CHAPTER I.

Fig. 1.

From this digression I pass on to the evolution of the compound microscope. The earliest workable form known was that designed by Eustachio Divini, who brought it to the notice of the Royal Society in 1668. It consisted of two plano-convex lenses, combined with their convex surfaces retained in apposition. His idea was subsequently improved upon by a London optician. Not long afterwards, Philip Bonnani published an account of his improved compound microscope; and we are certainly indebted to him for two or more forms of the movable horizontal microscopes, and for the compound condenser fitted with focussing gear for illuminating transparent objects by transmitted light. I must, however, pass by the many changes made in the structure and form of the instrument by the celebrated Dr. Culpeper, Scarlet, Cuff, and many other inventors.

Fig. 2.—Lieberkuhn’s Microscope.

Benjamin Martin’s Microscope.—Benjamin Martin, about 1742, was busily engaged in making improvements in the microscope, and I may say he was certainly the first to provide accurate results for determining the exact magnifying power of any object-lens, so that the observer might state the exact amplification in a certain number of diameters. He devised numerous improvements in the mechanism and optical arrangements of the instrument; the rack and pinion focussing adjustments; the inclining movements to the pillar carrying the stage; and the rectangular mechanical motions to the stage itself. He was familiar with the principles of achromatism, since it appears he produced an achromatic objective about 1759, and he is said to have sent an achromatic objective to the Royal Societyabout that date. But an ingeniously constructed microscope by Martin found its way to George the Third, the grandfather of our Queen, and afterwards came into the possession of the late Professor John Quekett, of the Royal College of Surgeons, who presented it to the Royal Microscopical Society of London. This microscope will ever associate Martin’s name with the earliest and best form of the instrument, even should he not receive full recognition as the inventor of theachromatic microscope. On this account I introduce a carefully made drawing of so singularly perfect a form of the early English microscope to the notice of my readers. (Fig. 3.) The description given of it by the late Professor Quekett is as follows:—“It stands about two feet in height, and is supported on a tripod base, A; the central part of the stem, B, is of triangular figure, having a rack at the back, upon which the stage, O, and frame, D, supporting the mirror, E, are capable of being moved up or down. The compound body, F, is three inches in diameter; it is composed of two tubes, the inner of which contains the eye-piece, and can be raised or depressed by rack and pinion, so as to increase or diminish the magnifying power. At the base of the triangular bar is a cradle joint, G, by which the instrument can be inclined by turning the screw-head, H (connected with an endless screw acting upon a worm-wheel). The arm, I, supporting the compound body, is supplied with a rack and pinion, K, by which it can be moved backwards and forwards, and a joint is placed below it, upon which the body can be turned into the horizontal position; another bar, carrying a stage and mirror, can be attached by a screw, L N, so as to convert it into a horizontal microscope. The stage, O, is provided with allthe usual apparatus for clamping objects, and a condenser can be applied to its under surface; the stage itself may be removed, the arm, P, supporting it, turned round on the pivot, C, and another stage of exquisite workmanship placed in its stead, the under surface of which is shown at Q.”

Fig. 3.—Martin’s Universal Microscope. 1782.

This stage is strictly a micrometer one, having rectangular movements and a fine adjustment, the movements being accomplished by the fine-threaded screws, the milled heads of which are graduated. The mirror, E, is a double one, and can be raised or depressed by rack and pinion; it is also capable of removal, and an apparatus for holding large opaque objects, such as minerals, can be substituted for it. The accessory instruments are very numerous, and amongst the more remarkable may be mentioned a tube, M, containing a speculum, which can take the place of the tube, R, and so form a reflecting microscope. The apparatus for holding animalcules or other live objects, which is represented at S, as well as a plate of glass six inches in diameter, with four concave wells ground in it, can be applied to the stage, so that each well may be brought in succession under the magnifying power. The lenses belonging to this microscope are twenty-four in number; they vary in focal length from four inches to one-tenth of an inch; ten of them are supplied with Lieberkuhns. A small arm, capable of carrying single lenses, can be supplied at T, and when turned over, the stage of the instrument becomes a single microscope; there are four lenses suitable for this purpose, their focal length varying from one-tenth to one-fortieth of an inch. The performance of all the lenses is excellent, and no pains appear to have been spared in their construction. There are numerous other pieces of accessory apparatus, all remarkable for the beauty of their workmanship.4

In addition to the movements described by Quekett, the body-tube with its support can be moved in an arc concentrically with the axis of the triangular pillar, on the top of which it is fitted with a worm-wheel and endless-screw mechanism, actuated by the screw-head, T, below. It must therefore be admitted that Martin led the way far beyond his contemporaries, both in the design and the evolution of the microscope. Furthermore, in his “New Elementsof Optics,” 1759, he dealt with the principle of achromatism, by the construction of an achromatic telescope.

At a somewhat later period there lived in London a philosophical instrument maker of some repute, George Adams, who published in 1746 a quarto book, entitled “Micrographia Illustrata, or the Knowledge of the Microscope Explained.” This work fairly well describes “the nature, uses, and magnifying powers of microscopes in general, together with full directions how to prepare, apply, examine, and preserve minute objects.” Adams’ book was the first of the kind published in this country, and it contributed in no small degree to the advancement of microscopical science. Adams writes: “We owe the construction of the variable microscope to the ingenuity and generosity of a noble person. The apparatus belonging to it is more convenient, more certain, and more extensive than that of any other at present extant; consequently, the advantage and pleasure attending the observations in viewing objects through it must be as extensive in proportion.” This is believed to apply to Martin’s several microscopes, and that especially constructed for the king, afterwards improved upon by Adams. Another early form of microscope, Wilson Simple Scroll (1746), stamped on the cover of this book, and has thus become familiar to microscopists, was also made by Adams.

We now closely approach a period fertile in the improvement of the microscope, and in the discoveries made by its agency. The chief of those among the honoured names of the time we find Trembley, Ellis, Baker, Adams, Hill, Swammerdam, Lyonet, Needham, and a few others. Adams somewhat sarcastically observes “that every optician exercises his talents in improving (as he calls it) the microscope, in other words, in varying its construction and rendering it different in form from that sold by his neighbour; or at the best rendering it more complex and troublesome to manage.” There were no doubt good reasons for these and other strictures upon inventors as well as makers of microscopes, even in the Adams’ day. In the year 1787 the “Microscopical Essays” of his son were published, in which he described all the instruments in use up to that period.

Looking back, and taking a general survey of the work of nearly two centuries in the history of the microscope, it cannot be said thateither in its optical or mechanical construction any great amount of progress was made. This in part may have arisen from the fact that no pressing need was felt for either delicate focussing or higher magnification. At all events, it was not until the application of achromatism to the instrument that new life was infused into its use, and a great impetus was given to its development, both optically and mechanically.

In the year 1823 a strong desire became manifest for improved forms of the instrument, in France by M. Selligue, by Frauenhofer in Munich, by Amici in Modena, by M. Chevalier in Paris, and by Dr. Goring, Mr. Pritchard, and Mr. Tully in London. The result was that in 1824 a new form of achromatic object-glass was constructed of nine-tenths of an inch focal length, composed of three lenses, and transmitting a pencil of eighteen degrees; and which, as regards accurate correction throughout the field, was for some years regarded as perfect.

Sir David Brewster was the first to suggest the great importance of introducing materials of a more highly refracting nature into the construction of lenses. He wrote: “There can be no essential improvement expected in the microscope unless from the discovery of some transparent substance which, like the diamond, combines a high refractive with a low dispersive power.” Having experienced the greatest difficulty in getting a small diamond cut into a prism in London, he did not conceive it practicable to grind, polish, and form it into a lens.

Mr. Pritchard, however, was led to make the experiment, and on the 1st of December, 1824, “he had the pleasure of first looking through a diamond microscope.” Dr. Goring also tried its performance on various objects, both as a single microscope and as an objective of a compound instrument, and satisfied himself of its superiority over other kinds of lenses. But here Mr. Pritchard’s labours did not end. He subsequently found that the diamond used had many flaws in it, which led him to abandon the idea of finishing it. Having been prevented from resuming his operations on this refractory material for a time he made a third attempt, and met with another unexpected defect; he found that some lenses, unlike the first, gave a double or triple image instead of a single one, in consequence of some of their parts being either harder or softer than others. Thesedefects were found to be due to polarisation. Mr. Pritchard having learned how to decide whether a diamond is fit for a magnifier or not, subsequently succeeded in making two planoconvex lenses of adamant; these proved to be perfect for microscopic purposes. “One of these, of one-twentieth of an inch in focal length, is now in the possession of his Grace the Duke of Buckingham; the other, of one-thirtieth of an inch focus, is in his own hands.”

“In consequence of the high refracting power of a diamond lens over a glass lens, the former material may be at least one-third as thin as that of the latter, and if the focal length of both be equal, say, one-eightieth of an inch, the magnifying power of the diamond lens will be 2,133 diameters, whereas that of glass will be only 800.” At a date (1812) before Brewster proposed diamond lenses he demonstrated a simple method of rendering both single and compound microscopes achromatic. “Starting,” he says, “with the principle that all objects, however delicate, are best seen when immersed in fluid, he placed an object on a slip of glass, and put above a drop of oil, having a greater dispersive power than the single concave lens, which formed the object-glass of the microscope. The lens was then made to touch the fluid, so that the surface of the fluid was formed into a concave lens, and if the radius of the outward surface was such as to correct the dispersion, we should have a perfect achromatic microscope.” Here we have the immersion system foreshadowed. Shortly after these experiments of Brewster’s were in progress, Dr. Goring is said to have discovered that the structure of certain bodies could be readily seen in some microscopes and not in others. These bodies he named test objects. He then examined these tests with the achromatic combinations of the Tullys, and was led to the discovery that “the penetrating power of the microscope depends upon its angle of aperture.”

“While these practical investigations were in progress,” writes Andrew Ross, “the subject of achromatism engaged the attention of some of the most profound mathematicians in England, Sir John Herschel, and Professors Airy and Barlow. Mr. Coddington and others contributed largely to the theoretical examination of the subject; and although the results of their labours were not applicable to the microscope, they essentially promoted its improvement.”

About this period (1812) Professor Amici, of Modena, was experimentallyengaged in the improvement of the achromatic object-glass, and he invented a reflecting microscope superior to those of Newton, Baker, or Smith, made as early as 1738, and long ago abandoned. In 1815 Amici made further experiments, and introduced the immersion system; while Frauenhofer, of Munich, about the same time constructed object-glasses for the microscope of a single achromatic lens, in which the two glasses, although placed in juxtaposition, were not cemented together.

Dolland, it has been said, introduced achromatic lenses; but although he constructed many achromatic telescopes, he did not apply the same principle to microscopes, and those which he sold were only modifications of the compound microscope of Cuff.

Dr. Wollaston employed a new form of combination in a microscope constructed for his own use, and by which “he was able to see distinctly the finest markings upon the scales of theLepismaandPodura, and upon those of the gnat’s wing.” His doublet is still employed, and to which I shall refer under “Simple Microscopes.”

Fig. 3a.—Sir David Brewster’s Microscope, of the early part of the century, recently presented to the British Museum.

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 beingREFRACTED.Refractiontakes 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 theangle of incidenceand theangle of refractionrespectively, and the first law of refraction is that these angles lie in the same plane, or theplane of refractionis the same as theplane 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 sinesof the angles ofincidence 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.

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 theabsolute 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 thecritical 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 sameare 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 thefacesof 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 thetwo 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 inFig. 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 termedconjugate foci. By afocusin general is meant a point in which a number of rays of light meet, and the rays which thus meet, taken collectively, are termeda pencil.Fig. 7represents 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 itsvirtual 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.

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 theaxis 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) areconvergentorpositivelenses. Lenses with blunt edges (thinner at the centre) aredivergentornegativelenses. 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 allconverge 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 theaxisof 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 theverte.

Parallel rays falling on adouble-convexlens are brought to a focus in the centre of its diameter; conversely, rays diverging from that point are rendered parallel. Hence the focus of adouble-convexlens will be at just half the distance, or half the length, of the focus of aplano-convexlens 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 theprincipal 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; andvice-versâ. The same principles apply equally to aplano-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 itsprincipal 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 thecentre, oroptical 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, inFig. 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, passthrough the principal focus, F, and its intersection with the secondary axis, A O, determines the position ofa, the focus conjugate to A. We can in like manner determine the position ofb, the focus conjugate to B, another point of the object; and the joining linea bwill then be the magnified image of the line A B. It is evident that ifa bwere 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.

The refracting influence of aconcavelens (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 thenegativefocus. This will be, for aplano-concavelens, at the distance of the diameter of the sphere of curvature; and for adouble-concave, in the centre of that sphere.

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

InFig. 14A B is the object anda bthe 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 pointsa bare 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 imagea 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 inFig. 15. The raysa a, which pass through the lens near its circumference, are seen to bemore refracted, or come to a focus at a shorter distance behind it than the raysb b, which pass through near its centre or axis, and areless refracted. The consequence of this defect of lenses with spherical curvatures, which is calledspherical 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 bythe 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 raysa 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 circumferenceb; 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.

Ifa l,a l′, for example (Fig. 16) be part of an ellipse whose greater axis is to the distance between its focif fas the index of refraction is to unity, then parallel raysr l′,r′′ lincident upon the elliptical surfacel′ a l, will be refracted by the single action of that surface into lines which would meet exactly in the farther focusf, if there were no second surface intervening betweenl a l′andf. But as every useful lens must have two surfaces, we have only to describe a circlel a′ l′roundfas a centre, for the second surface of the lensl′ l.

Fig. 16.—Converging Meniscus.

As all the rays refracted at the surfacel a l′converge accurately tof, and as the circular surfacel a′ l′is perpendicular to every one of the refracted rays, all these rays will go on tofwithout 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, theperfection 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 withachromatism, so that a combination of prisms, or lenses, is said to beachromaticwhen 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 aschromatic 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 spectrumwill 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.

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