The eyepiece of a telescope is merely an instrument for magnifying the image produced by the objective or mirror. If one looks through a telescope without its eyepiece, drawing the eye back from the focus to its ordinary distance of distinct vision, the image is clearly seen as if suspended in air, or it can be received on a bit of ground glass.

It appears larger or smaller than the object seen by the naked eye, in proportion as the focal length of the objective is larger or smaller than the distance to which the eye has to drop back to see the image clearly.

This real image, the quality of which depends on the exactness of correction of the objective or mirror, is then to be magnified so much as may be desirable, by the eyepiece of the instrument. In broad terms, then, the eyepiece is a simple microscope applied to the image of an object instead of the object itself.

And looking at the matter in the simplest way the magnifying power of any simple lens depends on the focal length of that lens compared with the ordinary seeing distance of the eye. If this be taken at 10 inches as it often conventionally is, then a lens of 1 inch focus brings clear vision down to an inch from the object, increases the apparent angle covered by the object 10 times and hence gives a magnifying power of 10.

But if the objective has a focal length of 100 inches the image, as we have just seen, is already magnified 10 times as the naked eye sees it, hence with an objective of 100 inches focus and a 1 inch eyepiece the total magnification is 100 diameters. And this expresses the general law, for if we took the normal seeing distance of the naked eye at some other value than 10 inches, say 12½ inches then we should have to reckon the image as magnified by 8 times so far as the objective inches is concerned, but 12½ times due to the 1 inch eyepiece, and so forth. Thus the magnifying power of any eyepiece is F/f where F is the focal length of the objective or mirror and f that of the eyepiece. The focal distance of the eye quite drops out of the reckoning.

All these facts appear very quickly if one explores the image from an objective with a slip of ground glass and a pocket lens. An ordinary camera tells the same story. A distant object which covers 1° will cover on the ground glass 1° reckoned on a radius equal to the focal length of the lens. If this is equal to the ordinary distance of clear vision, an eye at the same distance will see the image (or the distant object) covering the same 1°.

The geometry of the situation is as follows: Let o Fig. 5, Chap. 1, be the objective. This lens, as in an ordinary camera, forms an inverted image of an object A B at its focus, as at a b, and for any point, as a, of the image there is a corresponding point of the object lying on the straight line from A to that point through the center, c, of the objective.

A pair of rays 1, 2, diverging from the object point A pass through rim and center of o respectively and meet in A. After crossing at this point they fall on the eye lens e, and if a is nearly in the principal focus of e, the rays 1 and 2 will emerge substantially parallel so that the eye will unite them to form a clear image.

Now if F is the focal length of o, and f that of a, the object forming the image subtends at the center of the objective, o, an angle A c B, and for a distant object this will be sensibly the angle under which the eye sees the same object.

If L is the half linear dimension of the image, the eye sees half the object covering the angle whose tangent is L/F. Similarly half the image ab is seen through the eye lens e as covering a half angle whose tangent is L/f. Since the magnifying power of the combination, m, is directly as the ratio of increase in this tangent of the visual angle, which measures the image dimension

m = F/f, as before

Further, as all the light which comes in parallel through the whole opening of the objective forms a single conical beam concentrating into a focus and then diverging to enter the eye lens, the diameter of the cone coming through the eye lens must bear the same relation to the diameter of o, that f does to F.

Any less diameter of e will cut off part of the emerging light; any more will show an emergent beam smaller than the eye lens, which is generally the case. Hence if we call p the diameter of the bright pencil of light which we see coming through the eye lens then for that particular eye lens,

m = o/p

That is, f = pF/o which is quite the easiest way of measuring the focal length of an eyepiece.

Point the telescope toward the clear sky, focusing for a distant object so that the emergent pencil is sharply defined at the ocular, and then measure its diameter by the help of a fine scale and a pocket lens, taking care that scale and emergent pencil are simultaneously in sharp focus and show no parallax as the eye is shifted a bit. This bright circle of the emerging beam is actually the projection by the eye lens of the focal image of the objective aperture.

This method of measuring power is easy and rather accurate. But it leads to trouble if the measured diameter of the objective is in fact contracted by a stop anywhere along the path of the beam, as occasionally happens. Examine the telescope carefully with reference to this point before thus testing the power.[17]

The eye lens of Fig. 5 is a simple double convex one, such as was used by Christopher Scheiner and his contemporaries. With a first class objective or mirror the simple eye lens such as is shown in Fig. 98a is by no means to be despised even now. Sir William Herschel always preferred it for high powers, and speaks with evident contempt of observers who sacrificed its advantages to gain a bigger field of view. Let us try to fathom the reason for his vigorously expressed opinion, strongly backed up by experienced observers like the late T. W. Webb and Mr. W. F. Denning.

First of all a single lens saves about 10% of the light. Each surface of glass through which light passes transmits 95 to 96% of that light, so that a single lens transmits approximately 90%, two lenses 81% and so on. This loss may be enough to determine the visibility of an object. Sir Wm. Herschel found that faint objects invisible with the ordinary two lens eyepiece came to view with the single lens.

Probably the actual loss is less serious than its effect on seeing conditions. The loss is due substantially to reflection at the surfaces, and the light thus reflected is scattered close to, or into, the eye and produces stray light in the field which injures the contrast by which faint objects become visible.

In some eyepieces the form of the surfaces is such that reflected light is strongly concentrated where the eye sees it, forming a “ghost” quite bright enough greatly to interfere with the vision of delicate contrasts.

The single lens has a very small sharp field, hardly 10° in angular extent, the image falling off rapidly in quality as it departs from the axis. If plano-convex, as is the eye lens of common two-lens oculars, it works best with the curved side to the eye, i.e., reversed from its usual position, the spherical aberration being much less in this position.

Herschel’s report of better definition with a single lens than with an ordinary two lens ocular speaks ill for the quality of the latter then available. Of course the single lens gives some chromatic aberration, generally of small account with the narrow pencils of light used in high powers.

A somewhat better form of eye lens occasionally used is the so-called Coddington lens, really devised by Sir David Brewster. This, Fig. 98b, is derived from a glass sphere with a thick equatorial belt removed and a groove cut down centrally leaving a diameter of less than half the radius of the sphere. The focus is, for ordinary crown glass, 3/2 the radius of the sphere, and the field is a little improved over the simple lens, but it falls off rather rapidly, with considerable color toward the edge.

The obvious step toward fuller correction of the aberrations while retaining the advantages of the simple lens is to make the ocular achromatic, like a minute objective, thus correcting at once the chromatic and spherical aberrations over a reasonably large field. As the components are cemented the loss of light at their common surface is negligible. Figure 98c shows such a lens. If correctly designed it gives an admirably sharp field of 15° to 20°, colorless and with very little distortion, and is well adapted for high powers.

Still better results in field and orthoscopy can be attained by going to a triple cemented lens, similar to the objective of Fig. 57. Triplets thus constituted are made abroad by Zeiss, Steinheil and others, while in this country an admirable triplet designed by Professor Hastings is made by Bausch & Lomb.

Such lenses give a beautifully flat and sharp field over an angle of 20° to 30°, quite colorless and orthoscopic. Fig. 99a, a form used by Steinheil, is an excellent example of the construction and a most useful ocular. The late R. B. Tolles made such triplets, even down to ? inch focus, which gave admirable results.

A highly specialized form of triplet is the so-called monocentric of Steinheil Fig. 99b. Its peculiarity is less in the fact that all the curves are struck from the same center than in the great thickness of the front flint and the crown, which, as in some photographic lenses, give added facilities for flattening the field and eliminating distortion.

The monocentric eyepiece has a high reputation for keen definition and is admirably achromatic and orthoscopic. The sharp field is about 32°, rather the largest given by any of the cemented combinations. All these optically single lenses are quite free of ghosts, reduce scattered light to a minimum, and leave little to be desired in precise definition. The weak point of the whole tribe is the small field, which, despite Herschel’s opinion, is a real disadvantage for certain kinds of work and wastes the observer’s time unless his facilities for close setting are more than usually good.

Hence the general use of oculars of the two lens types, all of them giving relatively wide fields, some of them faultless also in definition and orthoscopy. The earliest form, Fig. 100, is the very useful and common one used by Huygens and bearing his name, though perhaps independently devised by Campani of Rome. Probably four out of five astronomical eyepieces belong to this class.

The Huygenian ocular accomplishes two useful results—first, it gives a wider sharp field than any single lens, and second it compensates the chromatic aberration, which otherwise must be removed by a composite lens. It usually consists of a plano-convex lens, convex side toward the objective, which is brought inside the objective focus and forms an image in the plane of a rear diaphragm, and a similar eye lens of shorter focus by which this image is examined.

Fig. 100 shows the course of the rays—A being the field lens, B the diaphragm and C the eye lens. Let 1, 2, be rays which are incident near the margin of A. Each, in passing through the lens, is dispersed, the blue being more refracted than the red. Both rays come to a general focus at B, and, crossing, diverge slightly towards C.

But, on reaching C, ray 1, that was nearer the margin and the more refracted because in a zone of greater pitch, now falls on C the nearer its center, and is less refracted than ray 2 which strikes C nearer the rim. If the curvatures of A and C are properly related 1 and 2 emerge from C parallel to each other and thus unite in forming a distinct image.

Now follow through the two branches of l marked l_r, and l_v, the red and violet components. Ray l_v, the more refrangible, strikes C nearer the center, and is the less refracted, emerging from C substantially parallel with its mate l_r, hence blending the red and violet images, if, being of the same glass, A and C have suitably related focal lengths and separation.

As a matter of fact the condition for this chromatic compensation is

d = (f + f')/2

where d is the distance between the lenses and f, f', their respective focal lengths. If this condition of achromatism be combined with that of equal refraction at A and C, favorable to minimizing the spherical aberration, we find f = 3f' and d = 2f'. This is the conventional Huygenian ocular with an eye lens ? the focus of the field lens, spaced at double the focus of the eye lens, with the diaphragm midway.

In practice the ratio of foci varies from 1:3 to 1:2 or even 1:1.5, the exact figure varying with the amount of overcorrection in the objective and under-correction in the eye that has to be dealt with, while the value of d should be adjusted by actual trial on the telescope to obtain the best color correction practicable. One cannot use any chance ocular and expect the finest results.

The Huygenian eyepieces are often referred to as “negative” inasmuch as they cannot be used directly as magnifiers, although dealing effectively with an image rather than an object. The statement is also often made that they cannot be used with cross wires. This is incorrect, for while there is noticeable distortion toward the edge of the wide field, to say nothing of astigmatism, in and near the center of the field the situation is a good deal better.

Central cross wires in the plane of the diaphragm are entirely suitable for alignment of the instrument, and over a moderate extent of field the distortion is so small that a micrometer scale in the plane of the diaphragm gives very good approximate measurements, and indeed is widely used in microscopy.

It should be noted that the achromatism of this type of eyepiece is compensatory rather than real. One cannot at the same time bring the images of various colors to the same size, and also to the same plane. As failure in the latter respect is comparatively unimportant, the Huygenian eyepiece is adjusted so far to compensate the paths of the various rays as to bring the colored images to the same size, and in point of fact the result is very good.

The field of the conventional form of Huygenian ocular is fully 40°, and the definition, particularly centrally, is very excellent. There are no perceptible ghosts produced, and while some 10% of light is lost by reflection in the extra lens it is diffused in the general field and is damaging only as it injures the contrast of faint objects. The theory of the Huygenian eyepiece was elaborately given by Littrow, (Memoirs R. A. S. Vol. 4, p. 599), wherein the somewhat intricate geometry of the situation is fully discussed.

Various modifications of the Huygenian type have been devised and used. Figure 101a is the Airy form devised as a result of a somewhat full mathematical investigation by Sir George Airy, later Astronomer Royal. Its peculiarity lies in the form of the lenses which preserve the usual 3:1 ratio of focal lengths. The field lens is a positive meniscus with a noticeable amount of concavity in the rear face while the eye lens is a “crossed” lens, the outer curvature being about ? of the inner curvature. The marginal field in this ocular is a little better than in the conventional Huygenian.

A commoner modification now-a-days is the Mittenzwey form, Fig. 101b. This is usually made with 2:1 ratio of focal lengths, and the field lens still a meniscus, but less conspicuously concave than in the Airy form. The eye lens is the usual plano-convex. It is widely used, especially abroad, and gives perhaps as large available field as any ocular yet devised, approximately 50°, with pretty good definition out to the margin.

Finally, we come to the solid eyepiece Fig. 102a, devised by the late R. B. Tolies nearly three quarters of a century ago, and and often made by him both for telescopes and microscopes. It is practically a Huygenian eyepiece made out of a single cylinder of glass with a curvature ratio of 1½:1 between the eye and the field lens. A groove is cut around the long lens at about ? its length from the vertex of the field end. This serves as a stop, reducing the diameter of the lens to about one-half its focal length.

It is in fact a Huygenian eyepiece free from the loss of light in the usual construction. It gives a wide field, more extensive than in the ordinary form, with exquisite definition. It is really a most admirable form of eyepiece which should be used far more than is now the case. The late Dr. Brashear is on record as believing that all negative eyepieces less than ¾ inch focus should be made in this form.

So far as the writer can ascertain the only reason that it is not more used is that it is somewhat more difficult to construct than the two lens form, for its curvatures and length must be very accurately adjusted. It is consequently unpopular with the constructing optician in spite of its conspicuous merits. It gives no ghosts, and the faint reflection at the eye end is widely spread so that if the exterior of the cylinder is well blackened, as it should be, it gives exceptional freedom from stray light. Still another variety of the Huygenian ocular sometimes useful is analogous to the compensating eyepiece used in microscopy. If, as commonly is the case, a telescope objective is over-corrected for color to correct for the chromatism of the eye in low powers, the high powers show strong over correction, the blue focus being longer than the red, and the blue image therefore the larger.

If now the field lens of the ocular be made of heavy flint glass and the separation of the lenses suitably adjusted, the stronger refraction of the field lens for the blue pulls up the blue focus and brings its image to substantially the dimensions of the red, so that the eye lens performs as if there were no overcorrection of the objective.

The writer has experimented with an ocular of this sort as shown in Fig. 102b and finds that the color correction is, as might be expected, greatly improved over a Mittenzwey ocular of the same focus (? inch). There would be material advantage in thus varying the ocular color correction to suit the power.

In the Huyghenian eyepiece the equivalent focal length F is given by,

F = 2ff'/(f + f')

where f and f' are the focal lengths of the field and eye lenses respectively. This assumes the normal spacing, d, of half the sum of the focal lengths, not always adhered to by constructors. The perfectly general case, as for any two combined lenses is,

F = ff₁/(f + f₁-d)

To obtain a flatter field, and particularly one free from distortion the construction devised by Ramsden is commonly used. This consists, Fig. 103, of two plano convex lenses of equal focal length, placed with their plane faces outward, at a distance equal to, or somewhat less than, their common focal length. The former spacing is the one which gives the best achromatic compensation since as before the condition for achromatism is

d = ½(f + f')

When thus spaced the plane surface of the field lens is exactly in the focus of the eye lens, the combined focus F is the same as that of either lens, since as just shown in any additive combination of two lenses

F = ff'/(f + f'-d)

and while the field is flat and colorless, every speck of dust on the field lens is offensively in view.

It is therefore usual to make this ocular in the form suggested by Airy, in which something of the achromatic correction is sacrificed to obviate this difficulty, and to obtain a better balance of the residual aberrations. The path of the rays is shown in Fig. 103. The lenses A and B are of the same focal length but are now spaced at ? of this length apart.

The two neighboring rays 1, 2, coming through the objective from the distant object meet at the objective focus in a point, a, of the image plane a b. Thence, diverging, they are so refracted by A and B as to leave the latter substantially parallel so that both appear to proceed from the point c, of the image plane c, d, in the principal focus of B.

From the ordinary equation for the combination, F = ¾ f. The combination focusses ¼ f back of the principal focus of the objective, and the position of the eye is ¼ F back of the eye lens, which is another reason for shortening the lens spacing. At longer spacing the eye distance is inconveniently reduced.

Thus constituted, the Ramsden ocular, known as “positive” from its capability for use as a magnifier of actual objects, gives a good flat field free from distortion over a field of nearly 35° and at some loss of definition a little more. It is the form most commonly used for micrometer work.

In all optical instruments the aberrations increase as one departs from the axis, so that angular field is rather a loose term depending on the maximum aberrations that can be tolerated.[18]

Of the Ramsden ocular there are many modifications. Sometimes f and f' are made unequal or there is departure from the simple plano-convex form. More often the lenses are made achromatic, thus getting rid of the very perceptible color in the simpler form and materially helping the definition. Figure 104a shows such an achromatic ocular as made by Steinheil. The general arrangement is as in the ordinary Ramsden, but the sharp field is slightly enlarged, a good 36°, and the definition is improved quite noticeably.

A somewhat analogous form, but considerably modified in detail, is the Kellner ocular, Fig. 104b. It was devised by an optician of that name, of Wetzlar, who exploited it some three quarters of a century since in a little brochure entitled “Das orthoskopische Okular,” as notable a blast of “hot air” as ever came from a modern publicity agent.

As made today the Kellner ocular consists of a field lens which is commonly plano-convex, plano side out, but sometimes crossed or even equiconvex, combined with a considerably smaller eye lens which is an over-corrected achromatic. The focal length of the field lens is approximately 7/4 F, that of the eye lens 4/3 F, separated by about ¾ F.

This ocular has its front focal plane very near the field lens, sometimes even within its substance, and a rather short eye distance, but it gives admirable definition and a usable field of very great extent, colorless and orthoscopic to the edge. The writer has one of 2? focus, with an achromatic triplet as eye lens, which gives an admirable field of quite 50°.

The Kellner is decidedly valuable as a wide field positive ocular, but it has in common with the two just previously described a sometimes unpleasant ghost of bright objects. This arises from light reflected from the inner surface of the field lens, and back again by the front surface to a focus. This focus commonly lies not far back of the field lens and quite too near to the focus of the eye lens for comfort. It should be watched for in going after faint objects with oculars of the types noted.

A decidedly better form of positive ocular is the modern orthoscopic as made by Steinheil and Zeiss, Fig. 105a. It consists of a triple achromatic field lens, a dense flint between two crowns, with a plano-convex eye lens of much shorter focus (? to ½) almost in contact on its convex side.

The field triplet is heavily over-corrected for color, the front focal plane is nearly ½ F ahead of the front vertex of the field lens, and the eye distance is notably greater than in the Kellner. The field is above 40°, beautifully flat, sharp, and orthoscopic, free of troublesome ghosts. On the whole the writer is inclined to rate it as the best of two-lens oculars.

There should also here be mentioned a very useful long relief ocular, often used for artillery sights, and shown in Fig. 105b. It consists like Fig. 104a, of a pair of achromatic lenses, but they are placed with the crowns almost in contact and are frequently used with a simple plano convex field lens of much longer focus, to render the combination more fully orthoscopic.

The field, especially with the field lens, is wide, quite 40° as apparent angle for the whole instrument, and the eye distance is roughly equal to the focal length. It is a form of ocular that might be very advantageously used in finders, where one often has to assume uncomfortable angles of view, and long relief is valuable.

Whatever the apparent angular field of an ocular may be, the real angular field of view is obtained by dividing the apparent field by the magnifying power. Thus the author’s big Kellner, just mentioned, gives a power of 20 with the objective for which it was designed, hence a real field of 2½°, while a second, power 65, gives a real field of hardly 0°40', the apparent field in this case being a trifle over 40°. There is no escaping this relation, so that high power always implies small field.

The limit of apparent field is due to increasing errors away from the axis, strong curvature of the field, and particularly astigmatism in the outer zones. The eye itself can take in only about 40° so that more than this, while attainable, can only be utilized by peering around the marginal field.

For low powers the usable field is helped out by the accommodation of the eye, but in oculars of short focus the curvature of field is the limiting factor. The radius of curvature of the image is, in a single lens approximately 3/2 F, and in the common two lens forms about ¾ F.

In considering this matter Conrady has shown (M. N. 78 445) that for a total field of 40° the sharpness of field fails at a focal length of about 1 inch for normal power of accommodation. The best achromatic combinations reduce this limit to about ½ inch.

At focal lengths below this the sharpest field is obtainable only with reduced aperture. There is an interesting possibility of building an anastigmatic ocular on the lines of the modern photographic lens, which Conrady suggests, but the need of wide field in high powers is hardly pressing enough to stimulate research.

Finally we may pass to the very simple adjunct of most small telescopes, the terrestrial ocular which inverts the image and shows the landscape right side up. Whatever its exact form it consists of an inverting system which erects the inverted image produced by the objective alone, and an eyepiece for viewing this erected image. In its common form it is composed of four plano-convex lenses arranged as in Fig. 106. Here A and B for the inverting pair and C and D a modified Huygenian ocular. The image from the objective is formed in the front focus of AB which is practically an inverted ocular, and the erected image is formed in the usual way between C and D.

The apparent field is fairly good, about 35°, and while slightly better corrections can be gained by using lenses of specially adjusted curvatures, as Airy has shown, these are seldom applied. The chief objection to this erecting system is its length, some ten times its equivalent focus. Now and then to save light and gain field, the erector is a single cemented combination and the ocular like Fig. 99a or Fig. 102a. Fig. 107 shows a terrestrial eyepiece so arranged, from an example by the late R. B. Tolles. When carefully designed an apparent field of 40° or more can be secured, with great brilliancy, and the length of the erecting system is moderate.

Very much akin in principle is the eyepiece microscope, such as is made by Zeiss to give variable power and a convenient position of the eye in connection with filar micrometers, Fig. 108. It is provided with a focussing collar and its draw tube allows varying power just as in case of an ordinary microscope. In fact eyepiece microscopes have long been now and then used to advantage for high powers. They are easier on the eye, and give greater eye distance than the exceedingly small eye lenses of short focus oculars, and using a solid eyepiece and single lens objective lose no more light than an ordinary Huygenian ocular. The erect resultant image is occasionally a convenience in astronomical use.

Quite analogous to the eyepiece microscope is the so-called “Davon” micro-telescope. Originally developed as an attachment for the substage of a microscope to give large images of objects at a little distance it has grown also into a separate hand telescope, monocular or binocular, for general purposes. The attachment thus developed is shown complete in Fig. 109. D is merely a well corrected objective set in a mount provided with ample stops. The image is viewed by an ordinary microscope or special eyepiece microscope A, as the case may be, furnished with rack focussing at A' and assembled with the objective by means of the carefully centered coupling C.

It furnishes a compact and powerful instrument, very suitable for terrestrial or minor astronomical uses, like the Tolles’ short-focus hand telescopes already mentioned. When properly designed telescopes of this sort give nearly the field of prism glasses, weigh much less and lose far less light for the same effective power and aperture. They also have under fairly high powers rather the advantage in the matter of definition, other things being equal.