TELESCOPE

TELESCOPE, an optical instrument for viewing distant objects; so named by compounding the Greek words tele far off, and skopein I look at or contemplate. This name is commonly appropriated to the larger sizes of the instrument, while the smaller are called PERSPECTIVE-GLASSES, SPY-GLASSES, OPERA-GLASSES. A particular kind, which is thought to be much brighter than the rest, is called a NIGHT-GLASS.

To what has been said already with respect to the inventor of this most noble and useful instrument in the article OPTICS, we may add the two following claims.

Mr Leonhard Digges, a gentleman of the 17th century of great and various knowledge, positively asserts in his Stratagems, and in another work, that his father, a military gentleman, had an instrument which he used in the field, by which he could bring distant objects near, and could know a man at the distance of three miles. He says, that when his father was at home he had often looked through it, and could distinguish the waving of the trees on the opposite side of the Severn. Mr Digges resided in the neighbourhood of Bristol.

Francis Fontana, in his Celestial Observations, published at Naples in 1646, says, that he was assured by a Mr Hardy, advocate of the parliament of Paris, a person of great learning and undoubted integrity, that on the death of his father, there was found among his things an old tube, by which distant objects were distinctly seen; and that it was of a date long prior to the telescope lately invented, and had been kept by him as a secret.

It is not at all improbable, that curious people, handling spectacle glasses, of which there were by this time great varieties, both convex and concave, and amusing themselves with their magnifying power and the singular effects which they produced in the appearances of things, might sometimes chance so to place them as to produce distinct and enlarged vision. We know perfectly, from the table and scheme which Sirturus has given us of the tools or dishes in which the spectacle-

Telescope makers fashioned their glasses, that they had convex lenses formed to spheres of 24 inches diameter, and of 11 inferior sizes. He has given us a scheme of a set which he got leave to measure, belonging to a spectacle-maker of the name of Rogette at Corunna in Spain; and he says that this man had tools of the same sizes for concave glasses. It also appears, that it was a general practice (of which we do not know the precise purpose) to use a convex and concave glass together. If any person should chance to put together a 24-inch convex and a 12-inch concave (wrought on both sides) at the distance of six inches, he would have distinct vision, and the object would appear of double size. Concaves of six inches were not uncommon, and one such combined with the convex of 24, at the distance of nine inches, would have distinct vision, and objects would be quadrupled in diameter. When such a thing occurred, it was natural to keep it as a curiosity, although the rationale of its operation was not in the least understood. We doubt not but that this happened much oftener than in these two instances. The chief wonder is, that it was not frequent, and taken notice of by some writer. It is pretty plain that Galileo's first telescope was of this kind, made up of such spectacle-glasses as he could procure; for it magnified only three times in diameter; a thing easily procured by such glasses as he could find with every spectacle-maker. And he could not but observe, in his trials of their glasses, that the deeper concaves and flatter convexes he employed, he produced the greater amplification; and then he would find himself obliged to provide a tool not used by the spectacle-makers, viz. either a much flatter tool for a convex surface, or a much smaller sphere for a concave; and, notwithstanding his telling us that it was by reflecting on the nature of refraction, and without any instruction, we are persuaded that he proceeded in this very way. His next telescope magnified but five times. Now the slightest acquaintance with the obvious laws of refraction would have directed him at once to a very small and deep concave, which would be much easier made, and have magnified more. But he groped his way with such spectacle-glasses as he could get, till he at last made tools for very flat object-glasses and very deep eye-glasses, and produced a telescope which magnified about 25 times. Sirturus saw it, and took the measures of it. He afterwards saw a scheme of it which Galileo had sent to a German prince at Inspruck, who had it drawn (that is, the circles for the tools) on a table in his gallery. The object-glass was a plano-convex, a portion of a sphere, of 24 inches diameter; the eye-glass was a double concave of two inches diameter; the focal distances were therefore 24 inches and one inch nearly. This must have been a very lucky operation, for Sirturus says it was the best telescope he had seen: and we know that it requires the very best work to produce this magnifying power with such small spheres. Telescopes continued to be made in this way for many years; and Galileo, though keenly engaged in the observation of Jupiter's satellites, being candidate for the prize held out by the Dutch for the discovery of the longitude, and therefore much interested in the advantage which a convex eye-glass would have given him, never made them of any other form. Kepler published his Dioptrics in 1611; in which he tells us, all that he or others had discovered of the law of refraction, viz. that in very

small obliquities of incidence, the angle of refraction Telescope was nearly one-third of the angle of incidence. This was indeed enough to have pointed out, with sufficient exactness, the construction of every optical instrument that we are even now possessed of; for this proportionality of the angles of incidence and refraction is assumed in the construction of the optical figure for all of them; and the deviation from it is still considered as the refinement of the art, and was not brought to any rule till 50 years after by Huyghens, and called by him ABERRATION. Yet even the sagacious Kepler seems not to have seen the advantage of any other construction of the telescope; he just seems to acknowledge the possibility of it: and we are surprised to see writers giving him as the author of the astronomical telescope, or even as hinting at its construction. It is true, in the last proposition he shows how a telescope may be made apparently with a convex eye-glass: but this is only a frivolous fancy; for the eye-glass is directed to be made convex externally, and a very deep concave on the inside; so that it is, in fact, a meniscus with the concavity prevalent. In the 86th proposition, he indeed shows that it is possible so to place a convex glass behind another convex glass, that an eye shall see objects distinct, magnified, and inverted; and he speaks very sagaciously on the subject. After having said that an eye placed behind the point of union of the first glass will see an object inverted, he shows that a small part only will be seen; and then he shows that a convex glass, duly proportioned and properly placed, will show more of it. But in showing this, he speaks in a way which shows evidently that he had formed no distinct notions of the manner in which this effect would be produced, only saying vaguely that the convergency of the second glass would counteract the divergency beyond the focus of the first. Had he conceived the matter with any tolerable distinctness, after seeing the great advantage of taking in a field greater in almost any proportion, he would have eagerly caught at the thought, and enlarged on the immense improvement. Had he but drawn one figure of the progress of the rays through two convex glasses, the whole would have been open to his view.

This step, so easy and so important, was reserved for Father Scheiner, as has been already observed in the article OPTICS; and the construction of this author, together with that of Janzen, are the models on which all refracting telescopes are now constructed; and in all that relates to their magnifying power, brightness, and field of vision, they may be constructed on Kepler's principle, that the angles of refraction are in a certain given proportion to the angles of incidence.

But after Huyghens had applied his elegant geometry to the discovery of Snellius, viz. the proportionality, not of the angles, but of the sines, and had ascertained the aberrations from the foci of infinitely slender pencils, the reasons were clearly pointed out why there were such narrow limits affixed by nature to the performance of optical instruments, in consequence of the indistinctness of vision which resulted from constructions where the magnifying power, the quantity of light, or the field of vision, were extended beyond certain moderate bounds. The theory of aberrations, which that most excellent geometer established, has enabled us to diminish this indistinctness arising from any of these causes; and this diminution

Telescope. diminution is the sole aim of all the different constructions which have been contrived since the days of Galileo and Scheiner.

THE description which has been already given of the various constructions of telescopes in the article OPTICS, is sufficient for instructing the reader in the general principles of their construction, and with moderate attention will show the manner in which the rays of light proceed, in order to ensure the different circumstances of amplification, brightness, and extent of field, and even distinctness of vision, in as far as this depends on the proper intervals between the glasses. But it is insufficient for giving us a knowledge of the improvements which are aimed at in the different departures from the original constructions of Galileo and Scheiner, the advantage of the double eye-glass of Huyghens, and the quintuple eye-glass of Dollond: still more is it insufficient for showing us why the highest degrees of amplification and most extensive field cannot be obtained by the mere proportion of the focal distances of the glasses, as Kepler had taught. In short, without the Huyghenian doctrine of aberrations, neither can the curious reader learn the limits of their performance, nor the artist learn why one telescope is better than another, or in what manner to proceed to make a telescope differing in any particular from those which he servilely copies.

Although all the improvements in the construction of telescopes since the publication of Huyghens's Dioptrics have been the productions of this island, and although Dr Smith of Cambridge has given the most elegant and perspicuous account of this science that has yet appeared, we do not recollect a performance in the English language (except the Optics of Emerson) which will carry the reader beyond the mere schoolboy elements of the science, or enable a person of mathematical skill to understand or improve the construction of optical instruments. The last work on this subject of any extent (Dr Priestley's History of Vision) is merely a parlour book for the amusement of half-taught dilettanti, but is totally deficient in the mathematical part, although it is here that the science of optics has her chief claim to pre-eminence, and to the name of a DISCIPLINA ACCURATA. But this would have been ultra crepidam; and the author would in all probability have made as poor a figure here as he has done in his attempts to degrade his species in his Commentaries on the Vibratiuncule of Hartley; motions which neither the author nor his amplifiers were able to understand or explain. We trust that our readers, jealous as we are of every thing that sinks us in the scale of nature's works, will pardon this transient ejaculation of spleen, when our thoughts are called to a system which, of absolute and unavoidable necessity, makes the DIVINE MIND nothing but a quivering of that matter of which it is the AUTHOR and unerring DIRECTOR. Sed missum faciamus.

We think therefore that we shall do the public some service, by giving such an account of this higher branch of optical science as will at least tend to the complete understanding of this noble instrument, by which our conceptions of the extent of almighty power, and wisdom, and beneficence, are so wonderfully enlarged. In the prosecution of this we hope that many general rules will emerge, by which artists who are not mathematicians may be enabled to construct optical instruments with

intelligence, and avoid the many blunders and defects Telescope. which result from mere servile imitation.

The general aim in the construction of a telescope is, to form, by means of mirrors or lenses, an image of the distant object, as large, as bright, and as extensive as is possible, consistently with distinctness; and then to view the image with a magnifying glass in any convenient manner. This gives us an arrangement of our subject. We shall first show the principles of construction of the object-glass or mirror, so as that it shall form an image of the distant object with these qualities; and then show how to construct the magnifying glass or eye-piece, so as to preserve them unimpaired.

This indistinctness which we wish to avoid arises from two causes; the spherical figures of the refracting and reflecting surfaces, and the different refrangibility of the differently coloured rays of light. The first may be called the SPHERICAL, and the second the CHROMATIC indistinctness; and the deviations from the foci, determined by an elementary theorem, given under OPTICS, may be called the SPHERICAL and the CHROMATIC aberrations.

The limits of a Work like this will not permit us to give any more of the doctrine of aberrations than is absolutely necessary for the construction of achromatic telescopes; and we must refer the reader for a general view of the whole to Euler's Dioptrics, and other works of that kind. Dr Smith has given as much as was necessary for the comparison of the merits of different glasses of similar construction, and this in a very plain and elegant manner.

We shall begin with the aberration of colour, because it is the most simple.

Let white or compounded light fall perpendicularly on the flat side PQ (fig. 1.) of a plano-convex lens PVQ, whose axis is CV and vertex V. The white ray $\backslash rho\; P$ falling on the extremity of the lens is dispersed by refraction at the point P of the spherical surface, and the red ray goes to the point r of the axis, and the violet ray to the point v. In like manner the white ray $qQ$ is dispersed by refraction at Q, the red ray going to r, and the violet to v. The red ray Pr crosses the violet ray Qv in a point D, and Qr crosses Pv in a point E; and the whole light refracted and dispersed by the circumference whose diameter is PQ, passes through the circular area, whose diameter is DE. Supposing that the lens is of such a form that it would collect red rays, refracted by its whole surface in the point r, and violet in the point v; then it is evident that the whole light which occupies the surface of the lens will pass through this little circle, whose diameter is DE. Therefore white light coming from a point so distant that the rays may be considered as parallel, will not be collected in another point or focus, but will be dispersed over the surface of that little circle; which is therefore called the circle of chromatic dispersion; and the radiant point will be represented by this circle. The neighbouring points are in like manner represented by circles; and these circles encroaching on and mixing with each other, must occasion haziness or confusion, and render the picture indistinct. This indistinctness will be greater in the proportion of the number of circles which are in this manner mixed together. This will be in the proportion of the room that is for them; that is, in proportion to the area of the circle, or in the duplicate proportion

Telescope. tion of its diameter. Our first business therefore is, to obtain measures of this diameter, and to mark the connection between it and the aperture and focal distance of the lens.

Let $i$ be to $r$ as the sine of incidence in glass to the sine of refraction of the red rays; and let $i$ be to $v$ as the sine of incidence to the sine of refraction of the violet rays. Then we say, that when the aperture $PQ$ is moderate, $v-r\; :\; v+r-2i\; =\; DE\; :\; PQ$, very nearly. For let $DE$, which is evidently perpendicular to $VR$, meet the parallel incident rays in $K$ and $L$, and the radii of the spherical surface in $G$ and $H$. It is plain that $GPK$ is equal to the angle of incidence on the posterior or spherical surface of the lens; and $GPr$ and $GPv$ are the angles of the refraction of the red and the violet rays; and that $GK$, $GD$, and $GE$, are very nearly as the sines of those angles, because the angles are supposed to be small. We may therefore institute this proportion $DE\; :\; KD\; =\; v-r\; :\; r-i$; then, by doubling the consequents $DE\; :\; 2KD\; =\; v-r\; :\; 2r-2i$. Also $DE\; :\; 2KD\; +\; DE\; =\; v-r\; :\; 2r-2i\; +\; v-r\; =\; v-r\; :\; r+v-2i$. But $2KD\; +\; DE$ is equal to $KL$ or $PQ$. Therefore we have $DE\; :\; PQ\; =\; v-r\; :\; r+v-2i$. $Q.\; E.\; D.$

Cor. 1. Sir Isaac Newton, by most accurate observation, found, that in common glass the sines of refraction of the red and violet rays were 77 and 78 where the sine of incidence was 50. Hence it follows, that $v-r$ is to $v+r-2i$ as 1 to 55; and that the diameter of the smallest circle of dispersion is $\backslash frac\{1\}\{55\}$th part of that of the lens.

2. In like manner may be determined the circle of dispersion that will comprehend the rays of any particular colour or set of colours. Thus all the orange and yellow will pass through a circle whose diameter is $\backslash frac\{1\}\{55\}$th of that of the lens.

3. In different surfaces, or plano-convex lenses, the angles of aberration $rPv$ are as the breadth $PQ$ directly, and as the focal distance $VF$ inversely; because any angle $DPE$ is as its subtense $DE$ directly and radius $DP$ inversely. N. B. We call $VF$ the focal distance, because at this distance, or at the point $F$, the light is most of all consolidated. If we examine the focal distance by holding the lens to the sun, we judge it to be where the light is drawn into the smallest spot.

When we reflect that a lens of $\backslash frac\{1\}\{2\}$ inches in diameter has a circle of dispersion $\backslash frac\{1\}\{55\}$th of an inch in diameter, we are surprised that it produces any picture of an object that can be distinguished. We should not expect greater distinctness from such a lens than would be produced in a camera obscura without a lens, by simply admitting the light through a hole of $\backslash frac\{1\}\{55\}$th of an inch in diameter. This, we know, would be very hazy and confused. But when we remark the superior vivacity of the yellow and orange light in comparison with the rest, we may believe that the effect produced by the confusion of the other colours will be much less sensible. But a stronger reason is, that the light is much denser in the middle of the circle of dispersion, and is exceedingly faint towards the margin. This, however, must not be taken for granted; and we must know distinctly the manner in which the light of different colours is distributed over the circle of chromatic dispersion, before we pretend to pronounce on the immense difference between the indistinctness arising from colour and that

arising from the spherical figure. We think this the Telescope. more necessary, because the illustrious discoverer of the chromatic aberration has made a great mistake in the comparison, because he did not consider the distribution of the light in the circle of spherical dispersion. It is therefore proper to investigate the chromatic distribution of the light with the same care that we bestowed on the spherical dispersion in OPTICS, and we shall then see that the superiority of the reflecting telescope is incomparably less than Newton imagined it to be.

Therefore let $EB$ (fig. 2.) represent a plano-convex Fig. 2. lens, of which $C$ is the centre and $Cr$ the axis. Let us suppose it to have no spherical aberration, but to collect rays occupying its whole surface to single points in the axis. Let a beam of white or compounded light fall perpendicularly on its plane surface. The rays will be so refracted by its curved surface, that the extreme red rays will be collected at $r$, the extreme violet rays at $w$, and those of intermediate refrangibility at intermediate points, $o,\; y,\; g,\; b,\; p,\; v$, of the line $rw$, which is nearly $\backslash frac\{1\}\{55\}$th of $rC$. The extreme red and violet rays will cross each other at $A$ and $D$; and $AD$ will be a section or diameter of the circle of chromatic dispersion, and will be about $\backslash frac\{1\}\{55\}$th of $EB$. We may suppose $wr$ to be bisected in $b$, because $wb$ is to $br$ very nearly in the ratio of equality (for $rb\; :\; rC\; =\; bA\; :\; cE\; =\; bA\; :\; cB\; =\; wb\; :\; wC$). The line $rw$ will be a kind of prismatic spectrum, red from $r$ to $o$, orange-coloured from $o$ to $y$, yellow from $y$ to $g$, green from $g$ to $b$, blue from $b$ to $p$, purple from $p$ to $v$, and violet from $v$ to $w$.

The light in its compound state must be supposed uniformly dense as it falls upon the lens; and the same must be said of the rays of any particular colour. Newton supposes also, that when a white ray, such as $cE$, is dispersed into its component coloured rays by refraction at $E$, it is uniformly spread over the angle $DEA$. This supposition is indeed gratuitous; but we have no argument to the contrary, and may therefore consider it as just. The consequence is, that each point of the spectrum is not only equally luminous, but also illuminates uniformly its corresponding portion of $AD$: that is to say, the coating (so to term it) of any particular colour, such as purple, from the point $p$, is uniformly dense in every part of $AD$ on which it falls. In like manner, the colouring of yellow, intercepted by a part of $AD$ in its passage to the point $y$, is uniformly dense in all its parts. But the density of the different colours in $AD$ is extremely different: for since the radiation in $w$ is equally dense with that in $p$, the density of the violet colouring, which radiates from $w$, and is spread over the whole of $AD$, must be much less than the density of the purple colouring, which radiates from $p$, and occupies only a part of $AD$ round the circle $b$. These densities must be very nearly in the inverse proportion of $wb^2$ to $p^2b^2$.

Hence we see, that the central point $b$ will be very intensely illuminated by the blue radiating from $p^2b^2$ and the green intercepted from $bg$. It will be more faintly illuminated by the purple radiating from $vp$, and the yellow intercepted from $gy$; and still more faintly by the violet from $wv$, and the orange and red intercepted from $yr$. The whole colouring will be a white, tending a little to yellowness. The accurate proportion of these

Telescope. these colourings may be computed from our knowledge of the position of the points $a,\; y,\; g$, &c. But this is of little moment. It is of more consequence to be able to determine the proportion of the total intensity of the light in $b$ to its intensity in any other point $I$.

For this purpose draw $rIR,\; IwW$, meeting the lens in $R$ and $W$. The point $I$ receives none of the light which passes through the space $RW$: for it is evident that $bI\; :\; CR\; =\; bA\; :\; CE\; =\; 1\; :\; 55$, and that $CR\; =\; CW$; and therefore, since all the light incident on $EB$ passes through $AB$, all the light incident on $RW$ passes through $Iz$ ($bz$ being made $=\; bI$). Draw $oIO,\; yIY,\; gIG,\; I\backslash rho\; P,\; I\backslash sigma\; V$. It is plain, that $I$ receives red light from $RO$, orange from $OY$, yellow from $YG$, green from $GE$, a little blue from $BP$, purple from $PV$, and violet from $VW$. It therefore wants some of the green and of the blue.

That we may judge of the intensity of these colours at $I$, suppose the lens covered with paper pierced with a small hole at $G$. The green light only will pass through $I$; the other colours will pass between $I$ and $b$, or between $I$ and $A$, according as they are more or less refrangible than the particular green at $I$. This particular colour converges to $g$, and therefore will illuminate a small spot round $I$, where it will be as much denser than it is at $G$ as this spot is smaller than the hole at $G$. The natural density at $G$, therefore, will be to the increased density at $I$, as $gI^2$ to $gG^2$, or as $g^2b^2$ to $g^2A^2$, or as $bI^2$ to $GA^2$. In like manner, the natural density of the purple coming to $I$ through an equal hole at $P$ will be to the increased density at $I$ as $bI^2$ to $CP^2$. And thus it appears, that the intensity of the differently coloured illuminations of any point of the circle of dispersion, is inversely proportional to the square of the distance from the centre of the lens to the point of its surface through which the colouring light comes to this point of the circle of dispersion. This circumstance will give us a very easy, and, we think, an elegant solution of the question.

Bisect $CE$ in $F$, and draw $FL$ perpendicular to $CE$, making it equal to $CF$. Through the point $L$ describe the hyperbola $KLN$ of the second order, that is, having the ordinates $EK,\; FL,\; RN$, &c. inversely proportional to the squares of the abscissæ $CE,\; CF,\; CR$, &c.; so that $FL\; :\; RN\; =\; \backslash frac\{1\}\{CF^2\}\; :\; \backslash frac\{1\}\{CR^2\}$, or $=\; CR^2\; :\; CF^2$, &c. It is evident that these ordinates are proportional to the densities of the severally coloured lights which go from them to any points whatever of the circle of dispersion.

Now the total density of the light at $I$ depends both on the density of each particular colour and on the number of colours which fall on it. The ordinates of this hyperbola determine the first; and the space $ER$ measures the number of colours which fall on $I$, because it receives light from the whole of $ER$, and of its equal $BW$. Therefore, if ordinates be drawn from any point of $ER$, their sum will be as the whole light which goes to $I$; that is, the total density of the light at $I$ will be proportional to the area $NREK$. Now it is known that $CE\; \backslash times\; EK$ is equal to the infinitely extended area lying beyond $EK$; and $CR\; \backslash times\; RN$ is equal to the infinitely extended area lying beyond $RN$. Therefore the area $NREK$ is equal to $CR\; \backslash times\; RN\; -\; CE\; \backslash times\; EK$. But $RN$

and $EK$ are respectively equal to $\backslash frac\{CF^2\}\{CR^2\}$ and $\backslash frac\{CF^2\}\{CE^2\}$. There- Telescope.

fore the density at $I$ is proportional to $CF^2\; \backslash times\; \backslash left(\; \backslash frac\{CR\}\{CR^2\}\; -\; \backslash frac\{CE\}\{CE^2\}\; \backslash right)\; =\; CF^2\; \backslash times\; \backslash left(\; \backslash frac\{1\}\{CR\}\; -\; \backslash frac\{1\}\{CE\}\; \backslash right)\; =\; CF^2\; \backslash times\; \backslash frac\{CE\; -\; CR\}\{CE\; \backslash times\; CR\}\; =\; CF^2\; \backslash times\; \backslash frac\{ER\}\{CE\; \backslash times\; CR\}\; =\; \backslash frac\{CF^2\}\{CE\}\; \backslash times\; \backslash frac\{ER\}\{CR\}$. But because $CF$ is $\backslash frac\{1\}\{2\}$ of $CE$, $\backslash frac\{CF^2\}\{CE\}$ is $=\; \backslash frac\{CF^2\}\{2CF\}\; =\; \backslash frac\{CF\}\{2\}$, a constant quantity.

Therefore the density of the light at $I$ is proportional to $\backslash frac\{ER\}\{CR\}$, or to $\backslash frac\{AI\}\{bI\}$ because the points $R$ and $I$ are similarly situated in $EC$ and $A\; b$.

Farther, if the semiaperture $CE$ of the lens be called $1$, $\backslash frac\{CF^2\}\{2\}$ is $=\; \backslash frac\{1\}\{2\}$, and the density at $I$ is $=\; \backslash frac\{AI\}\{8bI\}$.

Here it is proper to observe, that since the point $R$ has the same situation in the diameter $EB$ that the point $I$ has in the diameter $AD$ of the circle of dispersion, the circle described on $EB$ may be conceived as the magnified representation of the circle of dispersion. The point $F$, for instance, represents the point $f$ in the circle of dispersion, which bisects the radius $bA$; and $f$ receives no light from any part of the lens which is nearer the centre than $F$, being illuminated only by the light which comes through $EF$ and its opposite $BF$. The same may be said of every other point.

In like manner, the density of the light in $f$, the middle between $b$ and $A$, is measured by $\backslash frac\{EF\}\{CF\}$, which is $=\; \backslash frac\{EF\}\{CF\}$, or $1$. This makes the density at this point a proper standard of comparison. The density there is to the density at $I$ as $1$ to $\backslash frac\{AI\}\{bI\}$, or as $bI$ to $AI$; and this is the simplest mode of comparison. The density half way from the centre of the circle of dispersion is to the density at any point $I$ as $bI$ to $IA$.

Lastly, through $L$ describe the common rectangular hyperbola $kLn$, meeting the ordinates of the former in $k,\; L$, and $n$; and draw $kh$ parallel to $EC$, cutting the ordinates in $g,\; f,\; r$, &c. Then $CR\; :\; CE\; =\; Ek\; :\; Rn$, and $CR\; :\; CE\; -\; CR\; =\; Ek\; :\; Rn\; -\; Ek$, or $CR\; :\; RE\; =\; Ek\; :\; rn$, and $bI\; :\; IA\; =\; Ek\; :\; rn$. And thus we have a very simple expression of the density in any point of the circle of dispersion. Let the point be anywhere, as at $I$. Divide the lens in $R$ as $AD$ is divided in $I$, and then $rn$ is as the density in $I$.

These two measures were given by Newton; the first in his Treatise de Mundi Systemate, and the last in his Optics; but both without demonstration.

If the hyperbola $kLn$ be made to revolve round the axis $CQ$, it will generate a solid spindle, which will measure the whole quantity of light which passes through different portions of the circle of dispersion. Thus the solid produced by the revolution of $Lkf$ will measure all the light which occupies the outer part of the circle of dispersion lying without the middle of the radius. This space is $\backslash frac\{3\}\{4\}$th of the whole circle; but the quantity of light is but $\backslash frac\{1\}\{4\}$th of the whole.

Telecope. A still more simple expression of the whole quantity of light passing through different portions of the circle of chromatic dispersion may now be obtained as follows:

It has been demonstrated, that the density of the light at I is as $\backslash frac\{AI\}\{bI\}$, or as $\backslash frac\{ER\}\{CR\}$. Suppose the figure to turn round the axis. I or R describe circumferences of circles; and the whole light passing through this circumference is as the circumference, or as the radius, and as the density jointly. It is therefore as $\backslash frac\{ER\}\{CR\}\; \backslash times\; CR$, that is, as $ER$. Draw any straight line $Em$, cutting $RN$ in $s$, and any other ordinate $FL$ in $r$. The whole light which illuminates the circumference described by I is to the whole light which illuminates the centre $b$ as $ER$ to $EC$, or as $Rs$ to $Cm$. In like manner, the whole light which illuminates the circumference described by the point $f$ in the circle of dispersion is to the whole light which illuminates the centre $b$, as $Fx$ to $Cm$. The lines $Cm$, $RS$, $Fx$, are therefore proportional to the whole light which illuminates the corresponding circumferences in the circle of dispersion. Therefore the whole light which falls on the circle whole radius is $bI$, will be represented by the trapezium in $CRS$; and the whole light which falls on the ring described by $IA$, will be represented by the triangle $ESR$; and so of any other portions.

By considering the figure, we see that the distribution of the light is exceedingly unequal. Round the margin it has no sensible density; while its density in the very centre is incomparably greater than in any other point, being expressed by the asymptote of a hyperbola. Also the circle described with the radius $\backslash frac\{Ab\}\{2\}$ contains $\backslash frac\{1\}\{4\}$ths of the whole light. No wonder then that the confusion caused by the mixture of these circles of dispersion is less than one should expect; besides, it is evident that the most lively or impressive colours occupy the middle of the spectrum, and are there much denser than the rest. The margin is covered with an illumination of deep red and violet, neither of which colours are brilliant. The margin will be of a dark claret colour. The centre revives all the colours, but in a proportion of intensity greatly different from that in the common prismatic spectrum, because the radiant points $I,\; p,\; b,\; g$, &c. by which it is illuminated, are at such different distances from it. It will be white; but we apprehend not a pure white, being greatly overcharged with the middle colours.

These considerations show that the coloured fringes, which are observed to border very luminous objects seen on a dark ground through optical instruments, do not proceed from the object-glass of a telescope or microscope, but from an improper construction of the eyeglasses. The chromatic dispersion would produce fringes of a different colour, when they produce any at all, and the colours would be differently disposed. But this dispersion by the object-glass can hardly produce any fringes: its effect is a general and almost uniform mixture of circles all over the field, which produces an uniform haziness, as if the object were viewed at an improper distance, or out of its focus, as we vulgarly express it.

VOL. XX. Part I.

Telecope. We may at present form a good guess at the limit which this cause puts to the performance of a telescope. A point of a very distant object is represented, in the picture formed by the object-glass, by a little circle, whose diameter is at least $\backslash frac\{1\}\{100\}$th of the aperture of the object-glass, making a very full allowance for the superior brilliancy and density of the central light. We look at this picture with a magnifying eye-glass. This magnifies the picture of the point. If it amplify it to such a degree as to make it an object individually distinguishable, the confusion is then sensible. Now this can be computed. An object subtending one minute of a degree is distinguished by the duldest eye, even although it be a dark object on a bright ground. Let us therefore suppose a telescope, the object-glass of which is of six feet focal distance, and one inch aperture. The diameter of the circle of chromatic dispersion will be $\backslash frac\{1\}\{100\}$th of an inch, which subtends at the centre of the object-glass an angle of about 9$\backslash frac\{1\}\{2\}$ seconds. This, when magnified six times by an eye-glass, would become a distinguishable object; and a telescope of this length would be indistinct if it magnified more than six times, if a point were thus spread out into a spot of uniform intensity. But the spot is much less intense about its margin. It is found experimentally that a piece of engraving, having fine cross hatches, is not sensibly indistinct till brought so far from the limits of perfectly distinct vision, that this indistinctness amounts to 6' or 5' in breadth.—Therefore such a telescope will be sensibly distinct when it magnifies 36 times; and this is very agreeable to experience.

We come, in the second place, to the more arduous task of ascertaining the error arising from the spherical figure of the surfaces employed in optical instruments. — Suffice it to say, before we begin, that although geometers have exhibited other forms of lenses which are totally exempt from this error, they cannot be executed by the artist; and we are therefore restricted to the employment of spherical surfaces.

Of all the determinations which have been given of spherical aberration, that by Dr Smith, in his Optics, which is an improvement of the fundamental theorem of that most elegant geometer Huyghens, is the most perspicuous and palpable. Some others are more concise, and much better fitted for after use, and will therefore be employed by us in the prosecution of this article. But they do not keep in view the optical facts, giving the mind a picture of the progress of the rays, which it can contemplate and discover amidst many modifying circumstances. By ingenious substitutions of analytical symbols, the investigation is rendered expeditious, concise, and certain; but these are not immediate symbols of things, but of operations of the mind; objects sufficiently subtle of themselves, and having no need of substitutions to make us lose sight of the real subject; and thus our occupation degenerates into a process almost without ideas. We shall therefore set out with Dr Smith's fundamental Theorem.

Let $AVB$ (fig. 3.) be a concave spherical mirror, of which $C$ is the centre, $V$ the vertex, $CV$ the axis, and $F$ the focus of an infinitely slender pencil of parallel rays passing

Telescope passing through the centre. Let the ray $aA$, parallel to the axis, be reflected in $AG$, crossing the central ray $CV$ in $f$. Let $AP$ be the sine of the semi-aperture $AV$, $AD$ its tangent, and $CD$ its secant.

The aberration $Ff$ from the principal focus of central rays is equal to $\backslash frac\{1\}\{2\}$ of the excess $VD$ of the secant above the radius, or very near equal to $\backslash frac\{1\}\{2\}$ of $VP$, the versed sine of the semi-aperture.

For because $AD$ is perpendicular to $CA$, the points $C,\; A,\; D$, are in a circle, of which $CD$ is the diameter; and because $AF$ is equal to $Cf$, by reason of the equality of the angles $fAC,\; fCA$, and $CAa$, $f$ is the centre of the circle through $C,\; A,\; D$, and $fD$ is $\backslash frac\{1\}\{2\}\; CD$. But $FC$ is $\backslash frac\{1\}\{2\}\; CV$. Therefore $Ff$ is $\backslash frac\{1\}\{2\}$ of $VD$.

But because $DV\; :\; VP\; =\; DC\; :\; VC$, and $DC$ is very little greater than $VC$ when the aperture $AB$ is moderate, $DV$ is very little greater than $VP$, and $Ff$ is very nearly equal to $\backslash frac\{1\}\{2\}$ of $VP$.

Cor. 1. The longitudinal aberration is $=\; \backslash frac\{AV^2\}\{4CV\}$, for $VP$ is very nearly $=\; \backslash frac\{AV^2\}\{2CV\}$.

Cor. 2. The lateral aberration $FG$ is $=\; \backslash frac\{AV^2\}\{2CV^2\}$. For $FG\; :\; Ff\; =\; AP\; :\; Pf\; =\; AV\; :\; \backslash frac\{1\}\{2\}\; CV$ nearly, and therefore $FG\; =\; \backslash frac\{AV^2\}\{4CV\}\; \backslash times\; \backslash frac\{2\}\{CV\}\; =\; \backslash frac\{AV^2\}\{2CV^2\}$.

Fig. 4 or 5. Let $AVB$ (fig. 4 or 5.) be a spherical surface separating two refracting substances, $C$ the centre, $V$ the vertex, $AV$ the semi-aperture, $AP$ its sine, $PV$ its versed sine, and $F$ the focus of parallel rays infinitely near to the axis. Let the extreme ray $aA$, parallel to the axis, be refracted into $AG$, crossing $CF$ in $f$, which is therefore the focus of extreme parallel rays.

The rectangle of the sine of incidence, by the difference of the sines of incidence and refraction, is to the square of the sine of refraction, as the versed sine of the semi-aperture is to the longitudinal aberration of the extreme rays.

Call the sine of incidence $i$, the sine of refraction $r$, and their difference $d$.

Join $CA$, and about the centre $f$ describe the arch $AD$.

The angle $ACV$ is equal to the angle of incidence, and $CAf$ is the angle of refraction. Then, since the sine of incidence is to the sine of refraction as $VF$ to $CF$, or as $As$ to $Cf$, that is, as $Df$ to $Cf$, we have

by conversion $CF\; :\; FV\; =\; Cf\; :\; fD$
altern. conver. $CF\; -\; Cf\; :\; CV\; -\; CD\; =\; CF\; :\; CV$
or $Ff\; :\; VD\; =\; CF\; :\; CV,\; =\; r\; :\; d$.

Now $PV\; =\; \backslash frac\{AP^2\}\{CP\; +\; CV\}\; =\; \backslash frac\{AP^2\}\{2CV\}$ nearly, and $PD\; =\; \backslash frac\{AP^2\}\{fP\; +\; fV\}$
$=\; \backslash frac\{AP^2\}\{2fV\}$ nearly, $=\; \backslash frac\{AP^2\}\{2FV\}$ nearly. Therefore $PV\; :\; PD$

$=\; FV\; :\; CV$, and $DV\; :\; PV\; =\; CF\; :\; FV$ nearly.
We had above $Ff\; :\; VD\; =\; r\; :\; d$;
and now $VD\; :\; PV\; =\; CF\; :\; FV,\; =\; r\; :\; i$;
therefore $Ff\; :\; PV\; =\; r^2\; :\; di$.

and $Ff\; =\; \backslash frac\{r^2\}\{di\}\; \backslash times\; PV$. Q. E. D.

Telescope The aberration will be different according as the refraction is made towards or from the perpendicular; that is, according as $r$ is less or greater than $i$. They are in the ratio of $\backslash frac\{r^2\}\{d^2\}$ to $\backslash frac\{i^2\}\{d^2\}$, or of $r^2$ to $i^2$. The aberration therefore is always much diminished when the refraction is made from a rare into a dense medium. The proportion of the sines for air and glass is nearly that of 3 to 2. When the light is refracted into the glass, the aberration is nearly $\backslash frac\{1\}\{2\}$ of $VP$; and when the light passes out of glass into air, it is about $\backslash frac\{2\}\{3\}$ of $VP$.

Cor. 1. $Ff\; =\; \backslash frac\{r^2\}\{d^2\}\; \backslash times\; \backslash frac\{AP^2\}\{2CV\}$ nearly, and it is also $=\; \backslash frac\{r^2\}\{d^2\}\; \backslash times\; \backslash frac\{AP^2\}\{2FV\}$, because $PV\; =\; \backslash frac\{AP^2\}\{2CV\}$ nearly, and $i\; :\; d\; =\; FV\; :\; CV$.

Cor. 2. Because $fP\; :\; PA\; =\; Ff\; :\; FG$
or $FV\; :\; AV\; =\; Ff\; :\; FG$ nearly,
we have $FG$, the lateral aberration, $=\; Ff\; \backslash times\; \backslash frac\{AV\}\{FV\}\; =$
$\backslash frac\{r^2\}\{d^2\}\; \backslash times\; \backslash frac\{AV^2\}\{2FV^2\}\; =\; \backslash frac\{r^2\}\{i^2\}\; \backslash times\; \backslash frac\{AV^2\}\{2CV^2\}$.

Cor. 3. Because the angle $Faf$ is proportional to $\backslash frac\{FG\}\{FV\}$
very nearly, we have the angular aberration $Faf\; =\; \backslash frac\{r^2\}\{d^2\}$
$\backslash times\; \backslash frac\{AV^2\}\{2FV^2\}\; =\; \backslash frac\{r^2\}\{i^2\}\; \backslash times\; \backslash frac\{AV^2\}\{2CV^2\}$.

In general, the longitudinal aberrations from the focus of central parallel rays are as the squares of the apertures directly, and as the focal distances inversely; and the lateral aberrations are as the cubes of the apertures directly, and the squares of the focal distances inversely; and the angular aberrations are as the cubes of the aperture directly, and the cubes of the focal distances inversely.

The reader must have observed, that to simplify the investigation, some small errors are admitted. $VP$ and $PD$ are not in the exact proportion that we assumed them, nor is $Df$ equal to $FV$. But in the small apertures which suffice for optical instruments, these errors may be disregarded.

This spherical aberration produces an indistinctness of vision, in the same manner as the chromatic aberration does, viz. by spreading out every mathematical point of the object into a little spot in its picture; which spots, by mixing with each other, confuse the whole. We must now determine the diameter of the circle of diffusion, as we did in the case of chromatic dispersion.

Let a ray $\backslash beta\; a$ (fig. 6.) be refracted on the other side of the axis, into $a\backslash phi$, cutting $AsG$ in $H$, and draw the perpendicular $EH$. Call $AVa$, $aV\backslash alpha$, $Vf$ (or $Vf$, or $V\backslash phi$, which in this comparison may be taken as equal) $=\; f$, $Ff\; =\; b$, and $fE\; =\; \backslash phi\backslash alpha$.

$AV^2\; :\; aV^2\; =\; Ff\; :\; F\backslash phi$ (already demonstrated) and $F\backslash phi\; =\; \backslash frac\{a^2\}\{a^2\; -\; b^2\}\; b$, and $Ff\; -\; F\backslash phi$, (or $f\backslash phi$) $=\; b\; -\; \backslash frac\{a^2\}\{a^2\; -\; b^2\}\; b\; =\; \backslash frac\{a^2\; b\; -\; a^2\; b\}\{a^2\}$,
$=\; \backslash frac\{b\}\{a^2\}\; \backslash times\; a^2\; -\; a^2\; =\; \backslash frac\{b\}\{a^2\}\; \backslash times\; a^2\; +\; a^2\; -\; a^2$. Also $Pf\; :\; PA\; =\; fE$

$\underbrace{\text{Telescope.}}_{= fE : EH, \text{ or } f : a = x : \frac{ax}{f}, = EH. \text{ And } P\pi : P\phi} \\ = EH : E\phi, \text{ or } x : f = \frac{ax}{f} : \frac{ax}{x}, = E\phi. \text{ Therefore} \\ f\phi = \frac{ax}{a} + x, = \frac{a+x}{a}, = \frac{x}{a} \times a + x. \text{ Therefore } \frac{x}{a} \\ \times a + x = \frac{b}{a^2} \times a + x \times a - x, \text{ and } \frac{x}{a} = \frac{b}{a^2} \times a - x, \\ \text{and } x = \frac{b}{a^2} \times a (a - x). \text{ Therefore } x \text{ is greatest when} \\ a \times a - x \text{ is greatest; that is, when } x = \frac{1}{2}a. \text{ Therefore} \\ EH \text{ is greatest when } P\pi \text{ is equal to the half of } AP. \\ \text{When this is the case, we have at the same time } \frac{b}{a^2} \times a \\ (a - x) = \frac{b}{a^2} \times \frac{1}{2}a^2, \text{ and } x = \frac{1}{2}b, \text{ or } EH = \frac{1}{2}FG. \text{ That} \\ \text{is, the diameter of the circle of aberration through which} \\ \text{the whole of the refracted light must pass, is } \frac{1}{2} \text{ of the} \\ \text{diameter of the circle of aberration at the focus of pa-} \\ \text{rallel central rays. In the chromatic aberration it was} \\ \frac{1}{2}; \text{ so that in this respect the spherical aberration does} \\ \text{not create so great confusion as the chromatic.}$

We are now able to compare them, since we have now the measure of both the circles of aberration.

It has not been found possible to give more than four inches of aperture to an object glass of 100 feet focal distance, so as to preserve sufficient distinctness. If we compute the diameter of the circle EH corresponding to this aperture, we shall find it not much to exceed $\backslash frac\{1\}\{120,000\}$ of an inch. If we restrict the circle of chromatic dispersion to $\backslash frac\{1\}\{625\}$ of the aperture, which is hardly the fifth part of the whole dispersion in it, it is $\backslash frac\{1\}\{625\}$ of an inch, and is about 1900 times greater than the other.

The circle of spherical aberration of a plano-convex lens, with the plane side next the distant object, is equal to the circle of chromatic dispersion when the semi-aperture is about $15^\backslash circ$: For we saw formerly that EH is $\backslash frac\{1\}\{2\}$ of FG, and that FG is $=\; \backslash frac\{r^2\}\{2AC^2\}\; \backslash frac\{AP^2\}\{2AC^2\}$, and therefore $EG\; =\; \backslash frac\{r^2\}\{2AC^2\}\; \backslash times\; \backslash frac\{AP^2\}\{8AC^2\}$. This being made $=\; \backslash frac\{AP\}\{55\}$, gives us $AP\; =\; \backslash sqrt\{\backslash frac\{8^2\; AC^2\}\{55\; r^2\}\}$, which is nearly $\backslash frac\{AC\}\{4\}$, and corresponds to an aperture of $30^\backslash circ$ diameter, if $r$ be to $i$ as 3 to 2.

Sir Isaac Newton was therefore well entitled to say, that it was quite needless to attempt figures which should have less aberration than spherical ones, while the confusion produced by the chromatic dispersion remained uncorrected. Since the indistinctness is as the squares of the diameters of the circles of aberration, the disproportion is quite beyond our imagination, even when Newton has made such a liberal allowance to the chromatic dispersion. But it must be acknowledged, that he has not attended to the distribution of the light in the circle of spherical aberration, and has hastily supposed it to be like the distribution of the coloured light, indefinitely rare in the margin, and denser in the centre.

$\backslash underbrace\{\backslash text\{Telescope.\}\}\_\{\backslash text\{We\; are\; indebted\; to\; Father\; Boscovich\; for\; the\; elegant\; determination\; of\; this\; distribution,\; which\; we\; have\; given\; in\; the\; article\; OPTICS.\; From\; this\; it\; appears,\; that\; the\; light\; in\; the\; margin\; of\; the\; circle\; of\; spherical\; aberration,\; instead\; of\; being\; incomparably\; rarer\; than\; in\; the\; spaces\; between\; it\; and\; the\; centre,\; is\; incomparably\; denser.\; The\; indistinctness\; therefore\; produced\; by\; the\; intersection\; of\; these\; luminous\; circumferences\; is\; vastly\; great,\; and\; increases\; the\; whole\; indistinctness\; exceedingly.\; By\; a\; gross\; calculation\; which\; we\; made,\; it\; appears\; to\; be\; increased\; at\; least\; 500\; times.\; The\; proportional\; indistinctness\; therefore,\; instead\; of\; being\; 19:01to\; 1,\; is\; only\; 19003to\; 500,\; or\; nearly\; 7220\; to\; 1;\; a\; proportion\; still\; sufficiently\; great\; to\; warrant\; Newton\text{'}s\; preference\; of\; the\; reflecting\; telescope\; of\; his\; invention.\; And\; we\; may\; now\; observe,\; that\; the\; reflecting\; telescope\; has\; even\; a\; great\; advantage\; over\; a\; refracting\; one\; of\; the\; same\; focal\; distance,\; with\; respect\; to\; its\; spherical\; aberration:\; For\; we\; have\; seen\; (Cor.\; 2.)\; that\; the\; lateral\; aberration\; is$ \backslash frac\{r^2\}\{2\}\; \backslash frac\{AV^2\}\{2CV^2\}$.\; This\; for\; a\; plano-convex\; glass\; is\; nearly$ \backslash frac\{9\}\{4\}\; \backslash frac\{AV^2\}\{2CV^2\}$.\; And\; the\; diameter\; of\; the\; circle\; of\; aberration$

is one-fourth of this, or $\backslash frac\{9\}\{16\}\; \backslash times\; \backslash frac\{AV^2\}\{2CV^2\}$. In like manner,

the lateral aberration of a concave mirror is $\backslash frac\{AV^2\}\{2CV^2\}$; and

the diameter of the circle of dispersion is $\backslash frac\{AV^2\}\{8CV^2\}$; and

therefore if the surfaces were portions of the same sphere, the diameter of the circle of aberration of refracted rays would be to that of the circle of aberration of reflected rays as $\backslash sqrt\{2\}$ to $\backslash frac\{1\}\{2\}$, or as 9 to 4. But when the refracting and reflecting surfaces, in the position here considered, have the same focal distance, the radius of the refracting surface is four times that of the reflecting surface. The proportion of the diameters of the circles of spherical aberration is that of $9\; \backslash times\; 4^2$ to 4, or of 144 to 4, or 36 to 1. The distinctness therefore of the reflector is $36\; \backslash times\; 36$, or 1296 times greater than that of a plano-convex lens (placed with the plane side next the distant object) of the same breadth and focal distance, and will therefore admit of a much greater magnifying power. This comparison is indeed made in circumstances most favourable to the reflector, because this is the very worst position of a plano-convex lens. But we have not as yet learned the aberration in any other position. In another position the refraction and consequent aberration of both surfaces are complicated.

Before we proceed to the consideration of this very difficult subject, we may deduce from what has been already demonstrated several general rules and maxims in the construction of telescopes, which will explain (to such readers as do not wish to enter more deeply into the subject), and justify the proportion which long practice of the best artists has sanctioned.

Indistinctness proceeds from the commixture of the circles of aberration on the retina of the eye: For any one sensible point of the retina, being the centre of a circle of aberration, will at once be affected by the admixture of the rays of as many different pencils of light as there are sensible points in the area of that circle, and will convey to the mind a mixed sensation of as many

Telescope. visible points of the object. This number will be as the area of the circle of aberrations, whatever be the size of a sensible point of the retina. Now in vision with telescopes, the diameter of the circle of aberration on the retina is as the apparent magnitude of the diameter of the corresponding circle in the focus of the eye-glass; that is, as the angle subtended by this diameter at the centre of the eye-glass; that is, as the diameter itself directly, and as the focal distance of the eye-glass inversely. And the area of that circle on the retina is as the area of the circle in the focus of the eye-glass directly, and as the square of the focal distance of the eye-glass inversely. And this is the measure of the apparent indistinctness.

Cor. In all sorts of telescopes, and also in compound microscopes, an object is seen equally distinct when the focal distances of the eye-glasses are proportional to the diameters of the circles of aberration in the focus of the object-glass.

Here we do not consider the trifling alteration which well constructed eye-glasses may add to the indistinctness of the first image.

In refracting telescopes, the apparent indistinctness is as the area of the object-glass directly, and as the square of the focal distance of the eye-glass inversely. For it has been shown, that the area of the circle of dispersion is as the area of the object-glass, and that the spherical aberration is insignificant when compared with this.

Therefore, to make reflecting telescopes equally distinct, the diameter of the object-glass must be proportional to the focal distance of the eye-glass.

But in reflecting telescopes, the indistinctness is as the sixth power of the aperture of the object-glass directly, and as the fourth power of the focal distance of the object-glass and square of the focal distance of the eye-glass inversely. This is evident from the dimensions of the circle of aberration, which was found proportional

Therefore, to have them equally distinct, the cubes of the apertures must be proportional to the squares of the focal distance multiplied by the focal distance of the eye-glass.

By these rules, and a standard telescope of approved goodness, an artist can always proportion the parts of any instrument he wishes to construct. Mr Huyghens made one, of which the object-glass had 30 feet focal distance and three inches diameter. The eye-glass had 3.3 inches focal distance. And its performance was found superior to any which he had seen; nor did this appear owing to any chance goodness of the object-glass, because he found others equally good which were constructed on similar proportions. This has therefore been adopted as a standard.

It does not at first appear how there can be any difficulty in this matter, because we can always diminish the aperture of the object-glass or speculum till the circle of aberration is as small as we please. But by diminishing this aperture, we diminish the light in the duplicate ratio of the aperture. Whatever be the aperture, the brightness is diminished by the magnifying power, which spreads the light over a greater surface in the bottom of the eye. The apparent brightness must be as the square of the aperture of the telescope directly, and the square

of the amplification of the diameter of an object inversely. Objects therefore will be seen equally bright if the apertures of the telescopes be as the focal distances of the object-glasses directly, and the focal distances of the single eye-glass (or eye-glass equivalent to the eye-piece) inversely. Therefore, to have telescopes equally distinct and equally bright, we must combine these proportions with the former. It is needless to go farther into this subject, because the construction of refracting telescopes has been so materially changed by the correction of the chromatic aberration, that there can hardly be given any proportion between the object-glass and eye-glasses. Every thing now depends on the degree in which we can correct the aberrations of the object-glass. We have been able so far to diminish the chromatic aberration, that we can give very great apertures without its becoming sensible. But this is attended with so great an increase of the aberration of figure, that this last becomes a sensible quality. A lens which has 30° for its semi-aperture, has a circle of aberration equal to its chromatic aberration. Fortunately we can derive from the very method of contrary refractions, which we employ for removing the chromatic aberration, a correction of the other. We are indebted for this contrivance also to the illustrious Newton.

We call this Newton's contrivance, because he was the first who proposed a construction of an object-glass in which the aberration was corrected by the contrary aberrations of glass and water.

Huyghens had indeed supposed, that our all-wise Creator had employed in the eyes of animals many refractions in place of one, in order to make the vision more distinct; and the invidious detractors from Newton's fame have caught at this vague conjecture as an indication of his knowledge of the possibility of destroying the aberration of figure by contrary refractions. But this is very ill-founded. Huyghens has acquired sufficient reputation by his theory of aberrations. The scope of his writing in the passage alluded to, is to show that, by dividing any intended refraction into parts, and producing a certain convergence to or divergence from the axis of an optical instrument by means of two or three lenses instead of one, we diminish the aberrations four or nine times. This conjecture about the eye was therefore in the natural train of his thoughts. But he did not think of destroying the aberration altogether by opposite refractions. Newton, in 1669, says, that opticians need not trouble themselves about giving figures to their glasses other than spherical. If this figure were all the obstacle to the improvement of telescopes, he could show them a construction of an object-glass having spherical surfaces where the aberration is destroyed; and accordingly gives the construction of one composed of glass and water, in which this is done completely by means of contrary refractions.

The general principle is this: When the radiant point R (fig. 7.), or focus of incident rays, and its conjugate focus F of refracted central rays, are on opposite sides of the refracting surface or lens V, the conjugate focus f of marginal rays is nearer to R than F is. But when the focus of incident rays R' lies on the same side with its conjugate focus F' for central rays, R'f' is greater than R'F'.

Now fig. 8. represents the contrivance for destroying the colour produced at F, the principal focus of the convex

Telescope. convex lens V, of crown glass, by means of the contrary refraction of the concave lens v of Flint glass. The incident parallel rays are made to converge to F by the first lens. This convergence is diminished, but not entirely destroyed, by the concave lens v, and the focus is formed in F. F and F' therefore are conjugate foci of the concave lens. If F be the focus of V for central rays, the marginal rays will be collected at some point f nearer to the lens. If F be now considered as the focus of light incident on the centre of v, and F' be the conjugate focus, the marginal ray pF would be refracted to some point f' lying beyond F'. Therefore the marginal ray pf may be refracted to F, if the aberration of the concave be properly adjusted to that of the convex.

This brings us to the most difficult part of our subject, the compounded aberrations of different surfaces. Our limits will not give us room for treating this in the same elementary and perspicuous manner that we employed for a single surface. We must try to do it in a compendious way, which will admit at once the different surfaces and the different refractive powers of different substances. This must naturally render the process more complicated; but we hope to treat the subject in a way easily comprehended by any person moderately acquainted with common algebra; and we trust that our attempt will be favourably received by an indulgent public, as it is (as far as we know) the only dissertation in our language on the construction of achromatic instruments. We cannot but express our surprise at this indifference about an invention which has done so much honour to our country, and which now constitutes a very lucrative branch of its manufacture. Our artists infinitely surpass all the performances of foreigners in this branch, and supply the markets of Europe without any competition; yet it is from the writings on the continent that they derive their scientific instruction, and particularly from the dissertations of Clairaut, who has wonderfully simplified the analysis of optical propositions. We shall freely borrow from him, and from the writings of Abbé Boscovich, who has considerably improved the first views of Clairaut. We recommend the originals to the curious reader. Clairaut's dissertations are to be found in the Memoirs of the Academy of Paris, 1756, &c., those of Boscovich in the Memoirs of the Academy of Bologna, and in his five volumes of Opuscula, published at Bassano in 1785. To these may be added D'Alembert and Euler. The only thing in our language is the translation of a very imperfect work by Schaeffer.

is nearly equal to $\backslash frac\{MX\}\{2XG\}\; +\; XG$, and MH is nearly equal to $\backslash frac\{MX^2\}\{2XH\}\; +\; XH$.

PROP. I. Let the ray mM, incident on the spherical surface AM, converge to G; that is, let G be the focus of incident rays. It is required to find the focus F of refracted rays?

Let m express the ratio of the sine of incidence and refraction; that is, let m be to r as the sine of incidence to the sine of refraction in the substance of the sphere.

Then $MG\; :\; GS\; =\; \backslash sin.\; MSH\; :\; \backslash sin.\; SMG$,
and $m\; :\; 1\; =\; \backslash sin.\; SMG\; :\; \backslash sin.\; SMH$,
therefore $m\; \backslash times\; MG\; :\; GS\; =\; \backslash sin.\; MSH\; :\; \backslash sin.\; SMH$.
Now $S,\; MSH\; :\; S,\; SMH\; =\; MH\; :\; HS$. Therefore, finally,
$m\; MG\; :\; GS\; =\; MH\; :\; HS$.

Now let MS, the radius of the refracting surface, be called a. Let AG, the distance of the focus of incident rays from the surface, be called r. And let AH, the focal distance of refracted rays, be called x. Lastly, let the sine MX of the semi-aperture be called e. Observe, too, that a, r, x, are to be considered as positive quantities, when AS, AG, AH, lie from the surface in the direction in which the light is supposed to move. If therefore the refracting surface be concave, that is, having the centre on that side from which the light comes; or if the incident rays are divergent, or the refracted rays are divergent; then a, r, x, are negative quantities.

It is plain that $HS\; =\; x\; -\; a$; $GS\; =\; r\; -\; a$; also $\backslash Delta\; X\; =\; \backslash frac\{e^2\}\{2a\}$ nearly. $HX\; =\; a\; -\; \backslash frac\{e^2\}\{2a\}$. $GX\; =\; r\; -\; \backslash frac\{e^2\}\{2a\}$. Now add to HX and to GX their differences from MH and MG, which (by the Lemma) are $\backslash frac\{e^2\}\{2x\}$ and $\backslash frac\{e^2\}\{2r\}$. We get $MH\; =\; x\; -\; \backslash frac\{e^2\}\{2a\}\; +\; \backslash frac\{e^2\}\{2x\}$, and $MG\; =\; r\; -\; \backslash frac\{e^2\}\{2a\}\; +\; \backslash frac\{e^2\}\{2r\}$. In order to shorten our notation, make $k\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{r\}$. This will make $MG\; =\; r\; -\; \backslash frac\{ke^2\}\{2\}$.

Now substitute these values in the final analogy at the top of this column, viz. $MH\; :\; HS\; =\; m\; \backslash cdot\; MG\; :\; GS$; it becomes $x\; -\; \backslash frac\{e^2\}\{2a\}\; +\; \backslash frac\{e^2\}\{2x\}\; :\; x\; -\; a\; =\; m\; \backslash cdot\; r\; -\; \backslash frac\{ke^2\}\{2\}\; :\; r\; -\; a$ (or $a\; r\; k$), because $k\; =\; \backslash frac\{r-a\}\{ar\}$, and $a\; r\; k\; =\; r\; -\; a$. Now multiply the extreme and mean terms of this analogy. It is evident that it must give us an equation which will give us a value of x or AH, the quantity sought.

But this equation is quadratic. We may avoid the solution by an approximation which is sufficiently accurate, by substituting for x in the fraction $\backslash frac\{e^2\}\{2x\}$ (which is very small in all cases of optical instruments), an approximate very easily obtained, and very near the truth. This is the focal distance of an infinitely slender pencil of rays converging to G. This we know by the common optical theorem to be $\backslash frac\{a\; m\; r\}\{m\; -\; 1\; r\; -\; a\}$. Let this be called

Fig. 9. Lemma 1. In the right-angled triangle MXS (fig. 9.), of which one side MX is very small in comparison of either of the others; the excess of the hypotenuse MS, above the side XS, is very nearly equal to $\backslash frac\{MX^2\}\{2MS\}$ or to $\backslash frac\{MX^2\}\{2XS\}$. For if about the centre S, with the radius SM, we describe the semicircle AMO, we have $AX\; \backslash times\; XO\; =\; MX^2$. Now $AX\; =\; MS\; -\; SX$, and $XO$, is nearly equal to $2MS$ or $2XS$; on the other hand, MS is nearly equal to $XS\; +\; \backslash frac\{MX^2\}\{2XS\}$; and in like manner $MG$

called $\backslash phi$; if we substitute $k$ in place of $\backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{r\}$, this value

This gives us, by the by, an easily remembered expression (and beautifully simple) of the refracted focus of an infinitely slender pencil, corresponding to any distance $r$ of the radiant point. For since $\backslash phi\; =\; \backslash frac\{am\}\{m-ak\}$, $\backslash frac\{1\}\{\backslash phi\}$ must be $=\; \backslash frac\{m-ak\}\{am\}\; =\; \backslash frac\{m\}\{am\}\; -\; \backslash frac\{ak\}\{am\}\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{k\}\{m\}$. We may even express it more simply, by expanding $k$, and it becomes $\backslash frac\{1\}\{\backslash phi\}\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{ma\}\; -\; \backslash frac\{1\}\{mr\}$.

Now put this value of $\backslash frac\{1\}\{\backslash phi\}$ in place of the $\backslash frac\{1\}\{x\}$ in the analogy employed above. The first term of the analogy becomes $x\; =\; \backslash frac\{e^2\}\{2a\}\; +\; \backslash frac\{e^2\}\{2a\}\; -\; \backslash frac\{k\; e^2\}\{2m\}$, or $x\; =\; \backslash frac\{k\; e^2\}\{2m\}$. The analogy now becomes $x\; =\; \backslash frac\{k\; e^2\}\{2m\}\; :\; x\; a\; =\; m\; r\; -\; \backslash frac\{m\; k\; e^2\}\{2\}\; :\; a\; r\; k$. Hence we obtain the linear equation $m\; r\; x\; =\; \backslash frac\{m\; k\; e^2\; x\}\{2\}\; -\; m\; r\; a\; +\; \backslash frac\{m\; k\; a\; e^2\}\{2\}\; =\; a\; r\; k\; x\; -\; \backslash frac\{a\; r\; k\; e^2\}\{2m\}$; from which we finally deduce

We may simplify this greatly by attending to the elementary theorem in fluxions, that the fraction $\backslash frac\{x+\backslash dot\{x\}\}\{y+\backslash dot\{y\}\}$ differs from the fraction $\backslash frac\{x\}\{y\}$ by the quantity $\backslash frac\{y\backslash dot\{x\}-x\backslash dot\{y\}\}\{y^2\}$; this being the fluxion of $\backslash frac\{x\}\{y\}$. Therefore $\backslash frac\{x+\backslash dot\{x\}\}\{y+\backslash dot\{y\}\}\; =\; \backslash frac\{x\}\{y\}\; +\; \backslash frac\{y\backslash dot\{x\}-x\backslash dot\{y\}\}\{y^2\}$. Now the preceding formula is nearly in this situation. It may be written thus;

$\backslash frac\{m\; r\; a\; -\; (\backslash frac\{1\}\{2\}\; m\; a\; k\; e^2\; -\; \backslash frac\{a\; r\; k^2\; e^4\}\{2m\})\}\{m\; r\; -\; a\; r\; k\; -\; \backslash frac\{1\}\{2\}\; m\; k\; e^2\}$, when the last terms of the numerator and denominator are very small in comparison with the first, and may be considered as the $\backslash dot\{x\}$ and $\backslash dot\{y\}$, while $m\; r\; a$ is the $x$, and $m\; r\; -\; a\; r\; k$ is the $y$. Treating it in this way, it may be stated thus:

The first term $\backslash frac\{m\; r\; a\}\{r\; (m\; -\; a\; k)\}$, or $\backslash frac\{m\; a\}\{m\; -\; a\; k\}$, is evidently $\backslash phi$, the focal distance of an infinitely slender pencil.

Therefore the aberration is expressed by the second term, which we must endeavour to simplify. Telecope.

If we now perform the multiplication indicated by $(m\; r\; -\; a\; r\; k)\; \backslash times\; (m\; k\; a\; -\; \backslash frac\{a\; r\; k^2\}\{m\})$, it is plain that $-\; m\; r\; \backslash times\; m\; k\; a$ destroys the first term $m\; r\; a\; \backslash times\; m\; k$ of the numerator of our small fraction, and there remains of this numerator $(m\; a^2\; r\; k^2\; -\; a\; r^2\; k^2\; +\; \backslash frac\{a^2\; r^2\; k^4\}\{m\})\; \backslash frac\{1\}\{2\}\; e^2$, which is equal to $m^2\; a^2\; \backslash left(\; \backslash frac\{r\; k^2\}\{m\}\; -\; \backslash frac\{r^2\; k^2\}\{m^2\; a\}\; +\; \backslash frac\{r^2\; k^4\}\{m^3\}\; \backslash right)\; \backslash frac\{1\}\{2\}\; e^2$.

The denominator was $r^2\; (m\; -\; a\; k)^2$, and the fraction now becomes $\backslash frac\{m^2\; a^2\; \backslash left(\; \backslash frac\{r\; k^2\}\{m\}\; -\; \backslash frac\{r^2\; k^2\}\{m^2\; a\}\; +\; \backslash frac\{r^2\; k^4\}\{m^3\}\; \backslash right)\; \backslash frac\{1\}\{2\}\; e^2\}\{(m\; -\; a\; k)^2\; \backslash left(\; \backslash frac\{k^2\}\{m\; r\}\; -\; \backslash frac\{k^2\}\{m^2\; a\}\; +\; \backslash frac\{k^2\}\{m^3\}\; \backslash right)\; \backslash frac\{1\}\{2\}\; e^2\}$, which is evidently $=\; \backslash phi^2\; \backslash left(\; \backslash frac\{k^2\}\{m\; r\}\; -\; \backslash frac\{k^2\}\{m^2\; a\}\; +\; \backslash frac\{k^2\}\{m^3\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$. Now recollect that $k\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{r\}$. Therefore $\backslash frac\{k^2\}\{m^2\}\; =\; \backslash frac\{1\}\{m^2\}\; \backslash left(\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{r\}\; \backslash right)^2\; =\; \backslash frac\{k^2\}\{m^2\; a\}\; -\; \backslash frac\{k^2\}\{m^2\; r\}$. Therefore, instead of $-\backslash frac\{k^2\}\{m^2\; a\}$, write $-\backslash frac\{k^2\}\{m^2\; r\}$, and we get the fraction $\backslash phi^2\; \backslash left(\; \backslash frac\{k^2\}\{m^3\}\; -\; \backslash frac\{k^2\}\{m^2\}\; -\; \backslash frac\{k^2\}\{m^2\; r\}\; +\; \backslash frac\{k^2\}\{m\; r\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}\; =\; \backslash phi^2\; \backslash left(\; \backslash frac\{k^2\}\{m\}\; -\; \backslash frac\{m\; k^2\}\{m^2\}\; -\; \backslash frac\{m\; k^2\}\{m^2\; r\}\; +\; \backslash frac\{m^2\; k^2\}\{m^2\; r\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$, which is equal to $\backslash phi^2\; \backslash frac\{1-m\}\{m^2\}\; \backslash left(\; k^2\; -\; \backslash frac\{m\; k^2\}\{r\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$ and finally to $-\backslash phi^2\; \backslash frac\{m-1\}\{m^2\}\; \backslash left(\; k^2\; -\; \backslash frac\{m\; k^2\}\{r\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$.

Therefore the focal distance of refracted rays is $x\; =\; \backslash phi\; -\; \backslash phi^2\; \backslash frac\{m-1\}\{m^2\}\; \backslash left(\; k^2\; -\; \backslash frac\{m\; k^2\}\{r\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$.

This consists of two parts. The first $\backslash phi$ is the focal distance of an infinitely slender pencil of central rays, and the other $-\backslash phi^2\; \backslash frac\{m-1\}\{m^2\}\; \backslash left(\; k^2\; -\; \backslash frac\{m\; k^2\}\{r\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$ is the aberration arising from the spherical figure of the refracting surface.

Our formula has thus at last put on a very simple form, and is vastly preferable to Dr Smith's for practice.

This aberration is evidently proportional to the square of the semi-aperture, and to the square of the distance $\backslash phi$; but in order to obtain this simplicity, several quantities were neglected. The assumption of the equality of $AX$ to $\backslash frac\{e^2\}\{2a\}$ is the first source of error. A much more accurate value of it would have been $\backslash frac\{2\; a\; e^2\}\{4\; a^2\; +\; e^2\}$, for it is really $=\; \backslash frac\{e^2\}\{2a\; -\; AX\}$. If for $AX$ we substitute its approximated value $\backslash frac\{e^2\}\{2a\}$, we should have $AX\; =\; \backslash frac\{e^2\}\{2a\; -\; \backslash frac\{e^2\}\{2a\}\}\; =\; \backslash frac\{2\; a\; e^2\}\{4\; a^2\; -\; e^2\}$. To have used this value would not have much complicated the calculus; but it did not occur to us till we had finished the investigation, and it would have required the whole to be changed. The operation in page 246. col. 1. par. 3. is another source of error. But these errors are very inconsiderable when the aperture

Telescope. ture is moderate. They increase for the most part with an increase of aperture, but not in the proportion of any regular function of it; so that we cannot improve the formula by any manageable process, and must be contented with it. The errors are precisely the same with those of Dr Smith's theorem, and indeed with those of any that we have seen, which are not vastly more complicated.

As this is to be frequently combined with subsequent operations, we shorten the expression by putting $\backslash theta$ for $\backslash frac\{m-1\}\{m^3\}\; \backslash left(\; k^2\; -\; \backslash frac\{m\; k^2\}\{r\}\; \backslash right)\; \backslash frac\{c^2\}\{2\}$. Then $\backslash phi^2\; \backslash theta$ will express the aberration of the first refraction from the focal distance of an infinitely slender pencil; and now the focal distance of refracted rays is $f\; =\; \backslash phi\; -\; \backslash phi^2\; \backslash theta$.

If the incident rays are parallel, $r$ becomes infinite, and $\backslash theta\; =\; \backslash frac\{m-1\}\{m^3\}\; k^2\; \backslash frac\{c^2\}\{2\}$. But in this case $k$ becomes $\backslash frac\{1\}\{a\}$, and $\backslash frac\{1\}\{\backslash phi\}\; =\; \backslash frac\{m-1\}\{m\; a\}$, and $\backslash phi\; =\; \backslash frac\{m\; a\}\{m-1\}$, and $\backslash phi^2\; \backslash theta$ becomes $\backslash frac\{m^2\; c^2\}\{(m-1)^2\}\; \backslash times\; \backslash frac\{m-1\}\{m^3\}\; \backslash times\; \backslash frac\{1\}\{a^2\}\; \backslash times\; \backslash frac\{c^2\}\{2\}\; =\; \backslash frac\{c^2\}\{2(m-1)ma\}$. This is the aberration of extreme parallel rays.

We must now add the refraction of another surface.

Lemma 2. If the focal distance AG be changed by a small quantity Gg, the focal distance AH will also be changed by a small quantity Hh, and we shall have $m\; \backslash cdot\; AG^2\; :\; AH^2\; =\; Gg\; :\; Hh$.

Draw Mg, Mh, and the perpendiculars Gi, Hk. Then, because the fines of the angles of incidence are in a constant ratio to the fines of the angles of refraction, and the increments of these small angles are proportional to the increments of the fines, these increments of the angles are in the same constant ratio. Therefore,

We have the angle CMg to HMh as m to 1.

Now $Gg\; :\; Gi\; =\; AG\; :\; AM$,
and $Gi\; :\; Hk\; =\; m\; \backslash cdot\; AG\; :\; HA$,
and $Hk\; :\; Hh\; =\; MA\; :\; AH$;
therefore $Gg\; :\; Hh\; =\; m\; \backslash cdot\; AG^2\; :\; AH^2$.

The easiest and most perspicuous method for obtaining the aberration of rays twice refracted, will be to consider the first refraction as not having any aberration, and determine the aberration of the second refraction. Then conceive the focus of the first refraction as shifted by the aberration. This will produce a change in the focal distance of the second refraction, which may be determined by this Lemma.

Fig. 10. PROP. II. Let AM, BN (fig. 10.) be two spherical surfaces, including a refracting substance, and having their centres C and c in the line AG. Let the ray aA pass through the centres, which it will do without refraction. Let another ray mM, tending to G, be refracted by the first surface into MH, cutting the second surface in N, where it is further refracted into NI. It is required to determine the focal distance BI?

It is plain that the fine of incidence on the second surface is to the fine of refraction into the surrounding air as 1 to m. Also BI may be determined in relation to BH, by means of BH, Nx, Be, and $\backslash frac\{1\}\{m\}$, in the same way that AH was determined in relation to AG, by means of AG, MX, AC, and m.

Let the radius of the second surface be b, and let e still express the semi-aperture, because it hardly differs

from Nx). Also let a be the thickness of the lens. Telescope. Then observe, that the focal distance of the rays refracted by the first surface, (neglecting the thickness of the lens and the aberration of the first surface), is the distance of the radiant point for the second refraction, or is the focal distance of rays incident on the second surface. In place of r therefore we must take $\backslash phi$; and as we made $k\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{r\}$, in order to abbreviate the calculation, let us now make $l\; =\; \backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{\backslash phi\}$; and make $\backslash frac\{1\}\{f\}\; =\; \backslash frac\{1\}\{b\}\; -\; m\; l$, as we made $\backslash frac\{1\}\{\backslash phi\}\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{m\}$. Lastly, in place of $\backslash theta\; =\; \backslash frac\{m-1\}\{m^3\}\; \backslash left(\; k^2\; -\; \backslash frac\{m\; k^2\}\{r\}\; \backslash right)\; \backslash frac\{c^2\}\{2\}$, make $\backslash theta\; =\; \backslash left(\; \backslash frac\{1\}\{m\}\; -\; 1\; \backslash right)\; m^2\; \backslash left(\; l^2\; -\; \backslash frac\{l^2\}\{m\; \backslash phi\}\; \backslash right)\; \backslash frac\{c^2\}\{2\}\; =\; -\backslash frac\{m-1\}\{m\}\; \backslash left(\; m^2\; l^2\; -\; \backslash frac\{m^2\; l^2\}\{\backslash phi\}\; \backslash right)\; \backslash frac\{c^2\}\{2\}$.

Thus we have got an expression similar to the other, and the focal distance BI, after two refractions, becomes $BI\; =\; f\; -\; f^2\; \backslash theta$.

But this is on the supposition that BH is equal to $\backslash phi$, whereas it is really $\backslash phi\; -\; \backslash phi^2\; \backslash theta\; =\; a$. This must occasion a change in the value just now obtained of BI. The source of the change is twofold. 1st, Because in the value $\backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{\backslash phi\}$, we must put $\backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{\backslash phi\; -\; \backslash phi^2\; \backslash theta\; =\; a\}$, and because we must do the same in the fraction $\backslash frac\{m^2\; l^2\}\{\backslash phi\}$. In the second place, when the value of BH is diminished by the quantity $\backslash phi^2\; \backslash theta\; +\; a$, BI will suffer a change in the proportion determined by the 2d Lemma. The first difference may safely be neglected, because the value of $\backslash theta$ is very small, by reason of the coefficient $\backslash frac\{c^2\}\{2\}$ being very small, and also because the variation bears a very small ratio to the quantity itself, when the true value of $\backslash phi$ differs but little from that of the quantity for which it is employed. The chief change in BI is that which is determined by the Lemma. Therefore take from BI the variation of BH, multiplied by $\backslash frac\{m\; BI^2\}\{BH^2\}$, which is very nearly $=\; \backslash frac\{m\; f^2\}\{\backslash phi^2\}$. The product of this multiplication is $m\; f^2\; \backslash theta\; +\; \backslash frac\{m\; f^2\; a\}\{\backslash phi^2\}$. This being taken from f, leaves us

for the value of BI $f\; -\; \backslash frac\{f^2\; m\; a\}\{\backslash phi^2\}\; -\; f^2\; (m\; \backslash theta\; +\; \backslash theta\text{'})$.

In this value f is the focal distance of an infinitely slender pencil of rays twice refracted by a lens having no thickness, $a\; \backslash frac\{m\; f^2\}\{\backslash phi^2\}$ is the shortening occasioned by the thickness, and $f^2\; (m\; \backslash theta\; +\; \backslash theta\text{'})$ is the effect of the two aberrations arising from the aperture.

It will be convenient, for several collateral purposes, to exterminate from these formulae the quantities k, l and $\backslash phi$. For this purpose make $\backslash frac\{1\}\{n\}\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{b\}$. We have already $k\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{r\}$; and $\backslash frac\{1\}\{\backslash phi\}\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{m\; a\}\; +\; \backslash frac\{1\}\{m\; r\}$; and $l\; =\; \backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{\backslash phi\}\; =\; \backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{a\}\; +\; \backslash frac\{1\}\{m\; a\}\; -\; \backslash frac\{1\}\{m\; r\}$. Now for $\backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{a\}$ write $-\backslash frac\{1\}\{n\}$, and

Telescope. and we get $l\; =\; \backslash frac\{1\}\{ma\}\; -\; \backslash frac\{1\}\{mr\}\; -\; \backslash frac\{1\}\{n\}$. Therefore $\backslash frac\{1\}\{f\}\; =\; \backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{a\}$ (by construction, page 347. Prop. II.) becomes $\backslash frac\{1\}\{b\}\; -\; \backslash frac\{1\}\{a\}\; +\; \backslash frac\{1\}\{r\}\; +\; \backslash frac\{1\}\{n\}\; =\; \backslash frac\{1\}\{n\}\; +\; \backslash frac\{1\}\{r\}\; -\; \backslash frac\{1\}\{n\}\; -\; \backslash frac\{m-1\}\{n\}\; +\; \backslash frac\{1\}\{r\}$.

This last value of $\backslash frac\{1\}\{f\}$ (the reciprocal of the focus of a slender pencil twice refracted), viz. $\backslash frac\{m-1\}\{n\}\; +\; \backslash frac\{1\}\{r\}$, is the simplest that can be imagined, and makes $n$ as a substitute for $\backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{b\}$; a most useful symbol, as we shall frequently find in the sequel. It also gives a very simple expression of the focal distance of parallel rays, which we may call the principal focal distance of the lens, and distinguish it in future by the symbol $\backslash rho$; for the expression $\backslash frac\{1\}\{f\}\; =\; \backslash frac\{m-1\}\{n\}\; +\; \backslash frac\{1\}\{r\}$, becomes $\backslash frac\{1\}\{\backslash rho\}\; =\; \backslash frac\{m-1\}\{n\}$ when the incident light is parallel. And this gives us another very simple and useful measure of $f$; for $\backslash frac\{1\}\{f\}$ becomes $\backslash frac\{1\}\{\backslash rho\}\; +\; \backslash frac\{1\}\{r\}$. These equations $\backslash frac\{1\}\{f\}\; =\; \backslash frac\{m-1\}\{n\}\; +\; \backslash frac\{1\}\{r\}$, $\backslash frac\{1\}\{\backslash rho\}\; =\; \backslash frac\{m-1\}\{n\}$, and $\backslash frac\{1\}\{f\}\; =\; \backslash frac\{1\}\{\backslash rho\}\; +\; \backslash frac\{1\}\{r\}$, deserve therefore to be made very familiar to the mind.

We may also take notice of another property of $n$. It is half the radius of an isosceles lens, which is equivalent to the lens whose radii are $a$ and $b$: for suppose the lens to be isosceles, that is $a=b$; then $n\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{a\}$. Now the second $a$ is negative if the first be positive, or positive if the first be negative. Therefore $\backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{b\}\; =\; \backslash frac\{1+b\}\{a^2\}\; =\; \backslash frac\{a+a\}\{a^2\}\; =\; \backslash frac\{2\}\{a\}$, and $\backslash frac\{1\}\{n\}\; =\; \backslash frac\{2\}\{a\}$, and $n\; =\; \backslash frac\{a\}\{2\}$. Now the focal distance of this lens is $\backslash frac\{m-1\}\{n\}$, and so is that of the other, and they are equivalent.

But, to proceed with our investigation, recollect that we had $\backslash theta\; =\; \backslash frac\{m-1\}\{m^2\}\; \backslash left(\; k^2\; -\; \backslash frac\{m^2\; l^2\}\{r^2\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$. Therefore $m\backslash theta\; =\; \backslash frac\{m-1\}\{m\}\; \backslash left(\; k^2\; -\; \backslash frac\{m^2\; l^2\}\{r^2\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$. And $\backslash theta\text{'}$ was $=\; \backslash frac\{m-1\}\{m\}\; \backslash left(\; -m^2\; l^2\; +\; \backslash frac\{m^2\; l^2\}\{r^2\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$. Therefore $m\backslash theta\; +\; \backslash theta\text{'}$, the aberration (neglecting the thickness of the lens) is $f^2\; \backslash frac\{m-1\}\{m\}\; \backslash left(\; \backslash frac\{k^2\}\{m\}\; -\; \backslash frac\{k^2\}\{r^2\}\; -\; m^2\; l^2\; +\; \backslash frac\{m^2\; l^2\}\{r^2\}\; \backslash right)\; \backslash frac\{e^2\}\{2\}$.

If we now write for $k,\; l$, and $\backslash phi$, their values as determined above, performing all the necessary multiplications, and arrange the terms in such a manner as to collect in one sum the coefficients of $a,\; n$, and $r$, we shall find 4 terms for the value of $m\backslash theta$, and 10 for the value of $\backslash theta\text{'}$. The 4 are destroyed by as many with contrary signs in the value of $\backslash theta\text{'}$, and there remain 6 terms to express the value of $m\backslash theta\; +\; \backslash theta\text{'}$, which we shall express by one symbol $q$; and the equation stands thus:

The focal distance therefore of rays twice refracted, reckoned from the last surface, or BI, corrected for aberration, and for the thickness of the lens, is $f\; =\; f\; \backslash frac\{m\; \backslash kappa\}\{\backslash phi^2\}\; =\; f^2\; q$, consisting of three parts, viz. $f$, the focal distance of central rays; $f^2\; \backslash frac\{m\; \backslash kappa\}\{\backslash phi^2\}$, the correction for the thickness of the lens; and $f^2\; q$, the aberration.

The formula at the top of this column appears very complex, but is of very easy management, requiring only the preparation of the simple numbers which form the numerators of the fractions included in the parentheses. When the incident rays are parallel, the terms vanish which have $r$ in the denominator, so that only the three first terms are used.

We might here point out the cases which reduce the aberration expressed in the formula last referred to, to nothing; but as they can scarcely occur in the object-glass of a telescope, we omit it for the present, and proceed to the combination of two or more lenses.

Lemma 3. If AG be changed by a small quantity $Gg$, BI suffers a change $Ii$, and $Gg\; :\; Ii\; =\; AG^2\; :\; BI^2$. For it is well known that the small angles GMG and INI are equal; and therefore their subtenses Gk, In are proportional to MG, NI, or to AG, AI nearly, when the aperture is moderate. Therefore we have (nearly)

PROP. III. To determine the focal distance of rays refracted by two lenses placed near to each other on a common axis.

Let AM, BN (fig. 11.) be the surfaces of the first lens, and CO, DP be the surfaces of the second, and let $\backslash beta$ be the thickness of the second lens, and $\backslash delta$ the interval between them. Let the radius of the anterior surface of the second lens be $a\text{'}$, and the radius of its posterior surface be $b\text{'}$. Let $m\text{'}$ be to 1 as the sine of incidence to the sine of refraction in the substance of the second lens. Lastly, let $\backslash rho\text{'}$ be the principal focal distance of the second lens. Let the extreme or marginal ray meet the axis in L after passing through both lenses, so that DL is the ultimate focal distance, reckoned from the last surface.

It is plain that DL may be determined by means of $a\text{'},\; b\text{'},\; m\text{'},\; \backslash rho\text{'},$ and CI, in the same manner that BI was determined by means of $a,\; b,\; m,\; \backslash rho$, and AG.

The value of BI is $f\; =\; m\; \backslash kappa\; \backslash frac\{f^2\}\{\backslash phi^2\}\; =\; f^2\; q$. Take from this the interval $\backslash delta$, and we have $CI\; =\; f\; =\; m\; \backslash kappa\; \backslash frac\{f^2\}\{\backslash phi^2\}\; -\; \backslash delta\; =\; f^2\; q$. Let the small part $-m\; \backslash kappa\; \backslash frac\{f^2\}\{\backslash phi^2\}\; -\; \backslash delta\; =\; f^2\; q$ be neglected for the present, and let CI be supposed $=\; f$. As we formed $\backslash phi,\; f,\; q$, by means of $a,\; b,\; m,\; n$, and $r$,

Telescope. $r$, let us now form $\backslash phi,\; f,\; f\text{'}$, and $q\text{'}$, for the second lens, by means of $a\text{'},\; b\text{'},\; m\text{'},\; n\text{'},\; \backslash left(=\; \backslash frac\{1\}\{a\text{'}\}\; -\; \backslash frac\{1\}\{b\text{'}\}\backslash right)$, and $r\text{'}$. $\backslash phi$ will be the focal distance of a slender pencil refracted by the first surface, $f$ will be the focal distance of this pencil after two refractions, and $q\text{'}$ will be the coefficient of the aberration, neglecting the thickness and interval of the lenses.

Proceeding in this way, $DL$ will be $=\; f\text{'}\; -\; m\; \backslash beta\; \backslash frac\{f\text{'}^2\}\{\backslash phi\}\; -\; f\text{'}^2\; q$. But because $CI$ is really less than $f$, by the quantity $m\; \backslash alpha\; \backslash frac\{f\text{'}^2\}\{\backslash phi^2\}\; +\; \backslash beta\; +\; f\text{'}^2\; q\text{'}$, we must (by Lemma 3.) subtract the product of this quantity, multiplied by $\backslash frac\{DL\}\{B^2\}$, (which is nearly $\backslash frac\{f\text{'}^2\}\{\backslash phi^2\}$), from $f\text{'}\; -\; m\; \backslash beta\; \backslash frac\{f\text{'}^2\}\{\backslash phi\}\; -\; f\text{'}^2\; q$.

By this process we shall have

The first term $f\text{'}$ of this value of $DL$ is the focal distance of a slender pencil of central rays refracted by both lenses, neglecting their thickness and distance; the second term, $-f\text{'}^2\; \backslash left(\; \backslash frac\{m\; \backslash alpha\}\{\backslash phi^2\}\; +\; \backslash frac\{\backslash beta\}\{f\text{'}^2\}\; +\; \backslash frac\{m\text{'}\; \backslash beta\}\{\backslash phi\text{'}^2\}\; \backslash right)$ is the correction necessary for these circumstances; and the third term, $-f\text{'}^2\; (q\; +\; q\text{'})$, is the correction for the aperture 2c. And it is evident that $q\text{'}$ is a formula precisely similar to $q$, containing the same number of terms, and differing only by the $m\text{'},\; a\text{'},\; n\text{'},\; r\text{'}$, employed in place of $m,\; a,\; n$, and $r$.

It is also evident, that if there be a third lens, we shall obtain its focal distance by a process precisely similar to that by which we obtained $DL$; and so on for any number of lenses.

Thus have we obtained formulae by which the foci of rays are determined in the most general terms; and in such a manner as shall point out the connection of the curvatures, thicknesses and distances of the lenses, with their spherical aberrations, and with the final aberration of the compound lens, and give the aberrations in separate symbols, so that we can treat them by themselves, and subject them to any conditions which may enable us to correct one of them by another.

We also see in general, that the corrections for the thickness and distance of the lenses are exhibited in terms which involve only the focal distances of central rays, and have very little influence on the aberrations, and still less on the ratio of the aberrations of the different lenses. This is a most convenient circumstance; for we may neglect them while we are determining $q$ and $q\text{'}$, and in determining the ratio of the focal distances of the several lenses, on which the correction of the chromatic aberration chiefly depends. Therefore, in the construction of a compound lens for uniting the different colours, we may neglect this correction for the thickness and distance till the end of the process. When we apply it, we shall find that it chiefly affects the final focal distance, making it somewhat longer, but has hardly any influence either on the chromatic or spherical aberration. We do not hesitate to say, that the final formulae here given are abundantly accurate, while they

are vastly more manageable than those employed by Telescope Euler or D'Alembert. We have calculated trigonometrically the progress of the rays through one of the glasses, which will be given as an example, giving it a very extravagant aperture, that the errors of the formula might be very remarkable. We found the real aberration exceed the aberration assigned by the formula by no more than $\backslash frac\{1\}\{100\}$th part, a difference which is quite insignificant. The process here given derives its simplicity from the frequent occurrence of harmonic proportions in all optical theorems. This enabled Mr Clairaut to employ the reciprocals of the radii and distances with so much simplicity and generality.

We consider it as another advantage of Mr Clairaut's method, that it gives, by the way, formulae for the more ordinary questions in optics, which are of wonderful simplicity, and most easily remembered. The chief problems in the elementary construction of optical instruments relate to the focal distances of central rays. This determines the focal distances and arrangement of the glasses. All the rest may be called the refinement of optics; teaching us how to avoid or correct the indistinctness, the colours, and the distortions, which are produced in the images formed by these simple constructions. We shall mention a few of these formulae which occur in our process, and tend greatly to abbreviate it when managed by an experienced analyst.

Let $m$ be to 1 as the sine of incidence to the sine of refraction; let $a$ and $b$ be the radii of the anterior and posterior surfaces of a lens; let $r$ be the distance of the radiant point, or the focus of incident central rays, and $f$ the distance of the conjugate focus; and let $p$ be the principal focal distance of the lens, or the focal distance of parallel rays. Make $\backslash frac\{1\}\{n\}$ equal to $\backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{b\}$; let the same letters $a\text{'},\; b\text{'},\; r\text{'},$ &c. express the same things for a second lens; and $a\text{'}\text{'},\; b\text{'}\text{'},\; r\text{'}\text{'},$ &c. express them for a third; and so on. Then we have $\backslash frac\{1\}\{f\}\; =\; \backslash frac\{m-1\}\{n\}\; +\; \backslash frac\{1\}\{r\}$; $\backslash frac\{1\}\{f\text{'}\}\; =\; \backslash frac\{m\text{'}-1\}\{n\text{'}\}\; +\; \backslash frac\{1\}\{r\text{'}\}$; $\backslash frac\{1\}\{f\text{'}\text{'}\}\; =\; \backslash frac\{m\text{'}\text{'}-1\}\{n\text{'}\text{'}\}\; +\; \backslash frac\{1\}\{r\text{'}\text{'}\}$, &c.

Therefore when the incident light is parallel, and $r$ infinite, we have $\backslash frac\{1\}\{p\}\; =\; \backslash frac\{m-1\}\{n\}$; $\backslash frac\{1\}\{p\text{'}\}\; =\; \backslash frac\{m\text{'}-1\}\{n\text{'}\}$; $\backslash frac\{1\}\{p\text{'}\text{'}\}\; =\; \backslash frac\{m\text{'}\text{'}-1\}\{n\text{'}\text{'}\}$, &c.

And when several lenses are contiguous, so that their intervals may be neglected, and therefore $\backslash frac\{1\}\{f\}$, belonging to the first lens, becomes $\backslash frac\{1\}\{r\}$, belonging to the second, we have

Nothing can be more easily remembered than these formulae, how numerous so ever the glasses may be.

Having thus obtained the necessary analysis and formula,

Telecope. mula, it now remains to apply them to the construction of achromatic lenses; in which it fortunately happens, that the employment of several surfaces, in order to produce the union of the differently refrangible rays, enables us at the same time to employ them for correcting each other's spherical aberration.

In the article OPTICS we gave a general notion of the principle on which we may proceed in our endeavours to unite the differently refrangible rays. A white or compounded ray is separated by refraction into its component coloured rays, and they are diffused over a small angular space. Thus it appears, that the glass used by Sir Isaac Newton in his experiments diffused a white ray, which was incident on its posterior surface in an angle of $30^\backslash circ$, in such a manner that the extreme red ray emerged into air, making an angle of $50^\backslash circ\; 21\backslash frac\{1\}\{2\}\text{'}$ with the perpendicular; the extreme violet ray emerged in an angle of $51^\backslash circ\; 15\backslash frac\{1\}\{2\}\text{'}$; and the ray which was in the confines of green and blue, emerged in an angle of $50^\backslash circ\; 48\backslash frac\{1\}\{2\}\text{'}$. If the sine of the angle $30^\backslash circ$ of incidence be called 0.5, which it really is, the sine of the emergence of the red ray will be 0.77; that of the violet ray will be 0.78; and that of the intermediate ray will be 0.77$\backslash frac\{1\}\{2\}$, an exact mean between the two extremes. This ray may therefore be called the mean refrangible ray, and the ratio of 77$\backslash frac\{1\}\{2\}$ to 50, or of 1.55 to 1, will very properly express the mean refraction of this glass; and we have for this glass $m=1.55$. The sine of refraction, being measured on a scale, of which the sine of incidence occupies 100 parts, will be 154 for the red ray, 155 for the mean ray, and 156 for the violet ray. This number, or its ratio to unity, is commonly taken to represent the refractive power of the glass. There is some impropriety in this, unless we consider ratios as measured by their logarithms: for if $m$ be 1, the substance does not refract at all. The refractive power can be properly measured only by the refraction which it produces; that is, by the change which it makes in the direction of the light, or the angle contained between the incident and refracted rays. If two substances produce such deviations always in one proportion, we should then say that their refractive powers are in that proportion. This is not true in any substance; but the sines of the angles, contained between the refracted ray and the perpendicular, are always in one proportion when the angle of incidence in both substances is the same. This being a cognisable function of the real refraction, has therefore been assumed as the only convenient measure of the refractive powers. Although it is not strictly just, it answers extremely well in the most usual cases in optical instruments: the refractions are moderate; and the sines are very nearly as the angles contained between the rays and the perpendicular; and the real angles of refraction, or deflections of the rays, are almost exactly proportional to $m-1$. The most natural and obvious measure of the refractive powers would therefore be $m-1$. But this would embarrass some very frequent calculations; and we therefore find it best, on the whole, to take $m$ itself for the measure of the refractive power.

The separation of the red, violet, and intervening rays, has been called dispersion; and although this arises merely from a difference of the refractive power in respect of the different rays, it is convenient to distinguish this particular modification of the refractive power by a

name, and we call it the DISPERSIVE POWER of the refracting substance. Telecope.

It is susceptible of degrees; for a piece of flint-glass will refract the light, so that when the sine of refraction of the red ray is 77, the sine of the refraction of the violet ray is nearly 78$\backslash frac\{1\}\{2\}$; or if the sine of refraction of the red ray, measured on a particular scale, is 1.54, the sine of refraction of the violet ray is 1.57. The dispersion of this substance, being measured by the difference of the extreme sines of refraction, is greater than the dispersion of the other glass, in the proportion of 3 to 2.

But this alone is not a sufficient measure of the absolute dispersive power of a substance. Although the ratio of 1.54 to 1.56 remains constant, whatever the real magnitude of the refractions of common glass may be, and though we therefore say that its dispersive power is constant, we know, that by increasing the incidence and the refraction, the absolute dispersion is also increased. Another substance shows the same properties, and in a particular case may produce the same dispersion; yet it has not for this sole reason the same dispersive power. If indeed the incidence and the refraction of the mean ray be also the same, the dispersive power cannot be said to differ; but if the incidence and the refraction of the mean ray be less, the dispersive power must be considered as greater, though the actual dispersion be the same; because if we increase the incidence till it becomes equal to that in the common glass, the dispersion will now be increased. The proper way of conceiving the dispersion therefore is, to consider it as a portion of the whole refraction; and if we find a substance making the same dispersion with half the general refraction, we must say that the dispersive quality is double; because by making the refraction equal, the dispersion will really be double.

If therefore we take $m$ as a symbol of the separation of the extreme rays from the middle ray, $\backslash frac\{m\}\{m-1\}$ is the natural measure of the dispersive power. We shall express this in the Leibnizian notation, thus $\backslash frac\{dm\}\{m-1\}$, that we may avoid the indistinctness which the Newtonian notation would occasion when $m$ is changed for $m\text{'}$ or $m\text{'}\text{'}$.

It is not unusual for optical writers to take the whole separation of the red and violet rays for the measure of the dispersive power, and to compare this with the refracting power with respect to one of the extreme rays. But it is surely better to consider the mean refraction as the measure of the refracting power: and the deviation of either of the extremes from this mean is a proper enough measure of the dispersion, being always half of it. It is attended with this convenience, that being introduced into our computations as a quantity infinitely small, and treated as such for the ease of computation, while it is really a quantity of sensible magnitude; the errors arising from this supposition are diminished greatly, by taking one half of the deviation and comparing it with the mean refraction. This method has, however, this inconvenience, that it does not exhibit at once the refractive power in all substances respecting any particular colour of light; for it is not the ray of any particular colour that suffers the mean refraction. In common glass it is the ray which is in the confines of the yellow and blue; in flint-glass it is nearly the middle

Telecope. The blue ray; and in other substances it is a different ray. These circumstances appear plainly in the different proportions of the colours of the prismatic spectrum exhibited by different substances. This will be considered afterwards, being a great bar to the perfection of achromatic instruments.

The way in which an achromatic lens is constructed is, to make use of a contrary refraction of a second lens to destroy the dispersion or spherical aberration of the first.

The first purpose will be answered if $\backslash frac\{dm\}\{n\}$ be equal to $-\backslash frac\{dm\text{'}\}\{n\text{'}\}$. For, in order that the different coloured rays may be collected into one point by two lenses, it is only necessary that $\backslash frac\{1\}\{f\text{'}\}$, the reciprocal of the focal distance of rays refracted by both, may be the same for the extreme and mean rays, that is, that $\backslash frac\{m+d\; m-1\}\{n\}\; +\; \backslash frac\{m\text{'}+d\; m\text{'}-1\}\{n\text{'}\}\; +\; \backslash frac\{1\}\{r\}$ be of the same value with $\backslash frac\{m-1\}\{n\}\; +\; \backslash frac\{m\text{'}-1\}\{n\text{'}\}\; +\; \backslash frac\{1\}\{r\}$, which must happen if $\backslash frac\{dm\}\{n\}\; +\; \backslash frac\{dm\text{'}\}\{n\text{'}\}$ be = 0, or $\backslash frac\{dm\}\{n\}\; =\; -\backslash frac\{dm\text{'}\}\{n\text{'}\}$. This may be seen in another way, more comprehensible by such as are not versant in these difficulties. In order that the extreme colours which are separated by the first lens may be rendered parallel by the second; we have shown already that $n$ and $n\text{'}$ are proportional to the radii of the equivalent isosceles lenses, being the halves of these radii. They are therefore (in these small refractions) inversely proportional to the angles formed by the surfaces at the edges of the lenses. $n\text{'}$ may therefore be taken for the angle of the first lens, and $n$ for that of the second. Now the small refraction by a prism, whole angle (also small) is $n\text{'}$, is $m-1\; \backslash times\; n$. The dispersive power being now substituted for the refractive power, we have for this refraction of the prism $d\; m\; \backslash times\; n\text{'}$. This must be destroyed by the opposite refraction of the other prism $d\; m\text{'}\; \backslash times\; n$. Therefore $d\; m\; \backslash times\; n\text{'}\; =\; d\; m\text{'}\; \backslash times\; n$, or $\backslash frac\{dm\}\{n\}\; =\; -\backslash frac\{dm\text{'}\}\{n\text{'}\}$. In like manner, this effect will be produced by three lenses if $\backslash frac\{dm\}\{n\}\; +\; \backslash frac\{dm\text{'}\}\{n\text{'}\}\; +\; \backslash frac\{dm\text{'}\text{'}\}\{n\text{'}\text{'}\}$ be = 0, &c.

Lastly, the errors arising from the spherical figure, which we expressed by $-R^2(q+q\text{'})$ will be corrected if $q+q\text{'}$ be = 0. We are therefore to discover the adjustments of the quantities employed in the preceding formulae, which will insure these conditions. It will render the process more perspicuous if we collect into one view the significations of our various symbols, and the principal equations which we are to employ.

Also the following subsidiary values:

Also, in a double object-glass, the correction of spherical aberration requires $q+q\text{'}=0$.

And a triple object-glass requires $q+q\text{'}+q\text{'}\text{'}=0$. For the whole error is multiplied by $F^2$, and by $\backslash frac\{1\}\{2\}\; e^2$; and therefore the equation which corrects this error may be divided by $F^2\; \backslash frac\{1\}\{2\}\; e^2$.

This equation in the fourteenth line from the top of the column, giving the value of $q,\; q\text{'},\; q\text{'}\text{'}$, may be much simplified as follows: In the first place, they may be divided by $m,\; m\text{'},\; m\text{'}\text{'}$, by applying them properly to the terms within the parentheses, and expunging them from the denominator of the general factors $\backslash frac\{m-1\}\{m\},\; \backslash frac\{m\text{'}-1\}\{m\text{'}\}$,

$\backslash frac\{m\text{'}-1\}\{m\text{'}\}$. This does not alter the values of $q,\; q\text{'},\; q\text{'}\text{'}$. In the second place the whole equations may be afterwards divided by $m\text{'}-1$. This will give the values of $\backslash frac\{q\}\{m\text{'}-1\},\; \backslash frac\{q\text{'}\}\{m\text{'}-1\},\; \backslash frac\{q\text{'}\text{'}\}\{m\text{'}-1\}$, which will still be equal to nothing if $q+q\text{'}+q\text{'}\text{'}$ be equal to nothing.

This division reduces the general factor $\backslash frac\{m\text{'}-1\}\{m\text{'}\}$ of $q\text{'}$ to $\backslash frac\{1\}\{m\text{'}\}$. And in the equation for $q$ we obtain, in place of the general factor $\backslash frac\{m-1\}\{m\}$, the factor $\backslash frac\{m-1\}\{m\text{'}-1\}$, or $c$. This will also be the factor of the value of $q\text{'}\text{'}$ when the third lens is of the same substance with the first, as is generally

I i 2

Telescope. ly the case. And, in the third place, since the rays incident on the first lens are parallel, all the terms vanish from the value of $q$ in which $\backslash frac\{1\}\{r\}$ is found, and there remain only the three first, viz. $\backslash frac\{m^3\}\{n^3\}\; -\; \backslash frac\{2m^2+m\}\{an^2\}\; +\; \backslash frac\{m+2\}\{a^2n\}$.

Performing these operations, we have

Let us now apply this investigation to the construction of an object-glass; and we shall begin with a double lens.

Here we have to determine four radii $a,\; b,\; a\text{'},\; b\text{'}$. Make $n=1$. This greatly simplifies the calculus, by exterminating it from all the denominators. This gives

for the equation $\backslash frac\{dm\}\{n\}\; +\; \backslash frac\{dm\text{'}\}\{n\text{'}\}\; =\; 0$, the equation $dm\; +\; \backslash frac\{dm\text{'}\}\{n\text{'}\}\; =\; 0$, or $dm\; =\; -\backslash frac\{dm\text{'}\}\{n\text{'}\}$, and $\backslash frac\{1\}\{n\text{'}\}\; =\; -\backslash frac\{dm\}\{dm\text{'}\}$, $=\; -u$. Also we have $r\text{'}$, the focal distance of the light incident on the second lens, the same with the principal focal distance $\backslash rho$ of the first lens, (neglecting the interval, if any). Now $\backslash frac\{1\}\{\backslash rho\}\; =\; \backslash frac\{m-1\}\{n\}$, which in the present case is $=\; m-1$. Also $\backslash frac\{1\}\{\backslash rho\}$ is $=\; -u(m\text{'}-1)$, and $\backslash frac\{1\}\{p\}\; =\; m-1\; -\; u(m\text{'}-1)\; =\; u\text{'}$.

Make these substitutions in the values of $\backslash frac\{q\}\{m^2-1\}$ and $\backslash frac\{q\text{'}\}\{m^2-1\}$, and we obtain the following equation:

Arrange these terms in order, according as they are factors of $\backslash frac\{1\}\{a^3\}$, $\backslash frac\{1\}\{a^2\}$, $\backslash frac\{1\}\{a\}$, $\backslash frac\{1\}\{a^2\}$, or independent quantities. It puts on this form:

Let $A$ be the coefficient of $\backslash frac\{1\}\{a^3\}$, $B$ that of $\backslash frac\{1\}\{a^2\}$, $C$ that of $\backslash frac\{1\}\{a\}$, $D$ that of $\backslash frac\{1\}\{a^2\}$, and $E$ the sum of the independent quantity; that is, let $A\; =\; \backslash frac\{c(m+2)\}\{m\}$, $B\; =\; c(2m+1)$, $C\; =\; \backslash frac\{u(m\text{'}+2)\}\{m\text{'}\}$, $D\; =\; u^3(2m\text{'}+1)\; -\; \backslash frac\{4u(m\text{'}+1)(m-1)\}\{m\text{'}\}$, and $E\; =\; c\; m^3\; +\; u^3(3m\text{'}+1)(m-1)\; -\; u^3\; m^3\; -\; \backslash frac\{u(3m\text{'}+2)(m-1)^2\}\{m\text{'}\}$.

Our final equation becomes

The coefficients of this equation and the independent quantity are all known, from our knowledge of $m,\; m\text{'},\; d,\; m\text{'},\; d\text{'}$; and we are to find the values of $a$ and $a\text{'}$, and from them and $n=1$ to find the values of $b$ and $b\text{'}$.

But it is evidently an indeterminate equation, because there are two unknown quantities; so that there may be an infinity of solutions. It must be rendered determinate by means of some other conditions to which it may be subjected. These conditions must depend on some other circumstances which may direct our choice.

One circumstance occurs to us which we think of very great consequence. In the passage of light from one substance to another, there is always a considerable portion reflected from the posterior surface of the first and from the anterior surface of the last; and this reflection is more copious in proportion to the refraction. This loss of light will therefore be diminished by making the internal surfaces of the lenses to coincide; that is, by making $b=a\text{'}$. This will be attended with another advantage. If we put between the glasses a substance of nearly the same refracting power, we shall not only completely prevent this loss of light, but we shall greatly diminish the errors which arise from an imperfect polish of the surfaces. We have tried this, and find the effect very surprising. The lens being polished immediately after the figure has been given it, and while it was almost impervious to light by reason of its roughness, which was still sensible to the naked eye, performed as well as when finished in the finest manner.

N. B. This condition, by taking away one refraction, obliges us to increase those which remain, and therefore increases the spherical aberrations. And since our formulae do not fully remove those (by reason of the small quantities neglected in the process), it is uncertain whether this condition be the most eligible. We have, however, no direct argument to the contrary.

Let us see what determination this gives us.

In this case $\backslash frac\{1\}\{a\text{'}\}\; =\; \backslash frac\{1\}\{b\text{'}\}\; =\; \backslash frac\{1\}\{a\}\; -\; 1$. For because $\backslash frac\{1\}\{n\}\; =\; \backslash frac\{1\}\{a\}$ and $n=1$, we have $1\; +\; \backslash frac\{1\}\{b\}\; =\; \backslash frac\{1\}\{a\}$, and $\backslash frac\{1\}\{b\}\; =\; \backslash frac\{1\}\{a\}\; -\; 1$.

Therefore $\backslash frac\{1\}\{a^2\}\; =\; \backslash frac\{1\}\{a^2\}\; -\; \backslash frac\{2\}\{a\}\; +\; 1$. Therefore, in our final equation, put $\backslash frac\{1\}\{a^2\}\; -\; \backslash frac\{2\}\{a\}\; +\; 1$ in place of $\backslash frac\{1\}\{a^2\}$, and $\backslash frac\{1\}\{a\}\; -\; 1$ in place

Telecope place of $\backslash frac\{1\}\{a\}$, and it becomes $\backslash frac\{A-C\}\{a^2\}\; -\; \backslash frac\{B+D-2C\}\{a\}\; +\; E+D-C=0$.

Thus have we arrived at a common affected quadratic equation, where $\backslash frac\{1\}\{a\}$ is the unknown quantity. It has the common form $\backslash rho\; x^2\; +\; qx\; +\; r\; =\; 0$, where $\backslash rho$ is $A-C$, $q$ is equal to $2C-B-D$, $r$ is equal to $E+D-C$, and $x$ is equal to $\backslash frac\{1\}\{a\}$.

Divide the equation by $\backslash rho$, and we have $x^2\; +\; \backslash frac\{q\}\{\backslash rho\}x\; +\; \backslash frac\{r\}\{\backslash rho\}\; =\; 0$. Make $s\; =\; \backslash frac\{q\}\{\backslash rho\}$ and $t\; =\; \backslash frac\{r\}\{\backslash rho\}$, and we have $x^2\; +\; sx\; +\; t\; =\; 0$. This gives us finally $\backslash frac\{1\}\{a\}$, or $x\; =\; -\backslash frac\{1\}\{2\}s\; \backslash pm\; \backslash sqrt\{\backslash frac\{1\}\{4\}s^2\; -\; t\}$.

This value of $\backslash frac\{1\}\{a\}$ is taken from a scale of which the unit is half the radius of the isosceles lens which is equivalent to the first lens, or has the same focal distance with it. We must then find (on the same scale) the value of $b$, viz. $\backslash frac\{1\}\{a\}\; -\; 1$, which is also the value of $a\text{'}$. Having obtained $a\text{'}$, we must find $b\text{'}$ by means of the equation $\backslash frac\{1\}\{n\text{'}\}\; =\; \backslash frac\{1\}\{a\text{'}\}\; -\; \backslash frac\{1\}\{b\text{'}\}$, and therefore $\backslash frac\{1\}\{b\text{'}\}\; =\; \backslash frac\{1\}\{a\text{'}\}\; -\; \backslash frac\{1\}\{n\text{'}\}$. But $\backslash frac\{1\}\{n\text{'}\}\; =\; u$. Therefore $\backslash frac\{1\}\{b\text{'}\}\; =\; \backslash frac\{1\}\{a\text{'}\}\; +\; u$, $=\; \backslash frac\{1\}\{a\}\; +\; u\; -\; 1$.

Thus is our object-glass constructed; and we must determine its focal distance, or its reciprocal $\backslash frac\{1\}\{P\}$. This is $=\; m\; -\; 1\; -\; u(n\text{'}\; -\; 1)$.

All these radii and distances are measured on a scale of which $n$ is the unit. But it is more convenient to measure every thing by the focal distance of the compound object-glass. This gives us the proportion which all the distances bear to it. Therefore, calling $P$ unity, in order to obtain $\backslash frac\{1\}\{a\}$ on this scale, we have only to state

the analogy $m\; -\; 1\; -\; u(n\text{'}\; -\; 1)\; :\; 1\; =\; \backslash frac\{1\}\{a\}\; :\; \backslash frac\{1\}\{A\}$, and $A$ is the radius of our first surface measured on a scale of which $P$ is the unit.

If, in the formula which expresses the final equation for $\backslash frac\{1\}\{a\}$, the value of $s$ should be positive, and greater than $\backslash frac\{1\}\{2\}s^2$, the equation has imaginary roots; and it is not possible with the glasses employed, and the conditions assumed, to correct both the chromatic and spherical aberrations.

If $s$ is negative and equal to $\backslash frac\{1\}\{2\}s^2$, the radical part of the value is $=\; 0$, and $\backslash frac\{1\}\{a\}\; =\; -\backslash frac\{1\}\{2\}s$. But if it be negative or positive, but less than $\backslash frac\{1\}\{2\}s^2$, the equation has two real roots, which will give two constructions. That is to be preferred which gives the smallest curvature of the surfaces; because, since in our formulae which determine the spherical aberration some quantities are neglected, these quantities are always greater when a large

arch (that is, an arch of many degrees) is employed. No radius should be admitted which is much less than $\backslash frac\{1\}\{4\}$ of the focal distance.

All this process will be made plain and easy by an example.

Very careful experiments have shown, that in common crown-glass the fine of incidence is to the fine of refraction as 1.526 is to 1, and that in the generality of flint-glass it is as 1.604 to 1. Also that $\backslash frac\{d\; m\}\{d\; n\; d\}\; =\; 0.6054\; =\; u$. Therefore $m\; -\; 1\; =\; 0.526$; $m\text{'}\; -\; 1\; =\; 0.604$; $e\; =\; \backslash frac\{m-1\}\{m\text{'}-1\}\; =\; 0.87086$. By these numbers we can compute the coefficients of our final equation. We shall find them as follows:

The general equation (p. 252. col. 2. lin. 8.), when subjected to the assumed coincidence of the internal surfaces, is $\backslash frac\{A-C\}\{a^2\}\; -\; \backslash frac\{B+D-2C\}\{a\}\; +\; E+D-C=0$. $A-C$ is $=\; 0.652$; $B+D-2C$ is $=\; 0.283$; and $E+D-C$ is $=\; -0.020$; and the equation with numerical coefficients is $\backslash frac\{0.652\}\{a^2\}\; -\; \backslash frac\{0.283\}\{a\}\; -\; 0.020\; =\; 0$, which corresponds to the equation $\backslash rho\; x^2\; +\; qx\; +\; r\; =\; 0$. We must now make $s\; =\; \backslash frac\{q\}\{\backslash rho\}\; =\; \backslash frac\{0.283\}\{0.652\}\; =\; 0.434$, and $t\; =\; \backslash frac\{r\}\{\backslash rho\}\; =\; \backslash frac\{0.02\}\{0.652\}\; =\; 0.0307$.

This gives us the final quadratic equation $\backslash frac\{1\}\{a^2\}\; -\; \backslash frac\{0.434\}\{a\}\; -\; 0.0307\; =\; 0$. To solve this, we have $-\backslash frac\{1\}\{2\}s\; =\; -0.217$, and $\backslash frac\{1\}\{4\}s^2\; =\; 0.0471$. From this take $t$, which is $=\; -0.0307$ (that is, to 0.0471 add 0.0307), and we obtain 0.0778, the square root of which is $=\; 0.2789$. Therefore, finally, $\backslash frac\{1\}\{a\}\; =\; 0.2170\; \backslash pm\; 0.2789$, which is either 0.4959 or $-0.0619$. It is plain that the first must be preferred,

because the second gives a negative radius, or makes the first surface of the crown-glass concave. Now as the convergence of the rays is to be produced by the crown-glass, the other surface must become very convex, and occasion great errors in the computed aberration. We therefore retain 0.4959 for the value of $\backslash frac\{1\}\{a\}$, and $a$ is

To obtain $b$, use the equation $\backslash frac\{1\}\{b\}\; =\; \backslash frac\{1\}\{a\}\; -\; 1$, which gives $\backslash frac\{1\}\{b\}\; =\; -0.5041$, and therefore a convex surface. $b$ is

$a\text{'}$ is the same with $b$, and $\backslash frac\{1\}\{a\text{'}\}\; =\; -0.5041$.

To obtain $b\text{'}$, use the equation $\backslash frac\{1\}\{b\text{'}\}\; =\; \backslash frac\{1\}\{a\text{'}\}\; +\; u$. Now $u\; =\; 0.6054$, and $\backslash frac\{1\}\{a\text{'}\}\; =\; -0.5041$. The sum of these is

Telescope. 0.1013; and since it is positive, the surface is concave.
$b\; =\; \backslash frac\{1\}\{.1013\}\; =\; 9.872.$

Now to obtain all the measures in terms of the focal distance P, we have only to divide the measures already found by 6.2383, and the quotients are the measures wanted.

If it be intended that the focal distance of the object-glass shall be any number $n$ of inches or feet, we have only to multiply each of the above radii by $n$, and we have their lengths in inches or feet.

Thus we have completed the investigation of the construction of a double object-glass. Although this was intricate, the final result is abundantly simple for practice, especially with the assistance of logarithms. The only troublesome thing is the preparation of the numerical coefficients A, B, C, D, E of the final equation. Strict attention must also be paid to the positive and negative signs of the quantities employed.

We might propose other conditions. Thus it is natural to prefer for the first or crown-glass lens such a form as shall give it the smallest possible aberration. This will require a small aberration of the flint-glass to correct it. But a little reflection will convince us that this form will not be good. The focal distance of the crown-glass must not exceed one-third of that of the compound glass; these two being nearly in the proportion of $d\; m\text{'}\; -\; d\; m$ to $d\; m\text{'}$. Therefore if this form be adopted, and $a$ be made about $\backslash frac\{1\}\{3\}$th of $b$, it will not exceed $\backslash frac\{1\}\{3\}$th of $P$. Therefore, although we may produce a most accurate union of the central and marginal rays by opposite aberrations, there will be a considerable aberration of some rays which are between the centre and the margin.

It is absolutely impossible to collect into one point the whole rays (though the very remotest rays are united with the central rays), except in a very particular case, which cannot obtain in an object-glass; and the small quantities which are neglected in the formula which we have given for the spherical aberration, produce errors which do not follow any proportion of the aperture which can be expressed by an equation of a manageable form. When the aperture is very large, it is better not to correct the aberration for the whole aperture, but for about $\backslash frac\{1\}\{3\}$th of it. When the rays corresponding to this distance are made to coincide with the central rays by means of opposite aberrations, the rays which are beyond this distance will be united with some of those which are nearer to the centre, and the whole diffusion will be considerably diminished. Dr Smith has illustrated

this in a very perspicuous manner in his theory of his Telescope Catoptric Microscope.

But although we cannot adopt this form of an object-glass, there may be other considerations which may lead us to prefer some particular form of the crown-glass, or of the flint-glass. We shall therefore adapt our general equation $\backslash frac\{A\}\{a^2\}\; -\; \backslash frac\{B\}\{a\}\; -\; \backslash frac\{C\}\{a^2\}\; -\; \backslash frac\{D\}\{a\text{'}\}\; +\; E\; =\; 0$ to this condition.

Therefore let $h$ express this selected ratio of the two radii of the crown-glass, making $\backslash frac\{a\}\{b\}\; =\; h$ (remembering always that $a$ is positive and $b$ negative in the case of a double convex, and $h$ is a negative number).

With this condition we have $\backslash frac\{1\}\{b\}\; =\; \backslash frac\{h\}\{a\}$. But when we make $n$ the unit of our formula of aberration, $\backslash frac\{1\}\{b\}\; =\; \backslash frac\{1\}\{a\}\; -\; 1$.

Therefore $1\; =\; \backslash frac\{1\}\{a\}\; -\; \backslash frac\{h\}\{a\}$, and $\backslash frac\{1\}\{a\}\; =\; \backslash frac\{1\}\{1-h\}$. Now substitute this for $\backslash frac\{1\}\{a\}$ in the general equation, and change all the signs (which still preserves it $=\; 0$), and we obtain

By this equation we are to find $\backslash frac\{1\}\{a\text{'}\}$, or the radius of the anterior surface of the flint-glass. The equation is of this form $\backslash rho\; x^2\; +\; q\; x\; +\; r\; =\; 0$, and we must again make $x\; =\; \backslash frac\{1\}\{a\text{'}\}$ and $t\; =\; \backslash frac\{r\}\{\backslash rho\}$. Therefore $x\; =\; \backslash frac\{D\}\{C\}$, and $x\; =\; \backslash frac\{1\}\{C\}\; \backslash times\; \backslash left(\; \backslash frac\{B\}\{1-h\}\; -\; \backslash frac\{A\}\{(1-h)^2\}\; -\; E\; \backslash right)$. Then, finally,

It may be worth while to take a particular case of this condition. Suppose the crown-glass to be of equal convexities on both sides. This has some advantages: We can tell with precision whether the curvatures are precisely equal, by measuring the focal distance of rays reflected back from its posterior surface. These distances will be precisely equal. Now it is of the utmost importance in the construction of an object-glass which is to correct the spherical aberration, that the forms be precisely such as are required by our formulae.

In this case of a lens equally convex on both sides $\backslash frac\{1\}\{a\}\; =\; -\backslash frac\{1\}\{b\}\; =\; \backslash frac\{1\}\{2\}$. Substitute this value for $\backslash frac\{1\}\{a\}$ in the general equation $\backslash frac\{A\}\{a^2\}\; -\; \backslash frac\{B\}\{a\}\; -\; \backslash frac\{C\}\{a^2\}\; -\; \backslash frac\{D\}\{a\text{'}\}\; +\; E\; =\; 0$, and then $\backslash frac\{A\}\{a^2\}\; =\; \backslash frac\{A\}\{4\}$; $\backslash frac\{B\}\{a\}$ becomes $\backslash frac\{B\}\{2\}$. Now change all the signs, and we have $\backslash frac\{C\}\{a^2\}\; +\; \backslash frac\{D\}\{a\text{'}\}\; -\; E\; -\; \backslash frac\{A\}\{4\}\; +\; \backslash frac\{B\}\{2\}\; =\; 0$, by

which we are to find $a\text{'}$. This in numbers is $\backslash frac\{1.362\}\{a^2\}\; -\; \backslash frac\{0.526\}\{a\text{'}\}$

This gives two real roots, viz. 0.8874, and -0.5008. If we take the first, we shall have a convex anterior surface for the flint-glass, and consequently a very deep concave for the posterior surface. We therefore take the second or negative root -0.5008.

We find $\backslash frac\{1\}\{b\text{'}\}$ as before, by the equation $\backslash frac\{1\}\{b\text{'}\}\; =\; \backslash frac\{1\}\{a\text{'}\}\; +\; u\; =\; 0.1045$, which will give a large value of $b\text{'}$.

and $\backslash frac\{1\}\{p\}$ is the same as in the former case, viz. 0.1603.

Having all these reciprocals, we may find $a,\; b,\; a\text{'},\; b\text{'},$ and $P$; and then dividing them by $P$, we obtain finally

By comparing this object-glass with the former, we may remark, that diminishing $a$ a little increases $b$, and in this respect improves the lens. It indeed has diminished $b\text{'}$, but this being already considerable, no inconvenience attends this diminution. But we learn, at the same time, that the advantage must be very small; for we cannot diminish $a$ much more, without making it as small as the smallest radius of the object-glass. This proportion is therefore very near the maximum, or best possible; and we know that in such cases, even considerable changes in the radii will make but small changes in the result: for these reasons we are disposed to give a strong preference to the first construction, on account of the other advantages which we showed to attend it.

As another example, we may take a case which is very nearly the general practice of the London artists. The radius of curvature for the anterior surface of the convex crown-glass is $\backslash frac\{1\}\{2\}$ths of the radius of the posterior surface, so that $h\; =\; \backslash frac\{1\}\{2\}$. This being introduced into the determinate equation, gives

As another condition, we may suppose that the second or flint-glass is of a determined form.

This case is solved much in the same manner as the former. Taking $h$ to represent the ratio of $a\text{'}$ and $b\text{'}$, we have $\backslash frac\{1\}\{a\}\; =\; \backslash frac\{1\}\{1-h\}$. This value being substituted in the

general equation $\backslash frac\{A\}\{a^2\}\; -\; \backslash frac\{B\}\{a\}\; -\; \backslash frac\{C\}\{a^2\}\; -\; \backslash frac\{D\}\{a\}\; +\; E\; =\; 0$, gives us

the final equation $x^2\; +\; rx\; +\; s\; =\; 0$, $s\; =\; \backslash frac\{B\}\{A\}$, and $s\; =\; \backslash frac\{1\}\{A\}\; \backslash underbrace\{\backslash text\{Telescope.\}\}\_\{a\text{'}\}$

We might here take the particular case of the flint-glass being equally concave on both sides. Then because $\backslash frac\{1\}\{b\text{'}\}\; =\; -u$, and in the case of equal concavities $\backslash frac\{2\}\{a\}\; =\; \backslash frac\{1\}\{b\text{'}\}\; =\; -u$, it is sufficient to put $-\backslash frac\{1\}\{2\}u$ for $\backslash frac\{1\}\{a\text{'}\}$.

This being done, the equation becomes $\backslash frac\{A\}\{a\}\; -\; \backslash frac\{B\}\{a\}\; \backslash frac\{Cu^2\}\{4\}\; +\; \backslash frac\{Du\}\{2\}\; +\; E\; =\; 0$. This gives $s\; =\; \backslash frac\{B\}\{A\}$, and $s\; =\; \backslash frac\{1\}\{A\}\; \backslash times\; \backslash left(\; \backslash frac\{4Du\; -\; 2Cu^2\}\{8\}\; +\; E\; \backslash right)$.

We imagine that these cases are sufficient for showing the management of the general equation; and the example of the numerical solution of the first case affords instances of the only niceties which occur in the process, viz. the proper employment of the positive and negative quantities.

We have oftener than once observed, that the formula is not perfectly accurate, and that in very large apertures errors will remain. It is proper therefore, when we have obtained the form of a compound object-glass, to calculate trigonometrically the progress of the light through it; and if we find a considerable aberration, either chromatic or spherical, remaining, we must make such changes in the curvatures as will correct them.

We have done this for the first example; and we find, that if the focal distance of the compound object-glass be 100 inches, there remains of the spherical aberration nearly $\backslash frac\{1\}\{30\}$th of an inch, and the aberration of colour is over corrected above $\backslash frac\{1\}\{30\}$th of an inch. The first aberration has been diminished about 6 times, and the other about 30 times. Both of the remaining errors will be diminished by increasing the radius of the inner surfaces. This will diminish the aberration of the crown-glass, and will diminish the dispersion of the flint more than that of the crown. But indeed the remaining error is hardly worth our notice.

It is evident to any person conversant with optical discussions, that we shall improve the correction of the spherical aberration by diminishing the refractions. If we employ two lenses for producing the convergence of the rays to a real focus, we shall reduce the aberration to $\backslash frac\{1\}\{4\}$th. Therefore a better achromatic glass will be formed of three lenses, two of which are convex and of crown-glass. The refraction being thus divided between them, the aberrations are lessened. There is no occasion to employ two concave lenses of flint-glass; there is even an advantage in using one. The aberration being considerable, less of it will serve for correcting the aberration of the crown-glass, and therefore such a form may be selected as has little aberration. Some light is indeed lost by these two additional surfaces; but this is much more than compensated by the greater apertures which we can venture to give when the curvature of the surface is so much diminished. We proceed therefore to

It is plain that there are more conditions to be assumed:

Telescope. fumed before we can render this a determinate problem, and that the investigation must be more intricate. At the same time, it must give us a much greater variety of constructions, in consequence of our having more conditions necessary for giving the equation this determinate form. Our limits will not allow us to give a full account of all that may be done in this method. We shall therefore content ourselves with giving one case, which will sufficiently point out the method of proceeding. We shall then give the results in some other eligible cases, as rules to artists by which they may construct such glasses.

Let the first and second glasses be of equal curvatures on both sides; the first being a double convex of crown-glass, and the second a double concave of flint-glass.

Still making $n$ the unit of our calculus, we have in the first place $a\; =\; -b,\; =\; -a\text{'},\; =\; b\text{'}$. Therefore $\backslash frac\{1\}\{a\}\; =\; \backslash frac\{1\}\{b\}\; =\; -\backslash frac\{1\}\{n\}$.

$\backslash left(\backslash frac\{1\}\{a\}\; -\; \backslash frac\{1\}\{b\}\backslash right)$, or $\backslash frac\{1\}\{n\}\; =\; -\backslash frac\{1\}\{n\}\; =\; -1$. Therefore the equation $\backslash frac\{d\; m\}\{n\}\; +\; \backslash frac\{d\; m\text{'}\}\{n\text{'}\}\; +\; \backslash frac\{d\; m\text{'}\text{'}\}\{n\text{'}\text{'}\}\; =\; 0$ becomes $u\; -\; 1\; +\; \backslash frac\{u\}\{n\text{'}\text{'}\}\; =\; 0$, or $\backslash frac\{1\}\{n\}\; =\; \backslash frac\{1\}\{u\}\; -\; 1$. Let us call this value $u\text{'}$.

We have $\backslash frac\{1\}\{p\}\; =\; m\; -\; 1$; $\backslash frac\{1\}\{p\text{'}\}\; =\; -(m\text{'}\; -\; 1)$; $\backslash frac\{1\}\{p\text{'}\text{'}\}\; =\; u\text{'}$.
$(m\; -\; 1)\; \backslash cdot\; \backslash frac\{1\}\{p\}\; =\; \backslash frac\{1\}\{p\}\; +\; \backslash frac\{1\}\{p\text{'}\}\; +\; \backslash frac\{1\}\{p\text{'}\text{'}\}\; =\; m\; -\; m\text{'}\; +\; u\text{'}\; (m\; -\; 1)$.

And if we make $m\text{'}\; -\; m\; =\; C$, we shall have $\backslash frac\{1\}\{p\}\; =\; -C$,
$+\; u\text{'}\; (m\; -\; 1)$. Also $\backslash frac\{1\}\{r\}\; =\; m\; -\; 1$; $\backslash frac\{1\}\{r\text{'}\}\; =\; m\; -\; 1\; -\; (m\text{'}\; -\; 1)$, $=\; m\; -\; m\text{'}\; =\; -C$.

The equality of the two curvatures of each lens gives $\backslash frac\{1\}\{a\}\; =\; \backslash frac\{1\}\{2n\}$. Therefore $\backslash frac\{1\}\{a\}\; =\; -\backslash frac\{1\}\{b\}\; =\; -\backslash frac\{1\}\{a\text{'}\}\; =\; \backslash frac\{1\}\{b\text{'}\}$; and $\backslash frac\{1\}\{b\text{'}\text{'}\}\; =\; \backslash frac\{1\}\{a\text{'}\text{'}\}\; =\; \backslash frac\{1\}\{n\text{'}\text{'}\}\; =\; \backslash frac\{1\}\{u\text{'}\}$.

Substituting these values in the equation (p. 252. col. 2. par. 1.), we obtain the three formulae.

Now arrange these quantities according as they are coefficients of $\backslash frac\{1\}\{a\text{'}\text{'}\}$ and of $\backslash frac\{1\}\{a\text{'}^2\}$, or independent quantities. Let the coefficient of $\backslash frac\{1\}\{a\text{'}\text{'}\}$ be $A$, that of $\backslash frac\{1\}\{a\text{'}^2\}$ be $B$, and the independent quantity be $C$, we have

Our equation now becomes $\backslash frac\{A\}\{a\text{'}\text{'}\}\; -\; \backslash frac\{B\}\{a\text{'}^2\}\; +\; C\; =\; 0$.
This reduced to numbers, by computing the values of the coefficients, is $\backslash frac\{1.312\}\{a\text{'}\text{'}\}\; -\; \backslash frac\{1.207\}\{a\text{'}^2\}\; -\; 0.3257\; =\; 0$.

This, divided by 1.312, gives $s\; =\; -0.92$; and $t\; =\; -0.2482$; $-\backslash frac\{1\}\{2\}\; t\; =\; 0.46$; $\backslash frac\{1\}\{4\}\; t^2\; =\; 0.2116$; and $\backslash sqrt\{\backslash frac\{1\}\{2\}\; t^2\}\; =\; t\; =\; \backslash pm\; 0.6781$.

And, finally, $\backslash frac\{1\}\{a\text{'}\text{'}\}\; =\; 0.46\; \backslash pm\; 0.6781$.
This has two roots, viz. 0.2181 and $-1.1381$. The last would give a small radius, and is therefore rejected.

Now, proceeding with this value of $\backslash frac\{1\}\{a\text{'}\text{'}\}$ and the $\backslash frac\{1\}\{n\text{'}\text{'}\}$, we get the other radius $b\text{'}$, and then, by means of $u\text{'}$, we get the other radius which is common to the four surfaces. Then, by $\backslash frac\{1\}\{p\}\; =\; \backslash frac\{1\}\{p\text{'}\}\; =\; c\text{'}$, we get the value of $P$.

The radii being all on the scale of which $u$ is the unit, they must be divided by $P$ to obtain their value on the scale which has $P$ for its unit. This will give us

This is not a very good form, because the last surface has too great curvature.

We thought it worth while to compute the curvatures for a case where the internal surfaces of the lenses coincide, in order to obtain the advantages mentioned on a former occasion. The form is as follows:

The middle lens is a double concave of flint-glass; the last lens is of crown-glass, and has equal curvatures on both sides. The following table contains the dimensions of the glasses for a variety of focal distances. The first column contains the focal distances in inches; the second contains the radii of the first surface in inches; the third contains the radii of the posterior surface of the first lens and anterior surface of the second; and the fourth column has the radii of the three remaining surfaces.

We have had an opportunity of trying glasses of this construction, and found them equal to any of the same length, although executed by an artist by no means excellent in his profession as a glass-grinder. This very circumstance

Telecope circumstance gave us the opportunity of seeing the good effects of interposing a transparent substance between the glasses. We put some clear turpentine varnish between them, which completely prevented all reflection from the internal surfaces. Accordingly these telescopes were surprisingly bright; and although the roughness left by the first grinding was very perceptible by the naked eye before the glasses were put together, yet when joined in this manner it entirely disappeared, even when the glasses were viewed with a deep magnifier.

The aperture of an object-glass of this construction of 30 inches focal distance was 3 1/4 inches, which is considerably more than any of Mr Dollond's that we have seen.

If we should think it of advantage to make all the three lenses isosceles, that is, equally curved on both surfaces, the general equation will give the following radii:

This seems a good form, having large radii.

Should we choose to have the two crown-glass lenses isosceles and equal, we must make

This form hardly differs from the last.

Our readers will recollect that all these forms proceed on certain measures of the refractive and dispersive powers of the substances employed, which are expressed by $m$, $m\text{'}$, $d\; m$, and $d\; m\text{'}$: and we may be assured that the formulae are sufficiently exact, by the comparison (which we have made in one of the cases) of the result of the formula and the trigonometrical calculation of the progress of the rays. The error was but 1/20th of the whole, ten times less than another error, which unavoidably remains, and will be considered presently. These measures of refraction and dispersion were carefully taken; but there is great diversity, particularly in the flint-glass. We are well informed that the manufacture of this article has considerably changed of late years, and that it is in general less refractive and less dispersive than formerly. This must evidently make a change in the forms of achromatic glasses. The proportion of the focal distance of the crown-glasses to that of the flint must be increased, and this will occasion a change in the curvatures, which shall correct the spherical aberration. We examined with great care a parcel of flint-glass which an artist of this city got lately for the purpose of making achromatic object-glasses, and also some very white crown glass made in Leith; and we obtained the following measures:

We computed some forms for triple object glasses made of these glasses, which we shall subjoin as a specimen of the variations which this change of data will occasion.

If all the three lenses are made isosceles, we have

Or

If the middle lens be isosceles, the two crown-glass

lenses may be made of the same form and focal distance, Telecope and placed the same way. This will give us

N. B. This construction allows a much better form, if the measures of refraction and dispersion are the same that we used formerly. For we shall have

And this is pretty near the practice of the London opticians.

We may here observe, upon the whole, that an amateur has little chance of succeeding in these attempts. The diversity of glasses, and the uncertainty of the workman's producing the very curvatures which he intends, is so great, that the object-glass turns out different from our expectation. The artist who makes great numbers acquires a pretty certain guess at the remaining error; and having many lenses, intended to be of one form, but unavoidably differing a little from it, he tries several of them with the other two, and finding one better than the rest, he makes use of it to complete the set.

The great difficulty in the construction is to find the exact proportion of the dispersive powers of the crown and flint-glass. The crown is pretty constant; but there are hardly two pots of flint-glass which have the same dispersive power. Even if constant, it is difficult to measure it accurately; and an error in this greatly affects the instrument, because the focal distances of the lenses must be nearly as their dispersive powers. The method of examining this circumstance, which we found most accurate, was as follows:

The sun's light, or that of a brilliant lamp, passed through a small hole in a board, and fell on another board pierced also with a small hole. Behind this was placed a fine prism A (fig. 14.), which formed a spectrum ROV on a screen pierced with a small hole. Behind this was placed a prism B of the substance under examination. The ray which was refracted by it fell on the wall at D, and the distance of its illumination from that point to C, on which an unrefracted ray would have fallen, was carefully measured. This showed the refraction of that colour. Then, in order that we might be certain that we always compared the refraction of the same precise colour by the different prisms placed at B, we marked the precise position of the prism A when the ray of a particular colour fell on the prism B. This was done by an index AG attached to A, and turning with it, when we caused the different colours of the spectrum formed by A to fall on B. Having examined one prism B with respect to all the colours in the spectrum formed by A, we put another B in its place. Then bringing A to all its former positions successively, by means of a graduated arch HGK, we were certain that when the index was at the same division of the arch it was the very ray which had been made to pass through the first prism B in a former experiment. We did not solicitously endeavour to find the very extreme red and violet rays; because, although we did not learn the whole dispersions of the two prisms, we learned their proportions, which is the circumstance wanted in the construction of achromatic glasses. It is in vain to attempt this by measuring the spectrums themselves; for we cannot be certain of

Telescope selecting the very same colours for the comparison, because they succeed in an insensible gradation.

The intelligent reader will readily observe, that we have hitherto proceeded on the supposition, that when, by means of contrary refractions, we have united the extreme red and violet rays, we have also united all the others. But this is quite gratuitous. Sir Isaac Newton would, however, have made the same supposition; for he imagined that the different colours divided the spectrum formed by all substances in the proportions of a musical canon. This is a mistake. When a spectrum is formed by a prism of crown glass, and another of precisely the same length is formed by the side of it by a prism of flint-glass, the confine between the green and blue will be found precisely in the middle of the first spectrum, but in the second it will be considerably nearer to the red extremity. In short, different substances do not disperse the colours in the same proportion.

The effect of this irrationality (so to call it) of dispersion, will appear plainly, we hope, in the following manner: Let A (fig. 12.) represent a spot of white solar light falling perpendicularly on a wall. Suppose a prism of common glass placed behind the hole through which the light is admitted, with its refracting angle facing the left hand. It will refract the beam of light to the right, and will at the same time disperse this heterogeneous light into its component rays, carrying the extreme red ray from A to R, the extreme orange from A to O, the extreme yellow from A to Y, &c. and will form the usual prismatic spectrum ROYGBPC. If the whole length RC be divided into 1000 parts, we shall have (when the whole refraction AR is small) RO very nearly 125, RY=200, RG=335, RB=500, RP=667, RV=778, and RC=1000; this being the proportion observed in the differences of the sines of refraction by Sir Isaac Newton.

Perhaps a refracting medium may be found such, that a prism made of it would refract the white light from A', in the upper line of this figure, in such a manner that a spectrum R'O'Y'G'B'P'V'C' shall be formed at the same distance from A', and of the same length, but divided in a different proportion. We do not know that such a medium has been found; but we know that a prism of flint-glass has its refractive and dispersive powers so constituted, that if A'H' be taken about one-third of AR, a spot of white light, formed by rays falling perpendicularly at H', will be so refracted and dispersed, that the extreme red ray will be carried from H' to R', and the extreme violet from H' to C', and the intermediate colours to intermediate points, forming a spectrum resembling the other, but having the colours more constricted towards R', and more dilated towards C'; so that the ray which the common glass carried to the middle point B of the spectrum RC is now in a point B' of the spectrum R'C', considerably nearer to R'.

Dr Blair has found, on the other hand, that certain fluids, particularly such as contain the muriatic acid, when formed into a prism, will refract the light from H'' (in the lower line) so as to form a spectrum R''C'' equal to RC, and as far removed from A'' as RC is from A, but having the colours more dilated toward R'', and more constricted toward C'', than is observed in RC; so that the ray which was carried by the prism of common

glass to the middle point B is carried to a point B'', considerably nearer to C''.

Let us now suppose that, instead of a white spot at A, we have a prismatic spectrum AB (fig. 13.), and that the prism of common glass is applied as before, immediately behind the prism which forms the spectrum AB. We know that this will be refracted sidewise, and will make a spectrum ROYGBPC, inclined to the plane of refraction in an angle of 45°; so that drawing the perpendicular RC', we have RC'=C'C.

We also know that the prism of flint-glass would refract the spectrum formed by the first prism on EHF, in such a manner that the red ray will go to R, the violet to C, and the intermediate rays to points o, y, g, b, p, v, so situated that O'o is = R'O' of the other figure; Yy is = RY' of that figure, Gg = RG', &c. These points must therefore lie in a curve R o y g b p v C, which is convex toward the axis R'C'.

In like manner we may be assured that Dr Blair's fluid will form a spectrum R o' y' g' b' p' v' C, concave toward R'C'.

Let it be observed by the way, that this is a very good method for discovering whether a medium disperses the light in the same proportion with the prism which is employed for forming the first spectrum AB or EF. It disperses in the same or in a different proportion, according as the oblique spectrum is straight or crooked; and the exact proportion corresponding to each colour is had by measuring the ordinates of the curves R b C or R b' C.

Having formed the oblique spectrum RBC by a prism of common glass, we know that an equal prism of the same glass, placed in a contrary position, will bring back all the rays from the spectrum RBC to the spectrum AB, laying each colour on its former place.

In like manner, having formed the oblique spectrum R b C by a prism of flint-glass, we know that another prism of flint-glass, placed in the opposite direction, will bring all the rays back to the spectrum EHF.

But having formed the oblique spectrum RBC by a prism of common glass, if we place the flint-glass prism in the contrary position, it will bring the colour R back to E, and the colour C to F; but it will not bring the colour B to H, but to a point $\backslash bar\{B\}$, such that $B\backslash bar\{B\}$ is equal to $bH$, and $bB$ to $\backslash bar\{B\}H$. In like manner, the other colours will not be brought back to the straight line EHF, but to a curve $E\backslash bar\{B\}F$, forming a crooked spectrum.

In like manner, the fluids discovered by Dr Blair, when employed to bring back the oblique spectrum RBC formed by common glass, will bring its extremities back to E and F, and form the crooked spectrum $E\backslash bar\{B\}F$ lying beyond EHF.

This experiment evidently gives us another method for examining the proportionality of the dispersion of different substances.

Having, by common glass, brought back the oblique spectrum formed by common glass to its natural place AB, suppose the original spectrum at AB to contract gradually (as Newton has made it do by means of a lens), it is plain that the oblique spectrum will also contract, and so will the second spectrum at AB; and it will at last coalesce into a white spot. The effect will be equivalent to a gradual compression of the whole figure.

Telescope. figure, by which the parallel lines AR and BC gradually approach, and at last unite.

In like manner, when the oblique spectrum formed by flint-glass is brought back to EHF by a flint-glass prism, and the figure compressed in the same gradual manner, all the colours will coalesce into a white spot.

But when flint-glass is employed to bring back the oblique spectrum formed by common glass, it forms the crooked spectrum E & F. Now let the figure be compressed. The curve E & F will be doubled down on the line H & k, and there will be formed a compound spectrum H & k, quite unlike the common spectrum, being purple or claret-coloured at H by the mixture of the extreme red and violet, and green edged with blue at k by the mixture of the green and blue. The fluid prisms would in like manner form a spectrum of the same kind on the other side of H.

This is precisely what is observed in achromatic object-glasses made of crown-glass and flint: for the refraction from A to R corresponds to the refraction of the convex crown-glass; and the contrary refraction from R to E corresponds to the contrary refraction of the concave flint-glass, which still leaves a part of the still refraction, producing a convergence to the axis of the telescope. It is found to give a purple or wine-coloured focus, and within this a green one, and between these an imperfect white. Dr Blair found, that when the eye-glass was drawn out beyond its proper distance, a star was surrounded by a green fringe, by the green end of the spectrum, which crossed each other within the focus; and when the eye-glass was too near the object-glass, the star had a wine-coloured fringe. The green rays were ultimately most refracted. N. B. We should expect the fringe to be of a blue colour rather than a green. But this is easily explained: The extreme violet rays are very faint, so as hardly to be sensible; therefore when a compound glass is made as achromatic as possible to our senses, in all probability (nay certainly) these almost insensible violet rays are left out, and perhaps the extreme colours which are united are the red and the middle violet rays. This makes the green to be the mean ray, and therefore the most outstanding when the dispersions are not proportional.

Dr Blair very properly calls these spectrums, H & k and H' & k', secondary spectrums, and seems to think that he is the first who has taken notice of them. But Mr Clairault was too accurate a mathematician, and too careful an observer, not to be aware of a circumstance which was of primary consequence to the whole inquiry. He could not but observe that the success rested on this very particular, and that the proportionality of dispersion was indispensably necessary.

This subject was therefore touched on by Clairault; and fully discussed by Boscovich, first in his Dissertations published at Vienna in 1759; then in the Comment. Bononiensis; and, lastly, in his Opuscula, published in 1785. Dr Blair, in his ingenious Dissertation on Achromatic Glasses, read to the Royal Society of Edinburgh in 1793, seems not to have known of the labours of these writers; speaks of it as a new discovery; and exhibits some of the consequences of this principle in a singular point of view, as something very paradoxical and inconsistent with the usually received notions on these subjects. But they are by no means so. We are, however, much indebted to his ingenious researches, and his successful en-

deavours to find some remedy for this imperfection of Telescope. achromatic glasses. Some of his contrivances are exceedingly ingenious; but had the Doctor consulted these writers, he would have saved himself a good deal of trouble.

Boscovich shows how to unite the two extremes with the most outstanding colour of the secondary spectrum, by means of a third substance. When we have done this, the aberration occasioned by the secondary spectrums must be prodigiously diminished; for it is evidently equivalent to the union of the points H and k of our figure. Whatever cause produces this must diminish the curvature of the arches E & k and k & F: but even if these curvatures were not diminished, their greatest ordinates cannot exceed one-fourth of H & k; and we may say, without hesitation, that by uniting the mean or most outstanding ray with the two extremes, the remaining dispersion will be as much less than the uncorrected colour of Dollond's achromatic glass, as this is less than four times the dispersion of a common object-glass. It must therefore be altogether insensible.

Boscovich asserts, that it is not possible to unite more than two colours by the opposite refraction of two substances, which do not disperse the light in the same proportions. Dr Blair makes light of this assertion, as he finds it made in general terms in the vague and paltry extract made by Priestley from Boscovich in his Essay on the History of Optics; but had he read this author in his own dissertations, he would have seen that he was perfectly right. Dr Blair, however, has hit on a very ingenious and effectual method of producing this union of three colours. In the same way as we correct the dispersion of a concave lens of crown-glass by the opposite dispersion of a concave lens of flint-glass, we may correct the secondary dispersion of an achromatic convex lens by the opposite secondary dispersion of an achromatic concave lens. But the intelligent reader will observe, that this union does not contradict the assertion of Boscovich, because it is necessarily produced by means of three refracting substances.

The most essential service which the public has received at the hands of Dr Blair is the discovery of fluid mediums of a proper dispersive power. By composing the lenses of such substances, we are at once freed from the irregularities in the refraction and dispersion of flint-glass, which the chemists have not been able to free it from. In whatever way this glass is made, it consists of parts which differ both in refractive and dispersive power; and when taken up from the pot, these parts mix in threads, which may be disseminated through the mass in any degree of fineness. But they still retain their properties; and when a piece of flint-glass has been formed into a lens, the eye, placed in its focus, sees the whole surface occupied by glistening threads or broader veins running across it. Great rewards have been offered for removing this defect, but hitherto to no purpose. We beg leave to propose the following method: Let the glass be reduced to powder, and then melted with a great proportion of alkaline salt, so as to make a liquor sili-com. When precipitated from this by an acid, it must be in a state of very uniform composition. If again melted into glass, we should hope that it would be free from this defect; if not, the case seems to be desperate.

But by using a fluid medium, Dr Blair was freed from all this embarrassment; and he acquired another

Telescope. immense advantage, that of adjusting at pleasure both the refractive and dispersive powers of his lenses. In solid lenses, we do not know whether we have taken the curvatures suited to the refractions till our glass is finished; and if we have mistaken the proportions, all our labour is lost. But when fluids are used, it is enough that we know nearly the refractions. We suit our focal distances to these, and then select our curvatures, so as to remove the aberration of figure, preserving the focal distances. Thus, by properly tempering the fluid mediums, we bring the lens to agree precisely with the theory, perfectly achromatic, and the aberration of figure as much corrected as is possible.

Dr Blair examined the refractive and dispersive powers of a great variety of substances, and found great varieties in their actions on the different colours. This is indeed what every well informed naturalist would expect. There is no doubt now among naturalists about the mechanical connection of the phenomena of nature; and all are agreed that the chemical actions of the particles of matter are perfectly like in kind to the action of gravitating bodies; that all these phenomena are the effects of forces like those which we call attractions and repulsions, and which we observe in magnets and electrified bodies; that light is refracted by forces of the same kind, but differing chiefly in the small extent of their sphere of activity. One who views things in this way will expect, that as the actions of the same acid for the different alkalies are different in degree, and as the different acids have also different actions on the same alkali, in like manner different substances differ in their general refractive powers, and also in the proportion of their action on the different colours. Nothing is more unlikely therefore than the proportional dispersion of the different colours by different substances; and it is surprising that this inquiry has been so long delayed. It is hoped that Dr Blair will oblige the public with an account of the experiments which he has made. This will enable others to co-operate in the improvement of achromatic glasses. We cannot derive much knowledge from what he has already published, because it was chiefly with the intention of giving a popular, though not an accurate, view of the subject. The constructions which are there mentioned are not those which he found most effectual, but those which would be most easily understood, or demonstrated by the slight theory which is contained in the dissertation; besides, the manner of expressing the difference of refrangibility, perhaps chosen for its paradoxical appearance, does not give us a clear notion of the characteristic differences of the substances examined. Those rays which are ultimately most deflected from their direction, are said to have become the most refrangible by the combination of different substances, although, in all the particular refractions by which this effect is produced, they are less refracted than the violet light. We can just gather this much, that common glass disperses the rays in such a manner, that the ray which is in the centre of the green and blue occupies the middle of the prismatic spectrum; but in glasses, and many other substances, which are more dispersive, this ray is nearer to the ruddy extremity of the spectrum. While therefore the straight line RC' (Fig. 13.) terminates the ordinates O', YY', G', &c. which represent the dispersion of common glass, the ordinates which express the dispersions of these substances

are terminated by a curve passing through R and C', but lying below the line RC'. When therefore parallel heterogeneous light is made to converge to the axis of a convex lens of common glass, as happens at F in Fig. 6. C, the light is dispersed, and the violet rays have a shorter focal distance. If we now apply a concave lens of greater dispersive power, the red and violet rays are brought to one focus F'; but the green rays, not being so much refracted away from F', are left behind at F, and have now a shorter focal distance. But Dr Blair afterwards found that this was not the case with the muriatic acid, and some solutions in it. He found that the ray which common glass caused to occupy the middle of the spectrum was much nearer to the blue extremity when refracted by these fluids. Therefore a concave lens formed of such fluids which united the red and violet rays in F', refracted the green rays to F'.

Having observed this, it was an obvious conjecture, that a mixture of some of these fluids might produce a medium, whose action on the intermediate rays should have the same proportion that is observed on common glass; or that two of them might be found which formed spectra similarly divided, and yet differing sufficiently in dispersive power to enable us to destroy the dispersion by contrary refractions, without destroying the whole refraction. Dr Blair accordingly found a mixture of solutions of ammoniacal and mercurial salts, and also some other substances, which produced dispersions proportional to that of glass, with respect to the different colours.

And thus has the result of this intricate and laborious investigation corresponded to his utmost wishes. He has produced achromatic telescopes which seem as perfect as the thing will admit of; for he has been able to give them such apertures, that the incorrigible aberration arising from the spherical surfaces becomes a sensible quantity, and precludes farther amplification by the eye-glasses. We have examined one of his telescopes: The focal distance of the object-glass did not exceed 17 inches, and the aperture was fully 3½ inches. We viewed some single and double stars and some common objects with this telescope; and found, that in magnifying power, brightness, and distinctness, it was manifestly superior to one of Mr Dollond's of 42 inches focal length. It also gave us an opportunity of admiring the dexterity of the London artists, who could work the glasses with such accuracy. We had most distinct vision of a star when using an erecting eye-piece, which made this telescope magnify more than a hundred times; and we found the field of vision as uniformly distinct as with Dollond's 42 inch telescope magnifying 46 times. The intelligent reader must admire the nice figuring and centering of the very deep eye-glasses which are necessary for this amplification.

It is to be hoped that Dr Blair will extend his views to glasses of different compositions, and thus give us object-glasses which are solid; for those composed of fluids have inconveniences which will hinder them from coming into general use, and will confine them to the museums of philosophers. We imagine that antimonial glasses bid fair to answer this purpose, if they could be made free of colour, so as to transmit enough of light. We recommend this dissertation to the careful perusal of our readers. Those who have not made themselves much acquainted with the delicate and abstruse theory of aberrations, will find it exhibited in such a popular form

Telescope. form as will enable them to understand its general aim; and the well-informed reader will find many curious indications of inquiries and discoveries yet to be made.

We now proceed to consider the eye-glasses or glasses of telescopes. The proper construction of an eye-piece is not less essential than that of the object-glass. But our limits will not allow us to treat this subject in the same detail. We have already extended this article to a great length, because we do not know of any performance in the English language which will enable our readers to understand the construction of achromatic telescopes; an invention which reflects honour on our country, and has completed the discoveries of our illustrious Newton. Our readers will find abundant information in Dr Smith's Optics concerning the eye-glasses, chiefly deduced from Huyghen's fine theory of aberration (A). At the same time, we must again pay Mr Dollond the merited compliment of saying, that he was the first who made any scientific application of this theory to the compound eye-piece for erecting the object. His eye-pieces of five and six glasses are very ingenious reduplications of Huyghen's eye-piece of two glasses, and would probably have superseded all others, had not his discovery of achromatic object-glasses caused opticians to consider the chromatic dispersion with more attention, and pointed out methods of correcting it in the eye-piece without any compound eye-glasses. They have found that this may be more conveniently done with four eye-glasses, without sensibly diminishing the advantages which Huyghen showed to result from employing many small refractions instead of a lesser number of great ones. As this is a very curious subject, we shall give enough for making our readers fully acquainted with it, and content ourselves with merely mentioning the principles of the other rules for constructing an eye-piece.

Such readers as are less familiarly acquainted with optical discussions will do well to keep in mind the following consequences of the general focal theorem.

Fig. 15. If AB (fig. 15) be a lens, R a radiant point or focus of incident rays, and $a$ the focus of parallel rays coming from the opposite side; then,

1. Draw the perpendicular $a\text{'}a\text{'}$ to the axis, meeting the incident ray in $a\text{'}$, and $a\text{'}A$ to the centre of the lens. The refracted ray BF is parallel to $a\text{'}A$; for $Ra\text{'}\; :\; a\text{'}A\; =\; RB\; :\; BF\; (=RA\; :\; AF)$, which is the focal theorem.

2. An oblique pencil BP$b$ proceeding from any point P which is not in the axis, is collected to the point $f$, where the refracted ray BF cuts the line PA$f$ drawn from P through the centre of the lens; for $Pa\text{'}\; :\; a\text{'}A\; =\; PB\; :\; Bf$, which is also the focal theorem.

The Galilean telescope is susceptible of so little improvement, that we need not employ any time in illustrating its performance.

The simple astronomical telescope is represented in Telescope fig. 16. The beam of parallel rays, inclined to the axis, is made to converge to a point G, where it forms an image of the lowest point of a very distant object. These rays decussating from G fall on the eye-glass; the ray from the lowest point B of the object-glass falls on the eye-glass at $b$; and the ray from A falls on $a$; and the ray from the centre O falls on $o$. These rays are rendered parallel, or nearly so, by refraction through the eye-glass, and take the direction $bi$, $oi$, $ai$. If the eye be placed so that this pencil of parallel rays may enter it, they converge to a point of the retina, and give distinct vision of the lowest point of the object. It appears inverted, because the rays by which we see its lowest point come in the direction which in simple vision is connected with the upper point of an object. They come from above, and therefore are thought to proceed from above. We see the point as if situated in the direction $Io$. In like manner the eye placed at I, sees the upper point of the object in the direction $IP$, and its middle in the direction $IE$. The proper place for the eye is I; if brought much nearer the glass, or removed much farther from it, some, or the whole, of this extreme pencil of rays will not enter the pupil. It is therefore of importance to determine this point. Because the eye requires parallel rays for distinct vision, it is plain that F must be the principal focus of the eye-glass. Therefore, by the common focal theorem, $OF\; :\; OE\; =\; OE\; :\; OI$, or $OF\; :\; FE\; =\; OE\; :\; EI$.

The magnifying power being measured by the magnitude of the visual angle, compared with the magnitude of the visual angle with the naked eye, we have $\backslash frac\{OI\}\{OE\}$, or $\backslash frac\{OF\}\{OF\}$ for the measure of the magnifying

power. This is very nearly $=\; \backslash frac\{OE\}\{EI\}$, or $\backslash frac\{OF\}\{FI\}$.

As the line OE, joining the centres of the lenses, and perpendicular to their surfaces, is called the axis of the telescope, so the ray OG is called the axis of the oblique pencil, being really the axis of the cone of light which has the object-glass for its base. This ray is through its whole course the axis of the oblique pencil; and when its course is determined, the amplification, the field of vision, the apertures of the glasses, are all determined. For this purpose we have only to consider the centre of the object-glass as a radical point, and trace the process of a ray from this point through the other glasses: this will be the axis of some oblique pencil.

It is evident, therefore, that the field of vision depends on the breadth of the eye-glass. Should we increase this, the extreme pencil will pass through I, because O and I are still the conjugate foci of the eye-glass.

(A) While we thus repeatedly speak of the theory of spherical aberration as coming from Mr Huyghens, we must not omit giving a due share of the honour of it to Dr Barrow and Mr James Gregory. The first of these authors, in his Optical Lectures delivered at Cambridge, has given every proposition which is employed by Huyghens, and has even prosecuted the matter much further. In particular, his theory of oblique slender pencils is of immense consequence to the perfection of telescopes, by showing the methods for making the image of an extended surface as flat as possible. Gregory, too, has given all the fundamental propositions in his Optica Promota. But Huyghens, by taking the subject together, and treating it in a system, has greatly simplified it: and his manner of viewing the principal parts of it is incomparably more perspicuous than the performances of Barrow and Gregory.

Telescope. glafs. On the other hand, the angle resolved on for the extent or field of vision gives the breadth of the eye-glafs.

We may here observe, by the way, that for all optical instruments there must be two optical figures considered. The first shows the progress of a pencil of rays coming from one point of the object. The various foci of this pencil show the places of the different images, real or virtual. Such a figure is formed by the three rays $AG\; \backslash circ\; i$, $OG\; \backslash circ\; i$, $BG\; \backslash circ\; i$.

The second shows the progress of the axes of the different pencils proceeding through the centre of the object-glafs. The foci of this pencil of axes show the places where an image of the object-glafs is formed; and this pencil determines the field of vision, the apertures of the lenses, and the amplification or magnifying power. The three rays $OG\; \backslash circ\; i$, $OFEI$, $OHPI$, form this figure.

See also fig. 24. where the progress of both sets of pencils is more diversified.

The perfection of a telescope is to represent an object in its proper shape, distinctly magnified, with a great field of vision, and sufficiently bright. But there are limits to all these qualities; and an increase of one of them, for the most part, diminishes the rest. The brightness depends on the aperture of the object-glafs, and will increase in the same proportion (because $i$ will always be to $AB$ in the proportion of $EF$ to $FO$), till the diameter of the emergent pencil is equal to that of the pupil of the eye. Increasing the object-glafs any more, can send no more light into the eye. But we cannot make the emergent pencil nearly so large as this when the telescope magnifies much; for the great aperture of the object-glafs produces an indistinct image at $GF$, and its indistinctness is magnified by the eye-glafs.

A great field of vision is incompatible with the true shape of the object; for it is not strictly true that all rays flowing from $O$ are refracted to $I$. Those rays which go to the margin of the eye-glafs cross the axis between $E$ and $I$; and therefore they cross it at a greater angle than if they passed through $I$. Now had they really passed through $I$, the object would have been represented in its due proportions. Therefore since the angles of the marginal parts are enlarged by the aberration of the eye-glafs, the marginal parts themselves will appear enlarged, or the object appear distorted. Thus a chess-board viewed through a reading glass appears drawn out at the corners, and the straight lines are all changed into curves, as is represented in fig. 18.

The circumstance which most peremptorily limits the extent of field is the necessary distinctness. If the vision be indistinct, it is useless, and no other quality can compensate this defect. The distortion is very inconsiderable in much larger angles of vision than we can admit, and is unworthy of the attention paid to it by optical writers. They have been induced to take notice of it, because the means of correcting it in a considerable degree are attainable, and afford an opportunity of exhibiting their knowledge; whereas the indistinctness which accompanies a large field is a subject of most difficult discussion, and has hitherto baffled all their efforts to express by any intelligible or manageable formula.

Quaque tractata nuncere posse
Deserat relinquere.

This subject must, however, be considered. The image at $GF$ of a very remote object is not a plain surface perpendicular to the axis of the telescope, but is nearly spherical, having $O$ for its centre. If a number of pencils of parallel rays crossing each other in $I$ fall on the eye-glafs, they will form a picture on the opposite side, in the focus $F$. But this picture will by no means be flat, nor nearly so, but very concave towards $E$. Its exact form is of most difficult investigation. The elements of it are given by Dr Barrow; and we have given the chief of them in the article Optics, when considering the foci of infinitely slender pencils of oblique rays. Therefore it is impossible that the picture formed by the object-glafs can be seen distinctly in all its parts by the eye-glafs. Even if it were flat, the points $G$ and $H$ (fig. 16.) are too far from the eye-glafs when the middle $F$ is at the proper distance for distinct vision. When, therefore, the telescope is so adjusted that we have distinct vision of the middle of the field, in order to see the margin distinctly we must push in the eye-glafs; and having so done, the middle of the field becomes indistinct. When the field of vision exceeds 12 or 15 degrees, it is not possible by any contrivance to make it tolerably distinct all over; and we must turn the telescope successively to the different parts of the field that we may see them agreeably.

The cause of this indistinctness is, as we have already said, the shortness of the lateral foci of lateral and oblique pencils refracted by the eye-glafs. The oblique pencil $bG\; \backslash circ\; a$, by which an eye placed at $I$ sees the point $G$ of the image, is a cone of light, having a circular base on the eye-glafs; of which circle $a\; b$ is one of the diameters. There is a diameter perpendicular to this, which, in this figure, is represented by the point $c$. Fig. 17. represents the base of the cone as seen by an eye placed in the axis of the telescope, with the object-glafs as appearing behind it. The point $b$ is formed by a ray which comes from the lowest point $B$ of the object-glafs, and the point $a$ is illuminated by a ray from $A$. The point $c$ at the right hand of the circular base of this cone of light came from the point $C$ on the left side of the object-glafs; and the light comes to $d$ from $D$. Now the laws of optics demonstrate, that the rays which come through the points $c$ and $d$ are more convergent after refraction than the rays which come through $a$ and $b$. The analogies, therefore, which ascertain the foci of rays lying in planes passing through the axis do not determine the foci of the others. Of this we may be sensible by looking through a lens to a figure on which are drawn concentric circles crossed by radii. When the telescope is so adjusted that we see distinctly the extremity of one of the radii, we shall not see distinctly the circumference which crosses the extremity with equal distinctness, and vice versa. This difference, however, between the foci of the rays which come through $a$ and $b$, and those which come through $c$ and $d$, is not considerable in the fields of vision, which are otherwise admissible. But the same difference of foci obtains also with respect to the dispersion of light, and is more remarkable. Both d'Alembert and Euler have attempted to introduce it into their formula.

Telescope. multa; but they have made them useless for any practical purpose by their inextricable complication.

This must serve as a general indication of the difficulties which occur in the construction of telescopes, even although the object-glass were perfect, forming an image without the smallest confusion or distortion.

There is yet another difficulty or imperfection. The rays of the pencil $a\; G\; b$, when refracted through the eye-glass, are also separated into their component colours. The edge of the lens must evidently perform the office of a prism, and the white ray $G\; b$ will be so dispersed that if $b\; i$ be the path of its red ray, the violet ray, which makes another part of it, will take such a course $b\; n$ that the angle $i\; b\; n$ will be nearly $\backslash frac\{1\}\{4\}$th of $G\text{'}\; b\; r$. The ray $G\; a$ passing through a part of the lens whose surfaces are less inclined to each other, will be less refracted, and will be less dispersed in the same proportion very nearly. Therefore the two violet rays will be very nearly parallel when the two red rays are rendered parallel.

Hence it must happen, that the object will appear bordered with coloured fringes. A black line seen near the margin on a white ground, will have a ruddy and orange border on the outside and a blue border within: and this confusion is altogether independent on the object-glass, and is so much the greater as the visual angle $b\; i\; E$ is greater.

Such are the difficulties: They would be unfurmountable were it not that some of them are so connected that, to a certain extent, the diminution of one is accompanied by a diminution of the other. What are called the caustic curves are the geometrical loci of the foci of infinitely slender pencils. Consequently the point $G$ is very nearly in the caustic formed by a beam of light consisting of rays parallel to $I\; o$, and occupying the whole surface of the eye-glass, because the pencil of rays which are collected at $G$ is very small. Any thing therefore that diminishes the mutual inclination of the adjoining rays, puts their concourse farther off. Now this is precisely what we want: for the point $G$ of the image formed by the object-glass is already beyond the focus of the oblique slender pencil of parallel rays $i\; a$ and $i\; b$; and therefore, if we could make this focus go a little farther from $a$ and $b$, we shall bring it nearer to $G$, and obtain more distinct vision of this point of the object. Now let it be recollected, that in moderate refractions through prisms, two rays which are inclined to each other in a small angle are, after refraction, inclined to each other in the same angle. Therefore, if we can diminish the aberration of the ray $a\; i$, or $o\; i$, or $b\; i$, we diminish their mutual inclination; and consequently the mutual inclination of the rays $G\; a$, $G\; o$, $G\; b$, and therefore lengthen the focus, and get more distinct vision of the point $G$. Therefore we at once correct the distortion and the indistinctness: and this is the aim of Mr Huyghens's great principle of dividing the refractions.

The general method is as follows: Let $o$ be the object-glass (fig. 19.) and $E$ the eye-glass of a telescope, and $F$ their common focus, and $F\; G$ the image formed by the object-glass. The proportion of their focal distances is supposed to be such as gives as great a magnifying power as the perfection of the object-glass will admit. Let $B\; I$ be the axis of the emergent pencil. It

is known by the focal theorem that $G\; E$ is parallel to $B\; I$: therefore $B\; G\; E$ is the whole refraction or deflection of the ray $O\; H\; B$ from its former direction. Let it be proposed to diminish the aberrations by dividing this into two parts by means of two glasses $D$ and $e$, so as to make the ultimate angle of vision $b\; i\; e$ equal to $B\; I\; E$, and thus retain the same magnifying power and visible field. Let it be proposed to divide it into the parts $B\; G\; C$ and $C\; G\; E$.

From $G$ draw any line $G\; D$ to the axis towards $O$; and draw the perpendicular $D\; H$, cutting $O\; G$ in $H$; draw $H\; c$ parallel to $G\; C$, cutting $G\; D$ in $g$; draw $g\; f$ perpendicular to the axis, and $g\; e$ parallel to $G\; E$; draw $e\; b$ perpendicular to the axis; draw $D\; d$ parallel to $G\; C$, and $d\; d\text{'}$ perpendicular to the axis.

Then if there be placed at $D$ a lens whose focal distance is $D\; d$, and another at $e$ whose focal distance is $e\; f$, the thing is done. The ray $O\; H$ will be refracted into $H\; b$, and this into $b\; i$ parallel to $B\; I$.

The demonstration of this construction is so evident by means of the common focal theorem, that we need not repeat it, nor the reasons for its advantages. We have the same magnifying power, and the same field of vision; we have less aberration, and therefore less distortion and indistinctness; and this is brought about by a lens $H\; D$ of a smaller aperture and a greater focal distance than $B\; E$. Consequently, if we are contented with the distinctness of the margin of the field with a single eye-glass, we may greatly increase the field of vision; for if we increase $D\; H$ to the size of $E\; B$ we shall have a greater field, and much greater distinctness in the margin; because $H\; D$ is of a longer focal distance, and will bear a greater aperture, preserving the same distinctness at the edge. On this account the glass $H\; D$ is commonly called the Field-glass.

It must be observed here, however, that although the distortion of the object is lessened, there is a real distortion produced in the image $f\; g$. But this, when magnified by the glass $e$, is smaller than the distortion produced by the glass $E$, of greater aperture and shorter focus, on the undistorted image $G\; F$. But because there is a distortion in the second image $f\; g$, this construction cannot be used for the telescopes of astronomical quadrants, and other graduated instruments; because then equal divisions of the micrometer would not correspond to equal angles.

But the same construction will answer in this case, by taking the point $D$ on that side of $F$ which is remote from $O$ (fig. 20.). This is the form now employed in the telescopes of all graduated instruments.

The exact proportion in which the distortion and the indistinctness at the edges of the field are diminished by this construction, depends on the proportion in which the angle $B\; G\; E$ is divided by $G\; C$; and is of pretty difficult investigation. But it never deviates far (never $\backslash frac\{1\}\{4\}$th in optical instruments) from the proportion of the squares of the angles. We may, without any sensible error, suppose it in this proportion. This gives us a practical rule of easy recollection, and of most extensive use. When we would diminish an aberration by dividing the whole refraction into two parts, we shall do it most effectually by making them equal. In like manner, if we divide it into three parts by means of two additional

Telescope. additional glasses, we must make each $=\; \backslash frac\{1\}\{2\}d$ of the whole; and so on for a greater number.

This useful problem, even when limited, as we have done, to equal refractions, is as yet indeterminate; that is, susceptible of an infinity of solutions: for the point D, where the field-glass is placed, was taken at pleasure: yet there must be situations more proper than others. The aberrations which produce distortion, and those which produce indistinctness, do not follow the same proportions. To correct the indistinctness, we should not select such positions of the lens HD as will give a small focal distance to $b\; e$; that is, we should not remove it very far from F. Huygens recommends the proportion of 3 to 1 for that of the focal distances of the lens HD and $e\; b$, and says that the distance D e should be $=\; 2\; F\; e$. This will make $e\; i\; =\; \backslash frac\{1\}\{2\}\; F\; e$, and will divide the whole refraction into two equal parts, as any one will readily see by constructing the common optical figure. Mr Short, the celebrated improver of reflecting telescopes, generally employed this proportion; and we shall presently see that it is a very good one.

It has been already observed that the great refractions which take place on the eye-glasses occasion very considerable dispersions, and disturb the vision by fringing every thing with colours. To remedy this, achromatic eye-glasses may be employed, constructed by the rules already delivered. This construction, however, is incomparably more intricate than that of object-glasses: for the equations must involve the distance of the radiant point, and be more complicated: and this complication is immensely increased on account of the great obliquity of the pencils.

Most fortunately the Huyghenian construction of an eye-piece enables us to correct this dispersion to a great degree of exactness. A heterogeneous ray is dispersed at H, and the red ray belonging to it falls on the lens $b\; e$ at a greater distance from the centre than the violet ray coming from H. It will therefore be less refracted (ceteris paribus) by the lens $b\; e$; and it is possible that the difference may be such that the red and violet rays dispersed at H may be rendered parallel at $b$, or even a little divergent, so as to unite accurately with the red ray at the bottom of the eye. How this may be affected, by a proper selection of the places and figures of the lenses, will appear by the following proposition, which we imagine is new, and not inelegant.

Fig. 21. Let the compound ray OP (fig. 21.) be dispersed by the lens PC; and let PV, PR be its violet and red rays, cutting the axis in G and g. It is required to place another lens RD in their way, so that the emergent rays R r, V v, shall be parallel.

Produce the incident ray OP to Z. The angles ZPR, ZPV, are given, (and RPV is nearly $=\; \backslash frac\{ZPR\}\{27\}$) and

the intersections G and g with the axis. Let F be the focus of parallel red light coming through the lens RD in the opposite direction. Then (by the common optical theorem), the perpendicular $F\; e$ will cut PR in such a point e, that $e\; F$ will be parallel to the emergent ray R r, and to V v. Therefore if $e\; D$ cut PV in u, and $u\; f$ be drawn perpendicular to the axis, we shall have (also by the common theorem) the point f for the fo-

cus of violet rays, and $DF\; :\; D\; f\; =\; D\; e\; :\; D\; u\; =\; 28\; :\; 27$ nearly, or in a given ratio. Telescope.

The problem is therefore reduced to this, "To draw from a point D in the line CG a line $D\; e$, which shall be cut by the lines PR and PV in the given ratio."

The following construction naturally offers itself: Make $GM\; :\; gM$ in the given ratio, and draw MK parallel to Pg. Through any point D of CG draw the straight line PDK, cutting MK in K. Join GK, and draw $D\; e$ parallel to KG. This will solve the problem; and, drawing $e\; F$ perpendicular to the axis, we shall have F for the focus of the lens RD for parallel red rays.

The demonstration is evident: for MK being parallel to Pg, we have $GM\; :\; gM\; =\; GK\; :\; HK$, $=\; \backslash frac\{2\}\{27\}\; D\; :\; u\; D\; =\; \backslash frac\{2\}\{27\}\; D\; f\; /\; D$, in the ratio required.

This problem admits of an infinity of solutions; because the point D may be taken anywhere in the line CG. It may therefore be subjected to such conditions as may produce other advantages.

1. It may be restricted by the magnifying power, or by the division which we choose to make of the whole refraction which produces this magnifying power. Thus, if we have resolved to diminish the aberrations by making the two refractions equal, we have determined the angle R r D. Therefore draw GK, making the angle $MKG$ equal to that which the emergent pencil must make with the axis, in order to produce this magnifying power. Then draw MK parallel to Pg, meeting GK in K. Then draw PK, cutting the axis in D, and $D\; e$ parallel to GK, and $e\; F$ perpendicular to the axis. D is the place, and DF the focal distance of the eye-glass.

2. Particular circumstances may cause us to fix on a particular place D, and we only want the focal distance. In this case the first construction suffices.

3. We may have determined on a certain focal distance DF, and the place must be determined. In this case let

therefore given, and the place of F is determined; and since FD is given by supposition, D is determined.

The application of this problem to our purpose is difficult, if we take it in the most general terms; but the nature of the thing makes such limitations that it becomes very easy. In the case of the dispersion of light, the angle $GP\; g$ is so small that MK may be drawn parallel to PG without any sensible error. If the ray OP were parallel to CG, then G would be the focus of the lens PC, and the point M would fall on C; because the focal distance of red rays is to that of violet rays in the same proportion for every lens, and therefore $CG\; :\; C\; g\; =\; DF\; :\; D\; f$. Now, in a telescope which magnifies considerably, the angle at the object-glass is very small, and CG hardly exceeds the focal distance; and CG is to C g very nearly in the same proportion of 28 to 27. We may therefore draw through C (fig. 22.) a line CK parallel

Telescope. parallel to PG; then draw GK' perpendicular to the axis of the lenses, and join PK'; draw K'BE parallel to CG, cutting PK in B; draw BHI parallel to GK, cutting GK' in H. Join HD and PK. It is evident that CG is bisected in F, and that K'B = 2FD; also K'H : HG = K'B : BE, = CD : DG. Therefore DH is parallel to CK', or to PG. But because PF' = F'K', PD is = DB, and IH = HB. Therefore CD = HB, and FD = K'B, = 2FD; and FD is bisected in F.

That is, in order that the eye-glass RD may correct the dispersion of the field-glass PC, the distance between them must be equal to the half sum of their focal distances very nearly. More exactly, the distance between them must be equal to the half sum of the focal distance of the eye-glass, and the distance at which the field-glass would form an image of the object-glass. For the point G is the focus to which a ray coming from the centre of the object-glass is refracted by the field-glass.

This is a very simple solution of this important problem. Huyghens's eye-piece corresponds with it exactly. If indeed the dispersion at P is not entirely produced by the refraction, but perhaps combined with some previous dispersion, the point M (fig. 21.) will not coincide with C, (fig. 22.), and we shall have GC to GM, as the natural dispersion at P to the dispersion which really obtains there. This may destroy the equation

Thus, in a manner rather unexpected, have we freed the eye-glasses from the greatest part of the effect of dispersion. We may do it entirely by pushing the eye-glass a little nearer to the field-glass. This will render the violet rays a little divergent from the red, so as to produce a perfect picture at the bottom of the eye. But by doing so we have hurt the distinctness of the whole picture, because F is not in the focus of RD. We remedy this by drawing both glasses out a little, and the telescope is made perfect.

This improvement cannot be applied to the construction of quadrant telescopes, such as fig. 20. Mr Ramsden has attempted it, however, in a very ingenious way, which merits a place here, and is also instructive in another way. The field-glass HD is a plano-convex, with its plane side next the image GF. It is placed very near this image. The consequence of this disposition is, that the image GF produces a vertical image gf, which is much less convex towards the glass. He then places a lens on the point C, where the red ray would cross the axis. The violet ray will pass on the other side of it. If the focal distance of this glass be $fc$, the vision will be distinct and free from colour. It has, however, the inconvenience of obliging the eye to be close to the glass, which is very troublesome.

This would be a good construction for a magic-lantern, or for the object-glass of a solar microscope, or indeed of any compound microscope.

We may presume that the reader is now pretty familiar with the different circumstances which must be considered in the construction of an eye-piece, and proceed

to consider those which must be employed to erect the Telescope object.

This may be done by placing the lens which receives the light from the object-glass in such a manner, that a second image (inverted with respect to the first) may be formed beyond it, and this may be viewed by an eye-glass. Such a construction is represented in fig. 23. Plate DXXX. But, besides many other defects, it tinges the object prodigiously with colour. The ray $od$ is dispersed at $d$ into the red ray $dr$, and the violet $dv$, $v$ being farther from the centre than $r$, the refracted ray $vr\text{'}$ crosses $rr\text{'}$ both by reason of spherical aberration and its greater refrangibility.

But the common day telescope, invented by F. Rheits, has, in this respect, greatly the advantage of the one now described. The rays of compound light are dispersed at two points. The violet ray in its course falls without the red ray, but is accurately collected with it at a common focus, as we shall demonstrate by and by. Since they cross each other in the focus, the violet ray must fall within the red ray, and be less refracted than if it had fallen on the same point with the red ray. Had it fallen there it would have separated from it; but by a proper diminution of its refraction, it is kept parallel to it, or nearly so. And this is one excellence of this telescope: when constructed with three eye-glasses perfectly equal, the colour is sensibly diminished, and by using an eye-glass somewhat smaller, it may be removed entirely. We say no more of it at present, because we shall find its construction included in another, which is still more perfect.

It is evident at first sight that this telescope may be improved, by substituting for the eye-glass the Huyghenian double eye-glass, or field-glass and eye-glass represented in fig. 19. and 20.; and that the first of these may be improved and rendered achromatic. This will require the two glasses $ef$ and $gh$ to be increased from their present dimensions to the size of a field-glass, suited to the magnifying power of the telescope, supposing it an astronomical telescope. Thus we shall have a telescope of four eye-glasses. The three first will be of a considerable focal distance, and two of them will have a common focus at $b$. But this is considerably different from the eye-piece of four glasses which are now used, and are far better. We are indebted for them to Mr Dollond, who was a mathematician as well as an artist, and in the course of his research discovered resources which had not been thought of. He had not then discovered the achromatic object-glass, and was busy in improving the eye-glasses by diminishing their spherical aberration. His first thought was to make the Huyghenian addition at both the images of the day telescope. This suggested to him the following eye-piece of five glasses.

Fig. 24. represents this eye-piece, but there is not room for the object-glass at its proper distance. A pencil of rays coming from the upper point of the object is made to converge (by the object-glass) to G, where it would form a picture of that part of the object. But it is intercepted by the lens A $a$, and its axis is bent towards the axis of the telescope in the direction $ab$. At the same time, the rays which converged to G converge to $g$, and there is formed an inverted picture of the object at $gf$. The axis of the pencil is again refracted at

Telescope. $b$, crosses the axis of the telescope in $H$, is refracted again at $e$, at $d\text{'}$, and at $e$, and at last crosses the axis in $I$. The rays of this pencil, diverging from $g$, are made less diverging, and proceed as if they came from $g\text{'}$, in the line $Bg\text{'}$. The lens $cC$ causes them to converge to $g\text{'}$, in the line $G\text{'}\text{'}Cg\text{'}$. The lens $dD$ makes them converge still more to $G\text{'}\text{'}$, and there they form an erect picture $G\text{'}\text{'}F\text{'}\text{'}$; diverging from $G\text{'}\text{'}$, they are rendered parallel by the refraction at $e$.

At $H$ the rays are nearly parallel. Had the glass $Bb$ been a little farther from $A$, they would have been accurately so, and the object-glass, with the glasses $A$ and $B$, would have formed an astronomical telescope with the Huyghenian eye-piece. The glasses $C,\; D$, and $E$, are intended merely for bending the rays back again till they again cross the axis in $I$. The glass $C$ tends chiefly to diminish the great angle $BHb$; and then the two glasses $D$ and $E$ are another Huyghenian eye-piece.

The art in this construction lies in the proper adjustment of the glasses, so as to divide the whole bending of the pencil pretty equally among them, and to form the last image in the focus of the eye-glass, and at a proper distance from the other glass. Bringing $B$ nearer to $A$ would bend the pencil more to the axis. Placing $C$ farther from $B$ would do the same thing; but this would be accompanied with more aberration, because the rays would fall at a greater distance from the centres of the lenses. The greatest bending is made at the field-glass $D$; and we imagine that the telescope would be improved, and made more distinct at the edges of the field, by employing another glass of great focal distance between $C$ and $D$.

There is an image formed at $H$ of the object-glasses, and the whole light passes through a small circle in this place. It is usual to put a plate here pierced with a hole which has the diameter of this image. A second image of the object-glass is formed at $I$, and indeed wherever the pencils cross the axis. A lens placed at $H$ makes no change in any of the angles, nor in the magnifying power, and affects only the place where the images are formed. And, on the other hand, a lens placed at $f$, or $F\text{'}\text{'}$, where a real image is formed, makes no change in the places of the images, but affects the mutual inclination of the pencils. This affords a resource to the artist, by which he may combine properties which seem incompatible.

The aperture of $A$ determines the visible field and all the other apertures.

We must avoid forming a real image, such as $fg$, or $F\text{'}\text{'}G\text{'}\text{'}$, on or very near any glass. For we cannot see this image without seeing along with it every particle of dust and every scratch on the glass. We see them as making part of the object when the image is exactly on the glass, and we see them confusedly, and so as to confuse the object, when the image is near it. For when the image is on or very near any glass, the pencil of light occupies a very small part of its surface, and a particle of dust intercepts a great proportion of it.

It is plain that this construction will not do for the telescope of graduated instruments, because the microscope cannot be applied to the second image $fg$, on account of its being a little distorted, as has been observed of the Huyghenian eye-piece.

Also the interposition of the glass $C$ makes it difficult Telescope to correct the dispersion.

By proper reasoning from the correction in the Huyghenian eye-piece, we are led to the best construction of one with three glasses; which we shall now consider, taking it in a particular form, which shall make the discussion easy, and make us fully masters of the principles which lead to a better form. Therefore let $PA$ (fig. 25.) be the glass which first receives the light $Fig.\; 25.$ proceeding from the image formed by the object-glass, and let $OP$ be the axis of the extreme pencil. This is refracted into $PR$, which is again refracted into $Rr$ by the next lens $B$. Let $b$ be the focus of parallel rays of the second lens. Draw $PBr$. We know that $Ab\; :\; bB\; =\; PB\; :\; Br$, and that rays of one kind diverging from $P$ will be collected at $r$. But if $PR,\; PV$ be a red and a violet ray, the violet ray will be more refracted at $V$, and will cross the red ray in some intermediate point $g$ of the line $Rr$. If therefore the first image had been formed precisely on the lens $PA$, we should have a second image at $fg$ free from all coloured fringes.

If the refractions at $P$ and $R$ are equal (as in the common day telescope), the dispersion at $V$ must be equal to that at $P$, or the angle $gVr\; =\; VPR$. But we have ultimately $RPV\; :\; RrV\; =\; BC\; :\; AB$, ($=\; Bb\; :\; Ab$ by the focal theorem). Therefore $gVr\; :\; gVrV$, (or $gr\; :\; gV$, or $Cf\; :\; fB$) $=\; Bb\; :\; Ab$, and $AB\; :\; Ab\; =\; Rr\; :\; Rg$.

This shows by the way the advantage of the common day telescope. In this $AE\; =\; 2Ab$, and therefore $f$ is the place of the last image which is free from coloured fringes. But this image will not be seen free from coloured fringes through the eye-glass $Cr$, if $f$ be its focus: For had $gr,\; gv$ been both red rays, they would have been parallel after refraction; but $gv$ being a violet ray, will be more refracted. It will not indeed be so much deflected from parallelism as the violet ray, which naturally accompanies the red ray to $r$, because it falls nearer the centre. By computation its dispersion is diminished about $\backslash frac\{1\}\{4\}$th.

In order that $gv$ may be made parallel to $gr$ after refraction, the refraction at $r$ must be such that the dispersion corresponding to it may be of a proper magnitude. How to determine this is the question. Let the dispersion at $g$ be to the dispersion produced by the refraction at $r$ (which is required for producing the intended magnifying power) as $1$ to $9$. Make $9\; :\; 1\; =\; ff\; :\; fC$, $=\; fC\; :\; CD$, and draw the perpendicular $Dr\text{'}$ meeting the refracted ray $rr\text{'}$ in $r\text{'}$. Then we know by the common focal theorem, that if $f$ be the focus of the lens $Cr$, red rays diverging from $g$ will be united in $r\text{'}$. But the violet ray $gv$ will be refracted into $vv\text{'}$ parallel to $rr\text{'}$. For the angle $vr\text{'}r\; :\; vgr\; =$ (ultimately) $fC\; :\; CD\; =\; 9\; :\; 1$. Therefore the angle $vr\text{'}r$ is equal to the dispersion produced at $r$, and therefore equal to $r\text{'}v\text{'}$, and $vv\text{'}$ is parallel to $rr\text{'}$.

But by this we have destroyed the distinct vision of the image formed at $fg$, because it is no longer at the focus of the eye-glass. But distinct vision will be restored by pushing the glasses nearer to the object-glass. This makes the rays of each particular pencil more divergent after refraction through $A$, but scarcely makes any change in the directions of the pencils themselves. Thus the image comes to the focus $f$, and makes no sensible change in the dispersions.

In the common day telescope, the first image is formed in the anterior focus of the first eye-glass, and the second image is at the anterior focus of the last eye-glass. If we change this last for one of half the focal distance, and push in the eye-piece till the image formed by the object-glass is half way between the first eye-glass and its focus, the last image will be formed at the focus of the new eye-glass, and the eye-piece will be achromatic. This is easily seen by making the usual computations by the focal theorem. But the visible field is diminished, because we cannot give the same aperture as before to the new eye-glass; but we can substitute for it two eye-glasses like the former, placed close together. This will have the same focal distance with the new one, and will allow the same aperture that we had before.

On these principles may be demonstrated the correction of colour in eye-pieces with three glasses of the following construction.

Let the glasses A and B be placed so that the posterior focus of the first nearly coincides with the anterior focus of the second, or rather so that the anterior focus of B may be at the place where the image of the object-glass is formed, by which situation the aperture necessary for transmitting the whole light will be the smallest possible. Place the third C at a distance from the second, which exceeds the sum of their focal distances by a space which is a third proportional to the distance of the first and second, and the focal distance of the second. The distance of the first eye-glass from the object-glass must be equal to the product of the focal distance of the first and second divided by their sum.

Let O, A, B, C, the focal distances of the glasses, be O, a, b, c. Then make AB = a + b nearly; BC = b + c + $\backslash frac\{b^2\}\{b+c\}$; OA = $\backslash frac\{bc\}\{b+c\}$. The amplification

or magnifying power will be $\backslash frac\{ab\}\{ac\}$; the equivalent eye-

glass = $\backslash frac\{ac\}\{b\}$; and the field of vision = $3438\text{'}\; \backslash times$

Aperture of A
foc. dist. ob. gl.

These eye-pieces will admit the use of a micrometer at the place of the first image, because it has no distortion.

Mr Dollond was anxious to combine this achromaticism of the eye-pieces with the advantages which he had found in the eye-pieces with five glasses. This eye-piece of three glasses necessarily has a very great refraction at the glass B, where the pencil which has come from the other side of the axis must be rendered again convergent, or at least parallel to it. This occasions considerable aberrations. This may be avoided by giving part of this refraction to a glass put between the first and second, in the same way as he has done by the glass B put between A and C in his five glass eye-piece. But this deranges the whole process. His ingenuity, however, surmounted this difficulty, and he made eye-pieces of four glasses, which seem as perfect as can be desired. He has not published his ingenious investigation; and we observe the London artists work very much at random, probably copying the proportions of some of his

best glasses, without understanding the principle, and therefore frequently mistaking. We see many eye-pieces which are far from being achromatic. We imagine therefore that it will be an acceptable thing to the artists to have precise instructions how to proceed, nothing of this kind having appeared in our language, and the investigations of Euler, d'Alembert, and even Boscovich, being so abstruse as to be inaccessible to all but experienced analysts. We hope to render it extremely simple.

It is evident, that if we make the rays of different colours unite on the surface of the last eye-glass but one, commonly called the field-glass, the thing will be done, because the dispersion from this point of union will then unite with the dispersion produced by this glass alone; and this increased dispersion may be corrected by the last eye-glass in the way already shown.

Therefore let A, B (fig. 26.) be the stations which we have fixed on for the first and second eye-glasses, in order to give a proper portion of the whole refraction to the second glass. Let b be the anterior focus of B. Draw PB r through the centre of B. Make Ab : bB = AB : BK. Draw the perpendicular Kr, meeting the refracted ray in r. We know by the focal theorem, that red rays diverging from P will converge to r; but the violet ray PV, being more refracted, will cross Rr in some point g. Drawing the perpendicular fg, we get f for the proper place of the field-glass. Let the refracted ray Rr, produced backward, meet the ray OP coming from the centre of the object-glass in O. Let the angle of dispersion RPV be called $\backslash rho$, and the angle of dispersion at V, that is, rVv, be v, and the angle V r R be r.

It is evident that OR : OP = $\backslash rho\; :\; v$, because the dispersions are proportional to the sines of the refractions, which, in this case, are very nearly as the refractions themselves.

Let $\backslash frac\{OP\}\{OR\}$ (or $\backslash frac\{\backslash rho\; \backslash rho\}\{\backslash rho\; B\}$ or $\backslash frac\{\backslash rho\; B\}\{bB\}$) be made = m. Then v = m $\backslash rho$; also $\backslash rho\; :\; r\; =\; BK\; :\; AB$, = bB : Ab, and r = $\backslash rho$. $\backslash frac\{Ab\}\{bB\}$, or making $\backslash frac\{Ab\}\{bB\}\; =\; n$, r = n $\backslash rho$; therefore v : r = m : n, = $\backslash frac\{\backslash rho\; B\}\{bB\}\; :\; \backslash frac\{Ab\}\{bB\}\; =\; \backslash rho\; B\; :\; Ab$.

The angle RgV = gVr + grV = $\backslash rho\; \backslash cdot\; m\; +\; n$; and RgV : Rrv = Rr : Rg, or m + n : n = Rr : Rg, and Rg = Rr $\backslash frac\{n\}\{m+n\}$. But Rr is ultimately = BK = AB. $\backslash frac\{bB\}\{Ab\}\; =\; \backslash frac\{AB\}\{n\}$. Therefore Rg = $\backslash frac\{AB\}\{n\}\; \backslash times\; \backslash frac\{n\}\{m+n\}\; =\; \backslash frac\{n\}\{m+n\}$, and Bf = $\backslash frac\{AB\}\{m+n\}$.

This value of Bf is evidently = bB $\backslash times\; \backslash frac\{AB\}\{\backslash rho\; B\; +\; Ab\}$. Now bB being a constant quantity while the glass B is the same, the place of union varies with $\backslash frac\{AB\}\{\backslash rho\; B\; +\; Ab\}$. If we remove B a little farther from A, we increase AB, and $\backslash rho\; B$, and Ab, each by the same quantity. This evidently diminishes Bf. On the other hand, bringing B nearer to A increases Bf. If we keep the distance between the glasses the same, but increase the focal distance bB, we augment Bf, because this change aug-

Telescope. ments the numerator and diminishes the denominator of the fraction $\backslash frac\{bB\; \backslash times\; AB\}\{pB\; +\; Ab\}$.

In this manner we can unite the colours at what distance we please, and consequently can unite them in the place of the intended field-glass, from which they will diverge with an increased dispersion, viz. with the dispersion competent to the refraction produced there, and the dispersion $p\; \backslash times\; m+n$ conjoined.

It only remains to determine the proper focal distances of the field-glass and eye-glass, and the place of the eye-glass, so that this dispersion may be finally corrected.

This is an indeterminate problem, admitting of an infinity of solutions. We shall limit it by an equal division of the two remaining refractions, which are necessary in order to produce the intended magnifying power. This construction has the advantage of diminishing the aberration. Thus we know the two refractions, and the dispersion competent to each; it being nearly $\backslash frac\{1\}\{2\}$th of the refraction. Call this $q$. The whole dispersion at the field-glass consists of $q$, and of the angle $KgV$ of fig. 19. which we also know to be $=\; p\; \backslash times\; m+n$. Call their sum $s$.

Let fig. 27. represent this addition to the eye-piece. $Cg$ is the field-glass coming in the place of $fg$ of fig. 26. and $Rgw$ is the red ray coming from the glass $BR$. Draw $gs$ parallel to the intended emergent pencil from the eye-glass; that is, making the angle $Cgs$ with the axis correspond to the intended magnifying power. Bisect this angle by the line $gK$. Make $sg\; :\; gq\; =\; s\; :\; q$, and draw $qK$, cutting $Cg$ in $t$. Draw $tdD$, cutting $gK$ in $\backslash delta$, and the axis in $D$. Draw $\backslash delta\; d$ and $Dr$ perpendicular to the axis. Then a lens placed in $D$, having the focal distance $Dd$, will destroy the dispersion at the lens $ge$, which refracts the ray $gw$ into $gr$.

Let $gv$ be the violet ray, making the angle $vg\; r\; =\; s$. It is plain, by the common optical theorem, that $gr$ will be refracted into $rr\text{'}$ parallel to $\backslash delta\; d$. Draw $gD\text{'}$ meeting $rr\text{'}$, and join $vr\text{'}$. By the focal theorem two red rays $grg\text{'}$, will be united in $r\text{'}$. But the violet ray $gv$ will be more refracted, and will take the path $v\text{'}v\text{'}$, making the angle of dispersion $r\text{'}v\text{'}v\text{'}\; =\; q$, very nearly, because the dispersion at $v$ does not sensibly differ from that at $r$. Now, in the small angles of refraction which obtain in optical instruments, the angles $rr\text{'}v$, $rgv$ are very nearly as $gr$ and $rr\text{'}$, or as $gD$ and $Dr\text{'}$, or as $CD$ and $DT$; which, by the focal theorem, are as $Cd$ and $dD$; that is, $Dd\; :\; dc\; =\; rgv\; :\; rr\text{'}v$. But $Dd\; :\; dC\; =\; D\backslash delta\; :\; \backslash delta\; t$, $=\; sg\; :\; gq$, $=\; s\; :\; q$. But $rgv\; =\; s$; therefore $rr\text{'}v\; =\; q$, $=\; r\text{'}v\text{'}v\text{'}$, and $v\text{'}v\text{'}$ is parallel to $rr\text{'}$, and the whole dispersion at $g$ is corrected by the lens $Dr$. The focal distance $Ce$ of $Cg$ is had by drawing $Cx$ parallel to $Kg$, meeting $Rg$ in $x$, and drawing $xe$ perpendicular to the axis.

It is easy to see that this (not inelegant) construction is not limited to the equality of the refractions $wgr$, $Krr\text{'}$. In whatever proportion the whole refraction $wgr$ is divided, we always can tell the proportion of the dispersions which the two refractions occasion at $g$ and $r$, and can therefore find the values of $s$ and $q$. Indeed this solution includes the problem in p. 266. col. 2. par. ult.; but it had not occurred to us till the present occasion.

Our readers will not be displeased with this variety of Telescope. refractions.

The intelligent reader will see, that in this solution some quantities and ratios are assumed as equal which are not strictly so, in the same manner as in all the elementary optical theorems. The parallelism, however, of $ov\text{'}$ and $rr\text{'}$ may be made accurate, by pushing the lens $D$ nearer to $Cg$, or retiring it from it. We may also, by pushing it still nearer, induce a small divergence of the violet ray, so as to produce accurate vision in the eye, and may thus make the vision through a telescope more perfect than with the naked eye, where dispersion is by no means avoided. It would therefore be an improvement to have the eye-glass in a sliding tube for adjustment. Bring the telescope to distinct vision; and if any colour be visible about the edges of the field, shift the eye-glass till this colour is removed. The vision may now become indistinct: but this is corrected by shifting the place of the whole eye-piece.

We have examined trigonometrically the progress of a red and a violet ray through many eye-pieces of Dollond's and Ramsden's best telescopes; and we have found in all of them that the colours are united on or very near the field-glass; so that we presume that a theory somewhat analogous to ours has directed the ingenious inventors. We meet with many made by other artists, and even some of theirs, where a considerable degree of colour remains, sometimes in the natural order and often in the contrary order. This must happen in the hands of mere imitators, ignorant of principle. We presume that we have now made this principle sufficiently plain.

Fig. 28. represents the eye-piece of a very fine spy-glass by Mr Ramsden; the focal length of its object-glass is $8\backslash frac\{1\}\{2\}$ inches, with $1\backslash frac\{1\}\{2\}$th of aperture, $2^\{\backslash circ\}\; 05\text{'}$ of visible field, and 15.4 magnifying power. The distances and focal lengths are of their proper dimensions, but the apertures are $\backslash frac\{1\}\{2\}$ larger, that the progress of a lateral pencil might be more distinctly drawn. The dimensions are as follow:

Focal lengths $Aa=0.775$ $Bb=1.025$ $Cc=1.01$ $Dd=0.79$
Distances $AB=1.18$ $BC=1.83$ $CD=1.105$.

It is perfectly achromatic, and the colours are united, not precisely at the lens $Cg$, but about $\backslash frac\{1\}\{2\}$th of an inch nearer the eye-glass.

It is obvious that this combination of glasses may be used as a microscope; for if, instead of the image formed by the object-glass at $FG$, we substitute a small object, illuminated from behind, as in compound microscopes; and if we draw the eye-piece a very small way from this object, the pencils of parallel rays emergent from the eye-glass $D$ will become convergent to very distant points, and will there form an inverted and enlarged picture of the object, which may be viewed by a Huyghenian eye-piece; and we may thus get high magnifying powers without using very deep glasses. We tried the eye-piece of which we have given the dimensions in this way, and found that it might be made to magnify 180 times with very great distinctness. When used as the magnifier of a solar microscope, it infinitely surpasses every thing we have ever seen. The picture formed by a solar microscope is generally so indistinct, that it is fit only for amusing ladies; but with this magnifier it seemed

Telescope. ed perfectly sharp. We therefore recommend this to the artists as a valuable article of their trade.

The only thing which remains to be considered in the theory of refracting telescopes is the forms of the different lenses. Hitherto we have had no occasion to consider any thing but their focal distances; but their aberrations depend greatly on the adjustment of their forms to their situations. When the conjugate focuses of a lens are determined by the service which it is to perform, there is a certain form or proportion between the curvatures of their anterior and posterior surfaces, which will make their aberrations the smallest possible.

It is evident that this proportion is to be obtained by making the fluxion of the quantity within the parenthesis in the formula at the top of col. 2. p. 248. equal to nothing. When this is done, we obtain this formula for $a$, the radius of curvature for the anterior surface of a lens.

fine of incidence to the fine of refraction, and $r$ is the distance of the focus of incident rays, positive or negative, according as they converge or diverge, all measured on a scale of which the unit is $n$, = half of the radius of the equivalent isosceles lens.

It will be sufficiently exact for our purpose to suppose

As an example, let it be required to give the radii of curvature in inches for the eye-glass $b\; e$ of page 262. col. 1. par. 4. which we shall suppose of $1\backslash frac\{1\}\{2\}$ inches focal distance, and that $e\; c\; (=r)$ is $3\backslash frac\{1\}\{4\}$th inches.

The radius of curvature for the equivalent isosceles lens is 1.5, and its half is 0.75. Therefore $r\; =\; \backslash frac\{3\backslash frac\{1\}\{4\}\}\{0.75\}$,

These values are parts of a scale, of which the unit is 0.75 inches. Therefore

And here we must observe that the posterior surface is concave: for $b$ is a positive quantity, because $1-a$ is a positive quantity as well as $a$; therefore the centre of sphericity of both surfaces lies beyond the lens.

And this determination is not very different from the usual practice, which commonly makes this lens a plane convex with its flat side next the eye: and there will not be much difference in the performance of these two lenses; for in all cases of maxima and minima, even a pretty considerable change of the best dimensions does not make a sensible change in the result.

The same consideration leads to a rule which is very

simple, and sufficiently exact for ordinary situations. Telescope. This is to make the curvatures such, that the incident and emergent pencils may be nearly equally inclined to the surfaces of the lens. Thus in the eye-piece with five glasses, A and B should be most convex on their anterior sides; C should be most convex on the posterior side; D should be nearly isosceles; and E nearly plano-convex.

But this is not so easy a matter as appears at first sight. The lenses of an eye-piece have not only to bend the several pencils of light to and from the axis of the telescope; they have also to form images on the axes of these pencils. These offices frequently require opposite forms, as mentioned in par. 3. col. 2. p. 261. Thus the glass A fig. 28. should be most convex on the side next the object, that it may produce little distortion of the pencils. But it should be most convex next the eye, that it may produce distinct vision of the image FG, which is very near it. This image should have its concavity turned towards A, whereas it is towards the object-glass. We must therefore endeavour to make the vertical image $fg$ flatter, or even convex. This requires a glass very flat before and convex behind. For similar reasons the object-glass of a microscope and the simple eye-glass of an astronomical telescope should be formed the same way.

This is a subject of most difficult discussion, and requires a theory which few of our readers would relish; nor does our limits afford room for it. The artists are obliged to grope their way. The proper method of experiment would be, to make eye-pieces of large dimensions, with extravagant apertures to increase the aberrations, and to provide for each station A, B, C, and D, a number of lenses of the same focal distance, but of different forms: and we would advise making the trial in the way of a solar microscope, and to have two eye-pieces on trial at once. Their pictures can be formed on the same screen, and accurately compared; whereas it is difficult to keep in remembrance the performance of one eye-piece, and compare it with another.

We have now treated the theory of refracting telescopes with considerable minuteness, and have perhaps exceeded the limits which some readers may think reasonable. But we have long regretted that there is not any theory on this subject from which a curious person can learn the improvements which have been made since the time of Dr Smith, or an artist learn how to proceed with intelligence in his profession. If we have accomplished either of these ends, we trust that the public will receive our labours with satisfaction.

We cannot add any thing to what Dr Smith has delivered on the theory of reflecting telescopes. There appears to be the same possibility of correcting the aberration of the great speculum by the contrary aberration of a convex small speculum, that we have practised in the compound object-glass of an achromatic refracting telescope. But this cannot be, unless we make the radius of the convex speculum exceedingly large, which destroys the magnifying power and the brightness. This therefore must be given up. Indeed their performance, when well executed, does already surpass all imagination. Dr Herschel has found great advantages in what he calls the front view, not using a plane mirror to throw the pencils to one side. But this cannot

be practised in any but telescopes so large, that the loss of light, occasioned by the interposition of the observer's head, may be disregarded.

NOTHING remains but to describe the mechanism of some of the most convenient forms.

To describe all the varieties of shape and accommodation which may be given to a telescope, would be a task as trifling as prolix. The artists of London and of Paris have racked their inventions to please every fancy, and to suit every purpose. We shall content ourselves with a few general maxims, deduced from the scientific consideration of a telescope, as an instrument by which the visual angle subtended by a distant object is greatly magnified.

The chief consideration is to have a steady view of the distant object. This is unattainable, unless the axis of the instrument be kept constantly directed to the same point of it; for when the telescope is gently shifted from its position, the object seems to move in the same or in the opposite direction, according as the telescope inverts the object or shows it erect. This is owing to the magnifying power, because the apparent angular motion is greater than what we naturally connect with the motion of the telescope. This does not happen when we look through a tube without glasses.

All shaking of the instrument therefore makes the object dance before the eye; and this is disagreeable, and hinders us from seeing it distinctly. But a tremulous motion, however small, is infinitely more prejudicial to the performance of a telescope, by making the object quiver before us. A person walking in the room prevents us from seeing distinctly; nay, the very pulsation in the body of the observer, agitates the floor enough to produce this effect, when the telescope has a great magnifying power: For the visible motion of the object is then an imperceptible tremor, like that of an harpichord wire, which produces an effect precisely similar to optical indistinctness; and every point of the object is diffused over the whole space of the angular tremor, and appears coexistent in every part of this space, just as a harpichord wire does while it is sounding. The more rapid this motion is, the indistinctness is the more complete. Therefore the more firm and elastic and well bound together the frame-work and apertures of our telescope is, the more hurtful will this consequence be. A mounting of lead, were it practicable, would be preferable to wood, iron, or brass. This is one great cause of the indistinctness of the very finest reflecting telescopes of the usual construction, and can never be totally removed. In the Gregorian form, it is hardly possible to damp the elastic tremor of the small speculum, carried by an arm supported at one end only, even though the tube were motionless. We were witnesses of a great improvement made on a four-feet reflecting telescope, by supporting the small speculum by a strong plate of lead placed across the tube, and led by an adjusting screw at each end. But even the great mirror may vibrate enough to produce indistinctness. Refracting telescopes are free from this inconveniency, because a small angular motion of the object-glass round one of its own diameters has no sensible effect on the image in its focus. They are affected only by an angular motion of the axis of the telescope or of the eye-glasses.

This single consideration gives us great help towards

judging of the merits of any particular apparatus. We should study it in this particular, and see whether its form makes the tube readily susceptible of such tremulous motions. If it does, the firmer it is and the more elastic it is, the worse. All forms therefore where the tube is supported only near the middle, or where the whole immediately or remotely depend on one narrow joint, are defective.

Reasoning in this way, we say with confidence, that of all the forms of a telescope apparatus, the old fashioned simple stand represented in fig. 29. is by far the best, and that others are superior according as the disposition of the points of support of the tube approaches to this. Let the pivots A, B, be fixed in the lintel and sole of a window. Let the four braces terminate very near to these pivots. Let the telescope lie on the pin F, resting on the shoulder round the eye-piece, while the far end of it rests on one of the pins 1, 2, 3, &c.; and let the distance of these pins from F very little exceed the length of the telescope. The trembling of the axis, even when considerable, cannot affect the position of the tube, because the braces terminate almost at the pivots. The tremor of the brace CD does as little harm, because it is nearly perpendicular to the tube. And if the object-glass were close at the upper supporting pin, and the focus at the lower pin F, even the bending and trembling of the tube will have no effect on its optical axis. The instrument is only subject to horizontal tremors. These may be almost annihilated by having a slender rod coming from a hook's joint in the side of the window, and passing through such another joint close by the pin F. We have seen an instrument of this form, having AB parallel to the earth's axis. The whole apparatus did not cost 50 shillings, and we find it not in the least sensible manner affected by a storm of wind. It was by observations with this instrument that the tables of the motions of the Georgium Sidus, published in the Edinburgh Transactions, were constructed, and they are as accurate as any that have yet appeared. This is an excellent equatorial.

But this apparatus is not portable, and it is sadly deficient in elegance. The following is the best method we have seen of combining these circumstances with the indispensable requisites of a good telescope.

The pillar VX (fig. 30.) rises from a firm stand, and has a horizontal motion round a cone which completely fills it. This motion is regulated by a rack-work in the box at V. The screw of this rack-work is turned by means of the handle P, of a convenient length, and the screw may be disengaged by the click or detent V, when we would turn the instrument a great way at once. The telescope has a vertical motion round the joint Q placed near the middle of the tube. The lower end of the tube is supported by the stay OT. This consists of a tube RT, fastened to the pillar by a joint T, which allows the stay to move in a vertical plane. Within this tube slides another, with a stiff motion. This tube is connected with the telescope by another joint O, also admitting motion in a vertical plane. The side M of this inner tube is formed into a rack, in which works a pinion fixed to the top of the tube RT, and turned by the flat finger-piece R. The reader will readily see the advantages and the remaining defects of this apparatus. It is very portable, because the telescope is easily disengaged from it, and the legs and stay fold up. If the joint

Telescope. joint Q were immediately under A, it would be much freer from all tremor in the vertical plane. But nothing can hinder other tremors arising from the long pillar and the three springy legs. These communicate all external agitations with great vigour. The instrument should be set on a stone pedestal, or, what is better, a cask filled with wet sand. This pedestal, which necessity perhaps suggested to our scientific navigators, is the best that can be imagined.

Fig. 31. Fig. 31. is the stand usually given to reflecting telescopes. The vertical tube FBG is fastened to the tube by finger screws, which pass through the slits at F and G. This arch turns round a joint in the head of the divided pillar, and has its edge cut into an oblique rack, which is acted on by the horizontal screw, furnished with the finger-piece A. This screw turns in a horizontal square frame. This frame turns round a horizontal joint in the off-side, which cannot be seen in this view. In the side of this frame next the eye there is a finger-screw $a$, which passes through the frame, and presses on the round horizontal plate D. By screwing down this finger-screw, the frame is brought up, and presses the horizontal screw to the rack. Thus the elevation of the telescope is fixed, and may be nicely changed by the finger applied to A and turning this screw. The horizontal round plate D moves slightly round on another plate of nearly equal diameter. This under plate has a deep conical hollow socket, which is nicely fitted by grinding to a solid cone formed on the top of the great upright pillar, and they may be firmly fixed in any position by the finger-screw E. To the under plate is fastened a box $c$, containing a horizontal screw C, which always works in a rack cut in the edge of the upper plate, and cannot be disengaged from it. When a great vertical or horizontal motion is wanted, the screws $a$ and E are slackened, and by tightening them the telescope may be fixed in any position, and then any small movements may be given it by the finger plates A and C.

This stand is very subject to brisk tremor, either from external agitation of the pedestal, or from the immediate action of the wind; and we have seldom seen distinctly through telescopes mounted in this manner, till one end of the tube was pressed against something that was very steady and unelastic. It is quite astonishing what a change this produces. We took a very fine telescope made by James Short, and laid the tube on a great lump of soft clay, pressing it firmly down into it. Several persons, ignorant of our purpose, looked through it, and read a table of logarithms at the distance of 310 yards. We then put the telescope on its stand, and pointed it at the same object; none of the company could read at a greater distance than 235 yards, although they could perceive no tremor. They thought the vision as sharp as before; but the incontrovertible proof of the contrary was, that they could not read at such a distance.

If the round plates were of much greater dimensions; and if the lower one, instead of being fixed to the pillar, were supported on four stout pillars standing on another plate; and if the vertical arch had a horizontal axis turning on two upright frames firmly fixed to the upper plate—the instrument would be much freer from tremor. Such stands were made formerly; but being much

more bulky and inconvenient for package, they have gone into disuse. Telescope.

The high magnifying powers of Dr Herschel's telescopes made all the usual apparatus for their support extremely imperfect. But his judgement, and his ingenuity and fertility in resource, are as eminent as his philosophical ardour. He has contrived for his reflecting telescopes stands which have every property that can be desired. The tubes are all supported at the two ends. The motions, both vertical and horizontal, are contrived with the utmost simplicity and firmness. We cannot more properly conclude this article than with a description of his 40 feet telescope, the noblest monument of philosophical zeal and of princely munificence that the world can boast of.

Fig. 32. Fig. 32. represents a view of this instrument in a meridional situation, as it appears when seen from a convenient distance by a person placed to the south-west of it. The foundation in the ground consists of two concentric circular brick walls, the outermost of which is 42 feet in diameter, and the inside one 21 feet. They are two feet six inches deep under ground; two feet three inches broad at the bottom, and one foot two inches at the top; and are capped with paving stones about three inches thick, and twelve and three quarters broad. The bottom frame of the whole apparatus rests upon these two walls by twenty concentric rollers III, and is moveable upon a pivot, which gives a horizontal motion to the whole apparatus, as well as to the telescope.

The tube of the telescope, A, though very simple in its form, which is cylindrical, was attended with great difficulties in the construction. This is not to be wondered at; when its size, and the materials of which it is made, are considered. Its length is 39 feet four inches; it measures four feet ten inches in diameter; and every part of it is of iron. Upon a moderate computation, the weight of a wooden tube must have exceeded an iron one at least 3000 pounds; and its durability would have been far inferior to that of iron. It is made of rolled or sheet iron, which has been joined together without rivets, by a kind of seaming well known to those who make iron funnels for stoves.

Very great mechanical skill is used in the contrivance of the apparatus by which the telescope is supported and directed. In order to command every altitude, the point of support is moveable; and its motion is effected by mechanism, so that the telescope may be moved from its most backward point of support to the most forward, and, by means of the pulleys GG suspended from the great beam H, be set to any altitude, up to the very zenith. The tube is also made to rest with the point of support in a pivot, which permits it to be turned sidewise.

The concave face of the great mirror is 48 inches of polished surface in diameter. The thickness, which is equal in every part of it, remains now about three inches and a half; and its weight, when it came from the cast was 2118 pounds, of which it must have lost a small quantity in polishing. To put this speculum into the tube, it is suspended vertically by a crane in the laboratory, and placed on a small narrow carriage, which is drawn out, rolling upon planks, till it comes near the back of the tube; here it is again suspended.

suspended and placed in the tube by a peculiar apparatus.

The method of observing by this telescope is by what Dr Herschel calls the front view; the observer being placed in a seat C, suspended at the end of it, with his back towards the object he views. There is no small speculum, but the magnifiers are applied immediately to the first local image.

From the opening of the telescope, near the place of the eye-glass, a speaking pipe runs down to the bottom of the tube, where it goes into a turning joint; and after several other inflections, it at length divides into two branches, one going into the observatory D, and the other into the work-room E. By means of the speaking pipe the communications of the observer are conveyed to the assistant in the observatory, and the workman is directed to perform the required motions.

In the observatory is placed a valuable fiducial time-piece, made by Mr Shelton. Close to it, and of the same height, is a polar distance-piece, which has a dial-plate of the same dimensions with the time-piece: this piece may be made to show polar distance, zenith distance, declination or altitude, by setting it differently. The time and polar distance pieces are placed so that the assistants sit before them at a table, with the speaking-pipe rising between them; and in this manner observations may be written down very conveniently.

This noble instrument, with proper eye-glasses, magnifies above 6000 times, and is the largest that has ever been made. Such of our readers as wish for a fuller account of the machinery attached to it, viz. the stairs, ladders, and platform B, may have recourse to the second part of the Transactions of the Royal Society for 1795; in which, by means of 18 plates and 63 pages of letter-press, an ample detail is given of every circumstance relating to joiner's work, carpenter's work, and smith's work, which attended the formation and erection of this telescope. It was completed on August the 28th 1789, and on the same day was the sixth satellite of Saturn discovered.