cut by a diamond into two parts, which, when they were fixed in the same plane, refracted all the rays which passed through them equally; but one of them remaining fixed, and the other moving on a centre at a, according to the dotted line, would refract that portion of the rays which passed through it more than those which passed through the fixed part, and, being placed in the focus of the object-glass of a telescope, two images were formed of each object, by which its diameter could be measured. An index, and divided sector of a circle, served to measure the comparative refractions. The cross-hair micrometer, as described,' observes Mr. Watt, leaving me too much in the power of my assistants, where the distances were greater than permitted me to read off the number of chains on the rod myself, I thought of another about 1772, which consisted of a telescope with an object glass of a long focus, viz. three or four feet; this was placed in a tube with a slit in one side of it, nearly as long as the focus of the telescope, and the object-glass being fitted to a short tube, which slid from end to end of the slit, could be moved backwards and forwards by means of a piece of metal fixed to the short tube, and coming out through the slit; a glass of six to nine inches focus was all fixed in the outer tube, of the nature of what is called a field-glass, and to this was added an eye-glass, with a cross hair piece in its focus. 'Now it is evident that, if the object-glass be moved nearer the field-glass, their common focus will be shortened, and the image at the crosshairs diminished proportionally, until the glasses come into contact, when their common focus will be shorter than that of the field-glass alone; and two indexes fixed upon a rod being subtended by the cross hair at any given distance, the same rod with its indexes being removed nearer the observer, upon sliding the object-glass nearer the eye, they may again be subtended by the crosshairs, and a scale on the side of the tube will show the comparative distance they have been removed; and, the distance of the first object being known, that of the second will also be so. This scale could not, however, be a scale of equal parts, but one which could easily be laid down.' The common divided object-glass micrometer consists of two semi-lenses AB, fig. 3, of the same focal length, formed by dividing a convex lens into two equal parts, by a plane which passes through its axis. The centres of these semi-lenses are made to separate and approach each other, by means of a screw or pinion along the line A B; and the distance of their centres is measured upon a scale subdivided by a vernier, or, as in the wire micrometers, by a graduated head fixed upon the screw. If it be required to measure the angle subtended by two objects, M, N, the semi-lenses are separated till the two images of these objects are in contact, or till the image of M, formed by the semilens A, appears to be in contact with the image of N, formed by the semilens B. When this happens, the angle subtended by the objects is equal to the angle subtended by Å B, the distance of the centres of the semilenses at the point F, or the focus of the lenses where the contact of the image takes place. It is manifest that an image of M will be formed in the line A F, and at F the focus of rays diverging from M. In like manner an image of N will be formed in the line B F, and at F the focus of rays proceeding from the radiant point N. Hence it is obvious that the angle subtended at F, by M N, is the same as the angle subtended by A B at F. The angle AF B may be easily found trigonometrically, the sides A B and OF being known; but, as this angle is generally very small, it may, without any perceptible error, be considered as proportional to the subtense AB, or the distance between the centres of the semilenses. By determining, therefore, experimentally, the angle which corresponds to any distance A B of the semilenses, we may by simple proportion find the angle for any other dis tance. The new divided object glass micrometer contrived by Dr. Brewster consists of an achromatic object glass LL, fig. 4, having two semilenses A B, represented in fig. 5, moveable between it and its principal focus f. These semilenses are completely fixed, so that their centres are invariably at the same distance; but the angle subtended by the two images which they form is varied by giving them a motion along the axis Of of the lens LL. When the semilenses are close to L L the two images are much separated, and form a great angle; but, as the lenses are moved towards f, the centres of the images gradually approach each other, and the angle which they form is constantly increasing. By ascertaining, therefore, experimentally, the angle formed by the centres of the images, when the semilenses are placed close to L L, and also the angle which they subtend when the semilenses are at f, the other extremity of the scale, we have an instrument which will measure with the utmost accuracy all intermediate angles. In constructing this micrometer, for astronomical purposes, the semilenses may be made to move only along a portion of the axis Of, particularly if the instrument be intended to measure the diameters of the sun and moon, or any series of angles within given limits. By increasing the focal length of the semilenses, or by diminishing the distance between the centres, the angles may be made to vary with any degree of slowness, and of course each unit of the scale will correspond to a very small portion of the whole angle. The accuracy and magnitude of the scale, indeed, may be increased without limit; but it is completely unnecessary to carry this any farther than till the error of the scale is less than the probable error of observation. Let us now examine the theory of this micrometer, and endeavour to ascertain the nature of the scale for measuring the variations of the angle. For this purpose let LL, fig. 4, be the objectglass which forms an inverted image, mn, of the object M N, and let the semilenses A B, having their centres at an invariable distance, be interposed between the object-glass and its principal focus, in such a manner that their centres are equidistant from the axis Of. Now it is obvious that the size of the image m n is propor tional to the size of the object MN; and, as the angle subtended by M N depends upon its size, the magnitude of the image m n may, in the case of small angles, be assumed as a measure of the angle subtended by MN. As the rays which proceed from the point M are all converged to m, by means of the lens L L, the ray bA, which passes through the centre of the semilens A, must of course have the direction bm; and, as it suffers no refraction in passing through the centre of A, it will proceed in the same direction bAm, after emerging from the semilens, and will cross the axis at F. For the same reason the ray c B, proceeding from N, and passing through the centre of B, will cross the axis at F as it advances to n. If the distance of F from A and B happens to be equal to the focal length of the lenses A and B, when combined with LL, distinct images of M and N will be formed at F, and they will appear to touch one another; and the line mn, being the size of the image that would have been formed by the lens LL alone, will be a measure of the angle subtended by the points M N. If the point F, where the lines Am, Bn, cross the axis, should not happen to coincide with the focus of the lenses A, B, when combined with LL, then let this focus be at F', nearer A and B than F. Draw the lines A F′m, BF'n, then it is obvious that, if the angle subtended by M N were enlarged so as to be represented by n'm', instead of nm, or so that the lens LL alone would form an image of it equal to n'm', the point of intersection F would coincide with the focus F; so that in every position of the lenses A, B, with respect to LL, the points M N may always be made to subtend such an angle that, when they are placed before the telescope, the points FF will coincide, and consequently the mages of the points MN will be distinctly formed at F', and will be in contact. Whenever this happens the space nm will be a measure of the angle thus subtended by MN. Hence it follows that, whatever be the position of the semilenses A B, on the axis Of, the rays bA, c B, which pass through the centres of the semilenses, will cross the axis at some point F, corresponding with the focus of rays diverging from M N, and will mark out the size of the image n'm', and consequently the relative magnitude of the angle subtended by the two points M, N. From the equality of the vertical angles A F' B, n' F'm', and the parallelism of the lines A B, n' m', we shall have n' m' : AB=ƒF': GF', and calling ƒF'b, and considering that GF' F being the focal length of the semi Fb F + b lenses, we have n'm': ABb: Fb F+b n m2'AB+ and consequently A B x b F 2 X 3 10 n' m2 = 2 + = 2·6; from which it appears that, when bis in arithmetical progression, the angle n'm' varies at the same rate; and consequently the scale, which measures the variations of the angle subtended by the centres of the two images, is a scale of equal parts. This instrument undergoes a very singular change when constructed, as in fig. 6, so that the semilenses are outermost and immoveable, while another lens, LL, is made to move along the axis Gf. In this case a double image is formed as before, but the angle subtended by the centres of the images never suffers any change during the motion of the lens L L along the axis of the telescope. If the two images are in contact when the lens L L is close to the semilenses, they will continue in contact in every other position of LL; but the magnitude of the images is constantly increasing during the motion of L L towards f, the principal focus of the semilenses. The reason of this remarkable property will be understood from fig. 6, where M N are two objects placed at such an angle that the rays, passing through the centres A B of the semilenses, cross the axis at F, and will consequently be in contact. If the lens L L is removed to the position L'L', the rays M m, N n, which are incident upon it at the points m and n, having the same degree of convergency as before, will be refracted to F', the focus of the combined lenses for rays diverging from MN. Two distinct images of the object will therefore be formed at F', and these images will still be in contact. In like manner it may be shown, that whatever be the position of the lens LL between G and f, the rays Mƒ, Nf, will cross the axis at a point coinciding with the focus of the combined lenses, and will there form two images always in contact. Hence it follows that, though the magnifying power of the instrument is constantly changing with the position of the lens L L, yet the angle subtended by the centres of the two images never suffers the least variation. The circular micrometer is an instrument which has been, for many years past, much used on the continent, and is still held in high and deserved estimation there by astronomers of the first rank. From the simplicity of its construction, and the facility with which it may be used in any position of the telescope, it is frequently preferred, even in public and national observatories, to micrometers of a more complex nature, which require to be adjusted to the equatorial motion of the star. Moreover there is no necessity for its being illuminated; on which account it is peculiarly adapted to the observation of comets and small stars; and it is, indeed, to these two classes of the heavenly bodies that its application is now principally confined; although some astronomers of great eminence have considered that it is capable of equal accuracy to the wire Now, calling A B = 2, F = 10, and b = 1, 2, 3, micrometers. To voyagers and others it will successively, we shall obtain 2 x 1 n' m' = 2 + =2.2 10 2 x2 n' m2 = 2+ 2.4 10 prove of considerable advantage. The smallness of its size (not being much larger than a shilling) renders it very convenient for carriage; and the simplicity of its construction prevents any liability to injury. Astronomy is indeed more indebted to this little instrument than is perhaps generally known. Three out of the four new planets which have been discovered in the present century were discovered with very small telescopes, and their motions watched, and the elements of their orbits deduced from observations made with the circular micrometer only; whereby astronomers were enabled to look out for them with certainty at the time of their re-appearance at the next oppositions. And at the present day it is almost the only instrument used on the continent for the observation of comets. To the labors of Olbers, Bessel, and Frauenhofer, we are indebted for the high reputation which this instrument has attained. To the two former for the talent which they have displayed in explaining the analysis by which the observations may be reduced; and to the latter for the recent improvements which he has introduced in the construction of the instrument. M. Frauenhofer's micrometers are formed by means of very fine lines cut on glass with a diamond point in a peculiar manner, and placed in the focus of the telescope. One of these micrometers consists of concentric circular lines, drawn at unequal distances from each other; and the other consists of straight lines crossing each other at a given angle. The mode of cutting these lines has furnished M. Frauenhofer with a method of illuminating them, which, at the same time that it renders the lines visible, leaves the other part of the field of the telescope in darkness; so that the transits of the smallest stars may be observed by means of these micrometers, the lines appearing like so many silver threads suspended in the heavens. The circular mother-of-pearl micrometer, which Dr. Brewster has often used, both in determining small angles in the heavens and such as are subtended by terrestrial objects, is repre sented in fig. 7, which exhibits its appearance in the focus of the first eye-glass. The black ring which forms part of the figure is the diaphragm, and the remaining part is an annular portion of mother-of-pearl, having its interior circumference divided into 360 equal parts. The mother-ofpearl ring, which appears connected with the diaphragm, is completely separate from it, and is fixed at the end of a brass tube, which is made to move between the third eye-glass and the diaphragm, so that the divided circumference may be placed exactly in the focus of the glass next the eye. When the micrometer is thus fitted into the telescope, the angle subtended by the whole field of view, or by the diameter of the innermost circle of the micrometer, must be determined either by measuring a base, or by the passage of an equatorial star; and the angles subtended by any number of divisions or degrees will be found by a table constructed in the following manner : Let Amp n b, fig. 8, be the interior circumference of the micrometer scale, and let m n be the object to be measured. Bisect the arch m n in P, and draw Cm Cp Cn. The line Cp will be at right angles to mn, and therefore mn will be Consetwice the sine of half the arch m p n. quently A B m n Rad. Sine of mpn; therefore m n × R= sin. m pn x A B, and sin.m pn R -x A B, a formula by mp n = which the angle subtended by the chord of any number of degrees may be easily found. The first part of the formula, viz. sin.mpn is R constant, while A B varies with the size of the micrometer, and with the magnifying power which is applied. The following table, therefore, has been computed, which contains the value of the constant part of the formula for every degree or division of the scale: In order to find the angle subtended by any number of degrees we have only to multiply the constant part of the formula corresponding to that number in the table by A B, or the angle subtended by the whole field. Thus, if A B is 30', the angle subtended by 1° of the scale will be 30' x 0087 153", and the angle subtended by 40° will be 30′ x 342 = 10′ 15′′.6; and by making the calculation it will be found that, as the angle to be measured increases, the accuracy of the scale also increases; for, when the arch is only 1° or 2°, a variation of 1° produces a variation of about 16" in the angle; whereas, when the arch is between 170° and 180°, the variation of a degree does not produce a change of much more than 1" in the angle. The single-lens micrometer, contrived by Dr. Wollaston, must now be adverted to. Having had occasion to measure some very small wires, with a greater degree of accuracy than he was enabled to do by any instrument hitherto made use of for such purposes, he was led to contrive other means that might more effectually answer the end proposed. The instrument to which Dr. Wollaston had recourse is furnished with a sin ⚫8039 143 gle lens of about one-twelfth of an inch focal length. The aperture of such a lens is necessarily small; so that, when it is mounted in a plate of brass, a small perforation can be made by the side of it in the brass as near to its centre as one-twenty-fifth 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, or 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 compa ̈son may be made; and it may be seen with sufficient distinctness by the naked eye, with out any effort of nice adaptation, by reason of the smallness of the hole through which it is viewed. The construction that Dr. Wollaston chose for the scale is represented in fig. 9. It is composed of smal wires, about one-fiftieth of an inch in diameter, placed side by side, so as to form a scale of |