an instrument, by the help of which the apparent magnitudes of objects viewed through telescopes or microscopes are measured with great exactness.

The first micrometers were only mechanical contrivances for measuring the image of an object in the focus of the object-glass. Before these contrivances were thought of, astronomers were accustomed to measure the field of view in each of their telescopes, by observing how much of the moon they could see through it, the semidiameter being reckoned at 15 or 16 minutes; and other distances were estimated by the eye, comparing them with the field of view. Mr Gascoigne, an English gentleman, however, fell upon a much more exact method, and had a treatise on Optics prepared for the press; but he was killed during the civil wars, in the service of Charles I., and his manuscript was never found. His instrument, however, fell into the hands of Mr R. Townly, who says, that by the help of it he could mark above 40,000 divisions in a foot.

Mr Gascoigne's instrument being shown to Dr Hooke, he gave a drawing and description of it, and proposed several improvements in it, which may be seen in Phil. Trans. abr. Vol. I. p. 217. Mr Gascoigne divided the image of an object, in the focus of the object-glass, by the approach of two pieces of metal, ground to a very fine edge, in the place of which Dr Hooke would substitute two fine hairs stretched parallel to one another. Two other methods of Dr Hooke's, different from this, are described in his Posthumous Works, p. 497, 498. An account of several curious observations that Mr Gascoigne made by the help of his micrometer, particularly in the mensuration of the diameters of the moon and other planets, may be seen in the Phil. Trans. Vol. XLVIII. p. 190.

Mr Huygens, as appears by his System of Saturn, published in 1659, used to measure the apparent diameters of the planets, or any small angles, by first measuring the quantity of the field of view in his telescope; which, he says, is best done by observing the time which a star takes up in passing over it, and then preparing two or three long and slender brass plates, of various breadths, the sides of which were very straight, and converging to a small angle. In making use of these pieces of brass, he made them slide in two slots, that were made in the sides of the tube, opposite to the place of the image, and observed in what place it just covered the diameter of any planet, or any small distance that he wanted to measure. It was observed, however, by Sir Isaac Newton, that the diameters of planets, measured in this manner, will be larger than they should be, as all lucid objects appear to be, when they are viewed upon dark ones.

In the Ephemerides of the Marquis of Malvasia, published in 1662, it appears that he had a method of measuring small distances between fixed stars, and the diameters of the planets, and also of taking accurate draughts of the spots of the moon; and this was by a net of silver wire, fixed in the common focus of the object and eyeglass. He also contrived to make one of two stars to pass along the threads of this net, by turning it, or the telescope, as much as was necessary for that purpose; and he counted, by a pendulum-clock, beating seconds, the time that elapsed in its passage from one wire to another, which gave him the number of the minutes and seconds of a degree contained between the intervals of the wires of his net, with respect to the focal length of his telescope.

In 1666, Messrs Auzout and Picard published a description of a micrometer, which was nearly the same with that of the Marquis of Malvasia, excepting the method of dividing it, which they performed with more exactness by a screw. In some cases they used threads of silk, as being finer than silver wires. Dechales also recommends a micrometer consisting of fine wires, or silk threads, the distances of which were exactly known, disposed in the form of a net, as peculiarly convenient for taking a map of the moon.

M. de la Hire says, that there is no method more simple or commodious for observing the digits of an eclipse than a net in the focus of the telescope. These, he says, were generally made of silk threads; and that for this particular purpose six concentric circles had also been made use of, drawn upon oiled paper; but he advises to draw the circles on very thin pieces of glass with the point of a diamond. He also gives several particular directions to assist persons in the use of them. In another memoir he shows a method of making use of the same net for all eclipses, by using a telescope with two object-glasses, and placing them at different distances from one another.

M. Cassini invented a very ingenious method of ascertaining the right ascensions and declinations of stars, by fixing four cross hairs in the focus of the telescope, and turning it about its axis, so as to make them move in a line parallel to one of them. The difficulty there was in accomplishing this was entirely removed by a mechanical contrivance of Dr Bradley.

M. Lewenhoek's method of estimating the size of small objects was by comparing them with grains of sand, of which 100 placed in a line took up an inch. These grains he laid on the same plate with his objects. Micrometer objects, and viewed them at the same time. Dr Jurin's method was similar to this of M. Lewenhoek; for he found the diameter of a piece of fine silver-wire, by wrapping it as close as he could about a pin, and observing how many rings made an inch. For he used this wire in the same manner as Lewenhoek used his sand.

Mr Martin, in his Optics, recommends such a micrometer to a microscope as had been applied to telescopes; for he advises to draw a number of parallel lines on a piece of glass, with the fine point of a diamond, at the distance of \( \frac{1}{4} \) of an inch from one another, and to place it in the focus of the eye-glass. By this method Dr Smith contrived to take the exact draught of objects viewed by a double microscope. For he advises to get a lattice, made with small silver wires or small squares, drawn upon a plain glass by the strokes of a diamond, and to put it into the place of the image formed by the object-glass. Then by transferring the parts of the object, seen in the squares of the glass or lattice, upon similar corresponding squares, drawn on paper, the picture may be exactly taken. Mr Martin also introduced into compound microscopes another micrometer, consisting of a screw.

Dr Hooke used to look upon the magnified object with one eye, while, at the same time, he viewed other objects placed at the same distance with the other eye. In this manner he was able, by the help of a ruler, divided into inches and small parts, and laid on the pedestal of the microscope, to call, as it were, the magnified appearance of the object upon the ruler, and thereby exactly to measure the diameter which it appeared to have through the glass; which being compared with the diameter as it appeared to the naked eye, easily shewed the degree in which it was magnified. This, says Mr Baker, is a ready and good method for many objects; and he declares, from his own experience, that a little practice will render it exceedingly easy and pleasant.

We are obliged to Dr Hooke for an excellent method of viewing the sun without injuring our eyes. For this purpose he contrived that the rays should be reflected from one plane to another, till it was so much weakened that the eye might receive it with great safety and pleasure. This method is much preferable to that of looking at the sun through a smoky or coloured glass, which gives it a red and disagreeable hue. His discourse on this subject was read before the Royal Society June 28 1675.

These micrometers, however, have several considerable defects. In particular, it is not easy to measure with them objects that are in motion, or those which are too large to come within the field of view; so that the diameters of the sun and moon cannot well be measured with them to any great degree of exactness. Another method was found of measuring the apparent magnitude of the object, free from the inconveniences above mentioned, by means of a telescope furnished with two object-glasses. This ingenious method was hit upon about the same time both by Mr Servington Sauvery and the celebrated M. Bouguer.

In this instrument, both object-glasses are of equal focal distance, and placed one of them by the side of the other; so that the same eye-glass may serve for them both. By this means two distinct images of an object Micrometer are formed in the focus of the eye-glass; and since the distance of these images depends upon the distance at which the two object-glasses are placed from one another, it may be measured with great accuracy. Nor is it necessary that the whole disc of the sun or moon come within the field of view; since, if the images of a small part of the disc be formed by each object-glass, the whole diameter may easily be computed, by their position with respect to one another. For if the object be large, the images will approach towards, or perhaps even ly over one another. And the object-glasses being moveable, the two images may always be brought exactly to touch one another, and the diameter may be computed from the known distance of the centres of the two glasses.

Another advantage attending this instrument is, that by having a common micrometer in the focus of the eye-glass, when the two images of the sun or moon are made in part to cover one another, that part which is common to both the images may be measured with great exactness, as being viewed upon a ground that is only one half less luminous than itself; whereas, in general, the heavenly bodies are viewed upon a dark ground, and on that account are imagined to be larger than they really are. By a small addition to this instrument, provided it be of moderate length, M. Bouguer thought it very possible to measure angles of three or four degrees; which is of particular consequence in taking the distance of stars from the moon.

Mr Sauvery's paper containing a very particular description of his construction of this instrument was read at the Royal Society October 27, 1743, and M. Bouguer's account of his instrument which he called an heliometer is contained in the Memoirs of the Royal Academy of Sciences for the year 1748, p. 15.

A very great improvement was made in this kind of micrometer by Mr Dollond; for, instead of two complete object-glasses, he used only one, cut into two equal parts, one of them sliding by the other. Each half of this object-glass will give a separate and distinct image; and as the distance at which their centres are placed from one another may be exactly ascertained, the same uses may be made of them as of two entire object-glasses, and the application of them is much more commodious.

But the ground or reason of this new micrometer, as applied to the refractory or reflecting telescope, may be illustrated by figures, as follows:

Let ABCD represent any very distant object, as the sun, &c. and AB its diameter; also let EFGS represent the object-glass consisting of two segments EFG and ESG divided through the centre N in the right line EG. The angle under which it appears at the end of the telescope will be ANB equal to the angle KNL, under which the image KL is contained. Now, suppose the moveable segment EFG were by a mechanical contrivance drawn off to the position HI, the distance of their centres would be NO; and the two lines AN and BO passing through the centres N, O, of the segments, if produced, meet at the focus in L; and since BL and BK do also pass through the centres N and O, and the object being at an indefinitely great distance, the line OL will be parallel Micrometer to NK, and consequently the angle NLO is equal to the angle KNL or ANB; that is to say, the angle under which the object appears from the end of the telescope (or to the naked eye), is equal to the angle under which the distance between the two centres of the segments appear from the solar focus of the telescope.

And this will be the case in every distance of an object: for supposing the object AB were at some near distance from the telescope, and subtended the same angle ANB, the only consequence would be, that its image would be formed at a greater distance from the glass, suppose at MP; it would still be contained under the same angle MNP, equal to NLO, as before, upon the supposition that the segment HI and BO produced meets AP in the point P; that is to say, suppose that the segment HI is in such position that the moveable image QR formed by it, exactly coincides with the fixed image MP, formed by the segment ESG.

Concerning this vitreous micrometer we may farther observe, that its great excellency consists in this, that it depends solely in measuring the distance of the centres of the two segments, not only when applied alone at the end of a telescope, but even in conjunction with the object-glass of any common telescope; for, let EG and HI represent the two segments, as before, of a glass whose focal distance is very long, suppose, for instance, 50 feet; then, at a small distance from it, let AB represent the object-glass of a common long telescope, whose focal distance of parallel rays is CA, or its focus of very distant objects &c. Then this glass, combined with the foregoing segments, will have its focus shortened, and the common focus of both will be in point q. Then because the triangles ROQ and PNM are similar to the triangles rOq and pnm respectively; therefore the images RQ and PM will be similar, and alike posited to the two small images rq and pm; and therefore when these two images are in contact in the focus of the semi-lenses, they will likewise be in contact in the shortened compound focus. And as the centres N and O of the two semi-lenses GE and IH are separated farther from, or brought nearer to, each other, the images in either focus will be moved in similar manner; and when the centres N and O coincide, the images in each focus respectively will also coincide, or become one entire image; the difference in every case being only as to large and small, greater or lesser distance. Consequently, in the micrometer by which those two semi-lenses are moved by each other, the same turns of the screw which measures the angle OPN, and which brings the images into an exact contact in the single focus at Q, will be necessary for the same purpose in the compound focus also; so that by this means we have an opportunity of measuring the said angle OPQ, without being obliged to have so great and so unmanageable a length of the telescope.

However, the larger the focal distance of the lens AB is, the more distinct the contact of the images will appear; and because this is the point on which the whole perfection of this micrometer depends, it will be likewise necessary to have it so contrived, when applied to a telescope, that the centres NO may be equally distant from the axis of the telescope or centre of the aperture on either side; because, in this case, Micrometer, the point of contact in the two images will be just in the centre of the focus, and therefore the most distinct that it possibly can be.

But the application of this micrometer to refracting telescopes will be less convenient than when it is applied to a reflecting telescope; for if it be placed on the open end of the reflecting telescope, then will the rays that tend to form the larger images RQ and PM be incident upon the larger speculum AB, and from thence reflected to a compound focus, where the similar images rq and pm will be formed as before; the rays proceeding from these two images to the smaller speculum ab, will be reflected back through the hole of the larger, to form the images QR and PM, which likewise will still be in contact in the focus of the eye-glass DC, where it will be distinctly perceived by the eye at I. This contact will likewise be shewn in the focus of the eye-glass, if the centres O and N are properly disposed, as before-mentioned.

From what has been said, the general rationale of this micrometer will evidently appear; but one thing must not pass unregarded in an affair of such moment and consequence as the measuring these small angles in the science of astronomy. It has been customary to suppose, that the focus of a lens, or the local distance of rays parallel to its axis, is equal to the radius in a double and equally convex lens. But this is too great an error not to be noticed here; for in different sorts of glasses there is found a different refractive power, and the focus of parallel rays is at a different distance in each; but this distance in no sort of glass is equal to the radius, but falls short of it more or less. Now the foregoing demonstration regards the radius, and not the focal distance of parallel rays.

With regard to the planets, as Jupiter is the largest of all, and subtends an angle to the eye of 3' 12", the diameter of his image in the focus of a 50 foot glass will be about half an inch; and that will be the utmost distance to which the centres of the segments will be required to be separated for measuring the apparent diameters of the planets.

But for a heliometer, the diameter of the sun, being near 10 times as great as that of Jupiter, will require the centres of the segments in a glass of 40 or 50 feet focus to be removed from each other at least to the distance of four or five inches; and to take in the whole system of Jupiter's moons, the distance of the centres will be required much larger; and therefore, for such purposes, the segments of glasses of a less focal length must be used.

But, valuable as the object-glass micrometer undoubtedly is, some difficulties have been found in the use of it, owing to the alterations in the focus of the eye, which are apt to cause it to give different measures of the same angle at different times. For instance, in measuring the sun's diameter, the axes of the pencils of rays, which come through the two segments of the object-glass from contrary limbs of the sun, crossing one another at the focus of the telescope under an angle equal to that of the sun's diameter, the union of the limbs of the two images of the sun cannot appear perfect unless the eye be disposed to Micrometer see objects distinctly which are placed at the point of intersection. But if the eye be disposed to see objects distinctly, which are placed nearer the object-glass than the intersection is, the two limbs will appear separated by the interval of the axes of the pencils in that place; and if the eye be disposed to see objects distinctly, which are placed farther from the object-glass than the intersection is, the two limbs will appear to encroach upon each other by the distance of the axes of the pencils, after their crossing, taken at that place.

To explain this, let OV represent the centres of the two semicircular glasses of the object-glass micrometer, separated to the distance OV from each other, subtending the angle OaV, equal to the sun's diameter, at the point a, which is the common focus of the two pencils of rays having Oa and Va for their axes, namely, those proceeding from contrary sides of the sun, and passing through the contrary semi-circles; and let d be the eye-glass. It is evident, that if d be properly placed to give distinct vision of objects placed at the point a, the rays Oa, Va, as well as all the other rays belonging to those pencils, will be collected into one point upon the retina of the eye; and consequently, the two opposite limbs of the two images of the sun will seem to coincide, and the two images of the sun to touch one another externally. But if the state of the eye should alter, the place of the eye-glass remaining the same, the eye will be no longer disposed to see the image formed at the point a distinctly, but to see an object placed at e, nearer to or farther from the object-glass distinctly; and therefore an image will be formed on the retina exactly similar to the somewhat confused image formed by the rays on a plane perpendicular to their course at e. Consequently, as the two cones of solar rays, bO a, cVa, formed by the two semi-circles, are separated or encroach upon one another at this point of the axis by the distance ef, the two images of the sun will not seem to touch one another externally, but to separate or to encroach upon one another by the interval ef. The error hereby introduced into the measure of the sun's diameter will be the angle erf, subtended by ef at r the middle point between O and V, which is to ca or OaV, the sun's apparent diameter, as ee to er, or even to ar, on account of the smallness of ae with respect to ar.

These considerations concerning the cause of a principal error that has been found in the object-glass micrometer led to an inquiry, whether some method might not be found of producing two distinct representations of the sun, or any other object, which should have the axes of the pencils of rays, by which they are formed, diverging from one and the same point, or nearly so; and it occurred to Mr. Malkelyne, that this might be done by the refraction of a prism placed to receive part of the rays proceeding from the object, either before or after their refraction through the object-glass of a telescope. If the prism be placed without the object-glass, the rays that are refracted thro' it will make an angle with the rays that pass beside it equal to the refraction of the prism; and this angle will not be altered by the refraction of the object-glass afterwards. Consequently, two images of an object will be represented, and the prism so applied will enable us to measure the apparent diameter of any object, or any other angular distance which is equal to the refraction of the prism. But if the prism be placed within the object-glass, that is to say, between the object-glass and eye-glass, the angle measured by the instrument will vary according to the distance of the prism from the focus of the object-glass, bearing the same ratio to the refraction of the prism, as the distance of the prism from the focus bears to the focal length of the object-glass.

Let ACB (fig. 2.) represent the object-glass and d the eye-glass of a telescope, and PR a prism placed to intercept part of the rays coming from an object, suppose the sun, before they fall on the object-glass. The rays EE proceeding from the eastern limb of the sun, and refracted through the object-glass ACB without passing through the prism, will form the corresponding point of the sun's image at e; and the rays WW proceeding in like manner from the western limb of the sun will be refracted to form the corresponding point of the sun's image at W. But the rays zE, zE, zW, zW, proceeding in like manner from the eastern and western limbs of the sun, and falling on the prism PR, and thence refracted to the object-glass ACB, will, after refraction through it, form the correspondent points of the sun's image at zc, zW. Let the refraction of the prism be equal to the sun's apparent diameter: in this case, at whatever distance the prism be placed beyond the object-glass, the two images of the sun We, zW ze, will touch one another externally at the point ezW; for the rays zW, zW, proceeding from the western of the sun being inclined to the rays EE proceeding from the eastern limb in the angle of the sun's apparent diameter, will, after suffering a refraction in passing through the prism equal to the sun's apparent diameter, emerge from the prism and fall upon the object-glass parallel to the rays EE, and consequently will have their focus zW coincident with the focus e of the rays EE; and therefore the two images of the sun We, zW ze, will touch one another externally at the point ezW, and the instrument will measure the angle ECzW, and that only.

But if the prism be placed within the telescope, the angle measured by the instrument will be to the refraction of the prism as the distance of the prism from the focus of the object-glass is to the focal distance of the object-glass: or if two prisms be used to form the two images, with their refracting angles placed contrary ways, as represented in fig. 3. and 4., the angle measured will be to the sum of the refractions of the prisms, as the distance of the prisms from the focus of the object-glass is to the focal distance of the object-glass. For let ACB (fig. 3.) represent the object-glass, and d the eye-glass of a telescope, and PR, RS, two prisms interposed between them, with their refracting angles turned contrary ways, and the common sections of their refracting planes touching one another at R. The rays proceeding from an object, suppose the sun, will be deflected, by the refraction of the object-glass, to form an image of the sun at the focus; but part of them falling on one prism, and part on the other, will be thereby refracted contrary ways, so as to form two equal images We, zW ze, which, if the refractions of the prisms be of proper quantities, will touch one another externally at the point ezW. Let ECN be the axis of the pencil of rays EE proceeding from the sun's eastern limb; and WCO the axis of the pencil of rays WW proceeding from the sun's western limb; and the point N the place where the image of the sun's eastern limb would be formed, and the point O where that of the western limb would be formed, were not the rays diverted from their course by the refractions of the prisms. But by this means part of the rays EE, which were proceeding to N, falling on the prism PR, will be refracted to form an image of the sun's eastern limb at e, while others of the rays EE, which fall on the prism RS, will be refracted to form an image of the sun's eastern limb at 2e. In like manner, part of the rays WW, which were proceeding to form an image of the sun's western limb at O, falling on the prism RS, will be refracted to form an image of the sun's western limb at 2W coincident with e, the point of the image correspondent to the sun's eastern limb; while others of the rays WW, which fall on the prism PR, will be refracted to form the image of the sun's western limb at W. The two images We, 2W 2e, are supposed to touch one another externally at the point e2W. The ray EFR, which belongs to the axis ECN, and is refracted by the prism PR to e, undergoes the refraction NR, which (because small angles are proportional to their sines, and the sine of NR is equal to the sine of its supplement NRC), is to NCR as NC or Ce is to NR or Re. In like manner, the ray WGR, which belongs to the axis WCO, and is refracted by the prism RS to 2W or e, undergoes the refraction OR, which is to OC as OC or Ce is to RO or Re; therefore, by composition, ORN the sum of the refractions OR, NR, is to OCN the sum of the angles OC, NC, or the sun's apparent diameter, as Ce to Re; that is, as the focal distance of the object-glass to the distance of the prisms from the focus of the object-glass.

Or let the prisms PR, RS, be placed with their refracting angles P, S, turned from one another as in fig. 4: the refraction of the prism PR will transfer the image of the sun from ON to We, and the refraction of the prism RS will transfer the image ON to 2W 2e, the two images 2W 2e, We, touching one another externally at the point 2We. Let ECN, WCO, be the axes of the pencils of rays proceeding from the two extreme limbs of the sun, and N, O, the points where the images of the sun's eastern and western limbs would be formed by the object-glass, were it not for the refraction of the prisms; the ray EFR, which belongs to the axis ECN, and is refracted by the prism RS, to 2e, undergoes the refraction NR2e; and the ray WGR, which belongs to the axis WCO, and is refracted by the prism PR to W, undergoes the refraction ORW. Now NC2e, part of the angle measured, is to NR2e, the refraction of the prism RS, as RW to CW; and OCW, the other part of the angle measured, is to ORW, the refraction of the prism PR, in the same ratio of RW to CW: therefore OCN, the whole angle measured, is to ORN, the sum of the refractions of the two prisms, as RW to CW; that is, as the distance of the prisms from the focus of the object-glass to the focal distance of the object-glass.

When the prisms are placed in the manner represented in fig. 3, the point e of the image We is illuminated only by the rays which fall on the object-glass between A and F, and the point 2W only by the rays which fall on the object-glass between B and G. Now the angles CRF, CRG, equal to the refractions of the prisms, being constant, the spaces FC, CG, will increase in proportion as the distances RF, RG, increase, and the spaces AF, GB, diminish as much; and therefore the images at the point of mutual contact e2W will be each illuminated by half the rays which fall on the object-glass when the prisms are placed close to the object-glass, but will be enlightened less and less the nearer the prisms are brought to the focus of the object-glass.

But when the prisms are placed in the manner shown in fig. 4, the images at the point of contact, as the prisms are removed from the object-glass towards the eye-glass, will be enlightened with more than half the rays that fall on the object-glass, and will be most enlightened when the prisms are brought to the focus itself; for the point 2e of the image 2W 2e will be enlightened by all the rays EE that fall on the object-glass between B and B, and the point W of the image We will be enlightened by all the rays WW which fall on the object-glass between A and G. But the difference of the illuminations is not very considerable in achromatic telescopes, on account of the great aperture of the object-glass; as the greatest space FG is to the focal distance of the object-glass as the sum of the fines of the refractions of the prisms is to the radius.

There is a third way, and perhaps the best, of placing the prism, so as to touch one another along their sides which are at right angles to the common sections of their refracting planes. In this disposition of the prisms the images will be equally enlightened, namely, each with half the rays which fall on the object-glass, wherever the prisms be placed between the object-glass and eye-glass.

From what has been shewn it appears, that this instrument, which may be properly called the prismatic micrometer, will measure any angle that does not exceed the sum of the refractions of the prisms, excepting only very small angles, which cannot be taken with it on account of the vanishing of the pencils of rays at the juncture of the two prisms near the focus of the object-glass; that it will afford a very large scale, namely, the whole focal length of the object-glass, for the greatest angle measured by it; and that it will never be out of adjustment; as the point of the scale where the measurement begins (or the point of O) answers to the focus of the object-glass, which is a point for celestial objects, and a point very easily found for terrestrial objects. All that will be necessary to be done, in order to find the value of the scale of this micrometer, will be to measure accurately the distance of the prisms from the focus when the instrument is set to measure the apparent diameter of any object subtending a known angle at the centre of the object-glass, which may be easily found by experiment, as by measuring a base and the diameter of the object observed placed at the end of it, in the manner practised with other micrometers: for the angle subtended by this object will be to the angle subtended by a celestial object, or very remote land-object, when the distance of the prisms from the principal Micrometer cipal focus is the same as it was found from the actual focus in the terrestrial experiment, as the principal focal distance of the object-glass is to the actual focal distance in the said experiment.

It will probably be the best way in practice, instead of one prism to use two prisms, refracting contrary ways, and so divide the refraction between them (as represented in fig. 3, and 4). Achromatic prisms, each composed of two prisms of flint and crown-glass, placed with their refracting angles contrary ways, will undoubtedly be necessary for measuring angles with great precision by this instrument; and we can only add with pleasure, that it is found by experiment made with this instrument, as it was executed by Mr Dollond with achromatic prisms, ground with great care for this trial sometime ago, that the images, after refraction through the prisms, appear very distinct; and that observations of the apparent diameters of objects may be taken in the manner here proposed with ease and precision.

Two or more sets of prisms may be adapted to the same telescope, to be used each in their turn, for the more commodious measurement of different angles. Thus it may be very convenient to use one set of prisms for measuring angles not exceeding 36°; and consequently fit for measuring the diameters of the sun and moon, and the lucid parts and distances of the cusps in their eclipses; and another set of prisms to measure angles not much exceeding one minute, and consequently fit for measuring the diameters of all the other planets. This latter set of prisms will be the more convenient for measuring small angles, on account of a small imperfection attending the use of this micrometer, as before mentioned; namely, that angles cannot be measured with it when the prisms approach very near the focus of the object-glass, the pencils of rays being there lost at the point where the prisms touch one another.

Upon the principles that have been here explained, a prism placed within the telescope of an astronomical instrument, adjusted by a plumb-line or level, to receive all the rays that pass through the object-glass, may conveniently serve the purpose of a micrometer, and supercede the use both of the vernier scale and the external micrometer; and the instrument may then be always set to some even division before the observation. Thus the use of a telecopic level may be extended to measure with great accuracy the horizontal refractions, the depression of the horizon of the sea, and small altitudes and depressions of land-objects. Time and experience will doubtless suggest many other useful applications of this instrument.

But the greatest improvement which the micrometer hath yet received is from Dr Maskelyne, who hath invented a catoptric one. This, besides the advantage it derives from the principle of reflection, of not being disturbed by the heterogeneity of light, avoids every defect of other micrometers, and can have no aberration, nor any defect which arises from the imperfection of materials, or of execution, as the extreme simplicity of its construction requires no additional mirrors or glases to those required for the telescope; and the separation of the image being effected by the inclination of the two specula, and not depending on the focus of any lens or mirror, any alteration, in the eye of an observer, cannot affect the micrometer angle measured.

It has, peculiar to itself, the advantages of an adjustment to make the images coincide in a direction perpendicular to that of their motion; and also of measuring the diameter of a planet on both sides the zero, which will appear no inconsiderable advantage to observers who know how much easier it is to ascertain the contact of the external edges of two images than their perfect coincidence. A short explanation of the annexed drawings will make the construction and the properties of this micrometer obvious.

"I divided (says Mr Maskelyne) the small speculum of a reflecting telescope, of Caffegrain's construction, into two equal parts, by a plane across its centre; and by inclining the halves of the speculum to each other on an axis at right angles to the plane that separated them, I obtained two distinct images. The satisfaction I received on the first trial was checked by the apparent impossibility of reducing this principle to practice. The angular separation of the two images in this case being half the angular inclination of the two specula, it required an index of an unmanageable length to allow the quantity of one second of a degree to become visible. Some time afterwards, on revising the principle, I considered, that if both the halves of the mirror turned on their centre of curvature, there could be no alteration in their relative inclination to each other from their motion on this centre; and that any extent of scale might be obtained, by fixing the centre of motion at a proportional distance from the common centre of curvature. This will be better understood from the annexed figure.

"R (fig. 1.) represents the small speculum divided into two equal parts; one of which is fixed on the end of the arm B; the other end of the arm is fixed on a steel axis X, which crosses the end of the telescope C. The other half of the mirror R is fixed on the arm D, which arm at the other end terminates in a socket, that turns on the axis, X; both arms are prevented bending by the braces aa. G represents a double screw, having one part e cut into double the number of threads in an inch to that of the part g; the part e having 100 threads in one inch, and the part g 50 only. The screw e works in a nut F in the side of the telescope, while the part g turns in a nut H, which is attached to the arm B; the ends of the arms B and D, to which the mirrors are fixed, are separated from each other by the point of the double screw pressing against the stud b, fixed to the arm D, and turning in the nut H on the arm B. The two arms B and D are pressed against the direction of the double screw e.g by a spiral spring within the part n, by which means all shake or play in the nut H, on which the measure depends, is entirely prevented.

"From the difference of the threads on the screw at e and g, it is evident, that the progressive motion of the screw through the nut will be half the distance of the separation of the two halves of the mirror; and consequently the half mirrors will be moved equally in contrary directions from the axis of the telescope C.

"The wheel V fixed on the end of the double screw has its circumference divided into 100 equal parts, and numbered at every fifth division with 5, 10, &c., to 100; and the index I shows the motion of the screw with..." Micrometer with the wheel round its axis, while the number of revolutions of the screw is shewn by the divisions on the same index. The steel screw at R may be turned by the key S, and serves to incline the small mirror at right angles to the direction of its motion. By turning the finger-head T (fig. 2.) the eye-tube P is brought nearer or farther from the small mirror, to adjust the telescope to distinct vision; and the telescope itself hath a motion round its axis for the convenience of measuring the diameter of a planet in any direction. The inclination of the diameter measured with the horizon is shewn in degrees and minutes by a level and vernier on a graduated circle, at the breech of the telescope.

"The method of adjusting and using the catoptric micrometer is too obvious to require any explanation: it is only necessary to observe, that, besides the table for reducing the revolutions and parts of the screw to minutes, seconds, &c. it may require a table for correcting a very small error which arises from the eccentric motion of the half-mirrors. By this motion their centres of curvature will (when the angle to be measured is large) approach a little towards the large mirror: the equation for this purpose in small angles is insensible; but when angles to be measured exceed ten minutes, it should not be neglected. Or, the angle measured may be corrected by diminishing it in the proportion the versed sine of the angle measured, supposing the eccentricity radius, bears to the focal length of the small mirror.

"The telescope to which the catoptric micrometer is applied is of the Cassegrain construction. The great speculum is about 22 inches focus, and bears an aperture of 5½ inches, which is considerably larger than those of the same focal length are generally made: indeed, the apparent utility of this micrometer makes me wish to see the reflecting telescope meet with further improvements. I believe it would more tend to the advancement of the art of working mirrors, if writers on this subject, instead of giving us their methods of working imaginary parabolas, would demonstrate the properties of curves for mirrors, which, placed in a telescope, will shew images of objects perfectly free from aberration; or, what will yet be more useful in practice, of what forms specula might be made, that the aberration caused by one mirror may be corrected by that of the other. If mathematicians assume data which really exist, they must see, that when the two specula of a reflecting telescope are parabolas, they cause a very considerable aberration, which is negative, that is to say, the focus of the extreme ray is longer than those of the middle ones. If the large speculum is a parabola, the small one ought to be an ellipse; but when the small speculum is spherical, which is generally the case in practice, if concave, the figure of the large speculum ought to be an hyperbola; if convex, the large speculum ought to be an ellipse, to free the telescope from aberration.

"This will be easier understood by attending to the positions of the first and second images; when a curve is of such form that lines drawn from each image, and meeting in any part of the curve, make equal angles with the tangent to the curve at that point, it is evident, that such curve will be free from aberration.

"This is the property of a circle when the radiant Micrometer and image are in the same place; but, when they recede from each other, of an ellipse, of such form that the radiant and image are in the two focii, till, one distance becoming infinite the ellipse changes into a parabola, and to an hyperbola when the focus is negative; that is to say, when reflected rays diverge, and the focus is on the opposite side of the mirror.

"These principles made me prefer Cassegrain's construction of the reflecting telescope to either the Gregorian or Newtonian. In the former, errors caused by one speculum are diminished by those in the other.

"From a property of the reflecting telescope (which has not been attended to) that the apertures of the two specula are to each other very nearly in the proportion of their focal lengths, it follows, that their aberrations will be to each other in the same proportion; and these aberrations are in the same direction, if the two specula are both concave; or in contrary directions, if one speculum is concave, and the other convex.

"In the Gregorian construction, both specula being concave, the aberration at the second image will be the sum of the aberrations of the two mirrors; but in the Cassegrain construction, one mirror being concave, and the other convex, the aberration at the second image will be the difference between their aberrations. By assuming such proportions for the focii of the specula as are generally used in the reflecting telescope, which is about as 1 to 4, the aberration in the Cassegrain construction will be to that in the Gregorian as 3 to 5.

"I have mentioned these circumstances in hopes of recommending the demonstration of curves suited to the purposes of optics to the attention of mathematicians, which would be of great use to artists.

"I shall conclude with the description of a new micrometer suited to the principle of refraction; being sensible that both principles have their peculiar advantages. Though the former part of this paper proves my partiality to the principle of reflection applied to micrometers, yet the favourable opinion I have of the refracting telescope made me attentively consider some means of applying a micrometer to it, which might obviate the errors complained of in the former part of this paper.

"The application of any lens or medium between the object-glass and its focus must inevitably destroy the distinctness of the image; I therefore have employed for the micrometer-glass, one of the eye-glasses requisite in the common construction of the telescope; but if it should be found necessary to apply an additional eye-glass for the convenience of enlarging the scale, I am able thereby to correct both the colours and spherical aberration of the first eye-glass.

"This micrometer is applied to the erect eye-tube of a refracting telescope, and is placed in the conjugate focus of the first eye-glass; hence arises its great superiority to the object-glass micrometer. It has been before observed, that if a micrometer is applied at the object-glass, the imperfections of its glass are magnified by the whole power of the telescope; but in this position, the image being considerably magnified before it comes to the micrometer, any imperfection in its glass will be magnified only by the remaining eye- Micrometer glasses, which in any telescope seldom exceeds five or six times.

"By this position the size of the micrometer glass will not be the \(\frac{1}{2}\) part of the area which would be required if it was placed in the object-glass; and, notwithstanding this great disproportion of size, which is of great moment to the practical optician, the same extent of scale is preserved, and the images are uniformly bright in every part of the field of the telescope.

"Fig. 4. represents the glasses of a refracting telescope; \(x\) the principal pencil of rays from the object-glass \(O\); \(t\) and \(u\), the axis of two oblique pencils; \(a\), the first eye-glass; \(m\), its conjugate focus, or the place of the micrometer; \(b\) the second eye-glass; \(c\) the third; and \(d\) the fourth, or that which is nearest the eye. Let \(p\) be the diameter of the object-glass, \(e\) the diameter of a pencil at \(m\), and \(f\) the diameter of the pencil at the eye; it is evident, that the axis of the pencils from every part of the image will cross each other at the point \(m\); and \(e\), the width of the micrometer-glass, is to \(p\) the diameter of the object-glass as \(m\) is to \(g\), which is the proportion of the magnifying power at the point \(m\); and the error caused by an imperfection in the micrometer-glass placed at \(m\) will be to the error, had the micrometer been at \(O\), as \(m\) is to \(p\).

"Fig. 3. represents the micrometer; \(A\), a convex or concave lens divided into two equal parts by a plane across its centre; one of these semi-lenses is fixed in a frame \(B\), and the other in the frame \(E\); which two frames slide on a plate \(H\), and are pressed against it by thin plates \(a\); the frames \(B\) and \(E\) are moved in contrary directions by turning the button \(D\); \(L\) is a scale of equal parts on the frame \(B\); it is numbered from each end towards the middle with 10, 20, &c. There are two verniers on the frame \(E\), one at \(M\) and the other at \(N\), for the convenience of measuring the diameter of a planet, &c., on both sides the zero. The first division on both these verniers coincides at the same time with the two zeros on the scale \(L\); and, if the frame is moved towards the right, the relative motion of the two frames is shown on the scale \(L\) by the vernier \(M\); but if the frame \(B\) be moved towards the left, the relative motion is shown by the vernier \(N\).

"This micrometer has a motion round the axis of vision, for the convenience of measuring the diameter of a planet, &c., in any direction, by turning an endless screw \(F\), and the inclination of the diameter measured with the horizon is shown on the circle \(g\) by a vernier on the plate \(V\). The telescope may be adjusted to distinct vision by means of an adjusting screw, which moves the whole eye-tube with the micrometer nearer or farther from the object-glass, as telescopes are generally made; or the same effect may be produced in a better manner, without moving the micrometer, by sliding the part of the eye-tube \(m\) on the part \(n\), by help of a screw or pinion. The micrometer is made to take off occasionally from the eye-tube, that the telescope may be used without it."