this eBook is granted to the +public domain.</p> +</div> + +<p class="tac fs130 ls01em">The Philosophical Transactions of the Royal Society</p> + +<p class="surtitle">The Bakerian Lecture:</p> +<p><span class="pagenum" id="Page_23">23</span></p> + +<h1> +On the Mechanism of the Eye +</h1> + +<p class="tac ws02em fs120">by Thomas Young M.D. F.R.S.</p> + +<p class="tac mtb1em">Read November 27, 1800.</p> + + +<p class="mt2em ti0">I. In the year 1793, I had the honour of laying before the +Royal Society, some observations on the faculty by which the +eye accommodates itself to the perception of objects at different +distances.⁠<a id="FNanchor_1_1" href="#Footnote_1_1" class="fnanchor">1</a> The opinion which I then entertained, although it +had never been placed exactly in the same light, was neither so +new, nor so much forgotten, as was supposed by myself, and +by most of those with whom I had any intercourse on the subject. +Mr. <span class="smcap">Hunter</span>, who had long before formed a similar opinion, +was still less aware of having been anticipated in it, and +was engaged, at the time of his death, in an investigation of the +facts relative to it;⁠<a id="FNanchor_2_2" href="#Footnote_2_2" class="fnanchor">2</a> an investigation for which, as far as +physiology was concerned, he was undoubtedly well qualified. +Mr. <span class="smcap">Home</span>, with the assistance of Mr. <span class="smcap">Ramsden</span>, whose recent +loss this Society cannot but lament, continued the inquiry +which Mr. <span class="smcap">Hunter</span> had begun; and the results of his experiments +appeared very satisfactorily to confute the hypothesis of +the muscularity of the crystalline lens.⁠<a id="FNanchor_3_3" href="#Footnote_3_3" class="fnanchor">3</a> I therefore thought +it incumbent on me, to take the earliest opportunity of testifying +my persuasion of the justice of Mr. <span class="smcap">Home</span>’s conclusions, +which I accordingly mentioned in a Dissertation published at +<span class="pagenum" id="Page_24">24</span>Gottingen in 1796,⁠<a id="FNanchor_4_4" href="#Footnote_4_4" class="fnanchor">4</a> and also in an Essay presented last year +to this Society.⁠<a id="FNanchor_5_5" href="#Footnote_5_5" class="fnanchor">5</a> About three months ago, I was induced to +resume the subject, by perusing Dr. <span class="smcap">Porterfield</span>’s paper on +the internal motions of the eye;⁠<a id="FNanchor_6_6" href="#Footnote_6_6" class="fnanchor">6</a> and I have very unexpectedly +made some observations, which I think I may venture to say, +appear to be finally conclusive in favour of my former opinion, +as far as that opinion attributed to the lens a power of changing +its figure. At the same time, I must remark, that every person +who has been engaged in experiments of this nature, will +be aware of the extreme delicacy and precaution requisite, both +in conducting them, and in drawing inferences from them; and +will also readily allow, that no apology is necessary for the +fallacies which have misled many others, as well as myself, in +the application of those experiments to optical and physiological +determinations.</p> + + +<p class="mt2em ti0">II. Besides the inquiry respecting the accommodation of the +eye to different distances, I shall have occasion to notice some +other particulars relative to its functions; and I shall begin +with a general consideration of the sense of vision. I shall +then enumerate some dioptrical propositions subservient to my +purposes, and describe an instrument for readily ascertaining +the focal distance of the eye. On these foundations, I shall +investigate the dimensions and refractive powers of the human +eye in its quiescent state; and the form and magnitude of the +picture which is delineated on the retina, I shall next inquire, +how great are the changes which the eye admits, and what +degree of alteration in its proportions will be necessary for +these changes, on the various suppositions that are principally +<span class="pagenum" id="Page_25">25</span>deserving of comparison. I shall proceed to relate a variety of +experiments which appear to be the most proper to decide on +the truth of each of these suppositions, and to examine such +arguments as have been brought forwards, against the opinion +which I shall endeavour to maintain; and I shall conclude with +some anatomical illustrations of the capacity of the organs +of various classes of animals, for the functions attributed to +them.</p> + + +<p class="mt2em ti0">III. Of all the external senses, the eye is generally supposed +to be by far the best understood; yet so complicated and so +diversified are its powers, that many of them have been hitherto +uninvestigated; and on others, much laborious research has been +spent in vain. It cannot indeed be denied, that we are capable +of explaining the use and operation of its different parts, in a +far more satisfactory and interesting manner than those of the +ear, which is the only organ that can be strictly compared with +it; since, in smelling, tasting, and feeling, the objects to be examined +come almost unprepared into immediate contact with +the extremities of the nerves; and the only difficulty is, in conceiving +the nature of the effect produced by them, and its communication +to the sensorium. But the eye and the ear are +merely preparatory organs, calculated for transmitting the impressions +of light and sound to the retina, and to the termination +of the soft auditory nerve. In the eye, light is conveyed to +the retina, without any change of the nature of its propagation: +in the ear, it is very probable, that instead of the successive motion +of different parts of the same elastic medium, the small bones +transmit the vibrations of sound, as passive inelastic hard bodies, +obeying the motions of the air in their whole extent at the same +instant. In the eye, we judge very precisely of the direction of +<span class="pagenum" id="Page_26">26</span>light, from the part of the retina on which it impinges: in the +ear, we have no other criterion than the slight difference of motion +in the small bones, according to the part of the tympanum on +which the sound, concentrated by different reflections, first +strikes; hence, the idea of direction is necessarily very indistinct, +and there is no reason to suppose, that different parts of the +auditory nerve are exclusively affected by sounds in different +directions. Each sensitive point of the retina is capable of receiving +distinct impressions, as well of the colour as of the +strength of light; but it is not absolutely certain, that every +part of the auditory nerve is capable of receiving the impression +of each of the much greater diversity of tones that we can distinguish; +although it is extremely probable, that all the different +parts of the surface exposed to the fluid of the vestibule, are +more or less affected by every sound, but in different degrees +and succession, according to the direction and quality of the +vibration. Whether or no, strictly speaking, we can hear two +sounds, or see two objects, in the same instant, cannot easily be +determined; but it is sufficient, that we can do both, without the +intervention of any interval of time perceptible to the mind; +and indeed we could form no idea of magnitude, without a comparative, +and therefore nearly cotemporary perception of two +or more parts of the same object. The extent of the field of +perfect vision for each position of the eye, is certainly not very +great; but it will appear hereafter, that its refractive powers are +calculated to take in a moderately distinct view of a whole +hemisphere: the sense of hearing is equally perfect in almost +every direction.</p> + +<p><span class="pagenum" id="Page_27">27</span></p> + + +<p class="mt2em ti0">IV. DIOPTRICAL PROPOSITIONS.</p> + +<p class="tac mtb1em"><i>Proposition</i> I. <i>Phenomenon</i>.</p> + +<p>In all refractions, the ratio of the sine of the angle of +incidence to the sine of the angle of refraction is constant. +(<span class="smcap">Newton</span>’s Opt. I. Ax. 5. <span class="smcap">Smith</span>’s Opt. 13. <span class="smcap">Wood</span>’s Opt. 24.)</p> + +<p><i>Scholium</i> 1. We shall call it the ratio of <i>m</i> to <i>m</i> ⫧ 1, and +<i>m</i> ⫧ 1, <i>n</i>. In refractions out of air into water, <i>m</i> = 4 and <i>n</i> += 3, very nearly; out of air into glass, the ratio is nearly that +of 3 to 2.</p> + +<p><i>Scholium</i> 2. According to <span class="smcap">Barrow</span>, (<i>Lect. Opt</i>. ii 4.) <span class="smcap">Huygens</span>, +<span class="smcap">Euler</span>, (<i>Conject. phys. circa prop, soni et luminis. Opusc. t. ii.</i>) +and the opinion which I lately submitted to the Royal Society, +(Phil. Trans. for 1800. p. 128,) the velocity of light is the greater +the rarer the medium: according to <span class="smcap">Newton</span>, (Schol. Prop. +96. l. i. Princip. Prop. 10. p. 3. l. ii. Opt.) and the doctrine +more generally received, the reverse. On both suppositions, it +is always the same in the same medium, and varies in the ratio +of the sines of the angles. This circumstance is of use in facilitating +the computation of some very complicated refractions.</p> + +<p class="tac mtb1em"><i>Proposition</i> II. <i>Phenomenon</i>.</p> + +<p>If between two refracting mediums, a third medium, terminated +by parallel surfaces, be interposed, the whole refraction +will remain unchanged. (<span class="smcap">Newton</span>’s Opt. l. i. p. 2. Prop. 3. +<span class="smcap">Smith</span>, r. 399. <span class="smcap">Wood</span>, 105.)</p> + +<p><i>Corollary</i>. Hence, when the refractions out of two mediums +into a third are given, the refraction at the common surface of +these mediums may be thus found. Let the refractions given +<span class="pagenum" id="Page_28">28</span>be as <i>m</i>: <i>n</i>, and as <i>m</i>′: <i>n</i>′; then the ratio sought will be that of +<i>m n</i>′: <i>m</i>′ <i>n</i>. For instance, let the three mediums be glass, water, +and air; then <i>m</i> = 3, <i>n</i> = 2, <i>m</i>′ = 4, <i>n</i>′ = 3, <i>m n</i>′ = 9, and +<i>m</i>′ <i>n</i> = 8. If the ratios be 4: 3, and 13: 14, we have <i>m n</i>′: <i>m</i>′ <i>n</i> +:: 39: 56; and, dividing by 56—39, we obtain 2.3 and 3.3 +for <i>m</i> and <i>m</i> + 1, in Schol. 1, Prop. I.</p> + +<p class="tac mtb1em"><i>Proposition</i> III. <i>Problem</i>. (Plate II. <a href="#Pl.II">Fig. 1</a>.)</p> + +<p>At the vertex of a given triangle (CBA), to place a given refracting +surface (B), so that the incident and refracted rays may +coincide with the sides of the triangle (AB and BC.)</p> + +<p>Let the sides be called <i>d</i> and <i>e</i>; then in the base take, next to +<i>d</i> (or AB), a portion (AE) equal to <img style="vertical-align: -1.738ex; width: 9.336ex; height: 4.837ex;" src="images/1.svg" alt=" " data-tex="\(\dfrac{nd}{nd+me}\)">, or (AD =) <img style="vertical-align: -1.738ex; width: 9.336ex; height: 4.837ex;" src="images/2.svg" alt=" " data-tex="\(\dfrac{md}{md+ne}\)">; +draw a line (EB, or DB) to the vertex, and the surface must be +perpendicular to this line, whenever the problem is physically +possible. When <img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/3.svg" alt=" " data-tex="\(e\)"> becomes infinite, and parallel to the base, take +<img style="vertical-align: -1.577ex; width: 3.529ex; height: 4.676ex;" src="images/4.svg" alt=" " data-tex="\(\dfrac{nd}{m}\)"> or <img style="vertical-align: -1.577ex; width: 4.158ex; height: 4.676ex;" src="images/5.svg" alt=" " data-tex="\(\dfrac{md}{n}\)"> next to <img style="vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;" src="images/6.svg" alt=" " data-tex="\(d\)">, for the intersection of the radius of curvature.</p> + +<p class="tac mtb1em"><i>Proposition</i> IV. <i>Theorem</i>. (<a href="#Pl.II">Fig. 2</a>.)</p> + +<p>In oblique refractions at spherical surfaces, the line (AI, KL,) +joining the conjugate foci (A, I; K, L;) passes through the point +(G), where a perpendicular from the centre (H) falls on the +line (EF), bisecting the chords (BC, BD,) cut off from the incident +and refracted rays.</p> + +<p>Corollary 1. Let <i>t</i> and <i>u</i> be the cosines of incidence and refraction, +the radius being 1, and <i>d</i> and <i>e</i> the respective distances +of the foci of incident and refracted rays; then <img style="vertical-align: -1.738ex; width: 21.397ex; height: 4.837ex;" src="images/7.svg" alt=" " data-tex="\(e=\dfrac{mduu}{mdu-ndt-ntt}\)"></p> + +<p><i>Corollary</i> 2. For a plane surface, <img style="vertical-align: -1.738ex; width: 10.818ex; height: 4.837ex;" src="images/8.svg" alt=" " data-tex="\(e=\dfrac{mduu}{-ntt}\)">.</p> + +<p><span class="pagenum" id="Page_29">29</span></p> + +<p><i>Corollary</i> 3. For parallel rays, <img style="vertical-align: -0.186ex; width: 6.456ex; height: 1.756ex;" src="images/9.svg" alt=" " data-tex="\(d=\infty\)">, and <img style="vertical-align: -1.738ex; width: 13.287ex; height: 4.267ex;" src="images/10.svg" alt=" " data-tex="\(e=\dfrac{muu}{mu-nt}\)">.</p> + +<p><i>Scholium</i> 1. It may be observed, that the caustic by refraction +stops short at its cusp, not geometrically, but physically, the +total reflection interfering.</p> + +<p><i>Corollary</i> 4. Call <img style="vertical-align: -1.738ex; width: 9.216ex; height: 4.267ex;" src="images/11.svg" alt=" " data-tex="\(\dfrac{muu}{mu-nt}\)">, <i>b</i> and <img style="vertical-align: -1.738ex; width: 9.216ex; height: 4.683ex;" src="images/12.svg" alt=" " data-tex="\(\dfrac{ntt}{mu-nt}\)">, <i>c</i>; +then <img style="vertical-align: -1.738ex; width: 9.989ex; height: 4.837ex;" src="images/13.svg" alt=" " data-tex="\(e=\dfrac{bd}{d-c}\)">, +and <img style="vertical-align: -1.738ex; width: 13.725ex; height: 4.837ex;" src="images/14.svg" alt=" " data-tex="\(e-b=\dfrac{bc}{d-c}\)">; or, in words, the rectangle contained by the +focal lengths of parallel rays, passing and repassing any surface +in the same lines, is equal to the rectangle contained by the +differences between these lengths and the distances of any conjugate +foci.</p> + +<p><i>Corollary</i> 5. For perpendicular rays, <img style="vertical-align: -1.738ex; width: 24.431ex; height: 4.837ex;" src="images/15.svg" alt=" " data-tex="\(e=\dfrac{md}{d-n}=m+\dfrac{mn}{d-n}\)">; +or, if the radius be <img style="vertical-align: -1.738ex; width: 13.766ex; height: 4.837ex;" src="images/16.svg" alt=" " data-tex="\(a,e=\dfrac{mad}{d-na}\)">; and if <i>d</i> and <i>e</i> be given to find +the radius, <img style="vertical-align: -1.738ex; width: 13.55ex; height: 4.837ex;" src="images/17.svg" alt=" " data-tex="\(a=\dfrac{de}{md+ne}\)">.</p> + +<p><i>Corollary</i> 6. For rays perpendicular and parallel, <i>e</i> = <i>m</i>, or +<i>e</i> = <i>m a</i>.</p> + +<p><i>Corollary</i> 7. For a double convex lens, neglecting the thickness, +call the first radius <i>g</i>, the second <i>h</i>, and <img style="vertical-align: -2.016ex; width: 19.073ex; height: 5.115ex;" src="images/18.svg" alt=" " data-tex="\(e=\dfrac{ndgh}{dg+dh-ngh}\)">. +Hence <img style="vertical-align: -2.016ex; width: 18.145ex; height: 5.115ex;" src="images/19.svg" alt=" " data-tex="\(n=\dfrac{de}{d+e}\cdot \dfrac{g+h}{gh}\)">. and, +for parallel rays, <img style="vertical-align: -2.016ex; width: 10.215ex; height: 5.115ex;" src="images/20.svg" alt=" " data-tex="\(e=\dfrac{ngh}{g+h}\)">, and +<img style="vertical-align: -2.016ex; width: 13.207ex; height: 5.115ex;" src="images/21.svg" alt=" " data-tex="\(n=e\cdot \dfrac{g+h}{gh}\)">. If <img style="vertical-align: -1.738ex; width: 23.314ex; height: 4.837ex;" src="images/22.svg" alt=" " data-tex="\(g=h=a, e=\dfrac{nad}{2d-na}\)">; and for parallel rays <img style="vertical-align: -1.552ex; width: 7.621ex; height: 4.081ex;" src="images/23.svg" alt=" " data-tex="\(e=\dfrac{na}{2}\)">: +calling this principal focal length <i>b</i>, <img style="vertical-align: -1.738ex; width: 9.98ex; height: 4.837ex;" src="images/24.svg" alt=" " data-tex="\(e=\dfrac{bd}{d-b}\)">, as in +Cor. 4; whence we have the joint focus of two lenses; also, +<img style="vertical-align: -1.738ex; width: 9.98ex; height: 4.837ex;" src="images/25.svg" alt=" " data-tex="\(b=\dfrac{de}{d+e}\)">.</p> + +<p><i>Corollary</i> 8. In a sphere, <img style="vertical-align: -2.172ex; width: 23.798ex; height: 5.271ex;" src="images/26.svg" alt=" " data-tex="\(e=ma\cdot \dfrac{d+a}{2d-(m-2)a}\)">, for the distance +from the centre, and <img style="vertical-align: -1.552ex; width: 8.166ex; height: 4.081ex;" src="images/27.svg" alt=" " data-tex="\(b=\dfrac{ma}{2}\)">.</p> + +<p><span class="pagenum" id="Page_30">30</span></p> + +<p><i>Scholium</i> 2. In all these cases, if the rays converge, <i>d</i> must +be negative. For instance, to find the joint focus of two convex, +or concave lenses, the expression becomes, <img style="vertical-align: -1.738ex; width: 9.98ex; height: 4.837ex;" src="images/28.svg" alt=" " data-tex="\(e=\dfrac{bd}{b+d}\)">.</p> + +<p><i>Corollary</i> 9. In Cor. 3, the divisor becomes ultimately constant; +and, when the inclination is smalls the focus varies as <i>u u</i>.</p> + +<p><i>Corollary</i> 10. For parallel rays falling obliquely on a double +convex, or double concave lens, of inconsiderable thickness, the +radius being 1, <img style="vertical-align: -2.172ex; width: 16.179ex; height: 5.118ex;" src="images/29.svg" alt=" " data-tex="\(e=\dfrac{ntu}{2(mu-nt)}\)">; which varies ultimately as the +product of the cosines, or as <img style="vertical-align: -1.577ex; width: 12.492ex; height: 4.425ex;" src="images/30.svg" alt=" " data-tex="\(\dfrac{m+n}{nn}t+t^{2}\)">.</p> + +<p><i>Scholium</i> 3. In the double convex lens, the thickness diminishes +the effect of the obliquity near the axis; in the double +concave, it increases it.</p> + +<p><i>Scholium</i> 4. No spherical surface, excepting one particular +case, (<span class="smcap">Wood</span>, 155,) can collect an oblique pencil of rays, even +to a physical point. The oblique rays which we have hitherto +considered, are only such as lie in that section of the pencil +which is made by a plane passing through the centre and the +radiant point. They continue in this plane, notwithstanding the +refraction, and therefore will not meet the rays of the collateral +sections, till they arrive at the axis. The remark was made by +Sir <span class="smcap">Isaac Newton</span>, and extended by Dr. <span class="smcap">Smith</span>, (<span class="smcap">Smith</span> r. +493, 494:) it appears, however, to have been too little noticed. +(<span class="smcap">Wood</span>, 362.) The geometrical focus thus becomes a line, a +circle, an oval, or other figure, according to the form of the +pencil, the nature of the surface, and the place of the plane receiving +the image. Some of the varieties of the focal image of +a cylindrical pencil obliquely refracted are shown in Plate VI. +<a href="#Pl.VI">Fig. 28</a>.</p> + +<p><span class="pagenum" id="Page_31">31</span></p> + +<p><i>Corollary</i> 11. Hence the line joining the remoter conjugate +foci, will always pass through the centre. The distance +of the remoter focus of parallel rays will be expressed by +<img style="vertical-align: -1.738ex; width: 13.477ex; height: 4.267ex;" src="images/31.svg" alt=" " data-tex="\(f=\dfrac{m}{mu-nt}\)">; and the least circle of aberration will be at the +distance <img style="vertical-align: -2.172ex; width: 20.856ex; height: 5.606ex;" src="images/32.svg" alt=" " data-tex="\(\dfrac{1+u^{2}-2u^{4}}{(1+uu)\cdot (mu-nt)}\)"> dividing the length of aberration in +the ratio of the distance of its limits from the surface. In the +case of Cor. 10. <img style="vertical-align: -2.172ex; width: 16.369ex; height: 4.701ex;" src="images/33.svg" alt=" " data-tex="\(f=\dfrac{n}{2(mu-nt)}\)">.</p> + +<p>Corollary 12. This proposition extends also to reflected rays; +and, in that case, the line from the centre passes through the +point of incidence.</p> + +<p class="tac mtb1em"><i>Proposition</i> V. <i>Problem</i>.</p> + +<p>To find the place and magnitude of the image of a small +object, after refraction at any number of spherical surfaces.</p> + +<p><i>Construction</i>. (Plate II. <a href="#Pl.II">Fig. 3</a>.) From any point (B) in the +object (AB), draw lines to (C), the centre of the first surface, +and to (D), the focus of parallel rays coming in a contrary +direction: from the intersection of the second line (BD) with +the tangent (EF) at the vertex, draw a line (EH) parallel to +the axis, and it will cut the first line (BC) in (H), the first +image of the point (B). Proceed with this image as a new object, +and repeat the operation for each surface, and the last point +will be in the image required. For calculation, find the place +of the image by Cor. 5. Prop. IV. and its magnitude will be to +that of the object, as their respective distances from the centre.</p> + +<p><i>Corollary</i>. If a confused image be received on any given +plane, its magnitude will be determined by the line drawn from +the preceding image through the centre of the last surface.</p> + +<p><span class="pagenum" id="Page_32">32</span></p> + +<p class="tac mtb1em"><i>Proposition</i> VI. <i>Problem</i>.</p> + +<p>To determine the law by which the refraction at a spherical +surface must vary, so as to collect parallel rays to a perfect +focus.</p> + +<p><i>Solution</i>. Let <i>v</i> be the versed sine to the radius 1; then, at +each point without the axis, <i>n</i> remaining the same, <i>m</i> must +become <img style="vertical-align: -0.238ex; width: 12.254ex; height: 2.398ex;" src="images/34.svg" alt=" " data-tex="\(\sqrt{mm±2nv}\)">; and all the rays will be collected in +the principal focus.</p> + +<p><i>Corollary</i>. The same law will serve for a double convex lens, +in the case of equidistant conjugate foci, substituting <i>n</i> for <i>m</i>.</p> + +<p class="tac mtb1em"><i>Proposition</i> VII. <i>Problem</i>.</p> + +<p>To find the principal focus of a sphere, or lens, of which the +internal parts are more dense than the external.</p> + +<p><i>Solution</i>. In order that the focal distance may be finite, the +density of a finite portion about the centre must be equable: +call the radius of this portion <img style="vertical-align: -1.577ex; width: 2.127ex; height: 4.613ex;" src="images/35.svg" alt=" " data-tex="\(\dfrac{1}{l}\)">, that of the sphere being unity; +let the whole refraction out of the surrounding medium into this +central part, be as <i>m</i> to <i>n</i>; take <img style="vertical-align: -2.018ex; width: 17.679ex; height: 5.118ex;" src="images/36.svg" alt=" " data-tex="\(r=\dfrac{\log l}{\log m- \log n}\)">, and let the density +be supposed to vary every where inversely as the power <img style="vertical-align: -1.577ex; width: 2.127ex; height: 4.613ex;" src="images/37.svg" alt=" " data-tex="\(\dfrac{1}{r}\)"> +of the distance from the centre: then the principal focal distance +from the centre will be <img style="vertical-align: -1.738ex; width: 15.327ex; height: 4.774ex;" src="images/38.svg" alt=" " data-tex="\(\dfrac{r-1}{2}\cdot \dfrac{m}{nl-m}\)">. When <i>r</i> = 1, it becomes +<img style="vertical-align: -2.172ex; width: 19.488ex; height: 5.208ex;" src="images/39.svg" alt=" " data-tex="\(\dfrac{1}{2(\text{H.L}.m-\text{H.L}.n)}\)">. For a lens, deduct one fourth of the difference +between its axis and the diameter of the sphere of which its +surfaces are portions.</p> + +<p><i>Corollary</i>. If the density be supposed to vary suddenly at the +surface, <i>m</i> must express the difference of the refractions at the +<span class="pagenum" id="Page_33">33</span>centre and at the surface; and the focal distance, thus determined, +must be diminished according to the refraction at the surface.</p> + +<p class="tac mtb1em"><i>Proposition</i> VIII. <i>Problem</i>.</p> + +<p>To find the nearer focus of parallel rays falling obliquely on +a sphere of variable density.</p> + +<p><i>Solution</i>. Let <i>r</i> be as in the last proposition, <i>s</i> the sine of incidence, +<i>t</i> the cosine, and <i>e</i> the distance of the focus from the +point of emersion. Then <img style="vertical-align: -1.738ex; width: 11.4ex; height: 4.683ex;" src="images/40.svg" alt=" " data-tex="\(e=\dfrac{w-t}{2-tw}\)">, <i>w</i> being <img style="vertical-align: -2.76ex; width: 14.415ex; height: 5.796ex;" src="images/41.svg" alt=" " data-tex="\(=\dfrac{2}{(r-1)s^{\frac{r+1}{r-1}\cdot }}\)"> +<img style="vertical-align: -0.566ex; width: 49.549ex; height: 2.47ex;" src="images/42.svg" alt=" " data-tex="\((a\text{A}+b\text{B}+c\text{C}+...)+2a\text{A}+6b\text{B}s^{2}+10c\text{C}s^{4}+...\)"> +where <img style="vertical-align: -1.738ex; width: 10.127ex; height: 4.267ex;" src="images/43.svg" alt=" " data-tex="\(a=\dfrac{r}{r+1}\)">, <img style="vertical-align: -1.738ex; width: 11.032ex; height: 4.267ex;" src="images/44.svg" alt=" " data-tex="\(b=\dfrac{r}{3r-1}\)">, <img style="vertical-align: -1.738ex; width: 11.041ex; height: 4.267ex;" src="images/45.svg" alt=" " data-tex="\(c=\dfrac{r}{5r-3}\)">, <img style="vertical-align: -0.816ex; width: 33.197ex; height: 2.773ex;" src="images/46.svg" alt=" " data-tex="\(\text{A}=1, \text{B}=\frac{1}{2}\text{A}, \text{C}=\frac{3}{4}\text{B}, \text{D}=\frac{5}{6}\text{C}\)">. But, when <i>s</i> is large, the latter part of the series converges +somewhat slowly. The former part might be abridged +if it were necessary: but, since the focus in this case is always +very imperfect, it is of the less consequence to provide an easy +calculation.</p> + +<p><i>General Scholium</i>. The two first propositions relate to well +known phenomena; the third can hardly be new; the fourth +approaches the nearest to <span class="smcap">Maclaurin</span>’s construction, but is far +more simple and convenient; the fifth and sixth have no difficulty; +but the two last require a long demonstration. The one +is abridged by a property of logarithms; the other is derived from +the laws of centripetal forces, on the supposition of velocities +directly as the refractive densities, correcting the series for the +place of the apsis, and making the sine of incidence variable, +to determine the fluxion of the angle of deviation.</p> + +<p class="mt2em ti0">V. Dr. <span class="smcap">Porterfield</span> has employed an experiment, first +made by <span class="smcap">Scheiner</span>, to the determination of the focal distance +<span class="pagenum" id="Page_34">34</span>of the eye; and has described, under the name of an optometer, +a very excellent instrument, founded on the principle of the +phenomenon.⁠<a id="FNanchor_7_7" href="#Footnote_7_7" class="fnanchor">7</a> But the apparatus is capable of considerable +improvement; and I shall beg leave to describe an optometer, +simple in its construction, and equally convenient and accurate +in its application.</p> + +<p>Let an obstacle be interposed between a radiant point (R, +Plate II. <a href="#Pl.II">Fig. 4</a>,) and any refracting surface, or lens (CD), +and let this obstacle be perforated at two points (A and B) only. +Let the refracted rays be intercepted by a plane, so as to form +an image on it. Then it is evident, that when this plane (EF) +passes through the focus of refracted rays, the image formed +on it will be a single point. But, if the plane be advanced forwards +(to GH), or removed backwards (to IK), the small +pencils passing through the perforations, will no longer meet +in a single point, but will fall on two distinct spots of the plane +(G, H; I, K;) and, in either case, form a double image of the +object.</p> + +<p>Let us now add two more radiating points, (S and T, <a href="#Pl.II">Fig. 5</a>,) +the one nearer to the lens than the first point, the other more +remote; and, when the plane which receives the images passes +through the focus of rays coming from the first point, the images +of the second and third points must both be double (<i>s s</i>, <i>t t</i>;) +since the plane (EF) is without the focal distance of rays +coming from the furthest point, and within that of rays coming +from the nearest. Upon this principle. Dr. <span class="smcap">Porterfield</span>’s +optometer was founded.</p> + +<p>But, if the three points be supposed to be joined by a line, +and this line to be somewhat inclined to the axis of the lens, +<span class="pagenum" id="Page_35">35</span>each point of the line, except the first point (R, <a href="#Pl.II">Fig. 6</a>,) will +have a double image; and each pair of images, being contiguous +to those of the neighbouring radiant points, will form with them +two continued lines, and the images being more widely separated +as the point which they represent is further from the first +radiant point, the lines (<i>s t</i>, <i>s t</i>,) will converge on each side +towards (<i>r</i>) the image of this point, and there will intersect +each other.</p> + +<p>The same happens when we look at any object through two +pin holes, within the limits of the pupil. If the object be at the +point of perfect vision, the image on the retina will be single: +but, in every other case, the image being double, we shall appear +to see a double object: and, if we look at a line pointed nearly +to the eye, it will appear as two lines, crossing each other in the +point of perfect vision. For this purpose, the holes may be +converted into slits, which render the images nearly as distinct, +at the same time that they admit more light. The number may +be increased from two to four, or more, whenever particular +investigations render it necessary.</p> + +<p>The optometer may be made of a slip of card-paper, or of +ivory, about eight inches in length, and one in breadth, divided +longitudinally by a black line, which must not be too strong. +The end of the card must be cut as is shown in Plate III. <a href="#Pl.III">Fig. 7</a>, +in order that it may be turned up, and fixed in an inclined +position by means of the shoulder: or a detached piece, nearly +of this form, may be applied to the optometer, as it is here engraved. +A hole about half an inch square must be made in this +part; and the sides so cut as to receive a slider of thick paper, +with slits of different sizes, from a fortieth to a tenth of an inch +in breadth, divided by spaces somewhat broader; so that each +observer may choose that which best suits the aperture of his pupil. +<span class="pagenum" id="Page_36">36</span>In order to adapt the instrument to the use of presbyopic eyes, +the other end must be furnished with a lens of four inches focal +length; and a scale must be made near the line on each side +of it, divided from one end into inches, and from the other according +to the table here calculated from Cor. 7. Prop. IV, by +means of which, not only diverging, but also parallel and converging +rays from the lens are referred to their virtual focus. +The instrument is easily applicable to the purpose of ascertaining +the focal length of spectacles required for myopic or +presbyopic eyes. Mr. <span class="smcap">Cary</span> has been so good as to furnish +me with the numbers and focal lengths of the glasses commonly +made; and I have calculated the distances at which those +numbers must be placed on the scale of the optometer, so that +a presbyopic eye may be enabled to see at eight inches distance, +by using the glasses of the focal length placed opposite to the +nearest crossing of the lines; and a myopic eye with parallel +rays, by using the glasses indicated by the number that stands +opposite their furthest crossing. To facilitate the observation, +I have also placed these numbers opposite that point which +will be the nearest crossing to myopic eyes; but this, upon the +arbitrary supposition of an equal capability of change of focus +in every eye, which I must confess is often far from the truth. +It cannot be expected, that every person, on the first trial, +will fix precisely upon that power which best suits the defect +of his sight. Few can bring their eyes at pleasure to the state +of full action, or of perfect relaxation; and a power two or +three degrees lower than that which is thus ascertained, will be +found sufficient for ordinary purposes. I have also added to the +second table, such numbers as will point out the spectacles +necessary for a presbyopic eye, to see at twelve and at eighteen +inches respectively: the middle series will perhaps be the most +<span class="pagenum" id="Page_37">37</span>proper for placing the numbers on the scale. The optometer +should be applied to each eye; and, at the time of observing, the +opposite eye should not be shut, but the instrument should be +screened from its view. The place of intersection may be accurately +ascertained, by means of an index sliding along the scale. +The optometer is represented in Plate III. <a href="#Pl.III">Fig. 8 and 9</a>; and +the manner in which the lines appear, in <a href="#Pl.III">Fig. 10.</a></p> + +<p class="tac pt1"><i>Table</i> I. <i>For extending the scale by a lens of 4 inches focus:</i></p> + +<div class="center"> +<table id="table1"> +<tr class="bt"> +<td> + 4 +</td> +<td> +2.00 +</td> +<td> +11 +</td> +<td> +2.93 +</td> +<td> + 30 +</td> +<td> +3.52 +</td> +<td> + 200 +</td> +<td> +3.92 +</td> +<td> +-35 +</td> +<td> +4.51 +</td> +<td> +-12 +</td> +<td> +6.00 +</td> +</tr> +<tr> +<td> + 5 +</td> +<td> +2.22 +</td> +<td> +12 +</td> +<td> +3.00 +</td> +<td> + 40 +</td> +<td> +3.64 +</td> +<td> +∞ +</td> +<td> +4.00 +</td> +<td> +-30 +</td> +<td> +4.62 +</td> +<td> +-11 +</td> +<td> +6.29 +</td> +</tr> +<tr> +<td> + 6 +</td> +<td> +2.40 +</td> +<td> +13 +</td> +<td> +3.06 +</td> +<td> + 50 +</td> +<td> +3.70 +</td> +<td> +-200 +</td> +<td> +4.08 +</td> +<td> +-25 +</td> +<td> +4.76 +</td> +<td> +-10 +</td> +<td> +6.67 +</td> +</tr> +<tr> +<td> + 7 +</td> +<td> +2.55 +</td> +<td> +14 +</td> +<td> +3.11 +</td> +<td> + 60 +</td> +<td> +3.75 +</td> +<td> +-100 +</td> +<td> +4.17 +</td> +<td> +-20 +</td> +<td> +5.00 +</td> +<td> +-9.5 +</td> +<td> +6.90 +</td> +</tr> +<tr> +<td> + 8 +</td> +<td> +2.67 +</td> +<td> +15 +</td> +<td> +3.16 +</td> +<td> + 70 +</td> +<td> +3.78 +</td> +<td> +-50 +</td> +<td> +4.35 +</td> +<td> +-15 +</td> +<td> +5.45 +</td> +<td> +-9.0 +</td> +<td> +7.20 +</td> +</tr> +<tr> +<td> + 9 +</td> +<td> +2.77 +</td> +<td> +20 +</td> +<td> +3.33 +</td> +<td> + 80 +</td> +<td> +3.81 +</td> +<td> +-45 +</td> +<td> +4.39 +</td> +<td> +-14 +</td> +<td> +5.60 +</td> +<td> +-8.5 +</td> +<td> +7.56 +</td> +</tr> +<tr class="bb"> +<td> +10 +</td> +<td> +2.86 +</td> +<td> +25 +</td> +<td> +3.45 +</td> +<td> +100 +</td> +<td> +3.85 +</td> +<td> +-40 +</td> +<td> +4.44 +</td> +<td> +-13 +</td> +<td> +5.78 +</td> +<td> +-8.0 +</td> +<td> +8.00 +</td> +</tr> +</table> +</div> + +<p class="tac pt1"><i>Table</i> II. <i>For placing the numbers indicating the focal length of +convex glasses.</i></p> + +<div class="center"> +<table id="table2"> +<thead> +<tr> +<th> +Foc. +</th> +<th> +VIII. +</th> +<th> +XII. +</th> +<th> +XVIII. +</th> +<th> +Foc. +</th> +<th> +VIII. +</th> +<th> +XII. +</th> +<th> +XVIII. +</th> +<th> +Foc. +</th> +<th> +VIII. +</th> +<th> +XII. +</th> +<th> +XVIII. +</th> +</tr> +</thead> +<tr> +<td> + 0 +</td> +<td> + 8.00 +</td> +<td> +12.00 +</td> +<td> +18.00 +</td> +<td> +20 +</td> +<td> +13.33 +</td> +<td> + 30.00 +</td> +<td> +180.00 +</td> +<td> +8 +</td> +<td> +∞ +</td> +<td> +-24.00 +</td> +<td> +-14.40 +</td> +</tr> +<tr> +<td> +40 +</td> +<td> +10.00 +</td> +<td> +17.14 +</td> +<td> +32.73 +</td> +<td> +18 +</td> +<td> +14.40 +</td> +<td> + 36.00 +</td> +<td> +∞ +</td> +<td> +7 +</td> +<td> +-56.00 +</td> +<td> +-16.80 +</td> +<td> +-11.45 +</td> +</tr> +<tr> +<td> +36 +</td> +<td> +10.28 +</td> +<td> +18.00 +</td> +<td> +36.00 +</td> +<td> +16 +</td> +<td> +16.00 +</td> +<td> + 48.00 +</td> +<td> +-144.00 +</td> +<td> +6 +</td> +<td> +-24.00 +</td> +<td> +-12.00 +</td> +<td> +- 9.00 +</td> +</tr> +<tr> +<td> +30 +</td> +<td> +10.91 +</td> +<td> +20.00 +</td> +<td> +45.00 +</td> +<td> +14 +</td> +<td> +18.67 +</td> +<td> + 84.00 +</td> +<td> +- 63.00 +</td> +<td> +5 +</td> +<td> +-13.33 +</td> +<td> +- 8.57 +</td> +<td> +- 5.92 +</td> +</tr> +<tr> +<td> +28 +</td> +<td> +11.20 +</td> +<td> +21.00 +</td> +<td> +50.40 +</td> +<td> +12 +</td> +<td> +24.00 +</td> +<td> +∞ +</td> +<td> +- 36.00 +</td> +<td> +4.5 +</td> +<td> +-10.29 +</td> +<td> +- 7.20 +</td> +<td> +- 6.00 +</td> +</tr> +<tr> +<td> +26 +</td> +<td> +11.56 +</td> +<td> +22.29 +</td> +<td> +58.50 +</td> +<td> +11 +</td> +<td> +29.33 +</td> +<td> +-132.00 +</td> +<td> +- 28.29 +</td> +<td> +4.0 +</td> +<td> +- 8.00 +</td> +<td> +- 6.00 +</td> +<td> +- 5.14 +</td> +</tr> +<tr> +<td> +24 +</td> +<td> +12.00 +</td> +<td> +24.00 +</td> +<td> +72.00 +</td> +<td> +10 +</td> +<td> +40.00 +</td> +<td> +- 60.00 +</td> +<td> +- 22.50 +</td> +<td> +3.5 +</td> +<td> +- 6.22 +</td> +<td> +- 4.94 +</td> +<td> +- 4.34 +</td> +</tr> +<tr class="bb"> +<td> +22 +</td> +<td> +12.77 +</td> +<td> +20.40 +</td> +<td> +99.00 +</td> +<td> + 9 +</td> +<td> +72.00 +</td> +<td> +- 36.00 +</td> +<td> +- 18.00 +</td> +<td> +3.0 +</td> +<td> +- 4.80 +</td> +<td> +- 4.00 +</td> +<td> +- 3.6 +</td> +</tr> +</table> +</div> + + +<p class="tac pt1"><i>Table</i> III. <i>For concave glasses.</i></p> + +<div class="center"> +<table id="table3"> +<thead> +<tr> +<th> +Number. +</th> +<th> +Focus and<br>furthest<br>place. +</th> +<th> +Nearest<br>place. +</th> +<th> +Number. +</th> +<th> +Focus and<br>furthest<br>place. +</th> +<th> +Nearest<br>place. +</th> +<th> +Number. +</th> +<th> +Focus and<br>furthest<br>place. +</th> +<th> +Nearest<br>place. +</th> +</tr> +</thead> +<tr> +<td> +0 +</td> +<td> + +</td> +<td> +4.00 +</td> +<td> + 7 +</td> +<td> +8 +</td> +<td> +2.67 +</td> +<td> +14 +</td> +<td> +3.00 +</td> +<td> +1.71 +</td> +</tr> +<tr> +<td> +1 +</td> +<td> +24 +</td> +<td> +3.43 +</td> +<td> + 8 +</td> +<td> +7 +</td> +<td> +2.54 +</td> +<td> +15 +</td> +<td> +2.75 +</td> +<td> +1.63 +</td> +</tr> +<tr> +<td> +2 +</td> +<td> +18 +</td> +<td> +3.27 +</td> +<td> + 9 +</td> +<td> +6 +</td> +<td> +2.40 +</td> +<td> +16 +</td> +<td> +2.50 +</td> +<td> +1.54 +</td> +</tr> +<tr> +<td> +3 +</td> +<td> +16 +</td> +<td> +3.20 +</td> +<td> +10 +</td> +<td> +5 +</td> +<td> +2.22 +</td> +<td> +17 +</td> +<td> +2.25 +</td> +<td> +1.44 +</td> +</tr> +<tr> +<td> +4 +</td> +<td> +12 +</td> +<td> +3.00 +</td> +<td> +11 +</td> +<td> +4.5 +</td> +<td> +2.12 +</td> +<td> +18 +</td> +<td> +2.00 +</td> +<td> +1.33 +</td> +</tr> +<tr> +<td> +5 +</td> +<td> +10 +</td> +<td> +2.86 +</td> +<td> +12 +</td> +<td> +4.0 +</td> +<td> +2.00 +</td> +<td> +19 +</td> +<td> +1.75 +</td> +<td> +1.22 +</td> +</tr> +<tr class="bb"> +<td> +6 +</td> +<td> + 9 +</td> +<td> +2.77 +</td> +<td> +13 +</td> +<td> +3.5 +</td> +<td> +1.87 +</td> +<td> +20 +</td> +<td> +1.50 +</td> +<td> +1.02 +</td> +</tr> +</table> +</div> + + +<p><span class="pagenum" id="Page_38">38</span></p> +<p class="mt2em ti0">VI. Being convinced of the advantage of making every +observation with as little assistance as possible, I have endeavoured +to confine most of my experiments to my own eyes; +and I shall, in general, ground my calculations on the supposition +of an eye nearly similar to my own. I shall therefore +first endeavour to ascertain all its dimensions, and all its +faculties.</p> + +<p>For measuring the diameters, I fix a small key on each point +of a pair of compasses; and I can venture to bring the rings +into immediate contact with the sclerotica. The transverse +diameter is externally 98 hundredths of an inch.</p> + +<p>To find the axis, I turn the eye as much inwards as possible, +and press one of the keys close to the sclerotica, at the external +angle, till it arrives at the spot where the spectrum formed +by its pressure coincides with the direction of the visual axis, and, +looking in a glass, I bring the other key to the cornea. The +optical axis of the eye, making allowance of three hundredths +for the coats, is thus found to be 91 hundredths of an inch, from +the external surface of the cornea to the retina. With an eye +less prominent, this method might not have succeeded.</p> + +<p>The vertical diameter, or rather chord, of the cornea, is 45 +hundredths: its versed sine 11 hundredths. To ascertain the +versed sine, I looked with the right eye at the image of the +left, in a small speculum held close to the nose, while the left +eye was so averted that the margin of the cornea appeared as a +straight line, and compared the projection of the cornea with +the image of a cancellated scale held in a proper direction behind +the left eye, and close to the left temple. The horizontal +chord of the cornea is nearly 49 hundredths.</p> + +<p>Hence the radius of the cornea is 31 hundredths. It may +<span class="pagenum" id="Page_39">39</span>be thought that I assign too great a convexity to the cornea; +but I have corrected it by a number of concurrent observations, +which will be enumerated hereafter.</p> + +<p>The eye being directed towards its image, the projection of +the margin of the sclerotica is 22 hundredths from the margin +of the cornea, towards the external angle, and 27 towards the +internal angle of the eye: so that the cornea has an eccentricity +of one fortieth of an inch, with respect to the section of +the eye perpendicular to the visual axis.</p> + +<p>The aperture of the pupil varies from 27 to 13 hundredths; +at least this is its apparent size, which must be somewhat diminished, +on account of the magnifying power of the cornea, +perhaps to 25 and 12. When dilated, it is nearly as eccentric +as the cornea; but, when most contracted, its centre coincides +with the reflection of an image from an object held immediately +before the eye; and this image very nearly with the centre of +the whole apparent margin of the sclerotica: so that the cornea +is perpendicularly intersected by the visual axis.</p> + +<p>My eye, in a state of relaxation, collects to a focus on the +retina, those rays which diverge vertically from an object at the +distance of ten inches from the cornea, and the rays which +diverge horizontally from an object at seven inches distance. +For, if I hold the plane of the optometer vertically, the images +of the line appear to cross at ten inches; if horizontally, at seven. +The difference is expressed by a focal length of 23 inches. I +have never experienced any inconvenience from this imperfection, +nor did I ever discover it till I made these experiments; +and I believe I can examine minute objects with as much accuracy +as most of those whose eyes are differently formed. On +mentioning it to Mr. <span class="smcap">Cary</span>, he informed me, that he had +<span class="pagenum" id="Page_40">40</span>frequently taken notice of a similar circumstance; that many +persons were obliged to hold a concave glass obliquely, in order +to see with distinctness, counterbalancing, by the inclination of +the glass, the too great refractive power of the eye in the direction +of that inclination, (Cor. 10. Prop. IV.) and finding but +little assistance from spectacles of the same focal length. The +difference is not in the cornea, for it exists when the effect of +the cornea is removed by a method to be described hereafter. +The cause is, without doubt, the obliquity of the uvea, and of +the crystalline lens, which is nearly parallel to it, with respect +to the visual axis: this obliquity will appear, from the dimensions +already given, to be about 10 degrees. Without entering +into a very accurate calculation, the difference observed is found +(by the same corollary) to require an inclination of about 13 +degrees; and the remaining three degrees may easily be added, +by the greater obliquity of the posterior surface of the crystalline +opposite the pupil. There would be no difficulty in fixing the +glasses of spectacles, or the concave eye-glass of a telescope, in +such a position as to remedy the defect.</p> + +<p>In order to ascertain the focal distance of the lens, we must +assign its probable distance from the cornea. Now the versed +sine of the cornea being 11 hundredths, and the uvea being +nearly flat, the anterior surface of the lens must probably be +somewhat behind the chord of the cornea; but by a very inconsiderable +distance, for the uvea has the substance of a thin membrane, +and the lens approaches very near to it: we will therefore +call this distance 12 hundredths. The axis and proportions +of the lens must be estimated by comparison with anatomical +observations; since they affect, in a small degree, the determination +of its focal distance. M. <span class="smcap">Petit</span> found the axis +<span class="pagenum" id="Page_41">41</span>almost always about two lines, or 18 hundredths of an inch. +The radius of the anterior surface was in the greatest number +3 lines, but oftener more than less. We will suppose mine +to be <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/47.svg" alt=" " data-tex="\(3\frac{1}{4}\)">, or nearly <img style="vertical-align: -0.816ex; width: 2.595ex; height: 2.773ex;" src="images/48.svg" alt=" " data-tex="\(\frac{1}{10}\)"> of an inch. The radius of the +posterior surface was most frequently <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/49.svg" alt=" " data-tex="\(2\frac{1}{2}\)"> lines, or <img style="vertical-align: -0.816ex; width: 1.795ex; height: 2.773ex;" src="images/50.svg" alt=" " data-tex="\(\frac{2}{9}\)"> of an +inch.⁠<a id="FNanchor_8_8" href="#Footnote_8_8" class="fnanchor">8</a> The +optical centre will be therefore <img style="vertical-align: -0.912ex; width: 9.901ex; height: 2.869ex;" src="images/51.svg" alt=" " data-tex="\(\left( \frac{18\times 30}{30+22}= \right)\)"> about +one-tenth of an inch from the anterior surface: hence we have +22 hundredths, for the distance of the centre from the cornea. +Now, taking 10 inches as the distance of the radiant point, the +focus of the cornea will be 115 hundredths behind the centre +of the lens. (Cor. 5. Prop. IV.) But the actual joint focus is +(91 — 22 =) 69 behind the centre: hence, disregarding the +thickness of the lens, its principal focal distance is 173 hundredths. +(Cor. 7. Prop. IV.) For its refractive power in the +eye, we have (by Cor. 7. Prop. IV.) <i>n</i> = 13.5, and <i>m</i> = 14.5. +Calculating upon this refractive power, with the consideration +of the thickness also, we find that it requires a correction, +and comes near to the ratio of 14 to 13 for the sines. It +is well known that the refractive powers of the humours are +equal to that of water; and, that the thickness of the cornea is +too equable to produce any effect on the focal distance.</p> + +<p>For determining the refractive power of the crystalline lens +by a direct experiment, I made use of a method suggested to +me by Dr. <span class="smcap">Wollaston</span>. I found the refractive power of the +centre of the recent human crystalline to that of water, as 21 +to 20. The difference of this ratio from the ratio of 14 to 13, +ascertained from calculation, is probably owing to two circumstances. +The first is, that the substance of the lens being in +some degree soluble in water, a portion of the aqueous fluid +<span class="pagenum" id="Page_42">42</span>within its capsule penetrates after death, so as somewhat to +lessen the density. When dry, the refractive power is little inferior +to that of crown glass. The second circumstance is, the +unequal density of the lens. The ratio of 14 to 13 is founded +on the supposition of an equable density: but, the central part +being the most dense, the whole acts as a lens of smaller dimensions; +and it may be found by Prop. VII. that if the central +portion of a sphere be supposed of uniform density, refracting +as 21 to 20, to the distance of one half of the radius, and +the density of the external parts to decrease gradually, and at +the surface to become equal to that of the surrounding medium, +the sphere thus constituted, will be equal in focal length to a +uniform sphere of the same size, with a refraction of 16 to 15 +nearly. And the effect will be nearly the same, if the central +portion be supposed to be smaller than this, but the density +to be somewhat greater at the surface than that of the surrounding +medium, or to vary more rapidly externally than internally. +On the whole, it is probable that the refractive +power of the centre of the human crystalline, in its living state, +is to that of water nearly as 18 to 17; that the water imbibed +after death, reduces it to the ratio of 21 to 20; but that, on +account of the unequable density of the lens, its effect in the +eye is equivalent to a refraction of 14 to 13 for its whole size. +Dr. <span class="smcap">Wollaston</span> has ascertained the refraction out of air, into +the centre of the recent crystalline of oxen and sheep, to be +nearly as 143 to 100; into the centre of the crystalline of fish, +and into the dried crystalline of sheep, as 152 to 100. Hence, +the refraction of the crystalline of oxen in water, should be +as 15 to 14: but the human crystalline, when recent, is decidedly +less refractive.</p> + +<p><span class="pagenum" id="Page_43">43</span></p> + +<p>These considerations will explain the inconsistency of different +observations on the refractive power of the crystalline; +and, in particular, how the refraction which I formerly calculated, +from measuring the focal length of the lens,⁠<a id="FNanchor_9_9" href="#Footnote_9_9" class="fnanchor">9</a> is so much +greater than that which is determined by other means. But, +for direct experiments, Dr. <span class="smcap">Wollaston</span>’s method is exceedingly +accurate.</p> + +<p>When I look at a minute lucid point, such as the image of +a candle in a small concave speculum, it appears as a radiated +star, as a cross, or as an unequal line, and never as a perfect +point, unless I apply a concave lens inclined at a proper angle, +to correct the unequal refraction of my eye. If I bring the +point very near, it spreads into a surface nearly circular, and +almost equably illuminated, except some faint lines, nearly in a +radiating direction. For this purpose, the best image is a candle, +or a small speculum, viewed through a minute lens at some +little distance, or seen by reflection in a larger lens. If any +pressure has been applied to the eye, such as that of the finger +keeping it shut, the sight is often confused for a short time after +the removal of the finger, and the image is in this case spotty +or curdled. The radiating lines are probably occasioned by +some slight inequalities in the surface of the lens, which is very +superficially furrowed in the direction of its fibres: the curdled +appearance will be explained hereafter. When the point is further +removed, the image becomes evidently oval, the vertical +diameter being longest, and the lines a little more distinct than +before, the light being strongest in the neighbourhood of the +centre; but immediately at the centre there is a darker spot, +owing to such a slight depression at the vertex as is often +<span class="pagenum" id="Page_44">44</span>observable in examining the lens after death. The situation of +the rays is constant, though not regular; the most conspicuous +are seven or eight in number; sometimes about twenty fainter +ones may be counted. Removing the point a little further, the +image becomes a short vertical line; the rays that diverged horizontally +being perfectly collected, while the vertical rays are +still separate. In the next stage, which is the most perfect +focus, the line spreads in the middle, and approaches nearly to +a square, with projecting angles, but is marked with some +darker lines towards the diagonals. The square then flattens +into a rhombus, and the rhombus into a horizontal line unequally +bright. At every greater distance, the line lengthens, +and acquires also breadth, by radiations shooting out from it, but +does not become a uniform surface, the central part remaining +always considerably brightest, in consequence of the same flattening +of the vertex which before made it fainter. Some of +these figures bear a considerable analogy to the images derived +from the refraction of oblique rays, (Schol. 4. Prop. IV.) and +still more strongly resemble a combination of two of them in +opposite directions; so as to leave no doubt, but that both surfaces +of the lens are oblique to the visual axis, and co-operate +in distorting the focal point. This may also be verified, by +observing the image delineated by a common glass lens, when +inclined to the incident rays. (See Plate VI. <a href="#Pl.IV">Fig. 28—40</a>.)</p> + +<p>The visual axis being fixed in any direction, I can at the +same time see a luminous object placed laterally at a considerable +distance from it; but in various directions the angle +is very different. Upwards it extends to 50 degrees, inwards +to 60, downwards to 70, and outwards to 90 degrees. These +internal limits of the field of view nearly correspond with +<span class="pagenum" id="Page_45">45</span>the external limits formed by the different parts of the face, +when the eye is directed forwards and somewhat downwards, +which is its most natural position; although the internal limits +are a little more extensive than the external; and both are well +calculated for enabling us to perceive the most readily, such +objects as are the most likely to concern us. Dr. <span class="smcap">Wollaston</span>’s +eye has a larger field of view, both vertically and horizontally, +but nearly in the same proportions, except that it extends further +upwards. It is well known, that the retina advances further +forwards towards the internal angle of the eye, than towards +the external angle; but upwards and downwards its extent is +nearly equal, and is indeed every way greater than the limits of +the field of view, even if allowance is made for the refraction +of the cornea only. The sensible portion seems to coincide +more nearly with the painted choroid of quadrupeds: but the +whole extent of perfect vision is little more than 10 degrees; or, +more strictly speaking, the imperfection begins within a degree +or two of the visual axis, and at the distance of 5 or 6 degrees +becomes nearly stationary, until, at a still greater distance, vision +is wholly extinguished. The imperfection is partly owing to +the unavoidable aberration of oblique rays, but principally to +the insensibility of the retina: for, if the image of the sun +itself be received on a part of the retina remote from the axis, +the impression will not be sufficiently strong to form a permanent +spectrum, although an object of very moderate brightness +will produce this effect when directly viewed. It would probably +have been inconsistent with the economy of nature, to bestow a +larger share of sensibility on the retina. The optic nerve is at +present very large; and the delicacy of the organ renders it, +even at present, very susceptible of injury from slight irritation. +<span class="pagenum" id="Page_46">46</span>and very liable to inflammatory affections; and, in order to +make the sight so perfect as it is, it was necessary to confine that +perfection within narrow limits. The motion of the eye has a +range of about 55 degrees in every direction; so that the field +of perfect vision, in succession, is by this motion extended to +110 degrees.</p> + +<p>But the whole of the retina is of such a form as to receive +the most perfect image, on every part of its surface, that the +state of each refracted pencil will admit; and the varying density +of the crystalline renders that state more capable of delineating +such a picture, than any other imaginable contrivance could +have done. To illustrate this, I have constructed a diagram, +representing the successive images of a distant object filling the +whole extent of view, as they would be formed by the successive +refractions of the different surfaces. Taking the scale of +my own eye, I am obliged to substitute, for a series of objects +at any indefinitely great distance, a circle of 10 inches radius; +and it is most convenient to consider only those rays which pass +through the anterior vertex of the lens; since the actual centre +of each pencil must be in the ray which passes through the +centre of the pupil, and the short distance of the vertex of the +lens from this point, will always tend to correct the unequal +refraction of oblique rays. The first curve (Plate IV. <a href="#Pl.IV">Fig. 16</a>.) +is the image formed by the furthest intersection of rays refracted +at the cornea; the second, the image formed by the nearest intersection; +the distance between these, shows the degree of confusion +in the image; and the third curve, its brightest part. Such +must be the form of the image which the cornea tends to delineate +in an eye deprived of the crystalline lens; nor can any +external remedy properly correct the imperfection of lateral +<span class="pagenum" id="Page_47">47</span>vision. The next three curves show the images formed after +the refraction at the anterior surface of the lens, distinguished in +the same manner; and the three following, the result of all the +successive refractions. The tenth curve is a repetition of the +ninth, with a slight correction near the axis, at F, where, from +the breadth of the pupil, some perpendicular rays must fall. By +comparing this with the eleventh, which is the form of the retina, +it will appear that nothing more is wanting for their perfect +coincidence, than a moderate diminution of density in the lateral +parts of the lens. If the law, by which this density varies, were +more accurately ascertained, its effect on the image might be +calculated from the eighth proposition; but the operations would +be somewhat laborious: probably the image, thus corrected, +would approach very nearly to the form of the twelfth curve.</p> + +<p>To find the place of the entrance of the optic nerve, I fix +two candles at ten inches distance, retire sixteen feet, and direct +my eye to a point four feet to the right or left of the middle of +the space between them: they are then lost in a confused spot +of light; but any inclination of the eye brings one or the other +of them into the field of view. In <span class="smcap">Bernoulli</span>’s eye, a greater +deviation was required for the direction of the axis;⁠<a id="FNanchor_10_10" href="#Footnote_10_10" class="fnanchor">10</a> and the +obscured part appeared to be of greater extent. From the +experiment here related, the distance of the centre of the optic +nerve from the visual axis is found (by Prop. V.) to be 16 hundredths +of an inch; and the diameter of the most insensible part +of the retina, one-thirtieth of an inch. In order to ascertain the +distance of the optic nerve from the point opposite to the pupil, +I took the sclerotica of the human eye, divided it into segments, +from the centre of the cornea towards the optic nerve, and extended +it on a plane. I then measured the longest and shortest +<span class="pagenum" id="Page_48">48</span>distances from the cornea to the perforation made by the nerve, +and their difference was exactly one-fifth of an inch. To this +we must add a fiftieth, on account of the eccentricity of the +pupil in the uvea, which in the eye that I measured was not +great, and the distance of the centre of the nerve from the +point opposite the pupil will be 11 hundredths. Hence it appears, +that the visual axis is five hundredths, or one-twentieth of +an inch, further from the optic nerve than the point opposite the +pupil. It is possible that this distance may be different in different +eyes: in mine, the obliquity of the lens, and the eccentricity +of the pupil with respect to it, will tend to throw a direct +ray upon it, without much inclination of the whole eye; and it +is not improbable, that the eye is also turned slightly outwards, +if looking at any object before it, although the inclination is +too small to be subjected to measurement.</p> + +<p>It must also be observed, that it is very difficult to ascertain +the proportions of the eye so exactly as to determine, with certainty, +the size of an image on the retina; the situation, curvature, +and constitution of the lens, make so material a difference +in the result, that there may possibly be an error of almost one- +tenth of the whole. In order, therefore, to obtain some confirmation +from experiment, I placed two candles at a small distance +from each other, turned the eye inwards, and applied the +ring of a key so as to produce a spectrum, of which the edge +coincided with the inner candle; then, fixing my eye on the outward +one, I found that the spectrum advanced over two-sevenths +of the distance between them. Hence, the same portion of the +retina that subtended an angle of seven parts at the centre of +motion of the eye, subtended an angle of five at the supposed +intersection of the principal rays; (Plate III. <a href="#Pl.III">Fig. 11</a>.) and the +<span class="pagenum" id="Page_49">49</span>distance of this intersection from the retina was 637 thousandths. +This nearly corresponds with the former calculation; nor can +the distance of the centre of the optic nerve from the point of +most perfect vision be, on any supposition, much less than that +which is here assigned. And, in the eyes of quadrupeds, the +most strongly painted part of the choroid is further from the +nerve than the real axis of the eye.</p> + +<p>I have endeavoured to express in four figures, the form of +every part of my eye, as nearly as I have been able to ascertain +it; the first (Pl. V. <a href="#Pl.V">Fig. 17</a>.) is a vertical section; the second +(Fig. 18.) a horizontal section; the third and fourth are front +views, in different states of the pupil. (<a href="#Pl.V">Fig. 19 and 20</a>.)</p> + +<p>Considering how little inconvenience is experienced from so +material an inequality in the refraction of the lens as I have +described, we have no reason to expect a very accurate provision +for correcting the aberration of the lateral rays. But, as far as +can be ascertained by the optometer, the aberration arising from +figure is completely corrected; since four or more images of the +same line appear to meet exactly in the same point, which they +would not do if the lateral rays were materially more refracted +than the rays near the axis. The figure of the surfaces is sometimes, +and perhaps always, more or less hyperbolical⁠<a id="FNanchor_11_11" href="#Footnote_11_11" class="fnanchor">11</a> or elliptical: +in the interior laminæ indeed, the solid angle of the +margin is somewhat rounded off; but the weaker refractive +power of the external parts, must greatly tend to correct the +aberration arising from the too great curvature towards the +margin of the disc. Had the refractive power been uniform, it +might have collected the lateral rays of a direct pencil nearly as +well; but it would have been less adapted to oblique pencils of +<span class="pagenum" id="Page_50">50</span>rays; and the eye must also have been encumbered with a mass +of much greater density than is now required, even for the +central parts: and, if the whole lens had been smaller, it would +also have admitted too little light. It is possible too, that Mr. +<span class="smcap">Ramsden</span>’s observation,⁠<a id="FNanchor_12_12" href="#Footnote_12_12" class="fnanchor">12</a> on the advantage of having no reflecting +surface, may be well-founded: but it has not been demonstrated, +that less light is lost in passing through a medium +of variable density, than in a sudden transition from one part of +that medium to another; nor are we yet sufficiently acquainted +with the cause of this reflection, to be enabled to reason satisfactorily +on the subject. But, neither this gradation, nor any other +provision, has the effect of rendering the eye perfectly achromatic. +Dr. <span class="smcap">Jurin</span> had remarked this, long ago,⁠<a id="FNanchor_13_13" href="#Footnote_13_13" class="fnanchor">13</a> from observing +the colour bordering the image of an object seen indistinctly. +Dr. <span class="smcap">Wollaston</span> pointed out to me on the optometer, the red +and blue appearance of the opposite internal angles of the crossing +lines; and mentioned, at the same time, a very elegant experiment +for proving the dispersive power of the eye. He looks +through a prism at a small lucid point, which of course becomes +a linear spectrum. But the eye cannot so adapt itself as to make +the whole spectrum appear a line; for, if the focus be adapted +to collect the red rays to a point, the blue will be too much refracted, +and expand into a surface; and the reverse will happen +if the eye be adapted to the blue rays; so that, in either case, the +line will be seen as a triangular space. The observation is confirmed, +by placing a small concave speculum in different parts +of a prismatic spectrum, and ascertaining the utmost distances +at which the eye can collect the rays of different colours to a +focus. By these means I find, that the red rays, from a point at +<span class="pagenum" id="Page_51">51</span>12 inches distance, are as much refracted as white or yellow +light at 11. The difference is equal to the refraction of a lens +132 inches in focus. But the aberration of the red rays in a +lens of crown glass, of equal mean refractive power with the +eye, would be equivalent to the effect of a lens 44 inches in +focus. If, therefore, we can depend upon this calculation, the +dispersive power of the eye collectively, is one-third of the dispersive +power of crown glass, at an equal angle of deviation. 1 +cannot observe much aberration in the violet rays. This may +be, in part, owing to their faintness; but yet I think their aberration +must be less than that of the red rays. I believe it was +Mr. <span class="smcap">Ramsden</span>’s opinion, that since the separation of coloured +rays is only observed where there is a sudden change of density, +such a body as the lens, of a density gradually varying, would +have no effect whatever in separating the rays of different colours. +If this hypothesis should appear to be well-founded, we must +attribute the whole dispersion to the aqueous humour; and its +dispersive power will be half that of crown glass, at the same +deviation. But we have an instance, in the atmosphere, of a +very gradual change of density; and yet Mr. <span class="smcap">Gilpin</span> informs +me, that the stars, when near the horizon, appear very evidently +coloured. At a more favourable season of the year, it would +not be difficult to ascertain, by means of the optometer, the +dispersive power of the eye, and of its different parts, with greater +accuracy than by the experiment here related. Had the dispersive +power of the whole eye been equal to that of flint glass, +the distances of perfect vision would have varied from 12 inches +to 7 for different rays, in the same state of the mean refractive +powers.</p> + + +<p class="mt2em ti0">VII. The faculty of accommodating the eye to various +<span class="pagenum" id="Page_52">52</span>distances, appears to exist in very different degrees in different +individuals. The shortest distance of perfect vision in my eye, +is 26 tenths of an inch for horizontal, and 29 for vertical rays. +This power is equivalent to the addition of a lens of 4 inches +focus. Dr. <span class="smcap">Wollaston</span> can see at 7 inches, and with converging +rays; the difference answering to 6 inches focal length. Mr. +<span class="smcap">Abernethy</span> has perfect vision from 3 inches to 30, or a power +equal to that of a lens 3-1/3 inches in focus. A young lady of my +acquaintance can see at 2 inches and at 4; the difference being +equivalent to 4 inches focus. A middle aged lady at 3 and at 4; +the power of accommodation being only equal to the effect of a +lens of 12 inches focus. In general, I have reason to think, that +the faculty diminishes in some degree, as persons advance in life; +but some also of a middle age appear to possess it in a very small +degree. I shall take the range of my own eye, as being probably +about the medium, and inquire what changes will be necessary +in order to produce it; whether we suppose the radius of the +cornea to be diminished, or the distance of the lens from the +retina to be increased, or these two causes to act conjointly, or +the figure of the lens itself to undergo an alteration.</p> + +<p>1. We have calculated, that when the eye is in a state of +relaxation, the refraction of the cornea is such as to collect +rays diverging from a point ten inches distant, to a focus at +the distance of 13-2/3 tenths. In order that it may bring to the +same focus, rays diverging from a point distant 29 tenths, we +find (by Cor. 5, Prop. IV.) that its radius must be diminished +from 31 to 25 hundredths, or very nearly in the ratio of five +to four.</p> + +<p>2. Supposing the change from perfect vision at ten inches to +29 tenths, to be effected by a removal of the retina to a greater +<span class="pagenum" id="Page_53">53</span>distance from the lens, this will require, (by the same Corollary,) +an elongation of 135 thousandths, or more than one-seventh +of the diameter of the eye. In Mr. <span class="smcap">Abernethy</span>’s eye, an +elongation of 17 hundredths, or more than one-sixth, is requisite.</p> + +<p>3. If the radius of the cornea be diminished one-sixteenth, +or to 29 hundredths, the eye must at the same time be elongated +97 thousandths, or about one-ninth of its diameter.</p> + +<p>4. Supposing the crystalline lens to change its form; if it +became a sphere, its diameter would be 28 hundredths, and, its +anterior surface retaining its situation, the eye would have perfect +vision at the distance of an inch and a half. (Cor. 5 and +8, Prop. IV.) This is more than double the actual change. +But it is impossible to determine precisely how great an alteration +of form is necessary, without ascertaining the nature of the +curves into which its surfaces may be changed. If it were +always a spheroid more or less oblate, the focal length of each +surface would vary inversely as the square of the axis: but, if +the surfaces became, from spherical, portions of hyperbolic +conoids, or of oblong spheroids, or changed from more obtuse +to more acute figures of this kind, the focal length would vary +more rapidly. Disregarding the elongation of the axis, and +supposing the curvature of each surface to be changed proportionally, +the radius of the anterior must become about 24, and +that of the posterior 17 hundredths.</p> + + +<p class="mt2em ti0">VIII. I shall now proceed to inquire, which of these changes +takes place in nature; and I shall begin with a relation of experiments +made in order to ascertain the curvature of the cornea +in all circumstances.</p> + +<p>The method described in Mr. <span class="smcap">Home</span>’s Croonian Lecture for +<span class="pagenum" id="Page_54">54</span>1795,⁠<a id="FNanchor_14_14" href="#Footnote_14_14" class="fnanchor">14</a> appears to be far preferable to the apparatus of the +preceding year:⁠<a id="FNanchor_15_15" href="#Footnote_15_15" class="fnanchor">15</a> for a difference in the distance of two images +seen in the cornea, would be far greater, and more conspicuous, +than a change of its prominency, and far less liable to be +disturbed by accidental causes. It is nearly, and perhaps +totally impossible to change the focus of the eye, without some +motion of its axis. The eyes sympathize perfectly with each +other; and the change of focus is almost inseparable from a +change of the relative situation of the optic axes; so much, that +if I direct both my eyes at an object beyond their furthest focus, +I cannot avoid bringing that focus a little nearer: while one +axis moves, it is not easy to keep the other perfectly at rest; +and it is not impossible, that a change in the proportions of some +eyes, may render a slight alteration of the position of the axis +absolutely necessary. These considerations may partly explain +the trifling difference in the place of the cornea that was observed +in 1794. It appears that the experiments of 1795 were +made with considerable accuracy, and no doubt with excellent +instruments; and their failing to ascertain the existence of any +change, induced Mr. <span class="smcap">Home</span> and Mr. <span class="smcap">Ramsden</span> to abandon, +in great measure, the opinion which suggested them, and to +suppose, that a change of the cornea produces only one-third of +the effect. Dr. <span class="smcap">Olbers</span> of Bremen, who in the year 1780 +published a most elaborate dissertation on the internal changes +of the eye,⁠<a id="FNanchor_16_16" href="#Footnote_16_16" class="fnanchor">16</a> which he lately presented to the Royal Society, +had been equally unsuccessful in his attempts to measure this +change of the cornea, at the same time that his opinion was in +favour of its existence.</p> + +<p><span class="pagenum" id="Page_55">55</span></p> + +<p>Room was however still left for a repetition of the experiments; +and I began with an apparatus nearly resembling that +which Mr. <span class="smcap">Home</span> has described. I had an excellent achromatic +microscope, made by Mr. <span class="smcap">Ramsden</span> for my friend Mr. <span class="smcap">John +Ellis</span>, of five inches focal length, magnifying about 20 times. +To this I adapted a cancellated micrometer, in the focus of the +eye not employed in looking through the microscope: it was +a large card, divided by horizontal and vertical lines into +fortieths of an inch. When the image in the microscope was compared +with this scale, care was taken to place the head so that +the relative motion of the images on the micrometer, caused by +the unsteadiness of the optic axis, should always be in the direction +of the horizontal lines, and that there could be no error, +from this motion, in the dimensions of the image taken vertically. +I placed two candles so as to exhibit images in a vertical +position in the eye of Mr. <span class="smcap">König</span>, who had the goodness to +assist me; and, having brought them into the field of the microscope, +where they occupied 35 of the small divisions, I +desired him to fix his eye on objects at different distances in the +same direction: but I could not perceive the least variation in the +distance of the images.</p> + +<p>Finding a considerable difficulty in a proper adjustment of +the microscope, and being able to depend on my naked eye in +measuring distances, without an error of one 500th of an inch, I +determined to make a similar experiment without any magnifying +power. I constructed a divided eye-glass of two portions of +a lens, so small, that they passed between two images reflected +from my own eye; and, looking in a glass, I brought the apparent +places of the images to coincide, and then made the +change requisite for viewing nearer objects: but the images still +<span class="pagenum" id="Page_56">56</span>coincided. Neither could I observe any change in the images +reflected from the other eye, where they could be viewed with +greater convenience, as they did not interfere with the eye- +glass. But, not being at that time aware of the perfect sympathy +of the eyes, I thought it most certain to confine my observation +to the one with which I saw. I must remark that, by a +little habit, I have acquired a very ready command over the +accommodation of my eye, so as to be able to view an object +with attention, without adjusting my eye to its distance.</p> + +<p>I also stretched two threads, a little inclined to each other, +across a ring, and divided them by spots of ink into equal +spaces. I then fixed the ring, applied my eye close behind it, +and placed two candles in proper situations before me, and a +third on one side, to illuminate the threads. Then, setting a +small looking-glass, first at four inches distance, and next at +two, I looked at the images reflected in it, and observed at +what part of the threads they exactly reached across in each +case; and with the same result as before.</p> + +<p>I next fixed the cancellated micrometer at a proper distance, +illuminated it strongly, and viewed it through a pin-hole, by +which means it became distinct in every state of the eye; and, +looking with the other eye into a small glass, I compared the +image with the micrometer, in the manner already described. +I then changed the focal distance of the eye, so that the lucid +points appeared to spread into surfaces, from being too remote +for perfect vision; and I noted on the scale, the distance of their +centres; but that distance was invariable.</p> + +<p>Lastly, I drew a diagonal scale, with a diamond, on a looking- +glass, (Plate III. <a href="#Pl.III">Fig. 12</a>.) and brought the images into contact +with the lines of the scale. Then, since the image of the +<span class="pagenum" id="Page_57">57</span>eye occupies on the surface of a glass half its real dimensions, +at whatever distance it is viewed, its true size is always double +the measure thus obtained. I illuminated the glass strongly, +and made a perforation in a narrow slip of black card, which I +held between the images; and was thus enabled to compare +them with the scale, although their apparent distance was double +that of the scale. I viewed them in all states of the eye; +but I could perceive no variation in the interval between them.</p> + +<p>The sufficiency of these methods may be thus demonstrated. +Make a pressure along the edge of the upper eyelid with any small +cylinder, for instance a pencil, and the optometer will show that +the focus of horizontal rays is a little elongated, while that of +vertical rays is shortened; an effect which can only be owing to a +change of curvature in the cornea. Not only the apparatus here +described, but even the eye unassisted, will be capable of discovering +a considerable change in the images reflected from the cornea, +although the change be much smaller than that which is requisite +for the accommodation of the eye to different distances. +On the whole, I cannot hesitate to conclude, that if the radius +of the cornea were diminished but one-twentieth, the change +would be very readily perceptible by some of the experiments +related; and the whole alteration of the eye requires one-fifth.</p> + +<p>But a much more accurate and decisive experiment remains. I +take out of a small botanical microscope, a double convex lens, of +eight-tenths radius and focal distance, fixed in a socket one-fifth +of an inch in depth; securing its edges with wax, I drop into it a +little water, nearly cold, till it is three-fourths full, and then apply +it to my eye, so that the cornea enters halfway into the socket, +and is every where in contact with the water. (Plate III. <a href="#Pl.III">Fig. 13</a>.) +My eye immediately becomes presbyopic, and the refractive +<span class="pagenum" id="Page_58">58</span>power of the lens, which is reduced by the water to a focal +length of about 16 tenths, (Cor. 5. Prop. IV.) is not sufficient +to supply the place of the cornea, rendered inefficacious by the +intervention of the water; but the addition of another lens, of +five inches and a half focus, restores my eye to its natural state, +and somewhat more. I then apply the optometer, and I find +the same inequality in the horizontal and vertical refractions as +without the water; and I have, in both directions, a power of +accommodation equivalent to a focal length of four inches, as +before. At first sight indeed, the accommodation appears to +be somewhat less, and only able to bring the eye from the state +fitted for parallel rays to a focus at five inches distance; and +this made me once imagine, that the cornea might have some +slight effect in the natural state; but, considering that the artificial +cornea was about a tenth of an inch before the place of +the natural cornea, I calculated the effect of this difference, and +found it exactly sufficient to account for the diminution of the +range of vision. I cannot ascertain the distance of the glass +lens from the cornea to the hundredth of an inch; but the error +cannot be much greater, and it may be on either side.</p> + +<p>After this, it is almost necessary to apologize for having +stated the former experiments; but, in so delicate a subject, we +cannot have too great a variety of concurring evidence.</p> + + +<p class="mt2em ti0">IX. Having satisfied myself that the cornea is not concerned +in the accommodation of the eye, my next object was to inquire +if any alteration in the length of its axis could be discovered; +for this appeared to be the only possible alternative: and, considering +that such a change must amount to one-seventh of the +diameter of the eye, I flattered myself with the expectation of +submitting it to measurement. Now, if the axis of the eye +<span class="pagenum" id="Page_59">59</span>were elongated one-seventh, its transverse diameter must be +diminished one-fourteenth, and the semi-diameter would be +shortened a thirtieth of an inch.</p> + +<p>I therefore placed two candles so that when the eye was +turned inwards, and directed towards its own image in a glass, +the light reflected from one of the candles by the sclerotica +appeared upon its external margin, so as to define it distinctly +by a bright line; and the image of the other candle was seen in +the centre of the cornea. I then applied the double eye-glass, +and the scale of the looking-glass, in the manner already described; +but neither of them indicated any diminution of the +distance, when the focal length of the eye was changed.</p> + +<p>Another test, and a much more delicate one, was the application +of the ring of a key at the external angle, when the eye +was turned as much inwards as possible, and confined at the +same time by a strong oval iron ring, pressed against it at the +internal angle. The key was forced in as far as the sensibility +of the integuments would admit, and was wedged, by a moderate +pressure, between the eye and the bone. In this situation, +the phantom caused by the pressure extended within the field +of perfect vision, and was very accurately defined; nor did it, +as I formerly imagined, by any means prevent a distinct perception +of the objects actually seen in that direction; and a straight +line coming within the field of this oval phantom, appeared +somewhat inflected towards its centre; (Plate III. <a href="#Pl.III">Fig. 14</a>.) +a distortion easily understood by considering the effect of the +pressure on the form of the retina. Supposing now, the distance +between the key and the iron ring to have been, as it +really was, invariable, the elongation of the eye must have been +either totally or very nearly prevented; and, instead of an +<span class="pagenum" id="Page_60">60</span>increase of the length of the eye’s axis, the oval spot caused +by the pressure would have spread over a space at least ten +times as large as the most sensible part of the retina. But no +such circumstance took place: the power of accommodation +was as extensive as ever; and there was no perceptible change, +either in the size or in the figure of the oval spot.</p> + +<p>Again, since the rays which pass through the centre of the +pupil, or rather the anterior vertex of the lens, may, as already +observed, be considered as delineating the image; and, since +the divergence of these rays with respect to each other, is but +little affected by the refraction of the lens, they may still be +said to diverge from the centre of the pupil; and the image of +a given object on the retina must be very considerably enlarged, +by the removal of the retina to a greater distance from +the pupil and lens. (Cor. Prop. V⁠<a id="FNanchor_17_17" href="#Footnote_17_17" class="fnanchor">17</a>⁠.) To ascertain the real +magnitude of the image with accuracy, is not so easy as it at +first sight appears; but, besides the experiment last related, +which might be employed as an argument to this purpose, there +are two other methods of estimating it. The first is too hazardous +to be of much use; but, with proper precautions, it may be +attempted. I fix my eye on a brass circle placed in the rays of +the sun, and, after some time, remove it to the cancellated micrometer; +then, changing the focus of my eye, while the micrometer +remains at a given distance, I endeavour to discover +whether there is any difference in the apparent magnitude of the +spectrum on the scale; but I can discern none. I have not insisted +on the attempt; especially as I have not been able to make the +<span class="pagenum" id="Page_61">61</span>spectrum distinct enough without inconvenience; and no light +is sufficiently strong to cause a permanent impression on any +part of the retina remote from the visual axis. I therefore had +recourse to another experiment, I placed two candles so as +exactly to answer to the extent of the termination of the optic +nerve, and, marking accurately the point to which my eye was +directed, I made the utmost change in its focal length; expecting +that, if there were any elongation of the axis, the external +candle would appear to recede outwards upon the visible space. +(Plate III. <a href="#Pl.III">Fig. 15</a>.) But this did not happen; the apparent place +of the obscure part was precisely the same as before. I will +not undertake to say, that I could have observed a very minute +difference either way: but I am persuaded, that I should have +discovered an alteration of less than a tenth part of the whole.</p> + +<p>It may be inquired if no change in the magnitude of the +image is to be expected on any other supposition; and it will +appear to be possible, that the changes of curvature may be so +adapted, that the magnitude of the confused image may remain +perfectly constant. Indeed, to calculate from the dimensions +which we have hitherto used, it would be expected that the +image should be diminished about one-sixtieth, by the utmost +increase of the convexity of the lens. But the whole depends +on the situation of the refracting surfaces, and the respective +increase of their curvature, which, on account of the variable +density of the lens, can scarcely be estimated with sufficient +accuracy. Had the pupil been placed before the cornea, the +magnitude of the image must, on any supposition, have been +very variable: at present, this inconvenience is avoided by the +situation of the pupil; so that we have here an additional +instance of the perfection of this admirable organ.</p> + +<p><span class="pagenum" id="Page_62">62</span></p> + +<p>From the experiments related, it appears to be highly improbable +that any material change in the length of the axis +actually takes place; and it is almost impossible to conceive by +what power such a change could be effected. The straight +muscles, with the adipose substance lying under them, would +certainly, when acting independently of the socket, tend to +flatten the eye: for, since their contraction would necessarily +lessen the circumference or superficies of the mass that they +contain, and round off all its prominences, their attachment +about the nerve and the anterior part of the eye must therefore +be brought nearer together. (Plate V. <a href="#Pl.V">Fig. 21, 22</a>.) Dr. +<span class="smcap">Olbers</span> compares the muscles and the eye to a cone, of which +the sides are protruded, and would by contraction be brought +into a straight line. But this would require a force to preserve +the cornea as a fixed point, at a given distance from the origin +of the muscles; a force which certainly does not exist. In the +natural situation of the visual axis, the orbit being conical, the +eye might be somewhat lengthened, although irregularly, by +being forced further into it; but, when turned towards either +side, the same action would rather shorten its axis; nor is there +any thing about the human eye that could supply its place. +In quadrupeds, the oblique muscles are wider than in man; +and in many situations might assist in the effect. Indeed a +portion of the orbicular muscle of the globe is attached so near +to the nerve, that it might also co-operate in the action: and I +have no reason to doubt the accuracy of Dr. <span class="smcap">Olbers</span>, who +states, that he effected a considerable elongation, by tying threads +to the muscles, in the eyes of hogs and of calves; yet he does +not say in what position the axis was fixed; and the flaccidity +of the eye after death might render such a change very easy as +<span class="pagenum" id="Page_63">63</span>would be impossible in a living eye. Dr. <span class="smcap">Olbers</span> also mentions +an observation of Professor <span class="smcap">Wrisberg</span>, on the eye of a man +whom he believed to be destitute of the power of accommodation +in his life-time, and whom he found, after death, to have +wanted one or more of the muscles: but this want of accommodation +was not at all accurately ascertained. I measured, in +the human eye, the distance of the attachment of the inferior +oblique muscle from the insertion of the nerve: it was one-fifth +of an inch; and from the centre of vision not a tenth of an +inch; so that, although the oblique muscles do in some positions +nearly form a part of a great circle round the eye, their action +would be more fitted to flatten than to elongate it. We have +therefore reason to agree with <span class="smcap">Winslow</span>, in attributing to them +the office of helping to support the eye on that side where the +bones are most deficient: they seem also well calculated to +prevent its being drawn too much backwards by the action of +the straight muscles. And, even if there were no difficulty in +supposing the muscles to elongate the eye in every position, yet +at least some small difference would be expected in the extent +of the change, when the eye is in different situations, at an +interval of more than a right angle from each other; but the +optometer shews that there is none.</p> + +<p>Dr. <span class="smcap">Hosack</span> alleges that he was able, by making a pressure +on the eye, to accommodate it to a nearer object:⁠<a id="FNanchor_18_18" href="#Footnote_18_18" class="fnanchor">18</a> it does not +appear that he made use of very accurate means of ascertaining +the fact; but, if such an effect took place, the cause must have +been an inflection of the cornea.</p> + +<p>It is unnecessary to dwell on the opinion which supposes a +joint operation, of changes in the curvature of the cornea and +<span class="pagenum" id="Page_64">64</span>in the length of the axis. This opinion had derived very great +respectability, from the most ingenious and elegant manner in +which Dr. <span class="smcap">Olbers</span> had treated it, and from being the last result +of the investigation of Mr. <span class="smcap">Home</span> and Mr. <span class="smcap">Ramsden</span>. But +either of the series of experiments which have been related, +appears to be sufficient to confute it.</p> + + +<p class="mt2em ti0">X. It now remains to inquire into the pretensions of the +crystalline lens to the power of altering the focal length of the +eye. The grand objection to the efficacy of a change of figure +in the lens, was derived from the experiments in which those +who have been deprived of it have appeared to possess the +faculty of accommodation.</p> + +<p>My friend Mr. <span class="smcap">Ware</span>, convinced as he was of the neatness +and accuracy of the experiments related in the Croonian Lecture +for 1795, yet could not still help imagining, from the obvious +advantage all his patients found, after the extraction +of the lens, in using two kinds of spectacles, that there must, +in such cases, be a deficiency in that faculty. This circumstance, +combined with a consideration of the directions very judiciously +given by Dr. <span class="smcap">Porterfield</span>, for ascertaining the point in question, +first made me wish to repeat the experiments upon various +individuals, and with the instrument which I have above described +as an improvement of Dr. <span class="smcap">Porterfield</span>’s optometer: +and I must here acknowledge my great obligation to Mr. +<span class="smcap">Ware</span>, for the readiness and liberality with which he introduced +me to such of his numerous patients as he thought most +likely to furnish a satisfactory determination. It is unnecessary +to enumerate every particular experiment; but the universal +result is, contrary to the expectation with which I entered on +the inquiry, that in an eye deprived of the crystalline lens, the +<span class="pagenum" id="Page_65">65</span>actual focal distance is totally unchangeable. This will appear +from a selection of the most decisive observations.</p> + +<p>1. Mr. R. can read at four inches and at six only, with the +same glass. He saw the double lines meeting at three inches, +and always at the same point; but the cornea was somewhat +irregularly prominent, and his vision not very distinct; nor had +I, at the time I saw him, a convenient apparatus.</p> + +<p>I afterwards provided a small optometer, with a lens of less +than two inches focus, adding a series of letters, not in alphabetical +order, and projected into such a form as to be most legible +at a small inclination. The excess of the magnifying power +had the advantage of making the lines more divergent, and +their crossing more conspicuous; and the letters served for +more readily naming the distance of the intersection, and, at +the same time, for judging of the extent of the power of distinguishing +objects too near or too remote for perfect vision. +(Plate V. <a href="#Pl.V">Fig. 23</a>.)</p> + +<p>2. Mr. J. had not an eye very proper for the experiment; +but he appeared to distinguish the letters at <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/49.svg" alt=" " data-tex="\(2\frac{1}{2}\)"> inches, and +at less than an inch. This at first persuaded me, that he +must have a power of changing the focal distance: but I afterwards +recollected that he had withdrawn his eye considerably, +to look at the nearer letters, and had also partly closed his +eyelids, no doubt contracting at the same time the aperture of +the pupil; an action which, even in a perfect eye, always accompanies +the change of focus. The slider was not applied.</p> + +<p>3. Miss H. a young lady of about twenty, had a very narrow +pupil, and I had not an opportunity of trying the small optometer: +but, when she once saw an object double through the +slits, no exertion could make it appear single at the same distance. +<span class="pagenum" id="Page_66">66</span>She used for distant objects a glass of <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/52.svg" alt=" " data-tex="\(4\frac{1}{2}\)"> inches focus; +with this she could read as far off as 12 inches, and as near as +five: for nearer objects she added another of equal focus, and +could then read at 7 inches, and at <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/49.svg" alt=" " data-tex="\(2\frac{1}{2}\)">.</p> + +<p>4. <span class="smcap">Hanson</span>, a carpenter, aged 63, had a cataract extracted +a few years since from one eye: the pupil was clear and large, +and he saw well to work with a lens of <img style="vertical-align: -0.816ex; width: 2.927ex; height: 2.771ex;" src="images/53.svg" alt=" " data-tex="\(2\frac{3}{8}\)"> inches focus; and +could read at 8 and at 15 inches, but most conveniently at 11. +With the same glass, the lines of the optometer appeared always +to meet at 11 inches; but he could not perceive that they +crossed, the line being too strong, and the intersection too distant. +The experiment was afterwards repeated with the small +optometer: he read the letters from 2 to 3 inches; but the +intersection was always at <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/49.svg" alt=" " data-tex="\(2\frac{1}{2}\)"> inches. He now fully understood +the circumstances that were to be noticed, and saw the crossing +with perfect distinctness: at one time, he said it was a tenth of +an inch nearer; but I observed that he had removed his eye +two or three tenths from the glass, a circumstance which +accounted for this small difference.</p> + +<p>5. Notwithstanding <span class="smcap">Hanson</span>’s age, I consider him as a very +fair subject for the experiment. But a still more unexceptionable +eye was that of Mrs. <span class="smcap">Maberly</span>. She is about 30, and had +the crystalline of both eyes extracted a few years since, but +sees best with her right. She walks without glasses; and, with +the assistance of a lens of about four inches focus, can read +and work with ease. She could distinguish the letters of the +small optometer from an inch to <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/49.svg" alt=" " data-tex="\(2\frac{1}{2}\)"> inches; but the intersection +was invariably at the same point, about 19 tenths of an inch +distant. A portion of the capsule is stretched across the pupil, +and causes her to see remote objects double, when without her +<span class="pagenum" id="Page_67">67</span>glasses; nor can she, by any exertion, bring the two images +nearer together, although the exertion makes them more distinct, +no doubt by contracting the pupil. The experiment with +the optometer was conducted, in the presence of Mr. <span class="smcap">Ware</span>, +with patience and perseverance; nor was any opinion given to +make her report partial.</p> + +<p>Considering the difficulty of finding an eye perfectly suitable +for the experiments, these proofs may be deemed tolerably +satisfactory. But, since one positive argument will counterbalance +many negative ones, provided it be equally grounded +on fact, it becomes necessary to inquire into the competency of +the evidence employed to ascertain the power of accommodation +attributed, in the Croonian Lecture for 1794, to the eye of +<span class="smcap">Benjamin Clerk</span>. And it appears, that the distinction long +since very properly made by Dr. <span class="smcap">Jurin</span>, between distinct vision +and perfect vision, will readily explain away the whole of that +evidence.</p> + +<p>It is obvious that vision may be made distinct to any given +extent, by means of an aperture sufficiently small, provided at +the same time, that a sufficient quantity of light be left, while +the refractive powers of the eye remain unchanged. And it is +remarkable, that in those experiments, when the comparison +with the perfect eye was made, the aperture of the imperfect +eye only was very considerably reduced. <span class="smcap">Benjamin Clerk</span>, +with an aperture of <img style="vertical-align: -0.816ex; width: 2.595ex; height: 2.771ex;" src="images/54.svg" alt=" " data-tex="\(\frac{3}{40}\)"> of an inch, could read with the same +glass at <img style="vertical-align: -0.816ex; width: 2.927ex; height: 2.789ex;" src="images/55.svg" alt=" " data-tex="\(1\frac{7}{8}\)"> inch, and at 7 inches.⁠<a id="FNanchor_19_19" href="#Footnote_19_19" class="fnanchor">19</a> With an equal aperture, I +can read at <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/56.svg" alt=" " data-tex="\(1\frac{1}{2}\)"> inch and at 30 inches: and I can retain the state +of perfect relaxation, and read with the same aperture at <img style="vertical-align: -0.781ex; width: 2.927ex; height: 2.737ex;" src="images/57.svg" alt=" " data-tex="\(2\frac{1}{4}\)"> +inches; and this is as great a difference as was observed in +<span class="pagenum" id="Page_68">68</span><span class="smcap">Benjamin Clerk</span>’s eye. It is also a fact of no small importance, +that Sir <span class="smcap">Henry Englefield</span> was much astonished, as +well as the other observers, at the accuracy with which the +man’s eye was adjusted to the same distance, in the repeated +trials that were made with it.⁠<a id="FNanchor_20_20" href="#Footnote_20_20" class="fnanchor">20</a> This circumstance alone makes +it highly probable, that its perfect vision was confined within +very narrow limits.</p> + +<p>Hitherto I have endeavoured to shew the inconveniences +attending other suppositions, and to remove the objections to +the opinion of an internal change of the figure of the lens. +I shall now state two experiments, which, in the first place, come +very near to a mathematical demonstration of the existence of +such a change, and, in the second, explain in great measure its +origin, and the manner in which it is effected.</p> + +<p>I have already described the appearances of the imperfect +image of a minute point at different distances from the eye, in +a state of relaxation. For the present purpose, I will only +repeat, that if the point is beyond the furthest focal distance of +the eye, it assumes that appearance which is generally described +by the name of a star, the central part being considerably the +brightest. (Plate VI. F<a href="#Pl.VI">ig. 36—39</a>.) But, when the focal distance +of the eye is shortened, the imperfect image is of course +enlarged; and, besides this necessary consequence, the light is +also very differently distributed; the central part becomes faint, +and the margin strongly illuminated, so as to have almost +the appearance of an oval ring. (<a href="#Pl.VI">Fig. 41</a>.) If I apply the +slider of the optometer, the shadows of the slits, while the eye +is relaxed, are perfectly straight, dividing the oval either way +into parallel segments: (<a href="#Pl.VI">Fig. 42, 44</a>.) but, when the accommodation<span class="pagenum" id="Page_69">69</span> +takes place, they immediately become curved, and the +more so the further they are from the centre of the image, to +which their concavity is directed. (<a href="#Pl.VI">Fig. 43, 45</a>.) If the point +be brought much within the focal distance, the change of the +eye will increase the illumination of the centre, at the expense +of the margin. The same appearances are equally observable, +when the effect of the cornea is removed by immersion in water; +and the only imaginable way of accounting for the diversity, is +to suppose the central parts of the lens to acquire a greater +degree of curvature than the marginal parts. If the refraction +of the lens remained the same, it is absolutely impossible that +any change of the distance of the retina should produce a curvature +in those shadows, which, in the relaxed state of the eye, +are found to be in all parts straight; and, that neither the form +nor the relative situation of the cornea is concerned, appears +from the application of water already mentioned.</p> + +<p>The truth of this explanation is fully confirmed by the optometer. +When I look through four narrow slits, without exertion, +the lines always appear to meet in one point: but, when I +make the intersection approach me, the two outer lines meet +considerably beyond the inner ones, and the two lines of the +same side cross each other at a still greater distance. (Plate V. +<a href="#Pl.V">Fig. 24</a>.)</p> + +<p>The experiment will not succeed with every eye; nor can it +be expected that such an imperfection should be universal: but +one case is sufficient to establish the argument, even if no other +were found. I do not however doubt, that in those who have a +large pupil, the aberration may be very frequently observable. +In Dr. <span class="smcap">Wollaston</span>’s eye, the diversity of appearance is imperceptible; +but Mr. <span class="smcap">König</span> described the intersections exactly as +<span class="pagenum" id="Page_70">70</span>they appear to me, although he had received no hint of what I +had observed. The lateral refraction is the most easily ascertained, +by substituting for the slits a tapering piece of card, so +as to cover all the central parts of the pupil, and thus determining +the nearest crossing of the shadows transmitted through +the marginal parts only. When the furthest intersection was at +38, I could bring it to 22 parts with two narrow slits; but with +the tapered card only to 29. From these data we may determine +pretty nearly, into what form the lens must be changed, +supposing both the surfaces to undergo proportional alterations +of curvature, and taking for granted the dimensions already +laid down: for, from the lateral aberration thus given, we may +find (by Prop. III.) the subtangents at about one-tenth of an +inch from the axis; and the radius of curvature at each vertex, +is already determined to be about 21 and 15 hundredths of an +inch. Hence the anterior surface must be a portion of a hyperboloid, +of which the greater axis is about 50; and the posterior +surface will be nearly parabolical. In this manner the change +will be effected, without any diminution of the transverse diameter +of the lens. The elongation of its axis will not exceed +the fiftieth of an inch; and, on the supposition with which we +set out, the protrusion will be chiefly at the posterior vertex. +The form of the lens thus changed will be nearly that of Plate V. +<a href="#Pl.V">Fig. 26</a>; the relaxed state being nearly as represented in Fig. 25. +Should, however, the rigidity of the internal parts, or any other +considerations, render it convenient to suppose the anterior surface +more changed, it would still have room, without interfering +with the uvea; or it might even force the uvea a little forwards, +without any visible alteration of the external appearance of +the eye.</p> + +<p><span class="pagenum" id="Page_71">71</span></p> + +<p>From this investigation of the change of the figure of the +lens, it appears that the action which I formerly attributed to +the external coats, cannot afford an explanation of the phenomenon. +The necessary effect of such an action would be, to +produce a figure approaching to that of an oblate spheroid; +and, to say nothing of the inconvenience attending a diminution +of the diameter of the lens, the lateral refraction would +be much more increased than the central; nor would the +slight change of density, at an equal distance from the axis, +be at all equivalent to the increase of curvature: we must +therefore suppose some different mode of action in the power +producing the change. Now, whether we call the lens a +muscle or not, it seems demonstrable, that such a change of +figure takes place as can be produced by no external cause; +and we may at least illustrate it by a comparison with the +usual action of muscular fibres. A muscle never contracts, +without at the same time swelling laterally, and it is of no +consequence which of the effects we consider as primary. I +was induced, by an occasional opacity, to give the name of +membranous tendons to the radiations from the centre of the +lens; but, on a more accurate examination, nothing really analogous +to tendon can be discovered. And, if it were supposed +that the parts next the axis were throughout of a tendinous, and +therefore unchangeable nature, the contraction must be principally +effected by the lateral parts of the fibres; so that the coats +would become thicker towards the margin, by their contraction, +while the general alteration of form would require them to be +thinner; and there would be a contrariety in the actions of the +various parts. But, if we compare the central parts of each +surface to the belly of the muscle, there is no difficulty in +<span class="pagenum" id="Page_72">72</span>conceiving their thickness to be immediately increased, and to +produce an immediate elongation of the axis, and an increase +of the central curvature; while the lateral parts co-operate +more or less, according to their distance from the centre, and +in different individuals in somewhat different proportions. On +this supposition, we have no longer any difficulty in attributing +a power of change to the crystalline of fishes. M. <span class="smcap">Petit</span>, in a +great number of observations, uniformly found the lens of +fishes more or less flattened: but, even if it were not, a slight +extension of the lateral part of the superficial fibres would allow +those softer coats to become thicker at each vertex, and to form +the whole lens into a spheroid somewhat oblong; and here, the +lens being the only agent in refraction, a less alteration than in +other animals would be sufficient. It is also worthy of inquiry, +whether the state of contraction may not immediately add to +the refractive power. According to the old experiment, by +which Dr. <span class="smcap">Goddard</span> attempted to show that muscles become +more dense as they contract, such an effect might naturally +be expected. That experiment is, however, very indecisive, and +the opinion is indeed generally exploded, but perhaps too hastily; +and whoever shall ascertain the existence or non-existence +of such a condensation, will render essential service to physiology +in general.</p> + +<p>Dr. <span class="smcap">Pemberton</span>, in the year 1719, first systematically discussed +the opinion of the muscularity of the crystalline lens.⁠<a id="FNanchor_21_21" href="#Footnote_21_21" class="fnanchor">21</a> +He referred to <span class="smcap">Leeuwenhoek</span>’s microscopical observations; +but he so overwhelmed his subject with intricate calculations, +that few have attempted to develope it: and he grounded the +<span class="pagenum" id="Page_73">73</span>whole on an experiment borrowed from <span class="smcap">Barrow</span>, which with +me has totally failed; and I cannot but agree with Dr. <span class="smcap">Olbers</span> +in the remark, that it is easier to confute him than to understand +him. He argued for a partial change of the figure of the lens; +and perhaps the opinion was more just than the reasons adduced +for its support. <span class="smcap">Lobe′</span>, or rather <span class="smcap">Albinus</span>,⁠<a id="FNanchor_22_22" href="#Footnote_22_22" class="fnanchor">22</a> decidedly favours +a similar theory; and suggests the analogy of the lens to the +muscular parts of pellucid animals, in which even the best +microscopes can discover no fibres. <span class="smcap">Camper</span> also mentions +the hypothesis with considerable approbation.⁠<a id="FNanchor_23_23" href="#Footnote_23_23" class="fnanchor">23</a> Professor <span class="smcap">Reil</span> +published, in 1793, a Dissertation on the Structure of the Lens; +and, in a subsequent paper, annexed to the translation of my +former Essay in Professor <span class="smcap">Gren</span>’s Journal,⁠<a id="FNanchor_24_24" href="#Footnote_24_24" class="fnanchor">24</a> he discussed the +question of its muscularity. I regret that I have not now an +opportunity of referring to this publication; but I do not recollect +that Professor <span class="smcap">Reil</span>’s objections are different from those +which I have already noticed.</p> + +<p>Considering the sympathy of the crystalline lens with the +uvea, and the delicate nature of the change of its figure, there +is little reason to expect that any artificial stimulus would be +more successful in exciting a contractive action in the lens, than +it has hitherto been in the uvea; much less would that contraction +be visible without art. Soon after Mr. <span class="smcap">Hunter</span>’s death, I +pursued the experiment which he had suggested, for ascertaining +how far such a contraction might be observable. My apparatus +(Plate V. <a href="#Pl.V">Fig. 27</a>.) was executed by Mr. <span class="smcap">Jones</span>. It +consisted of a wooden vessel blacked within, which was to be +<span class="pagenum" id="Page_74">74</span>filled with cool, and then with warmer water: a plane speculum +was placed under it; a perforation in the bottom was filled with +a plate of glass; proper rings were fixed for the reception of +the lens, or of the whole eye, and also wires for transmitting +electricity: above these, a piece of ground and painted glass, +for receiving the image, was supported by a bracket, which +moved by a pivot, in connection with a scale divided into fiftieths +of an inch. With this apparatus I made some experiments, +assisted by Mr. <span class="smcap">Wilkinson</span>, whose residence was near +a slaughter-house: but we could obtain, by this method, no +satisfactory evidence of the change; nor was our expectation +much disappointed. I understand also, that another member +of this Society was equally unsuccessful, in attempting to produce +a conspicuous change in the lens by electricity.</p> + + +<p class="mt2em ti0">XI. In man and in the most common quadrupeds, the structure +of the lens is nearly similar. The number of radiations is of +little consequence; but I find that in the human crystalline there +are ten on each side, (Plate VI. <a href="#Pl.VI">Fig. 46</a>.) not three, as I once, +from a hasty observation, concluded.⁠<a id="FNanchor_25_25" href="#Footnote_25_25" class="fnanchor">25</a> Those who find any +difficulty in discovering the fibres, must have a sight very ill +adapted to microscopical researches, I have laboured with the +most obstinate perseverance to trace nerves into the lens, and +I have sometimes imagined that I had succeeded; but I cannot +positively go further than to state my full conviction of their +existence, and of the precipitancy of those who have absolutely +denied it. The long nerves, which are very conspicuous between +the choroid and sclerotic coats, divide each into two, +three, or more branches, at the spot where the ciliary zone +begins, and seem indeed to furnish the choroid with some fine +<span class="pagenum" id="Page_75">75</span>filaments at the same place. The branches often re-unite, with +a slight protuberance, that scarcely deserves the name of a +ganglion: here they are tied down, and mixed with the hard +whitish-brown membrane that covers the compact spongy substance, +in which the vessels of the ciliary processes anastomose +and subdivide. (Plate VI. <a href="#Pl.VI">Fig. 47</a>.) The quantity of the +nerves which proceeds to the iris, appears to be considerably +smaller than that which arrives at the place of division: hence +there can be little doubt that the division is calculated to supply +the lens with some minute branches; and it is not improbable, +from the appearance of the parts, that some fibres may pass to +the cornea; although it might more naturally be expected, that +the tunica conjunctiva would be supplied from without. But +the subdivisions which probably pass to the lens, enter immediately +into a mixture of ligamentous substance and of a tough +brownish membrane; and I have not hitherto been able to +develope them. Perhaps animals may be found in which this +substance is of a different nature; and I do not despair that, +with the assistance of injections, for more readily distinguishing +the blood vessels, it may still be possible to trace them in +quadrupeds. Our inability to discover them, is scarcely an +argument against their existence: they must naturally be delicate +and transparent; and we have an instance, in the cornea, +of considerable sensibility, where no nerve has yet been traced. +The capsule adheres to the ciliary substance, and the lens to +the capsule, principally in two or three points; but I confess, I +have not been able to observe that these points are exactly +opposite to the trunks of nerves; so that, probably, the adhesion +is chiefly caused by those vessels which are sometimes seen +passing to the capsule in injected eyes. We may, however, +<span class="pagenum" id="Page_76">76</span>discover ramifications from some of these points, upon and +within the substance of the lens, (Plate VI. <a href="#Pl.VI">Fig. 48</a>.) generally +following a direction near to that of the fibres, and sometimes +proceeding from a point opposite to one of the radiating lines of +the same surface. But the principal vessels of the lens appear +to be derived from the central artery, by two or three branches +at some little distance from the posterior vertex; which I +conceive to be the cause of the frequent adhesion of a portion +of a cataract to the capsule, about this point: they follow +nearly the course of the radiations, and then of the fibres; +but there is often a superficial subdivision of one of the radii, +at the spot where one of them enters. The vessels coming +from the choroid appear principally to supply a substance, +hitherto unobserved, which fills up the marginal part of the +capsule of the crystalline, in the form of a thin zone, and +makes a slight elevation, visible even through the capsule. +(<a href="#Pl.VI">Fig. 49—51</a>.) It consists of coarser fibres than the lens, but +in a direction nearly similar; they are often intermixed with +small globules. In some animals, the margin of the zone is +crenated, especially behind, where it is shorter: this is observable +in the partridge; and, in the same bird, the whole surface +of the lens is seen to be covered with points, or rather +globules, arranged in regular lines, (Plate VII. <a href="#Pl.VII">Fig. 52</a>.) so as +to have somewhat the appearance of a honeycomb, but towards +the vertex less uniformly disposed. This regularity is a sufficient +proof that there could be no optical deception in the appearance; +although it requires a good microscope to discover it distinctly: +but the zone may be easily peeled off under water, +and hardened in spirits. Its use is uncertain; but it may possibly +secrete the liquid of the crystalline; and it as much deserves the +<span class="pagenum" id="Page_77">77</span>name of a gland, as the greater part of the substances usually +so denominated. In peeling it off, I have very distinctly observed +ramifications, which were passing through it into the lens; +(Plate VI. <a href="#Pl.VI">Fig. 50</a>.) and indeed it is not at all difficult to +detect the vessels connecting the margin of the lens with its +capsule; and it is surprising that M. <span class="smcap">Petit</span> should have +doubted of their existence. I have not yet clearly discerned +this crystalline gland in the human eye; but I infer the existence +of something similar to the globules, from the spotted appearance +of the image of a lucid point already mentioned; for which +I can no otherwise account, than by attributing it to a derangement +of these particles, produced by the external force, and +to an unequal impression made by them on the surface of +the lens.</p> + +<p>In birds and in fishes, the fibres of the crystalline radiate +equally, becoming finer as they approach the vertex, till they +are lost in a uniform substance, of the same degree of firmness, +which appears to be perforated in the centre by a blood vessel. +(Plate VII. <a href="#Pl.VII">Fig. 53</a>.) In quadrupeds, the fibres at their angular +meeting are certainly not continued, as <span class="smcap">Leeuwenhoek</span> imagined, +across the line of division; but there does not appear to be any +dissimilar substance interposed between them, except that very +minute trunks of vessels often mark that line. But, since the +whole mass of the lens, as far as it is moveable, is probably +endued with a power of changing its figure, there is no need +of any strength of union, or place of attachment, for the fibres, +since the motion meets with little or no resistance. Every +common muscle, as soon as its contraction ceases, returns to +its natural form, even without the assistance of an antagonist; +and the lens itself, when taken out of the eye, in its capsule. +<span class="pagenum" id="Page_78">78</span>has elasticity enough to reassume its proper figure, on the +removal of a force that has compressed it. The capsule is +highly elastic; and, since it is laterally fixed to the ciliary zone, +it must co-operate in restoring the lens to its flattest form. If +it be inquired, why the lens is not capable of becoming less +convex, as well as more so, it may be answered, that the lateral +parts have probably little contractive power; and, if they had +more, they would have no room to increase the size of the disc, +which they must do, in order to shorten the axis; and the parts +about the axis have no fibres so arranged as to shorten it by +their own contraction.</p> + +<p>I consider myself as being partly repaid for the labour lost in +search of the nerves of the lens, by having acquired a more +accurate conception of the nature and situation of the ciliary +substance. It had already been observed, that in the hare and +in the wolf, the ciliary processes are not attached to the capsule +of the lens; and if by the ciliary processes we understand +those filaments which are seen detached after tearing +away the capsule, and consist of ramifying vessels, the observation +is equally true of the common quadrupeds, and I will +venture to say, of the human eye.⁠<a id="FNanchor_26_26" href="#Footnote_26_26" class="fnanchor">26</a> Perhaps this remark +has been made by others, but the circumstance is not generally +understood. It is so difficult to obtain a distinct view of +these bodies, undisturbed, that I am partly indebted to accident, +for having been undeceived respecting them: but, having once +made the observation, I have learnt to show it in an unquestionable +manner. I remove the posterior hemisphere of the +sclerotica, or somewhat more, and also as much as possible of +the vitreous humour, introduce the point of a pair of scissors +<span class="pagenum" id="Page_79">79</span>into the capsule, turn out the lens, and cut off the greater +part of the posterior portion of the capsule, and of the rest +of the vitreous humour. I next dissect the choroid and uvea +from the sclerotica; and, dividing the anterior part of the capsule +into segments from its centre, I turn them back upon the +ciliary zone. The ciliary processes then appear, covered with +their pigment, and perfectly distinct both from the capsule and +from the uvea; (Plate VII. <a href="#Pl.VII">Fig. 54</a>.) and the surface of the +capsule is seen shining, and evidently natural, close to the base +of these substances. I do not deny that the separation between +the uvea and the processes, extends somewhat further back +than the separation between the processes and the capsule; but +the difference is inconsiderable, and, in the calf, does not amount +to above half the length of the detached part. The appearance +of the processes is wholly irreconcileable with muscularity; and +their being considered as muscles attached to the capsule, is +therefore doubly inadmissible. Their lateral union with the +capsule, commences at the base of their posterior smooth surface, +and is continued nearly to the point where they are more +intimately united with the termination of the uvea; so that, +however this portion of the base of the processes were disposed +to contract, it would be much too short to produce any sensible +effect. What their use may be, cannot easily be determined: +if it were necessary to have any peculiar organs for secretion, +we might call them glands, for the percolation of the aqueous +humour; but there is no reason to think them requisite for this +purpose.</p> + +<p>The marsupium nigrum of birds, and the horse-shoe-like +appearance of the choroid of fishes, are two substances which +have sometimes, with equal injustice, been termed muscular. +All the apparent fibres of the marsupium nigrum are, as +<span class="pagenum" id="Page_80">80</span><span class="smcap">Haller</span> had very truly asserted, merely duplicatures of a +membrane, which, when its ends are cut off, may easily be +unfolded under the microscope, with the assistance of a fine hair +pencil, so as to leave no longer any suspicion of a muscular +texture. The experiment related by Mr. <span class="smcap">Home</span>,⁠<a id="FNanchor_27_27" href="#Footnote_27_27" class="fnanchor">27</a> can scarcely +be deemed a very strong argument for attributing to this substance +a faculty which its appearance so little authorises us to +expect in it. The red substance in the choroid of fishes, +(Plate VII. <a href="#Pl.VII">Fig. 55</a>.) is more capable of deceiving the observer; +its colour gives it some little pretension, and I began to examine +it with a prepossession in favour of its muscular nature. But, +when we recollect the general colour of the muscles of fishes, +the consideration of its redness will no longer have any +weight. Stripped of the membrane which loosely covers its +internal surface, (<a href="#Pl.VII">Fig. 56</a>.) it seems to have transverse divisions, +somewhat resembling those of muscles, and to terminate +in a manner somewhat similar; (<a href="#Pl.VII">Fig. 57</a>.) but, when +viewed in a microscope, the transverse divisions appear to be +cracks, and the whole mass is evidently of a uniform texture, +without the least fibrous appearance; and, if a particle of any +kind of muscle is compared with it, the contrast becomes very +striking. Besides, it is fixed down, throughout its extent, to +the posterior lamina of the choroid, and has no attachment +capable of directing its effect; to say nothing of the difficulty +of conceiving what that effect could be. Its use must remain, +in common with that of many other parts of the animal frame, +entirely concealed from our curiosity.</p> + +<p>The bony scales of the eyes of birds, which were long ago +described in the Philosophical Transactions by Mr. <span class="smcap">Ranby</span>,⁠<a id="FNanchor_28_28" href="#Footnote_28_28" class="fnanchor">28</a> +<span class="pagenum" id="Page_81">81</span>and by Mr. <span class="smcap">Warren</span>⁠<a id="FNanchor_29_29" href="#Footnote_29_29" class="fnanchor">29</a>⁠, + afterwards in two excellent Memoirs of +M. <span class="smcap">Petit</span> on the eye of the turkey and of the owl,⁠<a id="FNanchor_30_30" href="#Footnote_30_30" class="fnanchor">30</a> and lately +by Mr. <span class="smcap">Pierce Smith</span>,⁠<a id="FNanchor_31_31" href="#Footnote_31_31" class="fnanchor">31</a> + and Mr, <span class="smcap">Home</span>,⁠<a id="FNanchor_32_32" href="#Footnote_32_32" class="fnanchor">32</a> can, on any supposition, +have but little concern in the accommodation of the eye +to different distances: they rather seem to be necessary for the +protection of that organ, large and prominent as it is, and unsupported +by any strength in the orbit, against the various accidents +to which the mode of life and rapid motion of those animals +must expose it; and they are much less liable to fracture +than an entire bony ring of the same thickness would have been. +The marsupium nigrum appears to be intended to assist in +giving strength to the eye, to prevent any change in the +place of the lens by external force: it is so situated as to intercept +but little light, and that little is principally what would +have fallen on the insertion of the optic nerve; and it seems to +be too firmly tied to the lens, even to admit any considerable +elongation of the axis of the eye, although it certainly would +not impede a protrusion of the cornea.</p> + +<p>With respect to the eyes of insects, an observation of <span class="smcap">Poupart</span> +deserves to be repeated here. He remarks, that the eye +of the libellula is hollow; that it communicates with an air-vessel placed longitudinally in the trunk of the body; and that +it is capable of being inflated from this cavity: he supposes that +the insect is provided with this apparatus, in order for the +accommodation of its eye to the perception of objects at different +distances.⁠<a id="FNanchor_33_33" href="#Footnote_33_33" class="fnanchor">33</a> I have not yet had an opportunity of examining +<span class="pagenum" id="Page_82">82</span>the eye of the libellula; but there is no difficulty in supposing +that the means of producing the change of the refractive powers +of the eye, may be, in different classes of animals, as diversified +as their habits, and the general conformation of their +organs.</p> + +<p>I beg leave to correct here an observation in my former paper, +relative to the faint lateral radiations, which I supposed to proceed +from the margin of the iris.⁠<a id="FNanchor_34_34" href="#Footnote_34_34" class="fnanchor">34</a> I find, on further examination, +that they are occasioned by reflections from the eyelashes.</p> + + +<p class="mt2em ti0">XII. I shall now finally recapitulate the principal objects and +results of the investigation which I have taken the liberty of +detailing so fully to the Royal Society. First, the determination +of the refractive power of a variable medium, and its application +to the constitution of the crystalline lens. Secondly, the construction +of an instrument for ascertaining, upon inspection, the +exact focal distance of every eye, and the remedy for its imperfections. +Thirdly, to show the accurate adjustment of every +part of the eye, for seeing with distinctness the greatest possible +extent of objects at the same instant. Fourthly, to measure +the collective dispersion of coloured rays in the eye. Fifthly, +by immerging the eye in water, to demonstrate that its accommodation +does not depend on any change in the curvature of +the cornea. Sixthly, by confining the eye at the extremities +of its axis, to prove that no material alteration of its length can +take place. Seventhly, to examine what inference can be drawn +from the experiments hitherto made on persons deprived of the +lens; to pursue the inquiry, on the principles suggested by Dr. +<span class="smcap">Porterfield</span>; and to confirm his opinion of the utter inability +<span class="pagenum" id="Page_83">83</span>of such persons to change the refractive state of the organ. +Eighthly, to deduce, from the aberration of the lateral rays, +a decisive argument in favour of a change in the figure of the +crystalline; to ascertain, from the quantity of this aberration, +the form into which the lens appears to be thrown in my own +eye, and the mode by which the change must be produced in +that of every other person. And I flatter myself, that I shall +not be deemed too precipitate, in denominating this series of +experiments satisfactorily demonstrative.</p> + + +<hr class="chap x-ebookmaker-drop"> +<div class="chapter"> + <h2 class="nobreak" id="CORRECTIONS"> + CORRECTIONS. + </h2> +</div> + + + +<p><a href="#Page_28">Page 28</a>, line ii. Prop. III. <i>after e, insert</i> the base being unity.</p> + +<p><a href="#Page_30">Page 30</a>, line 8, Cor. 10. <i>for n t u, read n t t</i>; line 9, <i>for</i> product &c., <i>read</i> square +of the cosine of incidence.</p> + +<p><a href="#Page_31">Page 31</a>, line 5, Cor. 11. <i>for</i> <img style="vertical-align: -0.186ex; width: 12.357ex; height: 2.09ex;" src="images/58.svg" alt=" " data-tex="\(1+u^{2}-2u^{4}\)">, <i>read</i> <img style="vertical-align: -0.025ex; width: 4.412ex; height: 1.532ex;" src="images/59.svg" alt=" " data-tex="\(2 m u\)">.</p> + +<p><a href="#Page_31">Page 31</a>. Prop. V. Cor. See the note in p. 60.</p> + +<p><a href="#Page_33">Page 33</a>. Prop. VIII. By a mistake of a sign, the eighth proposition is rendered +erroneous; no use having been made of that proposition, it has been inserted without +proper revision. It ought to stand thus, with its demonstration:</p> + +<p><span class="smcap">Proposition</span> VIII. <span class="smcap">Problem</span>.</p> + +<p>To find the path of a ray of light falling obliquely on a sphere, of a refractive +density varying as any power of the distance from the centre.</p> + +<p>The refractive density, in the sense of these propositions, varies as the ratio of the +sines, and as the velocity of light in the medium. (Schol. 2. Prop. I.) Let the velocity +at the distance <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/60.svg" alt=" " data-tex="\(x\)"> be <img style="vertical-align: -0.025ex; width: 4.853ex; height: 4.888ex;" src="images/61.svg" alt=" " data-tex="\(x^{-\dfrac{1}{r}}\)">; then, considering the refractive force as a species of +attraction, we have, in Prop. 41. l. 1. Princip. <img style="vertical-align: -0.186ex; width: 16.305ex; height: 5.049ex;" src="images/62.svg" alt=" " data-tex="\(\sqrt{\text{ABFD}}=x^{-\dfrac{1}{r}}\)">, <img style="vertical-align: -0.439ex; width: 5.868ex; height: 2.032ex;" src="images/63.svg" alt=" " data-tex="\(Q=s\)">, the sine +of incidence, the radius being unity, <img style="vertical-align: -0.186ex; width: 8.987ex; height: 2.072ex;" src="images/64.svg" alt=" " data-tex="\(\text{Z}=s x^{-1}\)">, +<img style="vertical-align: -5.475ex; width: 25.942ex; height: 8.005ex;" src="images/65.svg" alt=" " data-tex="\(\text{D}c=\dfrac{s}{2xx\sqrt{x^{-\dfrac{2}{r}}-s^{2}x^{-2}}}\)"> +<img style="vertical-align: -1.552ex; width: 11.23ex; height: 6.415ex;" src="images/66.svg" alt=" " data-tex="\(=\dfrac{1}{2}sx^{\dfrac{1}{r}-2}\)">·<img style="vertical-align: -3.732ex; width: 19.117ex; height: 11.014ex;" src="images/67.svg" alt=" " data-tex="\(\left(1-s^{2}x^{\dfrac{2}{r}-2}\right)^{-\dfrac{1}{2}}\)">, +and the fluxion of the area described by the radius +<img style="vertical-align: -1.552ex; width: 14.347ex; height: 6.415ex;" src="images/68.svg" alt=" " data-tex="\(=-\dfrac{1}{2}s x^{\dfrac{1}{r}-2}ẋ\)">·<img style="vertical-align: -3.732ex; width: 19.117ex; height: 11.014ex;" src="images/69.svg" alt=" " data-tex="\(\left( 1-s^{2}x^{\dfrac{2}{r}-2} \right)^{-\dfrac{1}{2}}\)">. +Let the sine of the inclination to the radius +<span class="pagenum" id="Page_84">84</span>at each point be called <i>y</i>; then <img style="vertical-align: -0.464ex; width: 10.84ex; height: 5.327ex;" src="images/70.svg" alt=" " data-tex="\(y = s x^{\dfrac{1}{r}-1}\)">, +<img style="vertical-align: -1.577ex; width: 18.359ex; height: 6.44ex;" src="images/71.svg" alt=" " data-tex="\(ẏ = \dfrac{1-r}{r}s x^{\dfrac{1}{r}-2}ẋ\)">, and the fluxion +of the area = <img style="vertical-align: -1.738ex; width: 9.03ex; height: 4.267ex;" src="images/72.svg" alt=" " data-tex="\(\dfrac{r}{2r-2}ẏ\cdot\)"><img style="vertical-align: -0.566ex; width: 11.433ex; height: 5.404ex;" src="images/73.svg" alt=" " data-tex="\(\left( 1-yy \right)^{-\dfrac{1}{2}}\)">, of which the fluent is +<img style="vertical-align: -1.738ex; width: 8.741ex; height: 4.267ex;" src="images/74.svg" alt=" " data-tex="\(\dfrac{r}{2r-2}\text{Y}\)">, y being the sine of the arc Y; and the angle corresponding is +<img style="vertical-align: -1.738ex; width: 7.61ex; height: 4.267ex;" src="images/75.svg" alt=" " data-tex="\(\dfrac{r}{r-1}\text{Y}\)">. The value of that angle being +found for any two values of <i>x</i> or <i>y</i>, the difference is the intervening angle described +by the radius. This angle is therefore always to the difference of the inclinations as +<img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/76.svg" alt=" " data-tex="\(r\)"> to <img style="vertical-align: -0.186ex; width: 4.917ex; height: 1.692ex;" src="images/77.svg" alt=" " data-tex="\(r-1\)">, and the deviation is to that difference as <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/78.svg" alt=" " data-tex="\(1\)"> to <img style="vertical-align: -0.186ex; width: 4.917ex; height: 1.692ex;" src="images/77.svg" alt=" " data-tex="\(r-1\)">.</p> +<p>Corollary. Hence, in the passage to the apsis, and the return to the surface, the +deviation is always proportionate to the arc cut off by the incident ray produced: +therefore such a sphere could never collect parallel rays to any focus, the lateral density +being too small towards the surface.</p> + +<p><a href="#Page_33">Page 33</a>, line 20, <i>for</i> but the two last &c. <i>read</i> the seventh may either be deduced +from the eighth, or may be demonstrated independently of it.</p> + +<p><a href="#Page_42">Page 42</a>, line 18, <i>after</i> internally, <i>insert</i> Or, if a lens of equal mean dimensions, +and equal focal length, with the crystalline, be supposed to consist of two +segments of the external portion of such a sphere, the refractive density at the centre +of this lens must be as 18 to 17.</p> + +<p><a href="#Page_47">Page 47</a>, line 12, <i>for</i> calculated &c. <i>read</i> estimated by means of the eighth +proposition; and probably.</p> + +<p><a href="#Page_53">Page 53</a>, line 24, <i>for</i> 24, <i>read</i> 21; line 25, <i>for</i> 17, <i>read</i> 15.</p> + +<p><a href="#Page_61">Page 61</a>, line 21, <i>for</i> sixtieth, <i>read</i> fortieth..</p> + +<p><span class="pagenum"><a id="Page_85"></a>85</span></p> + + +<hr class="chap x-ebookmaker-drop"> +<div class="chapter"> + <h2 class="nobreak" id="EXPLANATION_OF_THE_FIGURES"> + EXPLANATION OF THE FIGURES. + </h2> +</div> + +<div class="explain"> +<figure class="figcenter illowe39_3750" id="Pl.II"> + <figcaption> + <p>Plate II</p> + </figcaption> + <img class="w100" src="images/072.jpg" alt=""> +</figure> + +<p>Plate II. Fig. 1. See <a href="#Page_28">Page 28</a>. Prop. III.</p> + +<p>Fig. 2. See Page 28. Prop. IV.</p> + +<p>Fig. 3. See <a href="#Page_31">Page 31</a>. Prop. V.</p> + +<p>Fig. 4—6. Relating to the optometer. See <a href="#Page_34">Page 34</a>.</p> + +<figure class="figcenter illowe39_3750" id="Pl.III"> + <figcaption> + <p>Plate III</p> + </figcaption> + <img class="w100" src="images/076.jpg" alt=""> +</figure> + +<p> Plate III. Fig. 7. The form of the ends of the optometer, +when made of card. The apertures in the shoulders are for +holding a lens: the square ends turn under, and are fastened +together.</p> + +<p>Fig. 8. The scale of the optometer. The middle line is +divided, from the lower end, into inches. The next column +shows the number of a concave lens requisite for a short- +sighted eye; by looking through the slider and observing the +number opposite to which the intersection appears when most +remote. By observing the place of apparent intersection when +nearest, the number requisite will be found in the other column, +provided that the eye have the average power of accommodation. +At the other end, the middle line is graduated for extending +the scale of inches by means of a lens four inches in +focus; the negative numbers implying that such rays as proceed +from them are made to converge towards a point on the +other side of the lens. The other column shows the focal length +of convex glasses required by those eyes to which the intersection +appears, when nearest, opposite the respective places of +the numbers.</p> + +<p>Fig. 9. A side view of the optometer, half its size.</p> + +<p>Fig. 10. The appearance of the lines through the slider.</p> + +<p>Fig. 11. Method of measuring the magnitude of an image +on the retina. See <a href="#Page_48">Page 48</a>.</p> + +<p><span class="pagenum" id="Page_86">86</span></p> + +<p>Fig. 12. Diagonal scale drawn on a looking-glass.</p> + +<p>Fig. 13. The method of applying a lens with water to the +cornea.</p> + +<p>Fig. 14. The appearance of a spectrum occasioned by pressure; +and the inflection of straight lines seen within the limits +of the spectrum.</p> + +<p>Fig. 15. An illustration of the enlargement of the image, +which would be the consequence of an elongation of the eye: +the images of the candles which, in one instance, fall on +the insertion of the nerve, falling, in the other instance, beyond +it.</p> + +<figure class="figcenter illowe39_3750" id="Pl.IV"> + <figcaption> + <p>Plate IV</p> + </figcaption> + <img class="w100" src="images/080.jpg" alt=""> +</figure> + +<p>Plate IV. Fig. 16. The successive forms of the image of a +large distant object, as it would be delineated by each refractive +surface in the eye; to show how that form at last coincides with +the retina. E G is the distance between the foci of horizontal +and vertical rays in my eye.</p> + +<figure class="figcenter illowe39_3750" id="Pl.V"> + <figcaption> + <p>Plate V</p> + </figcaption> + <img class="w100" src="images/084.jpg" alt=""> +</figure> + +<p>Plate V. Fig. 17. Vertical section of my right eye, seen from +without; twice the natural size.</p> + +<p>Fig. 18. Horizontal section, seen from above.</p> + +<p>Fig. 19. Front view of my left eye when the pupil is contracted; +of the natural size.</p> + +<p>Fig. 20. The same view when the pupil is dilated.</p> + +<p>Fig. 21. Outline of the eye and its straight muscles when +at rest.</p> + +<p>Fig. 22. Change of figure which would be the consequence +of the action of those muscles upon the eye, and upon the +adipose substance behind it.</p> + +<p>Fig. 23. Scale of the small optometer.</p> + +<p>Fig. 24. Appearance of four images of a line seen by my eye +when its focus is shortest.</p> + +<p><span class="pagenum" id="Page_87">87</span></p> + +<p>Fig. 25. Outline of the lens when relaxed; from a comparison +of M. <span class="smcap">Petit</span>’s measures with the phenomena of my own +eye, and on the supposition that it is found in a relaxed state +after death.</p> + +<p>Fig. 26. Outline of the lens sufficiently changed to produce +the shortest focal distance.</p> + +<p>Fig. 27. Apparatus for ascertaining the focal length of the +lens in water.</p> + +<figure class="figcenter illowe39_3750" id="Pl.VI"> + <figcaption> + <p>Plate VI</p> + </figcaption> + <img class="w100" src="images/088.jpg" alt=""> +</figure> + +<p>Plate VI. Fig. 28. Various forms of the image depicted by a +cylindrical pencil of rays obliquely refracted by a spherical surface, +when received on planes at distances progressively greater.</p> + +<p>Fig. 29. Image of a minute lucid object held very near to +my eye.</p> + +<p>Fig. 30. The same appearance when the eye has been +rubbed.</p> + +<p>Fig. 31—37. Different forms of the image of a lucid point +at greater and greater distances; the most perfect focus being +like Fig. 33, but much smaller.</p> + +<p>Fig. 38. Image of a very remote point seen by my right eye.</p> + +<p>Fig. 39. Image of a remote point seen by my left eye; being +more obtuse at one end, probably from a less obliquity of the +posterior surface of the crystalline lens.</p> + +<p>Fig. 40. Combination of two figures similar to the fifth +variety of Fig. 28; to imitate Fig. 38.</p> + +<p>Fig. 41. Appearance of a distant lucid point when the eye is +adapted to a very near object.</p> + +<p>Fig. 42, 44. Shadow of parallel wires in the image of a +distant point, when the eye is relaxed.</p> + +<p>Fig. 43, 45. The same shadows rendered curved by a +change in the figure of the crystalline lens.</p> + +<p><span class="pagenum" id="Page_88">88</span></p> + +<p>Fig. 46. The order of the fibres of the human crystalline.</p> + +<p>Fig. 47. The division of the nerves at the ciliary zone; the +sclerotica being removed. One of the nerves of the uvea is +seen passing forwards and subdividing. From the calf.</p> + +<p>Fig. 48. Ramifications from the margin of the crystalline +lens.</p> + +<p>Fig. 49. The zone of the crystalline faintly seen through the +capsule.</p> + +<p>Fig. 50. The zone raised from its situation, with the ramifications +passing through it into the lens.</p> + +<p>Fig. 51. The zone of the crystalline detached.</p> + +<figure class="figcenter illowe36_2500" id="Pl.VII"> + <figcaption> + <p>Plate VII</p> + </figcaption> + <img class="w100" src="images/092.jpg" alt=""> +</figure> + +<p>Plate VII. Fig. 52. The crenated zone, and the globules +regularly arranged on the crystalline of the partridge.</p> + +<p>Fig. 53. The order of the fibres in the lens of birds and +fishes.</p> + +<p>Fig. 54. The segments of the capsule of the crystalline +turned back, to show the detached ciliary processes. From +the calf.</p> + +<p>Fig. 55. Part of the choroid of the cod-fish, with its red +substance. The central artery hangs loose from the insertion +of the nerve.</p> + +<p>Fig. 56. The membrane covering this substance internally, +raised by the blow-pipe.</p> + +<p>Fig. 57. The appearance of the red substance, after the +removal of the membrane.</p> +</div> + +<h3>FOOTNOTES:</h3> +<div class="footnotes"> + +<div class="footnote"><p><a id="Footnote_1_1" href="#FNanchor_1_1" class="label">1</a> +Phil. Trans. for 1793, p. 169.</p></div> + +<div class="footnote"><p><a id="Footnote_2_2" href="#FNanchor_2_2" class="label">2</a> +Phil. Trans. for 1794, p. 21.</p></div> + +<div class="footnote"><p><a id="Footnote_3_3" href="#FNanchor_3_3" class="label">3</a> +Phil. Trans. for 1795, p. 1.</p></div> + +<div class="footnote"><p><a id="Footnote_4_4" href="#FNanchor_4_4" class="label">4</a> +De Corporis humani Viribus conservatricibus, p. 68.</p></div> + +<div class="footnote"><p><a id="Footnote_5_5" href="#FNanchor_5_5" class="label">5</a> +Phil. Trans. for 1800, p. 146.</p></div> + +<div class="footnote"><p><a id="Footnote_6_6" href="#FNanchor_6_6" class="label">6</a> +Edinb. Med. Essays, Vol. IV. p. 124.</p></div> + +<div class="footnote"><p><a id="Footnote_7_7" href="#FNanchor_7_7" class="label">7</a> +Edinb. Med. Ess. Vol. IV. p. 185.</p></div> + +<div class="footnote"><p><a id="Footnote_8_8" href="#FNanchor_8_8" class="label">8</a> +Mem. de I’Acad. de Paris, 1730. p. 6. Ed. Amst.</p></div> + +<div class="footnote"><p><a id="Footnote_9_9" href="#FNanchor_9_9" class="label">9</a> +Phil. Trans, for 1793. p. 174.</p></div> + +<div class="footnote"><p><a id="Footnote_10_10" href="#FNanchor_10_10" class="label">10</a> +Comm. Petrop. I. p. 314.</p></div> + +<div class="footnote"><p><a id="Footnote_11_11" href="#FNanchor_11_11" class="label">11</a> +<span class="smcap">Petit</span> Mem. del’Acad. 1725, p. 20.</p></div> + +<div class="footnote"><p><a id="Footnote_12_12" href="#FNanchor_12_12" class="label">12</a> +Phil.Trans. for 1795, p. 2.</p></div> + +<div class="footnote"><p><a id="Footnote_13_13" href="#FNanchor_13_13" class="label">13</a> +<span class="smcap">Smith</span>, e. 96.</p></div> + +<div class="footnote"><p><a id="Footnote_14_14" href="#FNanchor_14_14" class="label">14</a> +Phil. Trans, for 1796, p. 2.</p></div> + +<div class="footnote"><p><a id="Footnote_15_15" href="#FNanchor_15_15" class="label">15</a> +Phil. Trans, for 1795, p. 13.</p></div> + +<div class="footnote"><p><a id="Footnote_16_16" href="#FNanchor_16_16" class="label">16</a> +De Oculi Mutationibus internis. Gotting. 1780. 4°.</p></div> + +<div class="footnote"><p><a id="Footnote_17_17" href="#FNanchor_17_17" class="label">17</a> +This Corollary should stand thus. “If a confused image be received on any +given plane, it will be necessary, in order to determine its magnitude, to advert to the +aperture admitting the rays. If the aperture be supposed to be infinitely small, it may +be considered as a radiant point, in order to find the direction of the emergent rays.”</p></div> + +<div class="footnote"><p><a id="Footnote_18_18" href="#FNanchor_18_18" class="label">18</a> +Phil. Trans, for 1794. p. 212.</p></div> + +<div class="footnote"><p><a id="Footnote_19_19" href="#FNanchor_19_19" class="label">19</a> +Phil. Trans, for 1795. p. 9.</p></div> + +<div class="footnote"><p><a id="Footnote_20_20" href="#FNanchor_20_20" class="label">20</a> +Phil. Trans, for 1795. p. 8.</p></div> + +<div class="footnote"><p><a id="Footnote_21_21" href="#FNanchor_21_21" class="label">21</a> +De Facultate Oculi qua ad diversas Rerum distantias se accommodat. L. B. 1719. +Ap. Hall. Disp. Anat. IV. p. 301.</p></div> + +<div class="footnote"><p><a id="Footnote_22_22" href="#FNanchor_22_22" class="label">22</a> +De quibusdam Oculi Partibus, L. B. 1746. Ap. Hall. Disp. Anat. IV. p. 301.</p></div> + +<div class="footnote"><p><a id="Footnote_23_23" href="#FNanchor_23_23" class="label">23</a> +De Oculo Humano. L. B. 1742. Ap. Hall. Disp. Anat. VII. 2. p. 108, 109.</p></div> + +<div class="footnote"><p><a id="Footnote_24_24" href="#FNanchor_24_24" class="label">24</a> +1794. p. 352, 354.</p></div> + +<div class="footnote"><p><a id="Footnote_25_25" href="#FNanchor_25_25" class="label">25</a> +De Corp. Hum. Vir. Cons, p. 68.</p></div> + +<div class="footnote"><p><a id="Footnote_26_26" href="#FNanchor_26_26" class="label">26</a> +Vid. Hall. Physiol. V. p. 432. et <span class="smcap">Duverney</span>, ibi citat.</p></div> + +<div class="footnote"><p><a id="Footnote_27_27" href="#FNanchor_27_27" class="label">27</a> +Phil. Trans, for 1796. p. 18.</p></div> + +<div class="footnote"><p><a id="Footnote_28_28" href="#FNanchor_28_28" class="label">28</a> +Phil. Trans. Vol. XXXIII. p. 223. Abr. Vol. VII. p. 435.</p></div> + +<div class="footnote"><p><a id="Footnote_29_29" href="#FNanchor_29_29" class="label">29</a> +Phil. Trans. Vol. XXXIV. p. 113. Abr. Vol. VII. p. 437.</p></div> + +<div class="footnote"><p><a id="Footnote_30_30" href="#FNanchor_30_30" class="label">30</a> +Mem. del’Acad. 1735. p. 163. 1736, p. 166. Ed. Amst.</p></div> + +<div class="footnote"><p><a id="Footnote_31_31" href="#FNanchor_31_31" class="label">31</a> +Phil. Trans. for 1795. p. 263.</p></div> + +<div class="footnote"><p><a id="Footnote_32_32" href="#FNanchor_32_32" class="label">32</a> +Phil. Trans. for 1796. p. 14.</p></div> + +<div class="footnote"><p><a id="Footnote_33_33" href="#FNanchor_33_33" class="label">33</a> +Phil. Trans. Vol. XXII. p. 673. Abr. II. p. 762.</p></div> + +<div class="footnote"><p><a id="Footnote_34_34" href="#FNanchor_34_34" class="label">34</a> +Phil. Trans. for 1793. p. 178.</p></div> +</div> +<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 79069 ***</div> +</body> +</html> diff --git a/79069-h/images/072.jpg b/79069-h/images/072.jpg Binary files differnew file mode 100644 index 0000000..c876f94 --- /dev/null +++ b/79069-h/images/072.jpg diff --git a/79069-h/images/076.jpg b/79069-h/images/076.jpg Binary files differnew file mode 100644 index 0000000..a250ee5 --- /dev/null +++ b/79069-h/images/076.jpg diff --git a/79069-h/images/080.jpg b/79069-h/images/080.jpg Binary files differnew file mode 100644 index 0000000..6dc0f98 --- /dev/null +++ b/79069-h/images/080.jpg diff --git a/79069-h/images/084.jpg b/79069-h/images/084.jpg Binary files differnew file mode 100644 index 0000000..5787418 --- /dev/null +++ b/79069-h/images/084.jpg diff --git a/79069-h/images/088.jpg b/79069-h/images/088.jpg Binary files differnew file mode 100644 index 0000000..54af9ed --- /dev/null +++ b/79069-h/images/088.jpg diff --git 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