Jabez Hogg

Value of Inductive Science—Light: Its Propagation, Refraction, Reflection—Spherical and Chromatic Aberrations—Human Eye, formation of Images of External Objects in—Visual Angle increased—Abbe’s Theory of Microscopic Vision.

The advances made in physics and mechanics during the 17th and 18th centuries fairly opened the way to the attainment of greater perfection in all optical instruments. This has been particularly exemplified with reference to the invention of the microscope, as briefly sketched out in the previous chapter. Indeed, in the first half of the present century the microscope can scarcely be said to have held a position of importance among the scientific instruments in frequent use. Since then, however, the zoologist and botanist by its aid have laid bare the intimate structure of plants and animals, and thereby have opened up a vast kingdom of minute forms of life previously undreamt of; and in connection with chemistry a new science has been founded, that of bacteriology.

For these reasons it will be of importance to the student of microscopy to begin at the beginning, and it will be my endeavour to introduce to his notice such facts in physical optics as are closely associated with the formation of images, and, so to speak, systematise such stepping stones for work hereafter to be accomplished. Elementary principles only will be adduced, and without attempting to involve my readers in intricate mathematical problems, and which for the most part are unnecessary for the attainment of the object in view. I therefore pass at once to the consideration of the propagation of light through certain bodies.

The microscope, whether simple or compound, depends for its magnifying power on the influence exerted by lenses in altering the course of the rays of light passing through them being REFRACTED. Refraction takes place in accordance with two well-known laws of optics. When a ray of light passes from one transparent medium to another it undergoes a change of direction at the surface of separation, so that its course in the second medium makes an angle with its course in the first. This change of direction is a resultant of refraction. The broken appearance presented by a stick partly immersed in water, and viewed in an oblique position, is an illustration of the law of refraction. Liquids have a greater refractive power than air or gases. As a rule, with some few exceptions, the denser of the two substances has the greater refractive power; hence it is customary in enumerating some of the laws of optics to speak of the denser medium and the rarer medium. The more correct designation would be the more refractive and the less refractive.5

Fig. 4.—Law of Refraction.

Let R I (Fig. 4) be a ray incident at I on the surface of separation of two media, and let I S' be the course of the ray after refraction. Then the angles which R I and I S make with the normal are the angle of incidence and the angle of refraction respectively, and the first law of refraction is that these angles lie in the same plane, or the plane of refraction is the same as the plane of incidence. The law which connects the magnitudes of these angles, and which was discovered by Snell, a Dutch philosopher, can only be stated either by reference to a geometrical construction, or by using the language of trigonometry. Describe a circle about the point of incidence, I as a centre, and drop perpendiculars from the points where it cuts the rays on the normal. The law is that these perpendiculars, R' P', S' P, will have a constant ratio, or the sines of the angles of incidence and refraction are in a constant ratio; that is, so long as the media through which the ray first passes, and by which it is afterwards refracted, remain the same, and the light also of the same kind, then it is referred to as the law of sines.

Indices of Refraction.

The ratio of the sine of the angle of incidence to the sine of the angle of refraction, when a ray passes from one medium to another is termed the relative index of refraction. When a ray passes from vacuum into any medium, this ratio is always greater than unity, and is called the absolute index of refraction, or simply the index of refraction for the medium in question.

The absolute index of air is so small that it may be neglected in comparison with those of solids and liquids; but strictly speaking, the relative index for a ray passing from air into a given substance must be multiplied by the absolute index of the air, in order to obtain the true index of refraction.

Fig. 5.—Vision through a Glass Plate.

Critical Angle.—It will be seen from the law of sines that, when the incident ray is in the less refractive of the two media, to every possible angle of incidence there is a corresponding angle of refraction. The angle referred to is termed the critical angle, and is readily computed if the relative index of refraction be given. When the media are air and water, this angle is about 48° 30'. For air and ordinary kinds of glass its value varies from 38° to 41°.

The phenomenon of total reflection may be observed in several familiar instances. For example, if a glass of water, with a spoon in it, is held above the level of the eye, the under side of the surface is seen to shine like a mirror, and the lower part of the spoon is seen reflected in it. Effects of the same kind are observed when a ray of sunlight passes into an aquarium—on the other hand rays falling normally on a uniform transparent plate of glass with parallel faces keep their course; but objects viewed obliquely through the same are displaced from their true position. Let S (Fig. 5) be a luminous point which sends light to an eye not directly opposite to it, on the other side of a parallel plate. The emergent rays which enter the eye are parallel to the incident rays; but as they have undergone lateral displacement, their point of concourse is changed from S to S', and this is accordingly the image of S. The rays in such a case which compose the pencil that enters the eye will not exactly meet in any one point; there will be two focal lines, just as in the case of spherical mirrors. The displacement produced, as seen in the figure referred to above, increases with the thickness of the plate, its index of refraction, and the obliquity of incidence. This furnishes one of the simplest means of measuring the index of refraction of a glass substance, and is thus employed in Pichot’s refractometer (“Deschanel”).

Fig. 6.—Refraction through a Prism.

Refraction through a Prism.—A prism is a portion of a refracting medium bounded by two plane surfaces, inclined at a definite angle to one another. The two plane surfaces are termed the faces of the prism, and their inclination to one another is the refracting angle of the prism. A prism preserves the property of bending rays of light from their original course by refraction. A cylinder may be regarded as the limit of a prism whose sides increase in number and diminish in size indefinitely: it may also be regarded as a pyramid whose apex is removed to an indefinite distance.

Let S I (Fig. 6) be an incident ray in the plane of the principal section of the prism. If the external medium be air, or other substance of less refractive power than the prism, the ray on entering the same will be bent nearer to the normal, taking such a course as I E, and on leaving the prism will be bent away from the normal, taking the course E B. The effect of these two refractions is, therefore, to turn the ray away from the edge (or refracting angle) of the prism. In practice, the prism is usually so placed that I E, the path of the ray through the prism, makes equal angles with the two faces at which refraction occurs. If the prism is turned very far from this position, the course of the ray may be altogether different from that represented in the figure; it may enter at one face, be internally reflected at another, and come out at the third.

It is evident, therefore, that the minimum number of sides, i.e., the bounding faces, exclusive of the ends, which a prism can have is three. In this form, it constitutes a most valuable instrument of research in physical optics. A convex lens is practically merely a curved form of two prisms combined, their bases being brought into contact; on the other hand the concave lens is simply a reversal of the position of the apices brought into contact, as shown in Fig. 11. Both convex and concave lenses are therefore closely related to the prism.

Reflection.—The laws that govern the change of direction which a ray of light experiences when it strikes upon the surfaces of separation of two media and is thrown back into the same medium from which it approached is as follows:—When the reflecting surface is plain the direction of the reflected ray makes with the normal to the surface the same angle which the incident ray makes with the same normal; or, as it is usually expressed, the angles of reflection and incidence are equal. When the surfaces are curved the same law holds good. In all cases of reflection the energy of the ray is diminished, so that reflection must always be accompanied by absorption. The latter probably precedes the former. Most bodies are visible by light reflected from their surfaces, but before this takes place the light has undergone a modification, namely, that which imparts colour peculiar to the bodies viewed. When light impinges upon the surface of a denser medium part is reflected, part absorbed, and part refracted. But for a certain angle depending upon the refractive index of the refracting medium no refraction takes place. This angle is termed the angle of total reflection, since all the light which is not absorbed is wholly reflected.

Multiple images are produced by a transparent parallel plate of glass. If the glass be silvered at the back, as it usually is in the microscope-mirror, the second image is brighter than the first, but as the angle of incidence increases the first image gains upon the second; and if the luminous object be a lamp or candle, a number of images, one behind the other, will be visible to an eye properly placed in front. This is due to the fact that the reflecting power of a surface of glass increases with the angle of incidence.

Fig. 7.—Conjugate Foci of Curved Surfaces.

Concave Surfaces.—Rays of light proceeding from any given point in front of a concave spherical mirror, are reflected so as to meet in another point, and the line joining the two points passes through the centre of the sphere. The relation between them is or should be mutual, hence they are termed conjugate foci. By a focus in general is meant a point in which a number of rays of light meet, and the rays which thus meet, taken collectively, are termed a pencil. Fig. 7 represents two pencils of rays whose foci, S s, are conjugate, so that, if either of them be regarded as an incident pencil, the other will be the corresponding reflected pencil. Each point, in fact, sends a pencil of rays which converge, after reflection, to the conjugate focus. The principal focal distance is half the radius of curvature. But it will not escape attention that concave mirrors have two reflecting surfaces, a front and a back. This, however, does not practically disturb its virtual focus, since the achromatic condenser when brought into use collects and concentrates the light received from the mirror upon an object for the purpose of rendering it more distinctly visible to the eye when viewing an object placed on the stage of the microscope. The images seen in a plane mirror are always virtual, and any spherical mirror, whether concave or convex, is nearly equivalent to a plane mirror when the distance of the object from its surface is small in comparison with the radius of curvature.

Lenses.

Forms of Lenses.—A lens is a portion of a refracting medium bounded by two surfaces which are portions of spheres, having a common axis, termed the axis of the lens. Lenses are distinguished by different names, according to the nature of their surfaces.

Fig. 8.—Converging and Diverging Lenses.

Lenses with sharp edges (thicker at the centre) are convergent or positive lenses. Lenses with blunt edges (thinner at the centre) are divergent or negative lenses. The first group comprises:—(1) The bi-convex lens; (2) the plano-convex lens; (3) the convergent meniscus. The second group:—(4) The concave lens; (5) the plano-concave lens; (6) the divergent meniscus (Fig. 8).

Principal Focus.—A lens is usually a solid of revolution, and the axis of revolution is termed the principal axis of the lens. When the surfaces are spherical it is the line joining the centre of curvature.

From the great importance of lenses, especially convex lenses, in practical optics, it will be necessary to explain their properties somewhat at length.

Fig. 9.—Principal Focus of a Convex Lens.

Principal Focus of Convex Lens.—When rays which were originally parallel to the principal axis pass through a convex lens (Fig. 9), the effect of the two refractions which they undergo, one on entering and the other on leaving the lens, is to make them all converge approximately to one point F, which is called the principal focus. The distance A F of the principal focus from the lens is called the principal focal distance, or more briefly and usually, the focal length of the lens. The radiant point and its image after refraction are known as the conjugate foci. In every lens the right line perpendicular to the two surfaces is the axis of the lens. This is indicated by the line drawn through the several lenses, as seen in the diagram (Fig. 8). The point where the axis cuts the surface of the lens is termed the verte.

Parallel rays falling on a double-convex lens are brought to a focus in the centre of its diameter; conversely, rays diverging from that point are rendered parallel. Hence the focus of a double-convex lens will be at just half the distance, or half the length, of the focus of a plano-convex lens having the same curvature on one side. The distance of the focus from the lens will depend as much on the degree of curvature as upon the refracting power (termed the index of refraction) of the glass of which it may be formed. A lens of crown-glass will have a longer focus than a similar one of flint-glass; since the latter has a greater refracting power than the former. For all ordinary practical purposes we may consider the principal focus—as the focus for parallel rays is termed—of a double-convex lens to be at the distance of its radius, that is, in its centre of curvature; and that of a plano-convex lens to be at the distance of twice its radius, that is, at the other end of the diameter of its sphere of curvature. The converse of all this occurs when divergent rays are made to fall on a convex lens. Rays already converging are brought together at a point nearer than the principal focus; whereas rays diverging from a point within the principal focus are rendered still more diverging, though in a diminished degree. Rays diverging from points more distant than the principal focus on either side, are brought to a focus beyond it: if the point of divergence be within the circle of curvature, the focus of convergence will be beyond it; and vice-versÂ. The same principles apply equally to a plano-convex lens; allowance being made for the double distance of its principal focus; and also to a lens whose surfaces have different curvatures; the principal focus of such a lens is found by multiplying the radius of one surface by the radius of the other, and dividing this product by half the sum of the radii.

Fig. 10.—Principal Focus of Concave Lens.

In the case of a concave lens (Fig. 10), rays incident parallel to the principal axis diverge after passing through; and their directions, if produced backwards, would approximately meet in a point F; this is its principal focus. It is, however, only a virtual focus, inasmuch as the emergent rays do not actually pass through it, whereas the principal focus of a converging lens is real.

Fig. 11.—Principal Centre of Lens.

Optical Centre of a Lens.—Secondary Axes.—Let O and O' (Fig. 11) be the centres of the two spherical surfaces of a lens. Draw any two parallel radii, O I, O' E, to meet these surfaces, and let the joining line I E represent a ray passing through the lens. This ray makes equal angles with the normals at I and E, since these latter are parallel by construction; hence the incident and emergent rays S I, E R also make equal angles with the normals, and are therefore parallel. In fact, if tangent planes (indicated by the dotted lines in the figure) are drawn at I and E, the whole course of the ray S I E R will be the same as if it had passed through a plate bounded by these planes.

Let C be the point in which the line I E cuts the principal axis, and let R, R' denote the radii of the two spherical surfaces. Then from the similarity of the triangles O C I, O' C E, we have (O C)/(C O') = R'/R; which shows that the point C divides the line of centres O O' in a definite ratio depending only on the radii. Every ray whose direction on emergence is parallel to its direction before entering the lens, must pass through the point C in traversing the lens; and conversely, every ray which in its course through the lens traverses the point C, has parallel directions at incidence and emergence. The point C which possesses this remarkable property is called the centre, or optical centre, of the lens.

This diagram may also be taken to prove my former proposition, that the convex lens is practically a form of two prisms combined.

Fig. 12.—Conjugate Foci, one Real, the other Virtual.

Conjugate Foci, one Real, one Virtual.—When two foci are on the same side of the lens, one (the most distant of the two) must be virtual. For example, in Fig. 12, if S, S' are a pair of conjugate foci, one of them S being between the principal focus F and the lens, rays sent to the lens at a luminous point at S, will, after emergence, diverge as if from S'; and rays coming from the other side of the lens, if they converge to S' before incidence, will in reality be made to meet in S. As S moves towards the lens, S' moves in the same direction more rapidly; and they become coincident at the surface of the lens.

Formation of Real Images.—Let A B (Fig. 13) be an object in front of a lens, at a distance less than the principal focal length. It will have a real image on the other side of the lens. To determine the position of the image by construction, draw through any point A of the object a line parallel to the principal axis, meeting the lens in A'. The ray represented by this line will, after refraction, pass through the principal focus, F, and its intersection with the secondary axis, A O, determines the position of a, the focus conjugate to A. We can in like manner determine the position of b, the focus conjugate to B, another point of the object; and the joining line a b will then be the magnified image of the line A B. It is evident that if a b were the object, A B would be the image.

Fig. 13.—Real and Magnified Image.

The figures 12 and 13 represent the cases in which the distance of the object is respectively greater and less than twice the focal length of the lens.

The focal length of a lens is determined by the convexity of its surfaces and the refractive power of the material of which it is composed, being shortened either by an increase of refractive power, or diminution of the radii of curvature of the faces of the lens. The increase or decrease of spherical aberration is determined by the shape or curvature of the lens; it is less in the bi-convex than in other forms. When a lamp or other source of light is placed at the focus of the rays constituting that portion of its light which falls upon the lens, the light is so refracted as to become parallel. Should the source of light be brought nearer to the lens than the focus the refracted rays are still divergent, though not to the same extent; on the other hand, if the source be beyond the focus, the refracted rays are rendered convergent so as to meet at a point which is mathematically related to the distance of the luminous source from the focus. The former arrangement is that with which we are most familiar, since it is the ordinary magnifying glass.

Concave Lenses.

The refracting influence of a concave lens (Fig. 14) will be precisely the opposite of that of a convex. Rays which fall upon it in a parallel direction will be made to diverge as if from the principal focus, which is here called the negative focus. This will be, for a plano-concave lens, at the distance of the diameter of the sphere of curvature; and for a double-concave, in the centre of that sphere.

Fig. 14.—A Virtual Image formed by Concave Lens.

In Fig. 14 A B is the object and a b the image. Rays incident from A and B parallel to the principal axis will emerge as if they came from the principal focus F; hence, the points a b are determined by the intersections of the dotted lines in the figure with the secondary axis, O A, O B. An eye on the other side of the lens sees the image a b, which is always virtual, erect and diminished.

In the construction of the microscope, either simple or compound, the curvature of the lenses employed is usually spherical. Convergent lenses, with spherical curvatures, have the defect of not bringing all the rays of light which pass through them to one and the same focus. Each circle of rays from the axis of the lens to its circumference has a different focus, as shown in Fig. 15. The rays a a, which pass through the lens near its circumference, are seen to be more refracted, or come to a focus at a shorter distance behind it than the rays b b, which pass through near its centre or axis, and are less refracted. The consequence of this defect of lenses with spherical curvatures, which is called spherical aberration, is that a well-defined image or picture is not formed by them, for when the object is focussed, for the circumferential rays, the picture projected to the eye is rendered indistinct by a halo or confusion produced by the central rays falling in a circle of dissipation, before they have come to a focus. On the other hand, when placed in the focus of the central rays, the picture formed by them is rendered indistinct by the halo produced by the circumferential rays, which have already come to a focus and crossed, and now fall in a state of divergence, forming a circle of dissipation. The grosser defects of spherical aberration are corrected by cutting off the passage of the rays a a, through the circumferences of the lens, by means of a stop diaphragm, so that the central rays, b b, only are concerned in the formation of the image. This defect is reduced to a minimum, by using the meniscus form of lens, which is the segment of an ellipsoid instead of a sphere.

Fig. 15.—Spherical Aberration of Lens.

The ellipse and the hyperbola are forms of lenses in which the curvature diminishes from the central ray, or axis, to the circumference b; and mathematicians have shown that spherical aberration may be practically got rid of by employing lenses whose sections are ellipses or hyperbolas. The remarkable discovery of these forms of lenses is attributed to Descartes, who mathematically demonstrated the fact.

If a l, a l', for example (Fig. 16) be part of an ellipse whose greater axis is to the distance between its foci f f as the index of refraction is to unity, then parallel rays r l', r'' l incident upon the elliptical surface l' a l, will be refracted by the single action of that surface into lines which would meet exactly in the farther focus f, if there were no second surface intervening between l a l' and f. But as every useful lens must have two surfaces, we have only to describe a circle l a' l' round f as a centre, for the second surface of the lens l' l.

Fig. 16.—Converging Meniscus.

As all the rays refracted at the surface l a l' converge accurately to f, and as the circular surface l a' l' is perpendicular to every one of the refracted rays, all these rays will go on to f without suffering any refraction at the circular surface. Hence it should follow, that a meniscus whose convex surface is part of an ellipsoid, and whose concave surface is part of any spherical surface whose centre is in the farther focus, will have no appreciable spherical aberration, and will refract parallel rays incident on its convex surface to the farther focus.

Fig. 17.—Aplanatic Doublet.

The spherical form of lens is that most generally used in the construction of the microscope. If a true elliptical or hyperbolic curve could be ground, lenses would very nearly approach perfection, and spherical aberration would be considerably reduced. Even this defect can be further reduced in practice by observing a certain ratio between the radii of the anterior and posterior surfaces of lenses; thus the spherical aberration of a lens, the radius of one surface of which is six or seven times greater than that of the other, will be much reduced when its more convex surface is turned forward to receive parallel rays, than when its less convex surface is turned forwards. It should be borne in mind that in lenses having curvatures of the kind the object would only be correctly seen in focus at one point—the mathematical or geometrical axis of the lens.

Chromatic Aberration.—We have yet to deal with one of the most important of the phenomena of light, CHROMATIC ABERRATION, upon the correction of which, in convex lenses in particular, the perfection of the objective of the microscope so much depends. Chromatism arises from the unequal refrangibility and length of the different coloured rays of light that together go to make up white light; but which, when treated of in optics, is always associated with achromatism, so that a combination of prisms, or lenses, is said to be achromatic when the coloured rays arising from the dispersion of the pencil of light refracted through them are combined in due proportions as they are in perfectly white light.

A lens, however, of uniform material will not form a single white image, but a series of images of all colours of the spectrum, arranged at different distances, the violet being nearest, and the red the most remote, every other colour giving a blurred image; the superposition of these and the blending of the different elementary rays furnishing a complete explanation of the beautiful phenomenon of the rainbow. Sharpness of outline is rendered quite impossible in such a case, and this source of confusion is known as chromatic aberration.

In order to ascertain whether it is possible to remedy this evil by combining lenses of two different materials, Newton made some trials with a compound prism composed of glass and water (the latter containing a little sugar of lead), and he found it impossible by any arrangement of these two, or by other substances, to produce deviation of the transmitted light without separation into its component colours. If this ratio were the same for all substances, as Newton supposed, achromatism would be impossible; but, in fact, its value varies greatly, and is far greater for flint than for crown glass. If two prisms of these substances, of small refracting angles, be combined into one, with their edges turned in opposite directions, they will achromatise each other.

The chromatism of lenses may, however, be somewhat further reduced by stopping out the marginal rays, but as the most perfect correction possible is required when lenses are combined for microscopic uses, other means of correction are resorted to, as will be seen hereafter. I shall first proceed to show the deviations which rays of white light undergo in traversing a lens.

If parallel rays of light pass through a double-convex lens the violet rays, the most refrangible of them, will come to a focus at a point much nearer to the lens than the focus of the red rays, which are the least refrangible; and the intermediate rays of the spectrum will be focussed at points between the red and the violet. A screen held at either of these foci will show an image with prismatic fringes. The white light, A A'' (Fig. 18), falling on the marginal portion of the lens is so far decomposed that the violet rays are brought to a focus at C, and crossing there, diverge again and pass on to F F', while the red rays, B B'', do not come to a focus until they reach the point D, and cross the divergent violet rays, E E'. The foci of the intermediary rays of the spectrum (red, green, and blue) are intermediate between these extremes. The distance, C D, limiting the blue or violet, and the red is termed the longitudinal chromatic aberration of the lens. If the image be received upon a screen placed at C, violet will predominate and appear surrounded by a prismatic fringe, in which violet will predominate. If the screen be now shifted to D, the image will have a predominant red tint, surrounded by a series of coloured fringes in an inverted order to those seen in the former experiment. The line E E' joins the points of intersection between the violet and red rays, and this marks the mean focus, the point where the coloured rays will be least apparent.

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, of crown-glass whose index of refraction is 1·519, and dispersive power 0·036, and of flint-glass whose index of refraction is 1·589, and dispersive power 0·0393. The focal length of the convex crown-glass lens must be 4¹/₃ inches, and that of the concave flint-glass lens 7²/₃ inches, 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 the secondary 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 lens l l, on the prism A a C, which receives the rays F V, F R, at the same points, is interposed, these rays will unite at f, 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 F f. In like manner, the corresponding ray S F' will be refracted to f, and a white colourless image be the result.

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 a convex lens of crown-glass, and l l a concave one of flint-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.

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 made by 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.

The Human Eye.

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 termed the principal focus of 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 (the yellow spot, or fovea 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.

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 termed accommodation (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 termed spherical 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 the circumference 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.

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 by employing 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.

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 rays given off from a red-blue point sufficiently distant so that these rays may be regarded as parallel. The focus of the blue is at b, that of the red at r.

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 be in front of this point, at the focus of the blue rays for instance, the eye will perceive a blue 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 is behind the focus for red, the blue cone will be greater in diameter than the red, and we shall have a red circle of diffusion, larger in proportion as the retina is farther from the focus, but always surrounded by a blue ring M. If the blue-red point is five metres, or more, distant, the emmetropic7 eye will evidently see it more distinctly, i.e., as a small violet point; the hyperopic eye, whose retina is situated in front of the focus of its dioptric system, will see a blue circle, surrounded by red; the myopic eye, whose retina is behind its focus, will see a red circle, surrounded by blue. The size of these circles will be either larger or smaller when the principal focus of the eye is either in front of or behind 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 these unequal. 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.

The refractive anomaly alluded to is termed astigmatism (from the Greek, a privative, st??a, 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 sees a 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 the vertical meridian 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. A vertical line is seen distinctly when the horizontal meridian is adapted to its distance. It appears indistinct when 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 opposite effect of his stereoscope—conveys, in fact, the converse of relief produced 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 the fovea centralis of the retina of about 4 µ9 across, 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.

The Theory of Microscopical Vision.

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 ordinary dioptric method (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 to diffraction.

The phenomena of diffraction in general may be observed experimentally by plates of glass ruled with fine lines. Fig. 26 shows the appearance presented by a single candle-flame seen through such a 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

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 as Fig. 27, the appearance shown in Fig. 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 diaphragm opening 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.

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.

When by a diaphragm placed at the back of the objective, as in Fig. 28, we cover up all the diffraction spectra of Fig. 27a, and allow only the central rays to reach the image, the object will appear to be wholly 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 in Fig. 28a.

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

When a diaphragm such as that shown in Fig. 29 is placed at the back of the objective, so as to cut off each alternate one of the upper row of spectra in Fig. 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 in Fig. 29a.

In the same way, if we stop off all but the outer spectra, as in Fig. 30, the lines are apparently again doubled, as seen in Fig. 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 in Fig. 31, having sides parallel to this page, which gives the spectra shown in Fig. 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 at right angles to the diagonals.

On inserting the diaphragm, Fig. 32, and replacing the eye-piece, we find in the place of the old network the one shown in Fig. 32a, the squares being, however, smaller in the proportion of 1 : v2, as they should be in accordance with the theory propounded.

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

A great variety of appearances may be produced with the same arrangement of spectra. Any two adjacent spectra with the central beam (as b, c, a) will form equilateral triangles and give hexagonal markings. Or by stopping off all but g, c, e (or b, 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 v3 : 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 spectra g, c, or g, f, or b, 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 be predicted before 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 spectra b-g of Fig. 33 (identical with P. angulatum), giving the drawing copied in Fig. 34. The special feature was the small markings between the hexagons, which had not, before this time, been noticed on P. angulatum. On more closely scrutinizing a valve, stopping out the central beam and allowing the six spectra only 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 they ought to 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 shown that 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.

Lord Rayleigh’s Theory of the Formation of Optical Images, with Special Reference to the Microscope.12

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 luminous point, but a grating illuminated 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.

Definition of Aperture; Principles of Microscopic Vision.

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 term angle of aperture was 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 also that of numerical aperture, came under discussion, both being met by the majority of the Fellows of the Society and practical opticians by a non-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.13 The 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 term aperture. 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 its aperture, 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,” can emerge from 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 published in extenso in the “Journal of the Royal Microscopical Society,” and I am greatly indebted to Mr. Frank Crisp, LL.D., for an excellent resumÉ of 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 of opening, 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 its relative opening, that is, its opening in relation to its focal length. When this relative opening has been ascertained, it may be regarded as synonymous with that denoted in the telescope by absolute opening. 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 a greater number of rays in all from the same object, because it admits the same number as the latter from a smaller portion of the object. Thus, if the focal lengths of two lenses are as 2 : 1, and the first amplifies N diameters, the second will amplify 2 N with the same distance of the image, so that the rays which are collected to a given field of 1 mm. diameter of the image are admitted from a field of ¹/_N mm. in the first case, and of ¹/_2N mm. in the second. As the “opening” of the objective is estimated by the diameter (and not by the area) the higher power lens admits twice as 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 the diameter 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.

180° Oil Angle. (Numerical Aperture 1·52.)