Micrometer, from μικρός, small, and μέτρον, a measure, is the name of an instrument generally applied to telescopes and microscopes, for measuring small angular distances within the field of the former, or the size of small objects within that of the latter.

Previously to the invention of the telescope, astronomers experienced great difficulty in measuring small angles in the heavens; but we may safely infer from the observations of Hipparchus, that he had succeeded, either by the actual division of his instruments, or by estimation, in determining celestial arcs to one third of a degree.

When the telescope was applied by Galileo, and our countryman Harriot, to the examination of the solar spots, it does not appear that they executed their drawings from any other than estimated measures. This indeed seems quite certain in the case of Harriot, whose original sketches we have had an opportunity of inspecting. The elaborate solar observations of Scheiner made in 1611, with a telescope on a polar axis, and published in 1630 in his *Rosa Ursina*, though minutely laid down, and performed with great care, were certainly made without any instrument for subdividing the field of view.

As telescopic observations, however, multiplied, astronomers felt the necessity of having something more accurate than their eye for ascertaining minute distances in the heavens; and there can be no doubt that a micrometer was invented by our countryman Mr Gascoigne, previous to 1640, not long after the publication of the *Rosa Ursina*. According to the description of it which he addressed in a letter to Mr Oughtred, and to the account of one of Gascoigne's own instruments which Dr Hooke examined, its construction is as follows:—A small cylinder, stretching across the eye-tube of the telescope, is cut into a fine screw throughout one third of its length, the other two thirds being formed into a coarser screw, with threads at twice the distance. This compound screw is confined at both ends to its place, the fine part of it passing through a female screw in one bar, and the coarse part through a female screw in another bar, these two bars being grooved into each other, as in a sliding rule. Hence, if a nicely ground edge is fixed to one bar, and another to the other bar, so that these edges are accurately parallel, a motion of the screw round its axis will separate these two edges, and each edge will move with a different velocity. The parts of a revolution are measured by an index and divided face, at the coarse end of the screw, while the number of whole revolutions is measured by a graduated bar moved by the coarse screw. The fine screw serves the purpose of keeping the middle part of this variable field (or the opening between the edges) in the axis or line of collimation of the telescope; for while the coarse screw moves the edge which it carries from the other edge considered as fixed, the fine screw moves both the edges, and indeed the whole frame, in an opposite direction, with one half of the velocity, an effect which is produced by fixing its bar to the tube of the telescope. As Mr Gascoigne fell in the civil wars, near York, in 1644, before he had given any full account of his invention, and its application to as-

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1 See Phil. Trans. No. 29, p. 540, Nov. 1667; Lowthorp's Abridgment, vol. i. p. 226; and Costard's History of Astronomy.

Introduction. We are indebted to Mr Richard Townley, into whose hands one of the instruments fell, for the preservation of so valuable a relic. Mr Townley informs us that Mr Gascoigne had made use of his micrometer for some years before the civil wars, and had measured distances on the earth, determined the diameters of the planets, and endeavoured to find the moon's distance from two observations of her horizontal and meridional diameters. Mr Townley's instrument was of the size and weight of "an ordinary pocket watch." It marked 40,000 divisions in a foot, 2½ divisions corresponding to a second of space. Mr Townley had it improved by a common watchmaker. Flamsteed was presented with one of the instruments in 1670, by Sir Jonas Moore; but though he left three guineas with Mr Collins to get proper glasses made for it, he could not procure them till autumn 1671, when he began his observations with it at Derby, and continued them with it in 1671, 1672, 1673, and 1674. He informs us that Townley's improvement consisted in substituting one screw for two. He mentions also that Gascoigne had, in August 1640, measured with his micrometer the diameters of the sun and moon, and the relative distances of the stars in the Pleiades.

Dr Hooke made an important improvement in this micrometer, by substituting parallel hairs for the parallel edges of the brass plates; and Dr Pearson conjectures that he had adopted this construction in his zenith sector, by which he proposed, in his dispute with Hevelius, to measure single seconds.

It would appear, from the Ephemerides of the Marquis of Malvasia, published in the year 1662, that he had measured the distances of stars, and the diameters of the planets, and projected the lunar spots, by means of a reticle of silver wire fixed in the focus of the eye-glass of his telescope. In order to determine the distances of the wires which composed this network, he turned it round till a star moved along one of the wires, and having counted the number of seconds which the star took to pass over the different distances between the wires, he obtained a very accurate scale for all micrometrical purposes.

About the year 1666, MM. Auzout and Picard, unacquainted with what had been done by Gascoigne, published an account of a micrometer. Auzout's micrometer is said to have divided a foot into 24,000 or 30,000 parts. It resembled the Marquis of Malvasia's, with this difference, that the divisions were measured by a screw, and he sometimes employed fibres of silk in place of silver wires.

The celebrated Christian Huygens was also an early inventor of micrometrical methods; and the subject was prosecuted with great diligence and success by Cassini, Rüeumer, Bradley, Savary, Bouguer, Delond, Maskelyne, Ramsden, Sir W. Herschel, Troughton, Wollaston, Arago, Fraunhofer, and Amici.

In giving an account of the inventions and methods of these various authors, we shall adopt the following arrangement:

1. Description of wire-micrometers in which the wires are moved by one or more screws. 2. Description of wire-micrometers in which the angular distance of the wires is varied optically, by changing the magnifying power of the telescope. 3. Description of double-image micrometers in which two singly refracting lenses, semi-lenses, or prisms, are separated by screws. 4. Description of double-image micrometers in which the two images formed by two singly refracting lenses, semi-lenses, or prisms, are separated optically. 5. Description of double-image micrometers in which the two images are formed by double refraction. 6. Description of position-micrometers. 7. Description of the lamp-micrometer, and the lucid disc micrometer. 8. Description of fixed micrometers with an invariable scale. 9. Description of micrometers for microscopes.

CHAP. I.—DESCRIPTION OF WIRE-MICROMETERS IN WHICH THE WIRES ARE MOVED BY MEANS OF ONE OR MORE SCREWS.

The micrometer of Gascoigne, when furnished with Troughairs, as suggested by Dr Hooke, embodies the principle ten's wire of the best and most recent micrometers. Instruments of this construction have been made by all our eminent opticians; but we have no hesitation in saying, that the micrometer constructed by the late celebrated artist Mr Troughton combines all the ingenuity which has been displayed in this delicate and useful apparatus. This eye-piece, and micrometer attached to it, are shown in Plate CCCLVII. figs. 1, 2, and 3, where fig. 1 is a horizontal section in the direction of the axis of the telescope. The eye-piece AB consists of two plano-convex lenses A, B, of nearly the same focal length, and the two convex sides facing each other. They are placed at a distance less than the focal length of A, so that the wires of the micrometer, which must be distinctly seen, are beyond B. This arrangement gives a flat field, and prevents any distortion of the object. This eye-piece slides into the tube CD, which screws into the brass ring EF, through two openings, in which the oblong frame MW passes. A brass circle GH, fixed to the telescope by the screw I, has rack-teeth on its circumference, that receive the teeth of an endless screw W, which, being fixed by the arms XX to the oblong box MN, gives the latter and the eye-piece a motion of rotation round the axis of the telescope; and an index upon this box points out on the graduated circle upon GH, fig. 3, the angular motion of the eye-piece. The micrometer properly so called is shown in fig. 2, where K, L are two forks, each connected with a screw O and P, turned by the milled heads M and N. These forks are so fitted as to have no lateral shake. Two pins Q, R, with spiral springs coiled round them, pass loosely through holes in the forks K, L, so that when the forks are pressed by their screws towards Q and R, the spiral springs resist them, and consequently push them back when the screws are turned in the opposite direction.

Two fine hairs, or wires, or spiders' lines, S, T, are stretched across the forks, the one being fixed to the inner fork K, and the other to the outer fork L, so as to be perfectly parallel, and not to come in contact when they pass or eclipse one another, in which case they will appear as one line. A wire ST is stretched across the centre of the field, perpendicular to the parallel wires.

The most difficult part of this instrument in the execution, as well as the most important, is the screw or screws which move the forks. The threads must not only be at the same distance, but have their inclination equal all round. In the screw used by Troughton, there are about 103-6 threads in an inch. On the right hand of the line ST, fig. 2, is seen a scale, which indicates a complete revolution of either screw, the small round hole being the zero. This hole is bisected when the two lines appear as one.

In using this instrument, we separate the wires by their respective screws, till the object to be measured is exactly

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1 See Mr Baily's Account of the Reverend John Flamsteed, 1835, p. 24, 29, &c. 2 Phil. Trans. No. 21, p. 373, January 1666. Wire-Micrometers included between them. The number of revolutions and parts of a revolution necessary to bring the two wires into the position of zero, will then be a measure of the angle required, provided the value of a revolution has been previously ascertained with accuracy.

The easiest method of ascertaining the value of a revolution of the screw, according to the late Dr Pearson, who devoted much attention to this subject, is to ascertain how many revolutions and parts of one measure exactly the sun's vertical diameter in summer, when his altitude is such that the refraction of both limbs is almost the same. The sun's diameter in seconds being divided by that number, the quotient will be the value of a single revolution, the sun's diameter having been corrected by the difference between the refraction of his two limbs. The ordinary method of ascertaining the value of a revolution is, to observe accurately the time taken by an equatorial star, or a star of known declination reduced to the equator, to pass over the space between the wires when at a distance, and to convert this time into degrees, at the rate of 15° per hour. The number of degrees, minutes, and seconds, divided by the revolutions and parts of a revolution which are necessary to bring both wires into zero, will give the value of one revolution of the screw. The same thing may be done by measuring a base with great accuracy, and observing the space comprehended between the wires at that distance. The angular magnitude of this space, divided by the number of revolutions of the screws which bring the wires to zero, will be the value of each.

A most elegant and accurate method has been recently employed, we believe by Professor Gauss of Göttingen, for measuring the value of the revolutions of micrometer screws. He employs for this purpose a standard telescope, with a micrometer the value of whose scale has been accurately determined. Since the wires of a telescope-micrometer adjusted to distinct vision of the stars or planets are accurately in the focus of parallel rays falling on the object-glass, it follows, that rays issuing from the wires and falling on the inside of the object-glass, will emerge from it perfectly parallel. Now, if we place the object-glass of the standard telescope close or near to that of the first telescope, the parallel rays formed by those issuing from its wires will be refracted to the focus of the standard telescope, and a distinct image of the wires will be there formed. The observer, therefore, when he looks into the standard telescope, will see distinctly the wires of the first telescope, and, by means of his micrometer, he will be able to measure exactly the angular distance of these wires, at whatever distance they happen to be placed. This angular distance divided by the revolutions and parts of a revolution which are necessary to bring the wires of the first telescope to the zero of their scale, will give the value of one revolution of the screw, or of one unit of the scale on the right hand of the long wire ST; fig. 2.

The most essential parts of a micrometer are the parallel fibres, which require not only to be extremely fine, but of an uniform diameter throughout. Gascoigne, as we have seen, employed the edges of brass plates, Dr Hooke hairs, and subsequent astronomers wires and fibres of silk. Fontana, in 1773, recommended the spider's line as a substitute for wires, and he is said (we think erroneously) to have obtained them so fine as the 8000th part of a line. Mr Troughton had the merit of introducing the spider's line, which he found to be so fine, opaque, and elastic, as to answer all the purposes of practical astronomy. This distinguished artist, however, informed the writer of this article, that it was only the stretcher, or the long line which sustains the web, which possesses these useful properties. Sir David Brewster has employed the fibres of spun glass, which are bisected longitudinally with a fine transparent line about the 5000th of an inch in diameter. This central line increases with the diameter of the fibre, and diminishes with the refractive power of the glass. In cases of emergency, the fibres of melted sealing-wax may be advantageously employed, or, as recommended by Professor Wallace, the fibres of asbestos. We have found crystals of mesolite so minute and regular as to be well adapted for the same purpose.

The art of forming silver wire of extreme minuteness has been perfected by Dr Wollaston. Having placed a thin small platinum wire in the axis of a cylindrical mould, he wires-poured melted silver into the mould, so that the platinum wire formed the axis of the silver cylinder. The silver was now drawn out in the usual way, till its diameter was about the 300th of an inch, so that if the platinum wire was at first \(\frac{1}{300}\)th of the diameter of the silver cylinder, it will now be reduced to the 3000th part of an inch. The silver wire is now bent into the form of the letter U, and a hook being made at each of its ends, it is suspended by a gold wire in hot nitric acid. The silver is speedily dissolved by the acid, excepting at its ends, and the fine platinum wire which formed its axis remains untouched. In this way Dr Wollaston succeeded in forming wire \(\frac{1}{30000}\)th, \(\frac{1}{300000}\)th, and even \(\frac{1}{3000000}\)th of an inch in diameter. When the fibres are prepared, their ends are placed in parallel scratches or grooves drawn on the forks, or, in other cases, on the diaphragm or field bar, and fixed by a layer of bees' wax or varnish, or, what is more secure, by pinching them with a small screw-nail near their extremities. For a great deal of valuable practical information respecting the construction and use of the wire-micrometer, the reader is referred to the late Dr Pearson's Introduction to Practical Astronomy (vol. ii. p. 99, 110, 115, &c.), where valuable tables will be found for facilitating the application of the micrometer, both to celestial and terrestrial purposes. See also Sir John Herschel and Sir James South's Observations of 380 Double and Triple Stars (p. 22, 23), containing tables of the values of Troughton's screws.

**Chap. II.—Description of Wire-Micrometers in which the Angular Distance of the Wires is Varied Optically by Changing the Magnifying Power of the Telescope.**

MM. Römer and De la Hire first conceived the idea of varying the angular magnitude of the meshes of a net of silver wire fixed in the focus of the eye-glass of a telescope, for the purpose of measuring the digits of eclipses. This was done by a second lens moving between the wires and the object-glass. The late Mr Watt informed the writer of this article that he had used a similar principle, but had never published any account of it.

The plan of opening and shutting a pair of parallel wires optically instead of mechanically, and of using it as a general principle in micrometers, was first adopted by Sir David Brewster, and has been applied to a variety of methods of varying the magnifying power of the telescope.

The general principle will be readily understood from the annexed diagram, where AB, CD are two wires or lines of any kind permanently fixed in the focus of the eye-glass of a telescope. If the sun S's is in contact with the lower wire CD, it is obvious, that if we increase the magnifying power of the telescope by any optical means anterior to the wires, we may magnify or expand the sun's disc S's, till it becomes Ss, when its north or upper limb will exactly touch the upper wire AB. Now if the sun's diameter happens to be 31' Wire-Micrometer.

When its disc SS just fills the space between the wires AB, CD, the distance of the wires must have been 62', when, as at S'S', it fills only half that space. Hence the wires have been moved optically, so to speak, and have subtended all angles between 31' and 62'.

The methods of varying the magnifying power of the telescope used by Sir David Brewster, consist, 1, in varying the distance of the two parts of the achromatic eye-piece; and, 2, by varying the focal length of the principal object-glass by means of another object-glass, either convex or concave, moving between it and its principal focus.

The first of these methods is shown in fig. 2, where AB is the eye-piece with its four lenses; A, C, D, B, in their natural position. The part AFG, with the two lenses A, C, is fixed to the telescope, and a space is left between the tube AC and the outer tube AFG, to allow the moveable part DB of the eye-piece to get sufficiently near the lens C. The tube DB is moved out and in by a rack and pinion E. A scale is formed on the upper surface mn, and subdivided in the usual manner with a lens and vernier, which it is unnecessary to represent in the figure. The value of the divisions of the scale are determined by direct experiment. A motion of DB through a space of four inches will, generally speaking, double the magnifying power of the telescope.

The best method, however, of varying the magnifying power of the telescope is the second, which is shown in fig. 3, where O is the object-glass, f its principal focus, and L the second lens, which is moveable between O and f. Parallel rays RR, after being refracted by O, so that they would converge to f, are intercepted by L, which converges them to F, the focus of the combined lenses. The effect of the lens L is therefore to diminish the focal length of the object-glass, and consequently the magnifying power of the telescope, which will obviously be a minimum when the lens L is at l, and a maximum when it is at f'. The angle subtended by a pair of fixed wires will suffer an opposite change to the magnifying power, being a maximum when the lens L is at l, and a minimum when it is at f'. Hence the scale for measuring the variable angle of these wires may always be equal to the focal length of the object-glass O; and the inventor of the instrument has shown, both by theory and by experiment, that the scale is one of equal parts, the variations in the angle of the fixed wires being proportional to the variations in the position of the moveable lens.

When we wish to measure angles that do not suffer a great change, such as the diameters of the sun and moon, a scale less than the focal length of the object-glass will be sufficient. For example, if we take a lens L, which by a motion of ten inches varies the magnifying power from 40 to 35, then, if the angle of the wires is 29° when the lens L is at l, it will be 35° 9" when the lens is ten inches from l, or the magnifying power 35. We have, therefore, a scale of ten inches to measure a change of angle of 4° 9", Wire-Micrometer, so that every tenth of an inch will correspond to 3°-3, and crometers, every 100th of an inch to 1/4 of a second. Such a micrometer will serve to measure the diameters of the sun and moon at their various distances from the earth.

If we wish to measure the distances of some double stars, or the diameters of some of the smaller planets, with a telescope whose magnifying power varies from 300 to 240, by the motion of a lens over ten inches, place the parallel wires at a distance of 40°, which will be increased to 50° by the motion of the lens. Hence we have a scale of ten inches to measure ten seconds, or the tenth of an inch to measure one second, or the 100th of an inch to measure 1/10th of a second.

Several pairs of wires placed at different distances might be fixed upon the same diaphragm, or upon separate diaphragms, which could be brought into the focus when wanted; and the second pair of wires might be placed at such a distance that their least angle was equal to the largest angle of the first pair, and so on with the rest.

A wire-micrometer thus constructed is certainly free from almost all the sources of error which affect the common moveable wire-micrometer. The errors arising from the imperfection of the screw, the uncertainty of zero, and other causes, are avoided; and the wires are always equidistant from the centre of the field, so as to be equally affected by any optical imperfection in the telescope. The scale indeed may be formed by direct experiment, and the results will be as free from error as the experiments by which the scale was made.

When this micrometer is applied to a portable telescope, it becomes of great use in naval, military, or geodetical operations, and is employed in measuring distances, either by taking the angle subtended by a body of known dimensions, or by measuring the two angles subtended by a body of unknown dimensions from the two extremities of a known or measured base. For these purposes the telescope is fitted up without a stand, as shown in Plate CCCXLVII, fig. 4.

The principle of separating a pair of wires optically is singularly applicable to the Gregorian and Cassegrainian telescopes, where no additional lens or mirror is required. As the magnifying power of both these telescopes may be increased merely by increasing the distance of the eye-piece from the great speculum, and then re-adjusting the small speculum to distinct vision, we can thus vary the angle of a pair of fixed wires by making the eye-piece moveable. This will be easily comprehended from the annexed figure, where SS is the great speculum of a Gregorian reflector, AA the tube, M the small speculum, whose focus is G, and centre of curvature H. It is fixed to an arm MQ, moveable to and from SS in the usual way. The image R'R' is that formed by the speculum SS, and r'R' that formed by the small speculum. This last image being in the focus of the eye-glass E, will be seen distinct and magnified. If the eye-glass E is pulled out to E', then, in order that the object may be seen distinctly, the image r'R' must be brought into the position r'R', FP' being equal to EE'; but this can be done only by advancing the small speculum M to M', f and F' being now the conjugate foci of M. But by this process the magnifying power has been considerably increased, because the part of the whole magnifying power produced by M was equal to $\frac{MF}{MF'}$ where-as it is now $\frac{MF}{MF'}$, a much larger quantity. The angle subtended by the wires has therefore been diminished in the same proportion as the magnifying power has been increased. The scale, in this case, is not one of equal parts, but after the extreme points of it have been determined experimentally, the rest may be filled up either by calculation or direct experiment.

Dr Pearson has, with singular inaccuracy, stated that Sir David Brewster's "patent micrometer is not competent to measure very small angles, even if it had sufficient magnifying power." If he means the patent micrometer as made by Mr Harris, as a naval and military telescope for measuring distances, or as a coming-up glass, he is quite right, because the power of measuring small angles is not required for these practical purposes. But it is quite evident that the smallest angles can be measured by the micrometer when fitted up for astronomical purposes. We have only to use a pair of wires placed at a very small distance, or a pair of semi-lenses whose centres are placed at a very small distance, and then vary their angles till it becomes equal to the very small angle which we wish to measure.

**CHAP. III.—DESCRIPTION OF DOUBLE-IMAGE MICROMETERS IN WHICH TWO SIMPLY REFRACTING LENSES, SEMI-LENSES, OR PRISMS, ARE SEPARATED BY SCREWS.**

M. Röemer, the celebrated Danish astronomer, is said to have been the first who suggested the use of a double-image micrometer. He did this about 1678, but the idea does not seem to have been carried into effect, or known to his successors. Nearly seventy years afterwards, viz. in 1743, Mr Servington Savary, of Exeter, communicated to the Royal Society an account of a double-image micrometer; and five years afterwards, in 1748, the celebrated Bouguer proposed the very same construction, which he called a heliometer. This instrument consisted of two lenses, which could be separated and made to approach each other by a screw or other mechanical means. These lenses gave double images of every object; and when the two images of any object, such as the sun or moon, were separated till they exactly touched one another, the distance of the object-glasses afforded a measure of the solar or lunar diameter, after an experimental value of the divisions of the scale had been obtained.

As two complete lenses, however, must always have their least distance equal to the diameter of either, this instrument was incapable of measuring the diameters of small bodies. This obvious defect no doubt led John Dollond, in 1753, to the happy idea of the divided object-glass micrometer, in which the two halves of an object-glass are made to recede from the position in which they form a complete object-glass. When the centres of the two halves coincide, they obviously form one lens, and give only one image. When the centres are slightly separated the images will be slightly separated; and small objects may be brought into contact, and have the angles which they subtend accurately measured. The scale will, therefore, have a zero corresponding to the coincidence of the centres of the semi-lenses. The principle of this instrument will be understood from fig. 5, where H, E are two semi-lenses, whose centres are at H, E, and F their focus. If PQ be a circular object whose diameter is to be measured, or P, Q two points whose angular distance is to be determined, the lenses are to be separated till the two images x, z are in contact at F. As the rays QHF, PEF pass unrefracted through the centres H, E of the semi-lenses, the angle subtended by QP will be equal to the angle HFE, or that which the distance of the centres of the semi-lenses subtends at F. As the angles, therefore, are very small, they will vary as HE; and when the angles corresponding to any one distance of the centres is determined, those for any other distance will be ascertained by simple proportion.

Mr Dollond, who had not at this time invented the achromatic telescope, applied his micrometer to the object end of a reflecting telescope, as shown in Plate CCCXLVII. fig. 5, which represents the micrometer as seen from beyond the object end of the reflector. A piece of tube B, carrying the micrometer, slides into or over the tube A of the telescope, and is fastened to it by a screw. The tube B carries a wheel (not seen in the figure) formed of a ring racked at the outer edge, and fixed to the brass plate CC, so that a pinion moved by the handle D may turn it into any position. Two plates F, G are kept close to the plate CC by the rabbed bars H, H, but with so much play that they can move in contrary directions by turning the handle E, which drives a concealed pinion that works in the two racks seen in the highest part of the figure. As the two semi-lenses are fixed to the plates F, G, their centres will be separated by the action of the handle E, and their degree of separation is measured by a scale of five inches subdivided into 20ths of an inch, and read off by a vernier on the plate F, divided into 25 parts, corresponding to 24 of the scale, so that we can measure the separation of the semi-lenses to the $\frac{1}{32}$th of an inch. The vernier is seen to the right of H, and may be adjusted to the zero of the scale, or the position of the lenses when they give only one image, by means of the thumb-screw I, a motion of the vernier being permitted by the screws which fix it to the plate F passing through oblong holes.

In this construction, the micrometer is too far from the observer, and destroys the equilibrium of the telescope. The instrument itself, however, has more serious defects, as it has been found that the measures of the sun's diameter, taken by different observers, with the same instrument, and at the same time, differ so much as 12 or 15 seconds. This defect has been ascribed to the different states of the observers' eyes, according as they have a tendency to give distinct vision within or beyond the focal point, where the image is most perfect; in the former case the limbs being somewhat separated, and in the latter overlapping. M. Mosotti, in the *Effemeride* of Milan for 1821, has discovered the true cause of this defect, by a series of accurate experiments which he made with this micrometer attached to a Gregorian reflector of two feet in focal length. The focal length of the divided object-glass was 511$\frac{3}{35}$ inches, or 42 feet 7$\frac{1}{2}$ inches. M. Mosotti has shown that a diversity of measures will be obtained by the same observer, if, for the purpose of obtaining distinct vision, he gives a slight displacement to the small speculum by the adjusting screw. If the position of this speculum which gives distinct vision were a point, it would be easy to find that point; but as distinct vision may be obtained within a space of 10 or 12 thousandths of an inch, owing to aberration, every different observer will place the mirror at a different point within that range, and consequently obtain a measure corresponding to the image which he views. M. Mosotti recommends that the axis of the adjusting screw, which carries the small speculum, should carry a vernier connected with a scale on the outer surface of the tube A. By means of this vernier the observer is able to give a fixed position to the small speculum, so that he always views the same image, and is thus sure of obtaining the same measure of the same object, so far as the observation is concerned. M. Mosotti found also that the measures were affected by a change of temperature, which, by changing the length of the tube, displaced the small speculum. In his instrument this displacement amounted to 0.0075 of an inch, which, he has shown, corresponds to a change of focal length from 511.3357 to 514.84 inches; and that the error from this cause, upon a length of 30', will be 13" in excess.

The following is Dr Pearson's enumeration of the different sources of error in the divided object-glass micrometer when applied to reflectors.

1. A variation in the position of the small mirror when the eye estimates the point of distinct vision. 2. A displacement of the small mirror by change of temperature. 3. A change of focal distance when central and extreme rays are indiscriminately used. The amount of this error depends on the aberration of the semi-lenses. 4. A defect of adjustment, or of perfect figure, in the two specula, as they regard each other, the measures varying when taken in different directions.

In order to enable Dollond's micrometer to measure differences of declination and right ascension, Dr Maskelyne introduced the aid of cross wires, which he fixed in a moveable ring at the place where the double image is formed. One or both of the two planets or stars are referred to one or other of these lines, as will be seen in the annexed figure, which we take as an example, out of four cases.

Let ENWS be the field of view, NS the meridian, and EW the line of east and west; then, in order to obtain the difference of right ascension and declination of two stars, he opened the semi-lenses till he obtained double images of each star. He then turned round the micrometer till the two images of the first star passed over the vertical wire NS at the same instant, and having counted the time that elapsed till the two images of the other star passed over the same line, he had the difference of right ascension in time. By means of the screw which elevated his telescope, and partly by opening the semi-lenses, he made the north image of one star, and the south image of the other, as at A, B, describe in their motion the horizontal wire EW, and at that position of the semi-lenses the scale indicated the difference of declination.

A very important improvement upon the divided object-glass micrometer was made by Mr Dollond's son, who adapted it to a refracting telescope, and removed the different sources of error to which it had been found liable. This improvement consists both in the nature, form, and position of the semi-lenses. The semi-lenses are made concave, and consist of crown and flint glass, so as to give an achromatic image along with the object-glass of the telescope to which they are applied. These concave semi-lenses, of course, lengthen the focal distance of that object-glass. When a circular lens was bisected, as in the old construction, the metallic parts which held the semi-lenses obstructed the light in proportion to their separation; a defect of a serious nature in an instrument. In order to correct this evil, Mr J. Dollond substituted two long slices of glass cut from the diametral portion of a lens nearly six inches in diameter. Hence, in every position of these oblong semi-lenses, none of the metallic setting comes before the object-glass, and consequently the light is never obstructed, and is always of the same amount, whatever be the separation of the lenses. In the old construction, where the diameters of each lens slid along each other in contact, a part of the central portions having been removed by grinding the diameters smooth, the two images of an object never could coincide so as to give an accurate zero; but in the new construction, the space equal to what was removed by grinding is filled up with a brass scale and vernier, and the only evil of this is the loss of light corresponding to the thickness of this scale; but this trifling defect is amply compensated by the perfect coincidence of the images at zero.

This important instrument is shown in Plate CCCIVII., fig. 6, where the same letters are used as in fig. 5 to denote the analogous parts of the two instruments. The end of the telescope is shown at A, and B is the rim of brass, which, by sliding upon A, fixes the micrometer to the telescope. The frame CC', moved by teeth on its outer edge, carries one of the halves G of the lens, and a similar frame with teeth carries the other half F. The scale S, six inches long, is fastened like an edge-bar to CC', and each inch is subdivided into 20 parts, which are read off with a vernier of 25 parts, which is fastened as an edge-bar to the moveable frame that carries F. The two moveable frames are imbedded in a fixed plate HHP, screwed to the tube B of the micrometer, and having a circular hole in its middle equal to the diameter of the object-glass. The two semi-lenses are separated by turning the milled head to the right of A, which moves the frame CC', and then the other frame F through the medium of a concealed wheel and a concealed pinion. The mechanism for giving the rotatory motion is also concealed. The adjustment of the vernier to zero is effected by the screw I.

The property which the double-image micrometer possesses, of measuring angles in all directions, directed to it the attention of Ramsden and other eminent opticians. Ramsden accordingly communicated to the Royal Society of London, in 1777, an account of two instruments of this kind, under the name of the Dioptric and Catoptric Micrometers. In order to avoid the effects of aberration, Ramsden proposed, in his dioptric micrometer, to place two dioptric semi-lenses in the conjugate focus of the innermost lens of the erect eye-tube of a refracting telescope. In place of the imperfections of the lenses being magnified by the whole power of the telescope, they are magnified only about five or six times, and the size of the micrometer glass does not require to be 1/10th part of the area which is necessary in Dollond's instrument. This instrument is shown in Plate CCCIVII., fig. 7, where A is a convex or concave lens, bisected in the usual way. One of the semi-lenses is fixed in a frame B and the other in a similar frame E, both of which slide upon a plate H, against which they are pressed by thin plates a, a. The milled button D, by means of a pinion and rack, moves these frames in opposite directions; and the separation of the semi-lenses thus effected is measured by a scale of equal parts L on the frame B, the zero being in the middle, and the divisions read off by two verniers at M and N, carried by the frame E; the

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1 See Phil. Trans. vol. lix. 1779, p. 419. vernier M showing the relative motion of the two frames when the frame B moves to the right, and N when the frame B is moved to the left. An endless screw F gives the whole micrometer a motion round the axis of vision. This instrument being only the divided object-glass micrometer in miniature, and differently placed, the reader will have no difficulty in understanding its construction and use, from the details already given in the preceding pages.

Dr Pearson informs us, on the authority of Mr Troughton, that a Captain Countess, R.N., having accidentally broken the third lens of a terrestrial eye-piece of his telescope, observed the double images which it produced; and that this observation led to the contrivance of the coming-up glass that was first made by Nairne, with a double screw for separating the halves of the amplifying lens. Hence it is conjectured that Ramsden derived his idea of using a bisected lens for his dioptric micrometer and dynameter. The above facts may be quite true, but Ramsden certainly did not require any such hint, as it was a very natural transition from a bisected object-glass to a bisected eye-glass.

Dr Pearson also states that Mr George Dollond had constructed a dioptric micrometer almost the same as Ramsden's, without knowing anything of what Ramsden had proposed. We have no doubt that both these ingenious opticians were quite original in their ideas, for it will not be supposed that Captain Countess's broken lens furnished Mr Dollond with the idea of his contrivance. Dr Pearson has given a drawing and description of Mr Dollond's construction of the micrometer as made for Mr Davies Gilbert and himself. It does not appear that Mr Ramsden ever constructed it. The weight of this micrometer was found by Dr Pearson too great for an ordinary achromatic telescope.

Mr Ramsden likewise proposed a catoptric double-image micrometer, which, from being founded on the principle of reflection, is not disturbed by the heterogeneity of light, while he considered it as "avoiding every defect of other micrometers," having "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 glasses to those required for the telescope." It has also, peculiar to itself, the advantages of an adjustment to make the images coincide in a direction perpendicular to that of their motion. In order to effect these objects, Mr Ramsden divided the small speculum of a Cassegrainian reflector into two equal halves, and by inclining each half on an axis at right angles to the plane that separated them, he obtained two distinct images; but as their angular separation was only half the inclination of the specula, which would give only a small scale, he rejected this first idea, and separated the semi-specula by making them turn on their centre of curvature, any extent of scale being obtained by fixing the centre of motion at a proportional distance from the common centre of curvature. The mechanism necessary to effect this is shown in Plate CCCLVII, fig. 8, where A is the bisected speculum, one of the semi-specula being fixed on the inner end of the arm B, its outer end being fixed on a steel axis X extending across the mouth of the tube C. The other semi-speculum is fixed on the inner end of the arm D, its outer end terminating in a socket y, which turns upon the steel axis x. These arms are braced by the bars a, a'. A compound screw G, having its upper part cut into double the number of threads in an inch, viz. 100, to the lower part g, which has only 50, works with the handle in a nut F in the side of the tube, while the part g turns in a nut H fixed to the arm B. The point of the compound screw separates the ends of the arms B and D, and, pressing against the stud

A fixed to the arm D, turns in the nut H on the arm B. Double Image D against the direction of the double screw e.g., so as to prevent all shake or play in the nut H. The progressive motion of the screw through the nut will be half the distance of the semi-specula, so that these specula will be moved equally in opposite directions from the axis of the telescope.

A graduated circle V, divided into 100 parts on its cylindrical surface, is fixed on the upper end of the screw G, so as to cause it to separate the semi-specula. The fixed index I shows the parts of a revolution performed by the screw, while the number of whole revolutions of the screw is shown by the divisions of the same index. A steel screw K, moveable by a key, inclines the small speculum at right angles to the direction of its motion. Distinct vision is procured in the usual manner, and the telescope has a motion about its axis, in order to measure the diameter of a planet in any direction; and the angle of rotation in reference to the horizon is shown by a level, graduated circle, and vernier, at the eye end of the large tube.

A catoptric double-image micrometer has been suggested by Sir David Brewster as applicable to the Newtonian microscope. The plane mirror is bisected, and is made to form two images, either by giving each semi-speculum a motion round their common line of junction, or round a line perpendicular to that common line. The mechanism by which this may be effected does not require any description. If the micrometer is required for the sun or any luminous body, the small mirror may be made of parallel glass, which would have the advantage of not obstructing any of the light which enters the telescope, while it reflects enough for the purposes of distinct vision. We shall again have occasion to refer more particularly to this idea in the next section.

Professor Amici of Modena has described, in the Memoirs of the Italian Society, a new micrometer, which gives double images by means of semi-lenses separated by mechanical means; but as we have not access to this work, we shall draw our description of the instrument from one given by Dr Pearson, which is very far from being distinct, in so far at least as the construction of the semi-lenses, or bars of glass as they are called, are concerned. The semi-lenses seem to be portions of a large concave lens, separated in the usual manner, so as to give two distinct images of objects; but the peculiarity of the invention seems to consist in the lenses being placed between the object-glass of a telescope and its principal focus, the cone of rays being divided at a point about six inches before the place where the focal image is formed. Dr Pearson, who made experiments with one of these instruments, has hinted at the inconveniences which he experienced in using it.

CHAP. IV.—DESCRIPTION OF DOUBLE-IMAGE MICROMETERS IN WHICH THE TWO IMAGES FORMED BY TWO SIMPLY REFRACTING LENSES, SEMI-LENSES, OR PRISMS, ARE SEPARATED OPTICALLY.

In the year 1776 Dr Maskelyne constructed and used Maskelyne's prismatic micrometer, which he had contrived with lyne's view of getting rid of the sources of error to which he had found the divided object-glass micrometer liable. Having cut a prism or wedge of glass into two parts, so as to form two prisms of exactly the same refracting angle, he conceived the idea of fixing them together, so as to produce two images, and to vary the angle which these two images formed, by making the prisms move between the ob-

1 Introduction to Practical Astronomy, vol. ii. p. 192. Double-image micrometers.

The object-glass and its principal focus; so that the scale is equal to the whole focal length of the telescope. The two prisms may be placed in three ways, with their thin edges joined, with their square thick edges or backs joined, or with their sides or triangular edges joined. In the first position the double images will have only one half of the light which is incident on the object-lens when the prisms are close to it, and their degree of illumination will diminish as they approach the focus. In the second position they will, as before, have only one half of the incident light when close to the object-glass, but the illumination will gradually increase as the prisms advance to the focus. In the third case, the prisms being in a reverse position, the light will be the same in every part of the scale, each of them receiving half the rays which fall upon the object-glass. On this account Dr Maskelyne preferred this last arrangement.

In the instrument which Dr Maskelyne constructed, and which seemed to have had only a thirty-inch object-glass, the prisms were not achromatic, and consequently the touching limbs of a luminous body were affected with the prismatic colours. In the case of the sun, where all the rays might have been absorbed but the red, this was of little consequence; but in other cases it was a serious defect, which could be removed only by making the prisms achromatic; or it might have been diminished by making the prisms of fluor spar, in which the dispersion is very small. One of the Dollonds, accordingly, executed for Dr Maskelyne an achromatic prism, which performed well. It does not appear that Dr Maskelyne made any observations of value with this instrument.

A new divided object-glass micrometer has been constructed by Sir David Brewster, and described in his Treatise on New Philosophical Instruments. It consists of an achromatic object-glass LL, fig. 7, between which have an instrument which will measure with the greatest accuracy all angles between the two extreme ones. Another or more pair of semi-lenses may be used in the same telescope, and placed at smaller or greater distances, so that, by means of other scales adapted to them, we may obtain all angles that may be required. The lenses A, B may be concave or convex; and when a large scale is required, with a tenth of an inch to a second, or even greater, we have only to use semi-lenses of long foci, and the scale may be confined to the part of the tube nearest the focal point.

Sir David Brewster has proved, both from theory and experiment, that the scale is one of equal parts; so that, after having ascertained by experiment the two extreme angles, the whole scales may be completed by dividing the interval into any number of equal parts, and these subdivided, if necessary, by a vernier scale.

When the semi-lenses are placed without the object-

glass LL, and this object-glass moved towards f, as in the annexed figure, the angular distance of the images is invariable.

This instrument has been constructed for measuring distances, and as a coming-up glass for ascertaining whether a ship is approaching to or receding from the observer. In this form it constitutes part of fig. 8, Plate CCCLVII., the semi-lenses being made to screw into the same place as the second object-glass, and having a separate scale for themselves. In this form many of the instruments have been constructed by Tulley.

Among the optical micrometers, we may describe another invented by Sir David Brewster, and adapted solely micrometer to the Newtonian telescope. In order to get rid of the loss of light by the reflection of the small plane speculum, he uses an achromatic prism to reflect the light just as much out of the axis of the telescope as will allow the head of the observer to be applied to the eye-tube, without obstructing any of the light which enters the tube. By using two prisms, as in Maskelyne's instrument, and moving them along the axis of his telescope through a small distance, we shall obtain a good micrometer. The prisms may be separated mechanically, or a doubly refracting prism may be fixed upon the face of the single or achromatic prism used to turn aside the rays. The achromatism of a single glass prism may be corrected by the doubly refracting prism, a balance of refraction being left sufficient to turn aside the image to the observer's eye.

Chap. V.—Description of Double-Image Micrometers in which the Two Images are Formed by Double Refraction.

The happy idea of applying the two images formed by Roehon's double refraction to the construction of a micrometer unquestionably belongs to the Abbé Roehon; and though Dr meter Pearson has laboured to show that Dr Maskelyne's prismatic telescope was constructed before Roehon's, yet this does not in the smallest degree take away from the originality and priority of Roehon's invention; for the idea of varying the angle by the motion of the prisms can scarcely be viewed as an essential part of the invention. Although the double refraction of rock-crystal is small, yet, from its limpidity and hardness, the Abbé Rochon regarded it as superior to any other substance for making doubly refracting prisms. When he used one prism so cut that its refracting edge coincided with the axis of the prism, in which case its double refraction was the greatest, he found that the separation of the two images was too small to give the angles which he required. He therefore fell upon a most ingenious plan of doubling the amount of the double refraction of one prism, by using two prisms of rock-crystal, so cut out of the solid as to give each the same quantity of double refraction, and yet to double that quantity in the effect produced. This construction of the compound prism was so difficult, that M. Rochon informs us, that "he knew only one person, M. Narci, who was capable of giving rock-crystal the prismatic form in the proper direction for obtaining the double refractions necessary to the goodness of the micrometer." The method used by Narci seems to have been kept a secret, for in 1819 Dr Wollaston set himself to discover the method of constructing these compound prisms, and has described it in the Philosophical Transactions, but not in such a manner as to be very intelligible to those who are not familiar with such subjects. We conceive that the process may be easily understood from the following rule. Cut a hexagonal prism of quartz into two halves by a plane passing through or parallel to its axis. Grind and polish the two cut faces, and by means of Canada balsam cement the one upon the other, so that any line or edge in the one face may be perpendicular to the same line in the other. Cut and polish a face on each of the united portions, so that the common section of these faces with the cemented planes may be parallel to the axis of the crystal, while they are equally inclined to these planes, and the prism will be completed.

We shall now explain, by a diagram, a more simple and economical way of cutting these prisms, though the principle is exactly the same. Let AKGDBLHF be half of a hexagonal prism of quartz, the height of which, DF, is equal to half of its diameter AD. Bisect AD in C, and join CK, CG, and draw CE parallel to AB or DF. This line CE will be the axis of the prism. Grind and polish the section ABED, and cut off the prisms AKCBLE and DGCFHE, setting aside the intermediate similar prism KGCLHF. The faces ACEB, DCEF are square and equal, so that if we cement these faces together, making the line AB coincide with FE, AC will coincide with FD, CE with CF, and EB with CF. If we wish each prism to have an angle of 60°, we may take either GDFH or GCEH for the refracting face of it; we shall suppose the former. In this case we must grind and polish a face on the other prism ABL, which is accurately parallel to the face GDFH, and the compound prism will be completed. If 60° is too great, we must grind down the face GDFH till it has the desired inclination to DF, and grind and polish a face parallel to it on the other prism. The external faces, in short, to be made upon each prism, must be equally inclined to the cemented planes DCEF, ABEC, and have their common section DF parallel to the axis CE of the prism.

In place of cutting off the prism AKCBLE, we may cut off only the prism GCDHEF, leaving the intermediate one KGCLHE attached to AKCBLE, and proceed as before. The object of this is to leave enough of solid quartz at KL to give a face of the same breadth as GDFH. If the prisms required are small compared with the quartz crystal, we may obtain, by the first method, six prisms out of the crystal, or three pair of compound ones. On the other hand, if the required prism is large compared with the crystal of quartz, it may require one half of the crystal to make one prism, and the other half the other. Nay, it may be necessary to cut each individual prism out of separate crystals, the method of doing which is very obvious from the preceding description.

When the prism is completed, it is obvious that a ray of light incident perpendicularly on the face GHED will be perpendicular to the axis of the prism CE, and therefore the extraordinary ray will suffer the greatest deviation, viz. 17°; and the same is true of the other prism. But when the ray passes through both, it is found to have a deviation of 34', which is produced in the following manner:

Let AB be a line viewed through one of the prisms, with its refracting angle turned upwards; two images of it will be seen, viz. the extraordinary image at E, and the ordinary one at O.

If we now interpose the other prism with its refracting angle downwards, both these images E, O will be refracted downwards. But, owing to the transverse cutting of the prisms, the extraordinary image E, which was most raised, now suffers ordinary refraction, and is least depressed, so that in place of being refracted back to AB, it comes only to EO'. On the other hand, the ordinary image O, which suffered the least refraction, is now extraordinarily refracted, and, in place of reaching AB, is depressed to O'E'; and since the double refraction of each prism, as well as the angles of the prism, are equal, the angular distance of the images EO', O'E' formed by the combined prisms will be double of the distance EO, or 34'.

The same rule may be followed in cutting the prism out of the limpid and homogeneous topazes of New Holland, the principal axis of which coincides with the axis of the prism. When the crystals are amorphous, the cleavage planes will be a sufficient guide, as the above axis is always perpendicular to them. Such prisms are incomparably superior, as we have practically experienced, to those made of rock-crystal.

When a very large angle is required for any particular purposes, artificial crystals, such as carbonate of potash, &c. may be advantageously employed, the crystals being ground with oil, or any fluid in which they are not soluble. By cementing plates of parallel glass on their outer surfaces, they will be as permanent as rock-crystal.

Dr Pearson fitted up with one of Rochon's micrometers an achromatic telescope 33 inches in focal length, and having a magnifying power of 55. He applied it to two separate compound prisms, one of which had a constant angle of 32', and the other an angle only of 5', the vernier in the former case indicating seconds, and in the latter tenths of seconds. A drawing is given of the tube, with the prisms and scales, in Plate CCCXLVIII, fig. 9 and 10, as given by Dr Pearson. The tube is graduated from the

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1 In his first experiments Rochon corrected the dispersion of the rock-crystal prism by a similar prism placed in front of it, and having its exterior face perpendicular to the axis of the crystal, complete correction of the dispersion for the ordinary image.

2 Phil. Trans. 1829, p. 126. solar focus into two scales, one being placed on each side of the slit or opening cut along the middle of the tubes, to allow the sliding piece, shown separately in fig. 10, to move from the object-glass to the solar focus. This sliding piece holds the prism, the larger prism of 32° being shown as placed with its sliding piece in the tube, and the smaller prism of 5° being shown in the separate sliding piece, fig. 9. The two verniers of the scales are seen on each side of the two screws with milled heads, which pass through the slit, and serve to move the sliding piece to or from the object-glass when they are not too much tightened.

In his original memoir on the subject, published in the Journal de Physique for 1801, M. Rochon makes the following observations. "I ought not to omit, that in this new construction there are difficulties of execution not easy to surmount, which may have been one reason why these instruments, so useful to navigators, and in certain very nice astronomical observations, have not been adopted. This induced me at length to adopt Euler's method. In the construction of achromatic object-glasses I found I could increase or diminish the absolute effect of the double refraction within certain limits, by means of the interval between the glasses of different refracting powers; the separation of the images at the focus being so much the greater, as the interval is larger, when the flint-glass is the first of the object-glasses, and less when it is the second. Conformably to these new principles, I have had two telescopes with a doubly refracting medium constructed under my own inspection, which General Ganthame will employ for determining the position of his ships, and to find whether he be approaching any he may meet with at sea."

In 1812 M. Rochon constructed his doubly refracting micrometer in another form, from which he anticipated great advantages. He made a parallelopiped of rock-crystal, consisting of two prisms whose refracting angles were each about 30°, so that the angle which they gave was less than 30°, and the two images of the sun of course overlapped each other. The prisms being firmly united by mastic, he ground the parallelopiped into a convex lens, so that when combined with a concave one of flint-glass, it formed an achromatic object-glass with a focal length of about 3 decimetres, or nearly 12 inches. This object-glass separated the centres of the images of the sun about 28 minutes. "He then adapted to this object-glass a common micrometer, which measured angles of 10 minutes, and he had thus 3 decimetres and 10 minutes to complete the measure of the diameters of the sun or moon."

M. Arago appears to have been the first person to apply doubly refracting prisms to the eye-pieces of telescopes for the purpose of measuring very small angles. He explained his general method to the writer of this article in July 1814, and mentioned the results which he had obtained with it in measuring the diameters of the planets. We do not recollect distinctly how he varied the constant angle of the doubly refracting prism; but Dr Pearson and M. Biot state, that the constant angle was increased by placing the prism in an oblique direction as regards the line of vision; and that he determined the respective values of the angles thus increased by means of concentric circles placed vertically at a measured distance from the eye when looking through the prism; for as he knew the diameters of each circle, he could generally find one out of the number which would come into exact contact with its image, and thus give the value of the constant angle."

Dr Pearson has proposed an ocular crystal micrometer, and has given a drawing and description of the instrument. It is nothing more than M. Arago's ocular crystal prism, in which the constant angle is varied by Sir David Brewster's variable eye-piece already described. On the same principle, the angle of the prism may be varied by a convex or concave lens moving between the object-glass and its principal focus; but what would be still better, by pulling out or pushing in the eye-piece of a Gregorian or Cassegrainian telescope.

In 1821 Mr George Dollond communicated to the Royal Society an account of his spherical crystal micrometer, a very ingenious instrument, though we should think, one difficult to execute; and, at the same time, even when well executed, liable to error. Mr Dollond's improvement consists in making a sphere or lens from a piece of rock-crystal, and adapting it to a telescope in place of the usual eye-glass, as shown in Plate CCCLVII. fig. 11, where \(a\) is the sphere or lens, formed of rock-crystal, and placed in half holes, from which is extended the axis \(b\), with an attached index, the face of which is shown in fig. 12. This index registers the motion of the sphere on the graduated circle. The sphere \(a\) is so placed in the half holes, that when its natural axis (axis of double refraction, we presume) is parallel to the axis of the telescope, it gives only one image of the object. In a direction perpendicular to that axis, it must be so placed that when it is moved the separation of the images may be parallel to that motion. The method of acquiring this adjustment is by turning the sphere \(a\) in the half holes parallel to its own axis. A second lens \(d\) is introduced between the sphere and the primary image given by the object-glass, and its distance from the sphere should be in proportion to the magnifying power required. The magnifying powers engraven in fig. 12 are suited to an object-glass of 44 inches focal length. The following are the advantages of this construction, as stated by its inventor. 1. It is only necessary to select a piece of perfect crystal, and, without any knowledge of the angle that will give the greatest double refraction, to form the sphere of a proper diameter for the focal length required. 2. The angle may be taken on each side of zero, without reversing the eye-tube; and intermediate angles may be taken between zero and the greatest separation of the images, without exchanging any part of the eye-tube, it being only required to move the axis in which the sphere is placed. 3. It possesses the property of a common eye-tube and lens; for when the axis of the crystal is parallel to that of the object-glass, only one image will be formed, and that as distinctly as with any lens that does not refract doubly.

Dr Pearson had one of these instruments constructed by Mr Dollond, and applied to an achromatic object-glass 45-6 inches in focal length. He has shown that the scale is not one of equal parts, and has pointed out a method of determining the constant angle of the crystal.

Knowing from experience the imperfect structure of rock-crystal, especially in directions approaching to the axis, we dreaded that a spherical eye-glass of this material would not give perfect vision. Dr Pearson confirms this opinion by actual observation. He attempted to measure the diameter of Mars when about 9°, but its limits were... so imperfectly defined that no satisfactory observation could be made. We would therefore strongly recommend to Mr Dollond the substitution of limpid topaz from New Holland, in place of the rock-crystal.

**CHAP. VI.—DESCRIPTION OF POSITION MICROMETERS.**

A position micrometer is an instrument for measuring angles when a plane passing through the two lines which contain these angles is perpendicular to the axis of vision. Sir W. Herschel first proposed such an instrument for the purpose of verifying a conjecture, that the smaller of the two stars which compose a double star revolves round the larger one. Hence it became necessary to observe if a line joining the centres of any two stars always formed the same angle with the direction of its daily motion. After constructing the instrument which we are about to describe, and making a long series of observations, he verified his conjecture by the important discovery, that the double stars formed binary systems, in which the one revolved round the other.

The position micrometer used by Sir William Herschel in his earliest observations, viz. those made in 1779–1783, was made by Nairne, and was constructed as shown in Plate CCCLVII, fig. 13, which represents it when enclosed in a turned case of wood, and ready to be screwed into the eye-piece of the telescope. "A is a little box which holds the eye-glass. B is the piece which covers the inside work, and the box A screwed into it. C is the body of the micrometer, containing the brass work, showing the index-plate a projecting at one side, where the case is cut away to receive it. D is a piece having a screw b at the bottom, by means of which the micrometer is fastened to the telescope. To the piece C is given a circular motion, in the manner the horizontal motion is generally given to Gregorian reflectors, by the lower part going through the piece D, where it is held by the screw E, which keeps the two pieces C and D together, but leaves them at liberty to turn on each other. Fig. 14 is a section of the case containing the brass work, where may be observed the piece B hollowed out to receive the box A, which consists of two parts enclosing the eye-lens. This figure shows how the piece C passes through D, and is held by the ring E. The brass work, consisting of a hollow cylinder, a wheel and pinion, and index-plate, is there represented in its place. F is the body of the brass work, being a hollow cylinder with a broad rim C at its upper end; this rim is partly turned away to make a bed for the wheel d. The pinion e turns the wheel d, and carries the index-plate a. One of its pivots moves in the arm f, screwed on the upper part of c, which arm serves also to confine the wheel d to its place on c. The other pivot is held by the arm g fastened to F.

A section of the brass work is shown in fig. 15, where d is the wheel placed on the bed or socket of the rim of the cylinder cc, and is held down by the two pieces f, h, which are screwed on cc. The piece f projects over the centre of the index-plate to receive the upper pivot of the pinion mn, the fixed wire being fastened to cc, and the moveable wire op, fastened to the annular wheel dd. The index-plate a, milled on the edge, is divided into sixty parts, each subdivided into two. When the finger is drawn over the milled edge of the index-plate from q to r, the angle mio will open, and if drawn from r towards q, it will shut again. The case cc must have a sharp corner Micrometers, which serves as an index to point out the divisions on the index-plate.

We do not know the value of the divisions in the instrument used by Sir William Herschel; but in the position micrometer of the five-feet equatorial used by Sir John Herschel and Sir James South, in their observations on double stars, the position circle was large enough to show distinctly minutes of a degree by means of its vernier.

The position micrometer which we have now described has been greatly improved by Sir David Brewster; and the following account of these improvements, which is not susceptible of abridgment, is given in his own words.

In the position micrometer invented by Sir William Herschel, "the two wires always cross each other at position the centre of the field, and consequently their angular micrometric separation is produced uniformly by the motion of the ter. pinion. This very circumstance, however, though it renders it easy for the observer to read off the angle from the scale, is one of the greatest imperfections of the instrument. The observations must obviously be all made on one side of the centre of the field, as appears from fig. 16; and the use of the instrument is limited to those cases in which Ss is less than the radius SC. The greatest disadvantage of the instrument, however, is the shortness of the radius SC; for the error of observation must always diminish as the length of this radius increases. This disadvantage does not exist in measuring the angle of position of two stars S, s, for the distance Ss remains the same whatever be the length of SC; but in determining the angle which a line joining two stars forms with a line joining other two stars, or those which compose a double star (an observation which it may often be of great importance to make), and all other angles contained by lines whose apparent length is greater than SC, this imperfection is inseparable from the instrument. Nay, there are some cases in which the instrument completely fails; as, for instance, when we wish to measure the angles formed by two lines which do not meet in a focus, but only tend to a remote vertex. If the distance of the nearest extremities of these lines is greater than the chord of the angle which they form measured upon the radius SC, then it is impossible to measure that angle, for the wires cannot be brought to coincide with the lines by which it is contained. Nay, when the chord of the angle does exceed the distance between the nearest extremities, the position of the wires which can be brought into coincidence with the lines is so small as to lead to very serious errors in the result.

The new position micrometer which we propose to substitute for this instrument is free from the defects just noticed, and is founded on a beautiful property of the circle. If any two chords, AB, CD, fig. 12, intersect each other in the point O within the circle, the angle which they form at O will be equal to half the sum of the arches AC, BD; but if these chords do not intersect each other within the circle, but tend to any point O without the circle, when they would intersect each other if continued, as in fig. 13, then the angle which they form is equal to half the difference of the arches AC, BD; that is, calling φ the angle required, we have, in the first case, as shown in fig. 12,

\[ \varphi = \frac{AC + BD}{2} \]

and in the second case, as shown in fig. Hence, if the two wires AB, CD be placed in the focus of the first eye-glass of a telescope, the moveable one AB may be made to form every possible angle with the fixed one CD, and that angle may be readily found from the arches AC, BD.

The mechanism for measuring these arches is shown in Plate CCCLVIII, fig. 17, where the graduated circular head CD may be divided only into 180°, in order to save the trouble of halving the sum or the difference of the arches AC, BD; but as it would still be necessary to measure two arches before the angle could be ascertained, we have adopted another method, remarkable for its simplicity, and giving no more trouble than if the wires always intersected each other in the centre of the field.

Let AB, for example, fig. 13, be the fixed wire, and CD the moveable one, and let it be required to find at one observation the angle AOC or ϕ. Set the index of the vernier to zero when D coincides with B; and as C will be at e when D is at B, the arch eA will be a constant quantity, which we shall call b. Making AC = m, and BD = n, we have \( \frac{m + n}{2} \). But since the extremity C will move over the space Ce while D describes the space DB, these arches must be equal, consequently \( b = m - n \); hence, adding \( 2n \) to each side of the equation, we obtain \( b + 2n = m + n \), or \( \frac{b + n}{2} = \frac{m + n}{2} \), consequently \( \phi = \frac{b + n}{2} \).

Hence the angle AOC is equal to half the arch Ac added to the arch DB; or, since Ac is variable, the half of it is a constant quantity, and the angle required is equal to the sum of this constant quantity and the arch DB.

When the wires do not intersect each other, as in fig. 13, we have \( \frac{m - n}{2} \), and \( b = m + n \); hence, subtracting \( 2n \) from each side of the equation, we have \( b - 2n = m - n \), and, dividing by 2, we have \( \frac{b - n}{2} = \frac{m - n}{2} \), consequently \( \phi = \frac{b - n}{2} \). That is, the angle AOC is equal to the difference between half the arch Ac and the arch DB, or to a constant quantity diminished by the arch DB.

In finding the angle AOC, therefore, we have merely to observe the place of the index when the wires are in their proper position; and as the scale commences at B, or where D and B coincide, and is numbered both ways from B, the degree pointed out on the circular head CD, when increased or diminished by the constant quantity, will give the angle of the wires which is sought. The semicircle on each side of a diameter drawn through B is divided into 180°, the 180th degree being at the opposite end of that diameter.

We may simplify still further the method of reading off the angle AOC, so as to save the trouble even of recollecting the constant quantity, and of adding it to, or subtracting it from, the arch pointed out by the index of the vernier. This advantage is obtained by making the index of the vernier point to the constant quantity upon the part of the scale below B, fig. 12, when the points D, B coincide, or when the wire CD is in the dotted position eB; for it is obvious that if z marks the zero of the scale, and Bz is equal to the constant quantity, the arch Dz, which is pointed out by the index of the vernier, will be equal to \( \frac{b + n}{2} \), or the angle AOC. In like manner, in fig. 13, where the wires do not cross each other within the field, and where Bz is the constant quantity, the arch Dz marked Micrometers out by the index of the vernier is obviously equal to \( \frac{b - n}{2} \), or the angle AOC, which the wires tend to form at O.

By means of this adjustment, therefore, we are able to read off the angle AOC with the same facility as if the wires intersected each other in the centre of the field, when the arches are accurate measures of the angles at the centre.

The end of the eye-tube is represented in Plate CCCLVIII, fig. 17, where the circular head CD is divided into 360°, and subdivided by the vernier V. L is the level, and AB the part of the eye-piece which contains the diaphragm, with the fixed and moveable wires. The head CD, and the level L, are firmly fixed to the eye-tube T; and from the head CD there rises an annular shoulder, concentric with the tube, and containing the diaphragm, across which the fixed wire is stretched. This diaphragm, which is represented in fig 18, with the wire extended across, projects through the circle of brass EF. All these parts remain immoveable, while the outer tube AB, and the other half EF, of the circular head which contains the vernier V, have a rotatory motion upon the shoulder, which rises from CD. The tube AB is merely an outer case, to protect a little tube within it, which contains the eye-glass, and the moveable diaphragm, with its wire extended across it. The enclosed tube is screwed into the ring EF, and the outer tube is also screwed upon the same ring, so that by moving AB, a motion of rotation is communicated to the vernier V, and to the diaphragm and wire belonging to the inner tube, while the rest of the eye-piece, containing the other diaphragm with its wire, remains stationary. By these means the moveable wire is made to form every possible angle with the fixed wire, and the angle is determined by the method which we have already explained. The fixed wire is placed a good deal out of the centre of the diaphragm to which it belongs; and the diaphragm itself is placed in a cell, in which it can be turned round so as to adjust the wire to a horizontal line when the level is set. The moveable wire is likewise placed at a distance from the centre of its diaphragm, as shown in Plate CCCLVIII, fig. 19; but, by means of screws which pass through the inner tube into the edge of this diaphragm, it can be moved in a plane at right angles to the axis of the eye-piece, so that the moveable wire may be placed either in the centre of the field or at different distances from it.

The double-image micrometer affords a convenient method of measuring angles of position; and we believe this image micrometer was first made by Sir David Brewster.

In every instrument in which a double image of an object is formed by means of two semi-lenses with their centres at a distance, the one image appears to have a rotatory motion round the other when the telescope is turned about its axis. Thus, in Plate CCCLVIII, fig. 20, if A, B be images of two objects formed by the upper semi-lens when the common diameter of the semi-lenses is perpendicular to the horizon, and C, D the images of the same objects formed by the lower semi-lens; then, by turning the telescope about its axis, or the semi-lenses round in their tube, the image A will appear to move round C in the circle AaE, and the image B round D in the circle BaF, or in the opposite direction if the telescope, or the semi-lenses, are turned the other way. When the distance AC is equal to CD, as in fig. 21, the image A will pass over the image D; or if the telescope is turned in the opposite direction, the image B will pass over the image C. In like manner, when AC is greater or less than CD, the images will move as we have represented them in fig. 20 and 22. In all these cases the four images may be brought into one straight line; and when this takes place, the line which passes through all the images will uniformly form the same angle with the horizon, as the common diameter of the semi-lenses. It is very easy to ascertain with the utmost accuracy when the images form one straight line; but particularly in the case where AC, fig. 21, is equal to CD, for the image of A will then pass over D; and the coincidence of the images will mark the instant when the line which joins them is parallel to the common diameter of the lenses. Hence, as we obtain by this means the relation of the line joining the images to a fixed line in the instrument, the relation of this line to the horizon may be easily found by means of a level, and a divided circular head. If the image is a straight line (fig. 22), then the coincidence of the two images, so as to form one straight line, will indicate the parallelism of that line to the diameter of the semi-lenses.

In constructing a micrometer of this kind solely for the purpose of measuring angles when the eye is not at their vertex, either the object-glass or the third eye-glass might be made the divided lens. If the object-glass is divided, it should be so constructed that it may have a rotatory motion in its cell, by applying the hand to a milled circumference AB, Plate CCCLVIII. fig. 23. Connected with the tube TT of the telescope is a circular ring of brass CD divided into 360°; and the divisions upon this scale are pointed out by the index of a vernier v, which moves along with the semi-lenses. A level L is fixed to the plate AB, having its axis parallel to the common diameter of the lenses, and being adjusted to a horizontal line when the index points to the zero of the scale. In using the instrument, therefore, the observer turns round the semi-lenses by means of the projecting milled circumference AB, till the coincidence of the two images is distinctly perceived. The index of the vernier will then point out upon the graduated head the inclination of the line which is required.

When the telescope is long, this form of the instrument, though extremely simple, is not very convenient. The construction represented in Plate CCCLVIII. fig. 24, is in general to be preferred. This instrument consists of three tubes BL, LC, CA. At the extremity B of the first tube is placed the divided object-glass, and at the other extremity L is fixed the divided circular head EF. The tube CL, which remains always at rest, is fixed to the stand HI by means of the clasp and screw at H. The tube AC, which contains an eye-piece, moves within both the tubes CL and LB. The tube BL extends towards C within the tube CL, and round its circumference are cut a number of teeth, in which the endless screw G works, and thus gives a rotatory motion to the tube LB and the divided head EF. By this means the common diameter of the semi-lenses at B is made to form every possible angle with a horizontal line, which is indicated by a level above L, having its axis parallel to the common diameter of the semi-lenses. The index of the vernier v, fixed to the stationary tube CL, points out on the graduated head the angle required. When the instrument is constructed so that the semi-lenses are in the eye-piece, the graduated head must be placed in the eye-tube.

If the principle of this micrometer is applied to the double-image micrometer described in Chap. IV., in which a pair of fixed semi-lenses moves between the object-glass and its principal focus, we obtain an instrument which will measure at the same observation the angle subtended at the eye of the observer by a line joining two points, and likewise the angle which that line forms with the horizon or any other line. When the two images of the line which joins the two points are brought into contact by the motion of the semi-lenses along the axis of the tube, these images must necessarily be in the same straight line, so that the relation of that line to the horizon, or to any other given line, and the contact of the two images of it which determines its angular magnitude, are obtained simultaneously, without any additional observation or adjustment.

Dr Pearson has given some very useful details respecting the use of double-image micrometers in measuring angles of position.

When the brightness of the two stars which compose a double star is considerable, their discs may be drawn out into lines of light either by cylindrical refraction or reflection, and the coincidence of these lines will furnish the means of determining the angle of position.

A position micrometer upon a new principle has been proposed by Sir David Brewster. He expands the images of the two stars into luminous discs till they overlap each other. The southern limb of the lower disc is then made to move along the fixed wire of the position micrometer. A line joining the two points where the circumferences of the discs intersect each other, is obviously perpendicular to a line joining the centres of the stars, and will therefore form an angle with the fixed wire equal to the complement of the angle of position required. If we therefore make the moveable wire pass through the two points where the luminous discs intersect each other, the micrometer scale will give the complement of the angle of position.

**CHAP. VII.—DESCRIPTION OF THE LAMP MICROMETER AND THE LUCID-DISC MICROMETER.**

In measuring the distances and angle of position of double stars, Sir William Herschel encountered many practical difficulties which interfered with the accuracy of his results. The uncertainty of the real zero of his scale, the inflexion of light, the imperfections of the screws and divided bars and pinions, and the difficulty of obtaining fibres sufficiently minute for his purpose, but, above all, the disappearance of small stars by the illumination of the wire, led this eminent astronomer to the contrivance of his lamp micrometer, which, while it is exempt from these sources of error, has also the advantage of a large scale.

The lampmicrometer is represented in Plate CCCLVIII. fig. 25, where AB is the upright part of the stand, 9 feet high, upon which a wooden semicircle ghogg, 14 inches radius, may be fixed at different heights, by a peg p put into holes in the stand. An arm L, 30 inches long, moves round a pivot in the centre of the board by means of the handle P, which works by a string echo fastened to a hook at the back of L. The arm L is kept at any inclination to the horizon by the weight of the handle P, which is 10 feet long. A small slider b, about 3 inches long, moves along the front of L towards and from the centre at n, by means of the handle rD, which operates by a string passing over the pulley m, and returning by n to a barrel at r, while a second string bow, with a weight w, causes the slider b to return to the centre. The end of the arm L is shown on a large scale in fig. 26.

Two lamps a, b, figs. 27 and 28, 1½ inch high and 1½ inch deep, having sliding doors with small apertures made with a fine needle opposite the flame of a single cotton-thread wick, so that when the sliders are shut down, nothing is seen but two fine lucid points like stars of the third or fourth magnitude. The lamp a is placed at the centre u, so that its lucid point may occupy that centre, while b is hung on the slider S, so that its lucid point may be in a line with the lucid point of a when the arm L has a horizontal position.

A person, therefore, at a distance of ten feet, may govern In using this micrometer, Sir W. Herschel placed it ten feet from his left eye, while with his right he viewed a double star through his Newtonian reflector. By means of his left eye the double star is seen projected upon the micrometer, and he then placed his two lucid points at such a distance that they were exactly covered by the stars. The distance of the lucid points was the tangent of the magnified angles subtended by the stars to a radius of ten feet. This angle, therefore, being divided by the magnifying power of the telescope, gives the real angular distance of the centres of a double star. With a power of 460 the scale was a quarter of an inch for every second. Sir W. Herschel, in measuring the apparent diameter of a Lyra with this instrument, used a magnifying power of 6450. The magnified angle was $38^\circ 10''$, so that the real angle was $\frac{38^\circ 10''}{6450} = 0^\circ 355$, giving on this occasion a scale of no less than $8\frac{1}{2}$ inches to a second.

In using high magnifying powers, Sir W. Herschel employed another apparatus called the lucid-disc micrometer. A lucid disc made of oiled paper, or any other semi-transparent substance, was placed in the front slider of a lantern, and illuminated by a flame behind it. The lantern was then removed to a distance till the diameter of the disc appeared equal to that of a planet seen in the telescope, so that the angular diameter of the planet became known by dividing the angle subtended by the disc by the magnifying power of the telescope. The result was affected by the colour of the disc and the degree of illumination. The measure was always too small when the illumination was strong; and too great when a black disc was placed on an illuminated ground. Hence Sir William Herschel took a mean of the two as the true measure.

M. Schroeter of Lilienthal, in measuring the diameters of the four new planets, used a lucid-disc micrometer, which we presume was not much different from Sir W. Herschel's.

Dr Pearson constructed and used an analogous instrument, in which the left eye looked into a tube containing a system of lines upon a disc of glass, or a spider's line micrometer, so that the object seen with the right eye was projected against these divisions, and its angular diameter ascertained.

**CHAP. VIII.—DESCRIPTION OF FIXED MICROMETERS WITH AN INVARIABLE SCALE.**

The earliest fixed micrometer of which we have a distinct account is that of Huygens, who placed a circular diaphragm in the focus of the eye-glass of his telescope, and found its angular value to be seventeen and a quarter minutes by the time which a star took to pass across it. He then formed two or three long brass plates of different breadths so as to form wedges of different angles. In measuring the diameter of a planet with one of these, he made one of them slide through two slits in the opposite sides of the tube, so that the plane of the brass wedges touched the plane of the circular diaphragm; and having observed in what part of the wedge its breadth just covered the whole planet, he took this breadth in a pair of compasses, and having found what part it was of the whole aperture, he divided seventeen and a quarter minutes by this part, and obtained the diameter required. Sir Isaac Newton has stated, that the measures thus taken are all fixed micrometers in excess. Had Huygens used long wedge-shaped micrometers, slits or openings, he would by the same process have obtained measures which erred in defect, and the mean between these two measures would have given the true diameter of the planet.

The reticulum or fixed micrometer, composed of wires, Cassini's was invented and used by Cassini. It is shown in the annexed figure, where $ab$, $cd$, $ef$, $gh$, are four hairs or wires intersecting each other at right angles at $i$, in the focus of the eye-glass, $ab$, $cd$ being inclined $45^\circ$ to $ef$ and $gh$. In order to find with this apparatus the differences of the right ascension and declination of two stars, direct the telescope so that the first or preceding star may appear upon the wire $ab$, and turn round the tube till that star moves along $ab$. The time when this star reaches $i$ is carefully noted, and also the time when the following star reaches the wire $ed$. The interval between these times, converted into degrees, is the difference of right ascension required. To find the difference of their declinations, mark the time of the second or following star's arrival at the points $k$ and $l$ of the oblique wires $ef$, $gh$. The half of the interval between these times is the time in which the star describes the space $lm$ or $mk$, which, converted into degrees $a$, is the angular distance $lm$, which, multiplied by the cosine of the declination of the known or preceding star, gives an approximate difference, which, when applied to the declination of the known star, gives the approximate declination of the unknown or following star. If we now multiply the angular value of $lm$ by the cosine of this approximate declination, we shall have the correct difference of declination, which, applied to the declination of the known star, will give the true declination of the unknown star.

In using the above reticulum, Dr Bradley found his observations embarrassed by the crossing of all the wires at $i$, which hid the preceding star at the very point where it was required to be most distinctly seen. He therefore proposed the construction shown in the annexed figure, where the wires $hg$, $hi$, intersect one another perpendicularly at $f$, the centre of the rings. Two slender bars of brass $pg$, $og$ are fixed to the ring $abc$, and inclined each to the diameter $hg$ $26^\circ 34'$, half the angle of a rhomb whose greater diagonal is double of the lesser. Hence $fh$ and $fi$ will be each one half of $fg$, $ki = fy$, and $fm = nl = th$. To avoid the inconvenience of turning the telescope about its axis, Dr Bradley placed the ring $abc$, which carries the wires or brass bars in a groove in the fixed ring $ABC$, and, confining it laterally by three pieces of brass at $A$, $B$, and $C$, he employed an endless screw DEF to work in a toothed rack $de$, fixed to the inner ring $abc$. Let us now suppose the telescope so directed that a star is at $f$, and moving in any line $fg$; then, by turning the milled head D, the wire $fh$ will move round $f$ till it touches the star at $g$, and will then lie in the direction of the star's motion, while all other stars will move parallel to it. The mode of obtaining accurate results from this apparatus will be un-

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1 Phil. Trans. vol. Ixxii. p. 192. 2 Introd. Pract. Astron. vol. ii. p. 245. Fixed Micrometer.

Fixed Micrometer understood from the following description of an analogous contrivance described by Lacaille.

This eminent astronomer used a rhomb FMIL, the diagonals of which, FI, ML, must be exactly perpendicular to one another, and as two to one. The length FI was 15½ lines, the angle subtended by ML 1° 25' 55", and consequently FI 2° 50' 10", as determined by the passage of several of the equatorial stars. Now, as the vertical height of each triangle in the rhomb is equal to the short diagonal, the path of any star passing through the field in a line parallel to that diagonal will always cut off a similar triangle, and the distance of this path from the common apex will have the same ratio to the vertical height of the large triangle, one half FI, or ML, that the value of the diagonal in time has to the observed time of the passage. Hence the time which any star takes to pass from m to l, reduced to degrees, and multiplied by the cosine of the declination, is the difference of declination required. The difference of right ascension is obtained as before. With this simple apparatus, Lacaille observed the comparative right ascensions and declinations of the 1942 stars which are given in his Catalogus Australis Stelliferum. Lacaille constructed another reticulum, in which ML was one fourth of FI.

Mr Wollaston employed a reticulum similar to that shown in the annexed figure, AB being the horary line, and CD the equatorial line. The four squares being within the field, any of them may be used separately, so that observations made successively in a pair of them will check each other as well as the principal observation made in the large right-angled triangle. As the diagonals are equal to one another, they afford a larger passage of a star, and thus increase the accuracy of the observation.

M. Valz's reticule.

M. Valz of Nismes has proposed the reticulum shown in fig. 18, as possessing several advantages over others. It consists of three wires AB, CD, AD; and the equatorial one MN perpendicular to AB, CD. The arches AC, CM, MD, DB, &c. are all 60°. When the stars pass parallel to the equator, the angle formed at the vertex A or B will be 30°, and its cotangent = \(\sqrt{3}\) = 1:732; therefore, calling \(t\) the line of passage of one star from the first to the second wire, and \(e\) that of the other star, the difference of declination will be 1:732 (\(t - e\)), and the difference of right ascension will be had from the times of the stars passing the middle of the space between CD and AB, or the means of the times of passing CD and AB. Baron Zach observes, that this reticulum requires no illumination, that the values of its lines do not require to be known, and that it may be used out of exact adjustment to the parallel of declination, as the corrections for such want of adjustment are easily computed.

The fixed net-micrometer of Fraunhofer is shown in Plate CCCLVIII. fig. 29. The vertical and oblique lines shown in the figure are cut upon glass with fluoric acid or diamond, and the plate has a circular motion in its cell. These fixed lines are illuminated by light passing through the eye-tube micrometer and falling on their cut edges, so that no light passes down the tube. The lines parallel to \(e\) are adjusted perpendicular to a circle of declination, and in that position the oblique light of the lamp illuminates both the vertical and inclined lines. The mutual distances both of the vertical and inclined lines are known from the machine by which they were drawn, and hence the ratio of the times of transit of a star from the vertical to the inclined lines will enable us to determine the position of these in reference to a circle of declination. "The great number of lines," says Dr Pearson, "afford the means of making several observations, which, on an average, will give right ascensions and declinations equally exact, whether the differences of declination be great or small. When the difference of right ascension is small, as in the case of double stars, the transit of both stars cannot well be observed over the same individual line, but one of them may be observed at the first, and the other at the second line, alternately, till the observer is satisfied with his observation; and should the network experience an alteration of position from any cause, it will in all probability be detected before the computation is commenced. The ingenious artist contrived an engine by which he could cut straight parallel lines at distances so small as \(1/1000\) of units from each other, and to be crossed by other parallel lines at any given angle of declination; and in the net-micrometer he formed the parallel lines at such a distance from each other that the inclined intervals bear the same proportion to the vertical intervals that the cosine of the angle of inclination bears to radius, so that about as many transits will take place over the inclined as over the vertical lines. Five lines only are drawn at equal distances from each other, and the sixth line, including the fifth interval, is cut at the distance of one interstice and a half; yet the whole value of any number of intervals may always be known, provided it be noticed how many of the larger kind are included in the whole number. When the cell containing the disc is attached to a revolving graduated circle, the position of a line uniting two stars may also be measured, by first adjusting zero to the equatorial position of the line \(fg\), and then turning the divided disc round till all its parallels successively receive both the stars at the same instant as they pass through the field, which contemporary ingresses may be effected by repeated adjustments and subsequent trials, when the telescope is mounted on an equatorial stand properly rectified.

Another micrometer of Fraunhofer's is shown in Plate CCCLVIII. fig. 30, and consists of concentric circles cut upon glass, and illuminated like the preceding one. The inner and smallest circle is like a dot. Other five circles are seen in a telescope of five feet focal length, magnifying 110 times. With a power of 63, eight circles are visible; and with the lowest power of 45, eleven circular lines are visible. The observer may choose any circle he likes for observing the passage of the star, avoiding those in which it would pass near the centre or near the circumference. The observation, too, may be made with more rings than one. The following are the dimensions of the circles in one of these instruments:

| Diameter in Paris Inches | Diameter in Paris Inches | |--------------------------|--------------------------| | Circle 1. | 0·0038 | | | Circle 7. | | | 0·4126 | | 2. | 0·0243 | | | 8. | | | 0·5264 | | 3. | 0·0840 | | | 9. | | | 0·6388 | | 4. | 0·1678 | | | 10. | | | 0·7178 | | 5. | 0·2513 | | | 11. | | | 0·8012 | | 6. | 0·3590 |

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1 Zach's Correspondance Astronomique, tom. i. p. 353. 2 Introd. Pract. Astron. vol. ii. p. 143. By subtracting these numbers from one another, we obtain the breadths of the spaces between the rings.

Fraunhofer's suspended annular micrometer, which is a more perfect instrument than the preceding, is shown in Plate CCCLVIII. fig. 31, which represents a disc of glass, having a hole in its centre a little larger than the inner diameter of a metallic ring, so turned truly in a lathe. This ring is cemented on the glass disc, and when placed in the field of view of a telescope, the observer notes the instants when a star on the limb of a planet enters and emerges from each side of the ring. The only data required for computing the difference of right ascension and declination, are the times that an equatorial star takes to pass along the internal and external diameters of the ring. The passage of stars whose relative position is required, must be observed at a distance from the centre, as well as from the upper edge of the ring. The formulae of Bessel for using this and other circular micrometers will be found in Zach's Monatliche Correspondenz, vol. xvii. xxiv. and xxvi.

A simple and useful fixed micrometer, proposed by Mr. Cavallo, is a divided slip of thin mother-of-pearl stretching across the diaphragm of the telescope, and finely divided into 200ths of an inch. It is shown in Plate CCCLVIII. fig. 32, crossing the diaphragm of the telescope. The value of the division may be ascertained either by measuring a bar, or by the passage of an equatorial star across the field of view.

In a portable telescope this micrometer is very convenient, though it has the great disadvantage of shutting out the field of view. In telescopes upon stands it will measure only angles in one direction, unless there is a contrivance to turn it round. This micrometer has also the additional imperfection, that its divisions, from being unequally distant from the eye-glass, are not all seen with the same distinctness.

To remove these imperfections, Sir David Brewster proposed, in 1805, the circular mother-of-pearl micrometer, which is shown in Plate CCCLVIII. fig. 33, where the black ring which forms part of the figure is the diaphragm of the telescope, and the more luminous part is an annular portion of mother-of-pearl divided on its inner circumference into 360°. This divided circumference can be placed exactly in the focus of the eye-glass, and all its parts seen with perfect distinctness. In order to understand the use of the instrument, let ApB be the inner edge of the mother-of-pearl ring, and mn the object to be measured. Bisect the arch mn in p, and draw Cm, Cp, Cn, and we shall have AB : mn = rad. : sin. \(\frac{mpn}{2}\), and \(mpn = \sin.\)

\[\frac{1}{2} mpn \times AB,\] a formula by which the angle subtended by the chord of any number of degrees may be readily found. The first part of the formula is constant, while AB varies with the magnifying power employed.

**CHAP. IX.—DESCRIPTION OF MICROMETERS FOR MICROSCOPES.**

All the micrometers above described may be adapted to compound microscopes, where the eye-glass has a considerable focal length. A good micrometer, however, for single microscopes, which can be used with facility, and at the same time give accurate results, is still a desideratum. When the single lens is so minute, or when the first lens of a microscopic doublet or triplet almost touches the surface of the object, it is an extremely difficult matter to introduce any scale, or any minute body of known dimensions, with which the object may be compared. In some cases, when the object to be measured is minute, the seed of the *lycoperdon bovista* or *puff-ball* might be introduced, its diameter being about the \(\frac{1}{3500}\)th part of an inch; and when the object is less minute, the seed of *lycopodium* may be used, its diameter being \(\frac{1}{940}\)th of an inch. We may advantageously adopt, in some cases, the method of Dr. Jurin, who introduced into the field small pieces of silver or brass wire, whose diameter he had previously ascertained by coiling the wire round a cylinder, and observing how many breadths of the wire were contained in a given number of inches.

This method of introducing a substance of known dimensions may be carried much farther. We may use all the variety of hairs and wool which have a known diameter; and for this purpose Dr Young's tables of substances measured by the eriometer will be of great use. The following are a few of them:

| Substance | Diameter in Parts of an Inch | |----------------------------|------------------------------| | Lycoperdon bovista, seed of | 8500th of an inch. | | Smut of barley | 4600th ditto. | | Silk, fibre of (average) | 2500th ditto. | | Human blood, particles of (Bauer) | 2500th² ditto. | | Mole's fur | 1875th ditto. | | Goat's wool | 1575th ditto. | | Saxon wool | 1320th ditto. | | Farina of Laurestinus | 1100th ditto. | | Seed of lycopodium | 940th ditto. |

The distance of the fibres of the crystalline lens of fishes may also be advantageously used, and also the distance of the teeth which unite the fibres. For this purpose, the lens must be well dried, and perfectly hard, so that with a sharp knife we can detach minute portions of any of the lamina. The thinnest should be used; and as the fibres always taper to the pole, and the teeth become smaller in proportion as the fibres diminish, we must determine the distance of the fibres, and also those of the teeth, at both ends of the lamina, by the method described by Sir David Brewster in the Philosophical Transactions for 1833, p. 324. The larger lined scales of moths and butterflies may also be used, especially as we can measure the distance of the lines by the coloured spectra which these lines produce. These operations will require much dexterity on the part of the observer, and they are recommended only to those who cannot succeed in their measurements by other methods.

An excellent method of measuring microscopic objects is to project the image of the object against a divided scale, at a given distance from the eye. The scale must be seen either by the same eye which is looking into the microscope, or by the other eye. In the first case, the rays from the microscope will enter one side of the pupil, and the rays from the divided scale the other side; the aperture through which we look at the scale, and the aperture of the microscope, being at a distance less than the diameter of the pupil. When the right eye looks at the divided scale, the left, which looks into the microscope, will see the object projected against the scale, although it has no vision of the scale itself. This second method may be carried into effect in two ways. The scale may form no part

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1 Physico-Mathematical Dissertations, p. 45. 2 We consider this the best measure. 3 The late Mr. Pond has observed, that the pale, slender, double-headed scales of the *Poa* or *Pieris brassica*, which taper to a point, and terminate in a brush-like appendage, are of an invariable length, about \(\frac{1}{3500}\)th of an inch. of the instrument, and may be viewed by the naked eye; or it may form part of the instrument, like a binocular telescope, the left eye looking into one tube, viz. the microscope, while the right eye looks into another tube, in which a divided scale is magnified by an eye-lens.

Dr Wollaston has constructed and used a very ingenious micrometer on the first of the principles above mentioned, viz. when the object and the scale are viewed by the same eye; but its use is limited to microscopes with small lenses. When the lenses are larger, we have adopted another method, namely, to perforate the lens with a small hole in or near the centre, or, if it is thought better, near the margin of the lens. A slit extending from the margin of the lens may often be executed more easily.

The following is Dr Wollaston's own description of this instrument:

"This instrument," says Dr Wollaston, "is furnished with a single lens of about \( \frac{1}{2} \) th of an inch focal length. The aperture of each lens is necessarily small, so that when it is mounted on a plate of brass, a small perforation can be made by the side of it in the brass, as near to its centre as \( \frac{1}{2} \) th of an inch.

"When a lens thus mounted is placed before the eye for the purpose of examining any small object, the pupil is of sufficient magnitude for seeing distant objects at the same time through the adjacent perforation, so that the apparent dimensions of the magnified image might be compared with a scale of inches, feet, and yards, according to the distance at which it might be convenient to place it.

"A scale of smaller dimensions, attached to the instrument, will, however, be found preferable, on account of the steadiness with which the comparison may be made; and it may be seen with sufficient distinctness by the naked eye, without any effort of nice adaptation, by reason of the smallness of the hole through which it is viewed.

"The construction that I have chosen for the scale is represented in Plate CCCLVIII, fig. 33. It is composed of small wires about \( \frac{1}{2} \) th of an inch in diameter, placed side by side so as to form a scale of equal parts, which may be with ease counted by means of a certain regular variation of the lengths of the wires.

"The external appearance of the whole instrument is that of a common telescope consisting of three tubes. The scale occupies the place of the object-glass, and the little lens is situated at the smaller end, with a pair of plain glasses sliding before it, between which the subject of examination is to be included. This part of the apparatus is shown separately in fig. 35. It has a projection, with a perforation, through which a pin is inserted to connect it with a screw, represented at b, fig. 34. This screw gives lateral motion to the object, so as to make it correspond with any particular part of the scale. The lens has also a small motion of adjustment, by means of the cap c, fig. 36, which renders the view of the magnified object distinct.

"Before the instrument is completed, it is necessary to determine with precision the indications of the scale, which must be different, according to the distance to which the tube is drawn out. In my instrument, one division of the scale corresponds to \( \frac{1}{1000} \) th of an inch when it is at the distance of 16-6 inches from the lens; and since the apparent magnitude in small angles varies in the simple inverse ratio of this distance, each division of the same scale will correspond to \( \frac{1}{1000} \) th at the distance of 8-7 inches; and the intermediate fractions \( \frac{1}{1000}, \frac{1}{1000}, \ldots \) &c. are found by intervals of 1-66 inch, marked on the outside of the tube. The basis on which these indications were founded in this instrument was a wire, carefully ascertained to be \( \frac{1}{1000} \) th of an inch in diameter, the magnified image of which occupied fifty divisions of the scale when it was at the distance of 16-6 inches; and hence one division \( = \frac{1}{50 \times 200} = \frac{1}{10000} \). Since any error in the original estimate of this wire must pervade all subsequent measures derived from it, the substance employed was pure gold drawn till fifty-two inches in length weighed exactly five grains. If we assume the specific gravity of gold to be 19-36, a cylindrical inch will weigh 3837 grains; and we may hence infer the diameter of such a wire to be \( \frac{1}{1000} \) th of an inch, more nearly than can be ascertained by any other method.

"For the sake of rendering the scale more accurate, a similar method was, in fact, pursued with several gold wires of different sizes, weighed with equal care; and the subdivisions of the exterior scale were made to correspond with the average of their indications.

"In making use of this micrometer for taking the measure of any object, it would be sufficient, at any one accidental position of the tube, to note the number on the outside as denominator, and to observe the number of divisions and decimal parts which the subject of examination occupies on the interior scale as numerator of a fraction, expressing its dimensions in proportional parts of an inch; but it is preferable to obtain an integer as numerator, by sliding the tube inward or outward, till the image of the wire is seen to correspond with some exact number of divisions, not only for the sake of greater simplicity in the arithmetical computations, but because we can by the eye judge more correctly of actual coincidence than of the comparative magnitudes of adjacent intervals. The smallest quantity which the graduations of this instrument profess to measure, is less than the eye can really appreciate in sliding the tube inward or outward. If, for instance, the object measured be really \( \frac{1}{1000} \), it may appear \( \frac{1}{1000} \) or \( \frac{1}{999} \), in which case the doubt amounts to \( \frac{1}{999} \) th part of the whole quantity. But the difference is here exceedingly small in comparison to the extreme division of other instruments, where the nominal effect of its power is the same.

"A micrometer with a divided eye-glass may profess to measure as far as \( \frac{1}{1000} \) of an inch; but the next division is \( \frac{1}{1000} \) or \( \frac{1}{999} \); and though the eye may be able to distinguish that the truth lies between the two, it receives no assistance within one-half part of the larger measure."

The micrometer microscopes used for reading off the divisions on the graduated limb of astronomical instruments differ in no respect from the eye-pieces of telescopes fitted up with micrometers.

Notwithstanding the value of the methods described above, the want of a simple micrometer for microscopes of high power is felt by every person who has been practically occupied with this class of researches; and we cannot give a better proof of this than by adducing in support of our opinion the different measures that have been given by able and ingenious observers of the size of the particles of the human blood.

Dr Thomas Young..................1-6060th part of an inch. Dr Wollaston........................1-5000th ditto. MM. Prevost and Dumas.............1-4076th ditto. Captain Kater........................1-4000th ditto. M. Ehrenberg.......................1-3600th ditto. Messrs Hodgkin and Lister.........1-3000th ditto.

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1 Phil. Trans. 1813, p. 119. 2 In measuring the size of the fossil infusoria recently discovered by himself, M. Ehrenberg assumes a globule of human blood to be \( \frac{1}{1000} \) th of a line in diameter, or \( \frac{1}{1000} \) th of an inch, but of what inch is not mentioned. He does not state whether this mea- Sir David Brewster..............1-2556th part of an inch. Dr Jurin'.........................1-1940th ditto. Mr Bauer's best observation.....1-2500th ditto. next best.......................1-2000th ditto. worst observation..............1-1000th ditto.

The three measures of 1000, 2000, and 2500, have been recently given by Mr Bauer himself, as the different steps which he made towards what he conceives the best measure, viz. 1-2500th, which he obtained repeatedly with an improved achromatic microscope. As Dr Young obtained his measure eriometrically, namely, by measuring the diameter of the first red ring produced by looking through the blood at a luminous object, we cannot conceive it possible that he could have committed such a mistake as to make the diameter of that ring nearly thrice as great as it should be, according to Mr Bauer's results, or more than thrice as great as the concurring measures obtained by Jurin and Leeuwenhoek. The only explanation we can give is, that the particles of the blood must have an organized structure, or consist of portions separated by lines which have the magnitude assigned by Dr Young. In order to submit this explanation to the test of experiment, Sir David Brewster examined the particles of blood a few minutes after it was drawn, when dried by natural evaporation on a plate of glass. Each particle he found to consist of a dark rim, within which is a bright circle, then a darkish central spot, which spot in some globules may be resolved into a dark ring, a bright ring within this, and then a small central black spot. Here, then, is the cause of Dr Young's mistake. The red ring of light which he measured in the eriometer was not that which was due to the globule as a whole, but to the parts of the globule. Being anxious to obtain more complete evidence of this fact, we placed lycopodium powder beside the globule of blood, and found that the diameter of the globules was to that of the lycopodium seed as 5 to 18. We then compared the diameter of the red ring produced by the seed with the diameter of the red ring produced by division on steel, in which there were 1250 to the inch, as executed for us by the late Sir John Barton, and found the diameter of the seed to be the 697th of an inch. We compared it also with the ring produced by divisions of which there were 625 to the inch, and found its diameter the 717th part of an inch. The mean of these two is the 710th part of an inch, which, increased in the ratio of 5 to 18, gives the 2556th part of an inch as the measure of the diameter of the globules of blood, agreeing almost exactly with the recent measure of Mr Bauer.