The Human Eye.

Fig. 18.—Chromatic Aberration of Lens.

In the early part of this century the optical correction of chromatic aberration was partially brought about by combining a convex lens of crown-glass with a concave lens of flint-glass, in the proportion of which these two kinds of glass respectively refract and disperse rays of light; so that the one medium may by equal and contrary dispersion neutralise the dispersion caused by the other, without at the same time wholly neutralising its refraction. It is a curious fact that the media found most available for the purpose should be a combination of crown and flint-glass, ofcrown-glasswhose index of refraction is1·519, and dispersive power 0·036, and offlint-glasswhose index of refraction is 1·589, and dispersive power 0·0393. The focal length of the convex crown-glass lens must be 41⁄3inches, and that of the concave flint-glass lens 72⁄3inches, and the combined focal length 10 inches. The diagram (Fig. 19) shows how rays of light are brought to a focus, nearly free from colour. The small amount of residual colour in such a combination is termed thesecondary spectrum; the violet ray F Y, crossing the axis of the lens at V, and going to the upper end P of the spectrum, the red ray F B going to the lower end T. But as the flint-glass lensl l, on the prism AaC, which receives the rays F V, F R, at the same points, is interposed, these rays will unite atf, and form a small circle of white light, the ray S F being now refracted without colour from its primitive direction S F Y into the direction Ff. In like manner, the corresponding ray S F′ will be refracted tof, and a white colourless image be the result.

Fig. 19.—Correction of Chromatic Aberration.

The achromatic aplanatic objective constructed on the optical formula enunciated, did not meet all the difficulties experienced by the skilled microscopist, in obtaining resolution of the finest test objects, and whereby the intrinsic value of the objective (in his estimation) must stand or fall. There were other disturbing residuary elements besides those of the secondary spectrum, and which at a later period were met by the practical skill of the optician, who applied the screw-collar, and by means of which the back lens of the objective is made to approach the front lens, thus more accurately shortening the distance between the eye-piece, where the image is eventually formed, and the back lens of the objective.

In this diagram L L is aconvexlens ofcrown-glass, andl laconcaveone offlint-glass. A convex lens will refract a ray of light (S) falling at F on it exactly in the same manner as the prism A B C, whose faces touch the two surfaces of the lens at the points where the ray enters, and quits. The ray S F, thus refracted by the lens L L, or prism A B C, would have formed a spectrum (P T) on a screen or wall, had there been no other lens.

Fig. 20.—Virtual Image formed by Convex Lens.

Formation of Virtual Images.—The normal eye possesses a considerable power of adjusting itself to form a distinct image of objects placed at varying distances; the nearer, within a certain limit, the larger it appears, and the more distinctly the details are brought out. When brought within a distance of two or three inches, the images become blurred or quite indistinct, and when brought closer to the eye, cannot be seen at all, and it simply obstructs the light. Now the utility of a convex lens, when interposed between the object and the eye, consists in reducing the divergence of the rays forming the several pencils which issue from it, and send images to the retina in a state of moderate divergence, that is, as if they had issued from an object beyond the nearest point of distinct vision, and so that a more clearly defined image may reach the sensitive membrane of the eye. But, not only is the course of the several rays in each pencil altered as regards the rest, but the course of the pencils themselves is changed, so that they enter the eye under an angle corresponding with that under which they would have arrived from a larger object situated at a greater distance, and thus the picture formed by any object corresponds in all respects with one which would have been madeby the same object increased in its dimensions and viewed at the smallest ordinary distance of distinct vision. For instance, let an object A B (Fig. 20) be placed between a convex lens and its principal focus. Then the foci conjugate to the points A B are virtual, and their positions can be found by construction from the consideration that rays through A, B, parallel to the principal axis, will be refracted to F, the principal focus on the other side. The refracted rays, if produced backwards, must meet the secondary axis O A, O B in the required points. An eye placed on the other side of the lens will accordingly see a virtual image erect, magnified, and at a greater distance from the lens than the object. This is the principle of the simple microscope.

To gain a clear insight into the mode in which a single lens serves to magnify objects, it will be necessary to revert to the phenomena of ordinary vision. An eye free from any defect has a considerable power of adjusting itself to very considerable distances. One of the special functions of the eye is bringing the rays of light, by a series of dioptric mechanisms, to a perfect focus on its nervous sensitive layer, the retina. The eye in this respect has been compared to a photographic camera. But this is not quite correct. The retina is destined simply to receive the images furnished by the dioptric apparatus, and has no influence upon the formation of these images. The luminous rays are refracted by the dioptric apparatus; the images would be formed quite as well—indeed, even better in certain cases—if the retina were not there. The dioptric apparatus and its action are absolutely independent of the retina.

The same laws with regard to the passage of the rays of light into the human eye hold good, as those already enunciated in the previous pages. As to change of direction when rays are passing obliquely from a medium of low density to that of a higher density,i.e., it changes its course, and is bent towards the perpendicular. On leaving the denser for the rarer medium it is bent once more from the perpendicular. Again, by means of a convex lens, the rays of light from one source will be refracted so as to meet at a point termedthe principal focusof vision.

In the eye there are several surfaces separating the different media where refraction takes place. The refractive index of the aqueous humour and the tears poured out by the lachrymal gland is almost equal to that of the cornea. We may, therefore, speak of the refracting surfaces as three, viz.: Anterior surface of cornea, anterior surface of lens, and posterior surface of lens; and also of the refracting media as three—the aqueous humour, the lens, and vitreous humour. These several bodies are so adapted in the normal eye that parallel rays falling on the cornea are converged to a focus at the most sensitive spot (theyellow spot, orfovea centralis) in the retina, a point representing to the principal focus of the eye. A line drawn from this point through the centre of the cornea is called the optic axis of the eye-ball.

Fig. 21.—Nerve and Stellate Cell Layer of Cornea,6stained by chloride of gold; magnified 300 diameters.a, Nerve cells.b, Stellate cells.

Fig. 22.—Anterior section of Eye, showing changed form of lens during the act of accommodation, a voluntary action in the eye. M, Ciliary muscle; I, Iris; L, Lens; V, Vitreous Humour; A, Aqueous Humour; C, Cornea and optic axis.

But as we are able to form a distinct image of near objects, and as we notice when we turn our gaze from far to near objects there is a distinct feeling of muscular effort in the eyes, there must be some means whereby the eye can readily adapt itself for focussing near and distant objects. In a photographic camera the focus can be readily altered, either by changing the lenses, employing a lens of greater or less curvature, or by altering the distance of the screen from the lens. The last method is obviously impossible in the rigid eye-ball, and therefore the act of focussing for near and distant objects is associated with a change in the curvature of the lens, a faculty of the eye termedaccommodation(Fig. 22), a change chiefly accomplished by the ciliary (muscle) processes, which pull the lens forwards and inwards by virtual contracting power of the ciliary muscle, and by which its suspensory ligament is relaxed, and the front of the lens allowed to bulge forward. In every case, however, accommodation is associated with contraction of the iris, the special function of which is that of a limiting diaphragm (an iris-diaphragm),Fig. 23.

In an ordinary spherical bi-convex lens, as already pointed out, the rays of light passing through the periphery of the lens come to a focus at a nearer point than the rays passing through the central portion. In this way a certain amount of blurring of the image takes place, and which, in optical language, is termedspherical aberration. This defect of the eye is capable of correction in three possible ways, and which it may be well to repeat: 1. By making the refractive index of the lens higher at its centre than at its circumference; (2) By making the curvature of the lens less near thecircumference than at the centre; (3) By stopping out the peripheral rays of light by a diaphragm. The two latter methods are those resorted to in most optical instruments.

Fig. 23.—1. Equatorial section of Eyeball, showing Iris and Ciliary Processes, after washing away the pigment, × three diameters.2. Nerves of the Cornea of Kitten’s Eye, stained with iodine.3. Fibres or Tubules of Lens, × 250, seen to be made up of superimposed crenated layers, and is therefore not homogeneous in structure, but made up of a number of extremely fine tubules, whose curvatures are nearly spherical.

Fig. 23.—1. Equatorial section of Eyeball, showing Iris and Ciliary Processes, after washing away the pigment, × three diameters.

2. Nerves of the Cornea of Kitten’s Eye, stained with iodine.

3. Fibres or Tubules of Lens, × 250, seen to be made up of superimposed crenated layers, and is therefore not homogeneous in structure, but made up of a number of extremely fine tubules, whose curvatures are nearly spherical.

In the human eye an attempt is made to apply all these methods, but the most important is the third, that of applying the diaphragm formed by the iris, a circular semi-muscular curtain lying just in front of the anterior surface of the lens. The iris is also furnished with a layer of pigmental cells which effectually stop out all peripheral rays of light that otherwise would pass into the eye, creating circles of diffusion of a disturbing nature to perfect vision. This delicate membrane, then, is kept in constant action by a two-fold nerve supply, derived from five or six sources, which it is unnecessary to describe at length. But the eye, with all its marvellous adaptations, has an obvious defect, that of secondary or uncorrected chromatic aberration.

Chromatic Aberration of the Eye.—White light, as previously explained, is composed of different wave lengths; and accordingly as these undulations are either longer or shorter, so do they produce on the eye the impression of different colours. We have seen how a pencil of white light may, by means of a prism, be decomposed into a multi-coloured band. In an ordinary magnifying reading-glass these coloured fringes are always seen around the margins. In practical optics chromatic aberration is partially corrected byemploying two different kinds of glass in the construction of certain combined lenses. In the human eye chromatism cannot be corrected in this way; hence a blue light and a red light placed at the same distance from the eye appears to be unequally distant: the red light requiring greater accommodation in the eye than the blue, and this accordingly appears to be the nearer of the two.

This visual error may be experimentally shown and explained. There is a kind of glass which at first sight appears dark blue or violet, but which really contains a great deal of red. Take an ordinary microscope lamp, having a metal or opaque chimney, and drill a circular hole in it, about 3 mm. in diameter. This opening should be just at the height of the flame; cover it over with a piece of ground glass and a piece of the red-blue glass. Thus will be formed a luminous point whose light is composed of red and blue,i.e., of colours far apart from each other in the spectrum.

Fig. 24.—Chromatic Aberration of Eye, showing the wave differences of theblue and red rays of light(Landolt).

If rays coming from this point enter the eye, the blue rays (Fig. 24), being more strongly reflected than the red, will come to a focus sooner than the latter. The red rays, on the contrary, will be brought to a focus later than the blue, while the latter, past their focus, are diverging. Let A B C D (Fig. 24) be the section of a pencil of raysgiven off from a red-blue point sufficiently distant so that these rays may be regarded as parallel. The focus of the blue is atb, that of the red atr.

An eye is adapted to the distance of the luminous point when the circle of diffusion, received upon the retina, is at its minimum. This is the case when the sentient layer of the retina lies between the two foci E. In this case the point will appear as a small circle, composed of the two colours, that is to say—violet. If the retina bein frontof this point, at the focus of thebluerays for instance, the eye will perceive ablue point surrounded by a red circle, the latter being formed by the periphery of the luminous cone of red rays, which are focussed only after having passed the retina. The blue point will become a circle of diffusion larger in proportion as the retina is nearer the dioptric system, or as the focus for blue is farther behind it. But the blue circle will always be surrounded by a red ring. If, on the contrary, the retina isbehind the focus for red, the blue cone will be greater in diameter than the red, and we shall have ared circle of diffusion, larger in proportion as the retina is farther from the focus, but alwayssurrounded by a blue ringM. If the blue-red point is five metres, or more, distant, theemmetropic7eye will evidently see it more distinctly,i.e., as a smallviolet point; thehyperopiceye, whose retina is situatedin frontof the focus of its dioptric system, will see ablue circle, surrounded by red; themyopiceye, whose retina isbehindits focus, will see ared circle, surrounded by blue. The size of these circles will be either larger or smaller when the principal focus of the eye is either infrontof orbehind the retina.8

The refractive surfaces of a perfectly formed eye are very like an ellipsoid of revolution with two axes, one of which, the major axis of the ellipse, is at the same time the optic axis and that of rotation; the other is perpendicular to it, and is equal in all meridians. Eyes, however, perfectly constructed are rarely met with. The curvature of the cornea is nearly always greater in one meridian than in another. Its surfaces then cannot be regarded as entirely belonging to an ellipsoid of revolution, since the solid figure, of which the former would constitute a part, has not only two axes, but three, and theseunequal. This irregularity is not, however, always great enough to produce discomfort and it is therefore disregarded. But in other cases the difference of curvature in the different meridians of the eye attain to a higher degree, and vision falls far below the average.

Fig. 25.—Lines as seen by the Astigmatic.

The refractive anomaly alluded to is termedastigmatism(from the Greek, α privative, στιγμα, a point—inability to see a point). The way in which objects appear to such a person will mainly result from the way in which he seesa point. Take, for example, the vertical to be the most, and the horizontal to be the least, refractive meridian: place a vertical line (Fig. 25, I) at a stated distance before the eye, and the line will appear elongated, owing to the diffusion image of each of the points composing it. It will also seem to be somewhat broadened, as at II. If theverticalmeridian is adapted to the distance of the vertical, the line will appear very diffuse and broadened, as at III. All these little diffusion lines overlap each other, and give the line an elongated appearance. Hence a straight line is seen distinctly by an astigmatic eye only when the meridian to which it is perpendicular is perfectly adapted to its distance. Avertical lineis seen distinctly when thehorizontal meridianis adapted to its distance. It appearsindistinctwhen its image is formed by the vertical meridian. The way in which an astigmatic person sees points and lines led to the discovery of this remarkable irregularity in the refraction of the eye. The late Astronomer Royal, Sir George Airy, suffered for some years until, indeed, he discovered how it could be corrected. This anomaly of curvature of the refractive surfaces of the eye is now known to prevail largely among the more civilised races of mankind. It is, then, of very great importance when using high powers of the microscope. In most persons the visual power of both eyes is rarely quite equal; on the other hand, the mind exerts an important influence, dominates, as it were, the eye in the interpretation of visual sensations and images. An example of this is presented in Wheatstone’s pseudoscope, known to produce precisely the oppositeeffect of his stereoscope—conveys, in fact, theconverse of reliefproduced by the latter and better known instrument.

Visual Judgment.—The apparent size of an object is determined by the magnitude of the image formed on the retina, and this is inversely proportional to the distance. Thus the size of an image on the retina of an object two inches long at a distance of a foot, is equal to the image of an object four inches long at a distance of two feet. An object can be seen if the visual angle subtended by it is not less than sixty seconds. This is equivalent to an image on thefovea centralisof the retina of about 4 µ9across, and which corresponds to the diameter of a cone: so that while we have had under consideration the optical and physical conditions of human vision, we have likewise taken a lesson on the action of lenses used in the construction of the microscope.

It has been said that no comparison can be instituted between microscopic vision and macroscopic; that the images formed by minute objects are not delineated microscopically under ordinary laws of diffraction, and that the results are dioptrical. This assertion, however, cannot be accepted unconditionally, as will be seen on more careful examination of the late Professor Abbe’s masterly exposition of “The Microscopical Theory of Vision,” and also his subsequent investigations on the estimation of aperture and the value of wide-angled immersion objectives, published in the “Journal of the Royal Microscopical Society.”

The essential point in Abbe’s theory of microscopical vision is that the images of minute objects in the microscope are not formed exclusively on the ordinarydioptricmethod (that is, in the same way in which they are formed in the camera or telescope), but that they are largely affected by the peculiar manner in which the minute construction of the object breaks up the incident rays, giving rise todiffraction.

The phenomena of diffraction in general may be observed experimentally by plates of glass ruled with fine lines.Fig. 26shows the appearance presented by a single candle-flame seen through sucha plate, an uncoloured image of the flame occupying the centre, flanked on either side by a row of coloured spectra of the flame, which become dimmer as they recede from the centre. A similar phenomenon may be produced by dust scattered over a glass plate, and by other objects whose structure contains very minute particles, or the meshes of very fine gauze wire, the rays suffering a characteristic change in passing through such objects; that change consisting in the breaking up of a parallel beam of light into a group of rays, diverging with wide angle and forming a regular series of maxima and minima of intensity of light, due to difference of phase of vibration.10

Fig. 26.

In the same way, in the microscope, the diffraction pencil originating from a beam incident upon, for instance, a diatom, appears as a fan of isolated rays, decreasing in intensity as they are further removed from the direction of the incident beam transmitted through the structure, the interference of the primary waves giving a number of successive maxima of light with dark interspaces.

When a diaphragm opening is interposed between the mirror, and a plate of ruled lines placed upon the stage such asFig. 27, the appearance shown inFig. 27a, will be observed at the back of the objective on removing the eye-piece and looking down the tube of the microscope. The centre circles are the images of the diaphragmopening produced by the direct rays, while those on the other side (always at right angles to the direction of the lines) are the diffraction images produced by the rays which are bent off from the incident pencil. In homogeneous light the central and lateral images agree in size and form, but in white light the diffraction images are radially drawn out, with the outer edges red and the inner blue (the reverse of the ordinary spectrum), forming, in fact, regular spectra the distance separating each of which varies inversely as the closeness of the lines, being for instance with the same objective twice as far apart when the lines are twice as close.

Fig. 27.Fig. 27a.

Fig. 27.

Fig. 27a.

The influence of these diffraction spectra may be demonstrated by some very striking experiments, which show that they are not by any means accidental phenomena, but are directly connected with the image which is seen by the eye.

The first experiment shows that with the central beam, or any one of the spectral beams alone, only the contour of the object is seen, the addition of at least one diffraction spectrum being essential to the visibility of the structure.

Fig. 28.Fig. 28a.

Fig. 28.

Fig. 28a.

When by a diaphragm placed at the back of the objective, as inFig. 28, we cover up all the diffraction spectra ofFig. 27a, and allow only the central rays to reach the image, the object will appear to bewholly deprived of fine details, the outline alone will remain, and every delineation of minute structure will disappear, just as if the microscope had suddenly lost its optical power, as inFig. 28a.

This experiment illustrates a case of theobliterationof structure by obstructing the passage of the diffraction spectra to the eye-piece. The next experiment shows how the appearance of fine structure may becreatedby manipulating the spectra.

Fig. 29.Fig. 29a.

Fig. 29.

Fig. 29a.

When a diaphragm such as that shown inFig. 29is placed at the back of the objective, so as to cut off each alternate one of the upper row of spectra inFig. 27a, that row will obviously become identical with the lower one, and if the theory holds good, we should find the image of the upper lines identical with that of the lower. On replacing the eye-piece, we see that it is so, the upper set of lines are doubled in number, a new line appearing in the centre of the space between each of the old (upper) ones, and upper and lower set having become to all appearance identical, as seen inFig. 29a.

Fig. 30.Fig. 30a.

Fig. 30.

Fig. 30a.

In the same way, if we stop off all but the outer spectra, as inFig. 30, the lines are apparently again doubled, as seen inFig. 30a.

A case of apparent creation of structure, similar in principle to the foregoing, though more striking, is afforded by a network of squares,as inFig. 31, having sidesparallelto this page, which gives the spectra shown inFig. 31a, consisting of vertical rows for the horizontal lines and horizontal rows for the vertical ones. But it is readily seen that two diagonal rows of spectra exist at right angles to the diagonals of the squares, just as would arise from sets of lines in the direction of the diagonals, so that if the theory holds good we ought to find, on obstructing all the other spectra and allowing only the diagonal ones to pass to the eye-piece, that the vertical and horizontal lines have disappeared and are replaced by two new sets of lines atright angles to the diagonals.

Fig. 31.Fig. 31a.

Fig. 31.

Fig. 31a.

Fig. 32.Fig. 32a.

Fig. 32.

Fig. 32a.

On inserting the diaphragm,Fig. 32, and replacing the eye-piece, we find in the place of the old network the one shown inFig. 32a, the squares being, however, smaller in the proportion of 1 : √2, as they should be in accordance with the theory propounded.

An object such asPleurosigma angulatum, which gives six diffraction spectra arranged as inFig. 33, should, according to this theory, show markings in a hexagonal arrangement. For there will be one set of lines at right angles tob,a,e, another set at right angles toc,a,f, and a third at right angles tog,a,d. These three sets of lines will obviously produce the appearance shown inFig. 33a.

Fig. 33.Fig. 33a.

Fig. 33.

Fig. 33a.

Fig. 34.

A great variety of appearances may be produced with the same arrangement of spectra. Any two adjacent spectra with the centralbeam (asb,c,a) will form equilateral triangles and give hexagonal markings. Or by stopping off all butg,c,e(orb,d,f), we again have the spectra in the form of equilateral triangles; but as they are now further apart, the sides of the triangles in the two cases being as √3 : 1, the hexagons will be smaller and three times as numerous. Their sides will also be arranged at a different angle to those of the first set. The hexagons may be entirely obliterated by admitting only the spectrag,c, org,f, orb,f, etc., when new lines will appear at right angles, or obliquely inclined, to the median line. By varying the combinations of the spectra, therefore, different figures of varying size and positions are produced, all of which cannot, of course, represent the true structure. Not only, however, may the appearance of particular structure be obliterated or created, but it may even bepredictedbefore being seen under the microscope. If the position and relative intensity of the spectra in any particular case are given, the character of the resultant image, in some instances, may be worked out by mathematical calculations. A remarkable instance of such a prediction is to be found in the case recorded by Mr. Stephenson, where a mathematical student who had never seen a diatom, worked out the purely mathematical result of the interference of the six spectrab-gofFig. 33(identical withP. angulatum), giving the drawing copied inFig. 34. The special feature was the small markings between the hexagons, which had not, before this time, been noticed onP. angulatum. On more closely scrutinizing a valve, stopping out the central beam and allowing the six spectraonly to pass, the small markings were found actually to exist, though they were so faint they had previously escaped observation until the result of the mathematical deduction had shown that theyoughtto be seen.

These experiments seem to show that diffraction plays a very essential part in the formation of microscopical images, since dissimilar structures give identical images when the differences of their diffractive effect is removed, and conversely similar structures may give dissimilar images when their diffractive images are made dissimilar. Whilst a purely dioptric image answers point for point to the object on the stage, and enables a safe inference to be drawn as to the actual nature of that object, the visible indications of minute structure in a microscopical image are not always or necessarily conformable to the real nature of the object examined, so that nothing more can safely be inferred from the image as presented to the eye, than the presence in the object of such structural peculiarities as will produce the particular diffraction phenomena on which these images depend.

Further investigations and experiments led Abbe to discard so much of his theoretical conclusions relating to superimposed images having a distinct character as well as a different origin, and as to their capability of being separated and examined apart from each other. In a later paper he writes: “I no longer maintain in principle the distinction between the absorption image or direct dioptrical image and the diffraction image, nor do I hold that the microscopical image of an object consists of two superimposed images of different origin or a different mode of production. Thus it appears that both the absorption image and the diffraction image he held to be equally of diffraction origin; but while a lens of small aperture would give the former with facility, it would be powerless to reveal the latter, because of its limited capacity to gather in the strongly-deflected rays due to the excessively minute bodies the microscopical objective has to deal with.”11

Abbe’s theory of vision has been questioned by mathematicians, and since his death Lord Rayleigh went more deeply into the question of “the theory of the formation of optical images,” with special reference to the microscope and telescope. He has shownthat two lines cannot be fairly resolved unless their components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the aperture; also, that the measure of resolution is only possible with a square aperture, or one bounded by straight lines, parallel to the lines resolved.

Of the two methods adopted, that of Helmholtz’s consists in tracing the image representative of a mathematical point in the object, the point being regarded as self-luminous; that of Abbe’s the typical object was not, as we have seen, a luminouspoint, but agratingilluminated by plane waves of light. In the latter method, Lord Rayleigh argues that the complete representation of the object requires the co-operation of all the spectra which are focussed in the principal focal plane of the objective; when only a few are present the representation is imperfect, and wholly fails when there is only one. He then proceeds to show, by the aid of diagrams and mathematical formula, how the resolving power can be adduced.

On further criticism of the Abbe spectrum theory, he observes “that although the image ultimately formed may be considered to be due to the spectra focussed to a given point, the degree of conformity of the image to the object is another question. The consideration of the case of a very fine grating, which might afford no lateral spectra at all, shows the incorrectness of the usually accepted idea that if all the spectra are utilised the image will still be incomplete, so that the theory (originally promulgated by Abbe) requires a good deal of supplementing; while it is inapplicable when the incident light is not parallel, and when the object is, for example, a double point and not a grating. Even in the case of a grating, the spectrum theory is inapplicable, if the grating is self-luminous; for in this case no spectra can be formed since the radiations from the different elements of the grating have no permanent phase-relations.” For these reasons Lord Rayleigh advises that the question should be reconsidered from the older point of view, according to which the typical object is a point and not a grating. Such treatment will show that the theory of resolving power is essentially the same for all instruments.The peculiarities of the microscope, arising from the divergence-angles not being limited to be small, and from the different character of the illumination, are theoretically only differences of detail. These investigations can be extended to gratings, and the results so obtained confirm for the most part the conclusions of the spectrum theory.

Furthermore, that the function of the condenser in microscopic practice in throwing upon the object the image of the lamp-flame is to cause the object to behave, at any rate in some degree, as if it were self-luminous, and thus to obviate the sharply-marked interference bands which arise when permanent and definite phase-relations are permitted to exist between the radiations which issue from various points of the object. This is capable of mathematical proof; and in the case where the illumination is such that each point of the row or of the grating radiates independently, the limit to resolution is seen to depend only on the width of the aperture, and thus to be the same for all forms of aperture as for those of the rectangular. That Abbe’s theory of microscopic vision is fairly open to the criticisms passed on it by Lord Rayleigh must be taken for granted.

It must be well within the last half-century that the achromatic objective-glass for the microscope was brought to perfection and its value became generally recognised. Prior to the discovery of the achromatic principle in the construction of lenses it was assumed that the formation of the microscopic image took place (as we have already seen) on ordinary dioptric principles. As the image is formed in the camera or telescope, so it was said to be in the microscope. This belief existed, it will be remembered, at a time when dry objectives only were in favour and the use of the termangle of aperturewas misunderstood, when it was supposed that the different media with diffraction-indices were used; and the angle of the radiant pencil was believed not only to admit of a comparison of two apertures in the same medium, but likewise to admit of a standard of comparison when the media were entirely different in their refractive qualities.

It was during my tenure of office as secretary of the Royal Microscopical Society (1867 to 1873), that the aperture question, and alsothat ofnumerical aperture, came under discussion, both being met by the majority of the Fellows of the Society and practical opticians by anon-possumus.

Opticians alleged, that is, before the value of aperture became fully recognised (1860), that the achromatic objective had reached a stage of perfection, beyond which it was not possible to go; indeed, not only opticians, but physicists of high standing, as Professor Helmholtz, who made many important contributions to the theory of the microscope, and who, after duly weighing all the known physical laws on which the formation of images can be explained, emphatically stated that in his opinion “the limit of possible improvement of the microscope as an instrument of discovery had been very nearly reached.” A quarter of a century ago I ventured to throw a doubt upon so questionable a statement. I determined, if possible, to submit the aperture question to an exhaustive examination. My views were accordingly submitted to two of the highest authorities in this country—Sir George Airy, the then Astronomer Royal, and Sir George Stokes, Professor of Physics at Cambridge University—both of whom agreed with me that the possible increase of aperture would be attended with great advantage to the objective, and open the way to an extension of power resolution in the microscope.13The discussion afterwards took a warm turn, as will be seen on reference to “The Monthly Microscopical Journals” of 1874, 1875 and 1876.

The confusion into which the aperture question at this period had lapsed was no doubt due to the fact that its opponents had not yet grasped the true meaning of the termaperture. It was believed to be synonymous with “angular aperture,” much in use at the time. It will, however, appear quite unaccountable that even the older opticians should have confounded the latter with the former; and so entirely disregarded the fact that the angles of the pencil of light admitted by the objective cannot serve as a measure of itsaperture,and that high refractive media can greatly reduce the value length of waves of light.

When the medium in which the objective works is the same as air, it is not that a comparison can be made by the angles of the radiant pencils only, but by their sines. For example, if two dry objectives admit pencils of 60° and 180°, their real apertures are not as 1 : 3, but as 1 : 2 only. Aperture in fact is computed by mathematicians by tracing the rays from the back focus through the system of lenses to the front focus, the front focus being the point at which the whole cone of rays converge as free as may be from aberration. If the front focus be in air, no pencil greater than 82°, “double the angle of total reflection,” canemergefrom the plane front of the lens; and, obviously, if no greater cone can emerge to a focus one way, neither can any greater cone enter the body of the lens from the radiant. This angle, then, of 82°, must be regarded as the limit for dry lenses or objectives.

This limit, it will be seen on more careful examination, is very nearly the maximum angle that can be computed for a lens to have a front focus in air. This can be proved by the consideration of the angle of the image of rays, as they are radiated from the object itself in balsam: for although this angle of image rays viewed as nascent from a self-luminous object capable of scattering rays in all directions, may be 180° in the substance of the balsam and cover-glass, of the 180° only 82° of the central portion will emerge into air—all rays beyond this limit are internally reflected at the cover-glass. This cone, then, of 82° becomes 180° in air, and a large part must necessarily be lost by reflection at the first incidence on the plane front of the lens. But with a formula permitting the use of a water medium between the front lens and the cover-glass, the aperture of the image rays may reach 126°—double the critical angle from glass to water; and with an oil medium, the aperture will be found to be limited only by the form of the front lens that can be constructed by the optician.

To sum up, then, the effect of the immersion system, greatly assists in the correction of aberration, gives increased magnification and angular aperture, increase of working distance between the objective and object, and renders admissible the use of the thicker glass-cover.

The aperture question would in all probability have remained unsolved many years longer (ten or twelve years elapsed after I brought the question under discussion before opticians gave way), but for the fortunate circumstance that the eminent mathematical and practical optician, Professor Abbe, of Jena, was about to visit London. This came off in the early part of the seventies, when the late Mr. John Mayall and myself had the good fortune to interview him. The subject discussed was naturally the increase of aperture and the theory of microscopical vision. He readily at our request undertook to re-investigate the question in all its bearings on the microscope. It is almost unnecessary to add that the conclusions he came to, and the results obtained, have proved of inestimable value to the microscopist and practical optician, and it may well seem necessary to explain somewhat at greater length the conclusions the learned Professor came to, and by the adoption of which the microscope has been placed on a more scientific basis than it had before attained to. Several papers were publishedin extensoin the “Journal of the Royal Microscopical Society,” and I am greatly indebted to Mr. Frank Crisp, LL.D., for an excellentresuméof Abbe’s Monograph.14

The essential step in the consideration of aperture is, as I have said, to understand clearly what is meant by the term. It will at once be recognised that its definition must necessarily refer to its primary meaning ofopening, and must, in the case of an optical instrument, define its capacity for receiving rays from the object, and transmitting them to the image received at the eye-piece.

In the case of the telescope-objective, its capacity for receiving and transmitting rays is necessarily measured by the expression of its absolute diameter or “opening.” No such absolute measure can be applied in the case of the microscope objective, the largest constructed lenses of which having by no means the largest apertures, being, in fact, the lower powers of the instrument, whose apertures are for the most part but small. The capacity of a microscope objective for receiving and transmitting rays is, however, as will be seen, estimated by itsrelativeopening, that is, its opening in relation toits focal length. When this relative opening has been ascertained, it may be regarded as synonymous with that denoted in the telescope byabsoluteopening. That this is so will be better appreciated by the following consideration:—

In a single lens, the rays admitted within one meridional plane evidently increase as the diameter of the lens (all other circumstances remaining the same), and in the microscope we have, at the back of the lens, the same conditions to deal with as are in front in the case of the telescope; the larger or smaller number of emergent rays will therefore be measured by the clear diameter, and as no rays can emerge that have not first been admitted, this will give the measure of the admitted rays under similar circumstances.

If the lenses compared have different focal lengths but the same clear “openings,” they will transmit the same number of rays to equal areas of an image at a definite distance, because they would admit the same number if an object were substituted for the image; that is, if the lens were used as a telescope-objective. But as the focal lengths are different, the amplification of the images is different also, and equal areas of these images correspond to different areas of the object from which the rays are collected. Therefore, the higher power lens with the same opening as the lower power, will admit agreaternumber of rays in all from the same object, because it admits thesamenumber as the latter from asmallerportion of the object. Thus, if the focal lengths of two lenses are as 2 : 1, and the first amplifiesNdiameters, the second will amplify 2Nwith the same distance of the image, so that the rays which are collectedtoa given field of 1 mm. diameter of the image are admittedfroma field of1/Nmm. in the first case, and of1/2Nmm. in the second. As the “opening” of the objective is estimated by the diameter (and not by the area) the higher power lens admitstwiceas many rays as the lower power, because it admits the same number from a field of half the diameter, and, in general, the admission of rays by the same opening, but different powers, must be in the inverse ratio of the focal lengths.

In the case of the single lens, therefore, its aperture is determined by the ratio between the clear opening and the focal length. The same considerations apply to the case of a compound objective, substituting, however, for the clear opening of the single lens thediameter of the pencil at its emergence from the objective, that is, the clear utilised diameter of the back lens. All equally holds good whether the medium in which the objective is placed is the same in the case of the two objectives or different, as an alteration of the medium makes no difference in the power.

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