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

1. The first TELESCOPIC micrometers were only mechanical contrivances for measuring the image of an object in the focus of the object-glas. Before these contrivances were thought of, astronomers were accustomed to measure the field of view in each of their telescopes, by observing how much of the moon they could see through it, the semidiameter being reckoned at 15 or 16 minutes; and other distances were estimated by the eye, comparing them with the field of view. Mr Galcoigne, an English gentleman, however, fell upon a much more accurate method before the year 1641, and had a Treatise on Optics prepared for the press; but he was killed during the civil wars in the service of Charles I, and his manuscript was never found. His instrument, however, fell into the hands of Mr Towne*, who says, that by the help of it he could mark above 40,000 divisions in a foot.

2. Mr Galcoigne's instrument being shown to Dr Hooke, he gave a drawing and description of it, and proposed several improvements†. Mr Galcoigne divided the image of an object in the focus of the object-glas, by the approach of two pieces of metal ground to a very fine edge, in the place of which Dr Hooke would substitute two fine hairs stretched parallel to one another.

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

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

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

6. M. de la Hire says, that there is no method more Dela Hire's simple or commodious for observing the digits of an micrometric-eclipse than a net in the focus of the telescope. These, he says, were generally made of filken threads; and that for this particular purpose fix concentric circles had also been made use of, drawn upon oiled paper; but he advises to draw the circles on very thin pieces of glass with the point of a diamond. He also gives several particular directions to assist persons in the use of them.

7. Construction of Different Micrometers. The first we Common shall describe is the common micrometer. Let ABCD be a section of the telescope at the principal focus of the object-glas, or where the wires are situated, which are placed in a short tube containing the eye-glas, and may be turned into any position by turning that tube; mn is a fine wire extended over its centre; vw, xy, are two parallel wires well defined, and perpendicular to mn; vw is fixed, and xy moves parallel to it by means of a screw, which carries two indexes over a graduated plate, to show the number of revolutions and parts of a revolution which it makes. Now to measure any angle, we must first ascertain the number of revolutions and parts of a revolution corresponding to some known angle, which may be thus done: 1st, Bring the inner edges of the wires exactly to coincide, and set each index to 0; turn the screw, and separate the wires to any distance; and observe the time a star m is in passing along the wire mn from one vertical wire to the other: for that time, turned into minutes and seconds of a degree, will be the angle answering to the number of revolutions, or the angle corresponding to the distance. Thus, if d=col. of the star's declination, we have 15'dm, the angle corresponding to this distance; and hence, by proportion, we find the angle answering to any other. 2dly, Set up an object of a known diameter, or two objects at a given distance, and turn the screw till the vertical wires become tangents to the object, or till their opening just takes in the distance of the two objects upon the wire mn; then from the diameter, or distance of the two objects from each other, and their distance from the glas, calculate the angle, and observe the number of revolutions and parts corresponding. 3dly, Take the diameter of the sun on any day, by making the wires tangents to the opposite limbs, and find, from the nautical almanac, his diameter on that day. Here it will be best to take the upper and lower limbs of the sun when on the meridian, as he has then no motion perpendicular to the horizon. If the edges do not coincide when the indexes stand at 0, we must allow for the error. Instead of making a proportion, it is better to have a table calculated to show the angle correspond- ing to every revolution and parts of a revolution. But the observer must remember, that when the micrometer is fixed to telescopes of different focal lengths, a new table must be made. The whole system of wires is turned about in its own plane, by turning the eyetube round with the hand, and by that means the wire \( m n \) can be thrown into any position, and consequently angles in any position may be measured. Dr Bradley added a small motion by a rack and pinion to set the wires more accurately in any position.

8. But the micrometer, as now contrived, is of use, not only to find the angular distance of bodies in the field of view at the same time, but also of those which, when the telescope is fixed, pass through the field of view successively; by which means we can find the difference of their right ascensions and declinations. Let \( A a, B b, C c \), be three parallel and equidistant wires, the middle one bisecting the field of view; HOR a fixed wire perpendicular to them passing through the centre of the field; and \( F f, G g \), two wires parallel to it, each moveable by a micrometer forew, as before, so that they can be brought up to HOR, or a little beyond. Then to find the angular distance of two objects, bring them very near to \( B b \), and in a line parallel to it, by turning about the wires, and bring one upon HOR, and by the micrometer screw make \( F f \) or \( G g \) pass through the other; then turn the screw till that wire coincides with HOR, and the arc which the index has passed over shows their angular distance. If the objects be further remote than you can carry the distance of one of the wires \( F f, G g \) from HOR, then bring one object to \( F f \) and the other to \( G g \); and turn each micrometer screw till they meet, and the sum of the arcs passed over by each index gives their angular distance. If the objects be two stars, and one of them be made to run along HOR, or either of the moveable wires as occasion may require, the motion of the other will be parallel to these wires, and their difference of declinations may be observed with great exactness; but in taking any other distances, the motion of the stars being oblique to them, it is not quite so easy to get them parallel to \( B b \); because if one star be brought near, and the eye be applied to the other to adjust the wires to it, the former star will have gotten a little away from the wire. Dr Bradley, in his account of the use of this micrometer, published by Dr Mackeyne in the Philosophical Transactions for 1772, thinks the best way is to move the eye backwards and forwards as quick as possible; but it seems to be best to fix the eye at some point between, by which means it takes in both at once sufficiently well defined to compare them with \( B b \). In finding the difference of declinations, if both bodies do not come into the field of view at the same time, make one run along the wire HOR, as before, and fix the telescope and wait till the other comes in, and then adjust one of the moveable wires to it, and bring it up to HOR, and the index gives the difference of their declinations. The difference of time between the passage of the star at either of the cross moveable wires, and the transit of the other star over the cross fixed wire (which represents a meridian), turned into degrees and minutes, will give the difference of right ascension. The star has been here supposed to be bifected by the wire; but if the wire be a tangent to it, allowance must be made for the breadth of the wire, provided the adjustment be made for the coincidence of the wires. In observing the diameters of the sun, moon, or planets, it may perhaps be most convenient to make use of the outer edges of the wires, because they appear most distinct when quite within the limb; but if there should be any sensible inflection of the rays of light in passing by the wires, it will be best avoided by using the inner edge of one wire and the outward edge of the other; for by that means the inflection at both limbs will be the same way, and therefore there will be no alteration of the relative position of the rays passing by each wire. And it will be convenient in the micrometer to note at what division the index stands when the moveable wire coincides with HOR; for then you need not bring the wire when a star is upon it up to HOR, only reckon from the division at which the index then stands to the above division.

9. With a micrometer thus adapted to a telescope, Mr S. Savery of Exeter proposed a new way of measuring the difference between the greatest and least apparent diameters of the sun, although the whole of the sun was not visible in the field of view at once. The method we shall briefly describe. Place two object-glasses instead of one, so as to form two images whose limbs shall be at a small distance from each other; or instead of two perfect lenses, he proposed to cut a single lens into four parts of equal breadths by parallel lines, and to place the two segments with their straight sides against each other, or the two middle frustums with their opposite edges together; in either case, the two parts which before had a common centre and axis, have now their centres and axes separated, and consequently two images will be formed as before by two perfect lenses. Another method in reflectors was to cut the large concave reflector through the centre, and by a contrivance to turn up the outer edges whilst the straight ones remained fixed; by which means the axis of the two parts became inclined, and formed two images. Two images being formed in this manner, he proposed to measure the distance between the limbs when the diameters of the sun were the greatest and least, the difference of which would be the difference of the diameters required. Thus far we are indebted to Mr Savery for the idea of forming two images; and the admirable uses to which it was afterwards applied, we shall next proceed to describe.

10. The divided object-glass micrometer, as now made, was contrived by the late Mr John Dollond, and by him adapted to the object-end of a reflecting telescope, and has been since by the present Mr P. Dollond his son applied with equal advantage to the end of an achromatic telescope. The principle is this: The object-glass is divided into two segments in a line drawn through the centre; each segment is fixed in a separate frame of brass, which is moveable, so that the centres of the two segments may be brought together by a handle for that purpose, and thereby form one image of an object; but when separated they will form two images, lying in a line passing through the centre of each segment; and consequently the motion of each image will be parallel to that line, which can be thrown into any position by the contrivance of another handle to turn the glass about in its own plane. The brafs-work carries a vernier to measure the distance of the centres of the two segments. Now let E and H be the centres of the two segments, F their principal focus, and PQ two distant objects in FE, FH, produced, or the opposite limbs of the same object PBQD; then the images of P and Q, formed by each segment, or the images of the opposite limbs of the object PBQD, coincide at F; hence two images m x F, n x F of that object are formed, whose limbs are in contact; therefore the angular distance of the points P and Q is the same as the angle which the distance EH subtends at F, which, as the angles supposed to be measured are very small, will vary as EH extremely nearly; and consequently if the angle corresponding to one interval of the centres of the segments be known, the angle corresponding to any other will be found by proportion. Now to find the interval for some one angle, take the horizontal diameter of the sun on any day, by separating the images till the contrary limbs coincide, and read off by the vernier the interval of their centres, and look into the nautical almanac for the diameter of the sun on that day, and you have the corresponding angle. Or if greater exactness be required than from taking the angle in proportion to the distances of their centres, we may proceed thus:—Draw FG perpendicular to EH, which therefore bisects it; then one half EH, or EG, is the tangent of half the angle EFH; hence, half the distance of their centres is to the tangent of half the angle corresponding to that distance as half any other distance of the centres is to the tangent of half the corresponding angle (A).

11. From this the method of measuring small angles is manifest; for we consider P, Q either as two objects whose images are brought together by separating the two segments, or as the opposite limbs of one object PBQA, whose images, formed by the two segments E, H, touch at F; in the former case, EH gives the angular distance of the two objects; and in the latter, it gives the angle under which the diameter of the object appears. In order to find the angular distance of two objects, therefore, separate the segments till the two images which approach each other coincide; and to find the diameter of an object, separate the segments till the contrary limbs of the images touch each other, and read off the distance of the centres of the segment from the vernier (B), and find the angle as directed in the last article. Hence appears one great superiority in this above the wire micrometer; as, with the one any diameter of an object may be measured with the same ease and accuracy; whereas with the other we cannot with accuracy measure any diameter, except that which is at right angles to the direction of motion.

12. But, besides these two uses to which the instrument seems to well adapted, Dr Mafelyne * has shown, how it may be applied to find the difference of right ascensions and declinations. For this purpose, two wires at right angles to each other, bisecting the field of view, must be placed in the principal focus of the eye-glass, and moveable about in their own plane.—Let HCR c be the field of view, HR and C c the two wires; turn the wires till the westermolt star (which is the best, having further to move) run along ROH; Fig. 4. then separate the two segments, and turn about the micrometer till the two images of the same star lie in the wire C c; and then, partly by separating the segments, and partly by raising or depressing the telescope, bring the two innermost images of the two stars to appear and run along ROH, as a, b, and the vernier will give the difference of their declinations; because, as the two images of one of the stars coincided with C c, the image of each star was brought perpendicularly upon HR, or to HR in their proper meridian. And, for the same reason, the difference of their times of passing the wire CO c will give their difference of right ascensions. These operations will be facilitated, if the telescope be mounted on a polar axis. If two other wires KL, MN, parallel to C c, be placed near H and R, the observation may be made on two stars whose difference of meridians is nearly equal to HR the diameter of the field of view, by bringing the two images of one of the stars to coincide with one of these wires. If two stars be observed whose difference of declinations is well settled, the scale of the micrometer will be known.

13. It has hitherto been supposed, that the images of the two stars can be both brought into the field of view at once upon the wire HOR: but if they cannot, fet the micrometer to the difference of their declinations as nearly as you can, and make the image which comes first run along the wire HOR, by elevating or depressing the telescope; and when the other star comes in, if it do not also run along HOR, alter

(A) If the object is not distant let f be the principal focus; then \( \frac{f}{f'} : FG :: FG : FK \) (FG being produced to meet a line joining the apparent places of the two objects P, Q.), : dividendo, \( fG : FG :: GK : FK \), and alternando, \( fG : GK :: FG : FK :: \) (by similar triangles) \( EH : PQ \), hence \( \frac{EH}{fG} = \frac{PQ}{GK} \), therefore the angle subtended by EH at f= the angle subtended by PQ at G ; and consequently, as \( fG \) is constant, the angle measured at G is, in this case, also proportional to EH. The instrument is not adapted to measure the angular distance of bodies, one of which is near and the other at a distance, because their images would not be formed together.

(B) To determine if there be any error in the adjustment of the micrometer scale, measure the diameter of any small well-defined object, as Jupiter's equatorial diameter, or the longest axis of Saturn's ring, both ways, that is, with o on the vernier to the right and left of o on the scale, and half the difference is the error required. This error must be added to or subtracted from all observations, according as the diameter measured with o on the vernier, when advanced on the scale, is less or greater than the diameter measured the other way. And it is also evident, that half the sum of the diameters thus measured gives the true diameter of the object. the micrometer till it does, and half the sum of the numbers shown by the micrometer at the two separate observations of the two stars on the wire HOR will be the difference of their declinations. That this should be true, it is manifestly necessary that the two segments should recede equally in opposite directions; and this is effected by Mr Dollond in his new improvement of the object-glas micrometer.

14. The difference of right ascensions and declinations of Venus or Mercury in the sun's disk and the sun's limb may be thus found. Turn the wires so that the north limb n of the sun's image AB, or the north limb of the image V of the planet, may run along the wire RH, which therefore will then be parallel to the equator, and consequently Cc a secondary to it; then separate the segments, and turn about the micrometer till the two images Vv of the planet pass Cc at the same time, and then by separating the segments, bring the north limb of the northernmost image V of the planet to touch HR, at the time the northernmost limb n of the southernmost image AB of the sun touches it, and the micrometer shows the difference of declinations of the northernmost limbs of the planet and sun, for the reason formerly given (Art. 11.) we having brought the northernmost limbs of the two innermost images V and AB to HR, these two being manifestly interior to v and the northernmost limb N of the image PQ. In the same manner we take the difference of declinations of their southernmost limbs; and half the difference of the two measures (taking immediately one after another) is equal to the difference of the declinations of their centres, without any regard to the sun's or planet's diameters, or error of adjustment of the micrometer; for as it affects both equally, the difference is the same as if there were no error: and the difference of the times of the transits of the eastern or western limbs of the sun and planet over Cc gives the difference of their right ascensions.

15. Instead of the difference of right ascensions, the distance of the planet from the sun's limb, in lines parallel to the equator, may be more accurately observed thus: Separate the segments, and turn about the wires and micrometer, so as to make both images V, v, run along HR, or so that the two intersections I, T of the sun's image may pass Cc at the same time. Then bring the planet's and sun's limbs into contact, as at V, and do the same for the other limb of the sun, and half the difference gives the distance of the centre of the planet from the middle of the chord on the sun's disk parallel to the equator, or the difference of the right ascensions of their centres, allowing for the motion of the planet in the interval of the observations, without any regard to the error of adjustment, for the same reason as before. For if you take any point in the chord of a circle, half the difference of the two segments is manifestly the distance of the point from the middle of the chord; and as the planet runs along HR, the chord is parallel to the equator.

In like manner, the distances of their limbs may be measured in lines perpendicular to the equator, by bringing the micrometer into the position already described, (Art. 13.), and instead of bringing V to HR, separate the segments till the northernmost limbs coincide as at V; and in the same manner make their southernmost images to coincide, and half the difference of the two measures, allowing for the planet's motion, gives the difference of the declinations of their centres.

Hence the true place of a planet in the sun's disc may at any time of its transit be found; and consequently the nearest approach to the centre and the time of ecliptic conjunction may be deduced, although the middle should not be observed.

16. But however valuable the object-glas micrometer undoubtedly is, difficulties sometimes have been found in its use, owing to the alteration of the focus of the eye, which will cause it to give different measures of the same angle at different times. For instance, in measuring the sun's diameter, the axis of the pencil coming through the two segments from the contrary limbs of the sun, as PF, QF, fig. 3. crossing one another in the focus F under an angle equal to the sun's femidiameter, the union of the limbs cannot appear perfect, unless the eye be disposed to see objects distinctly at the place where the images are formed; for if the eye be disposed to see objects nearer to or further off than that place, in the latter case the limbs will appear separated, and in the former they will appear to lap over (c). This imperfection led Dr Mafkelyne to inquire, whether some method might not be found of producing two distinct images of the sun, or any other object, by bringing the axis of each pencil to coincide, or very nearly so, before the formation of the images, by which means the limbs when brought together would not be liable to appear separated from any alteration of the eye; and this he found would be effected by the refraction of two prisms, placed either without or within the telescope; and on this principle, placing the prisms within, he constructed a new micrometer, and had one executed by Mr Dollond, which upon trial answered as he expected. The construction is as follows.

17. Let AB be the object-glas; ab the image, sup-Dr Mafkelyne of the fun, which would have been formed in kelyne's the principal focus Q; but let the prisms PR, SR be placed to intercept the rays, and let EF, WG, be two rays proceeding from the eastern and western limbs of the fun, converging, after refraction at the lens, to a and b; and suppose the refraction of the prisms to be such, that in fig. 8. the ray EFR, after refraction at

(c) For if the eye can see distinctly an image at F, the pencils of rays, of which PF, QF are the two axes, diverging from F, are each brought to a focus on the retina at the same point; and therefore the two limbs appear to coincide: but if we increase the refractive power of the eye, then each pencil is brought to a focus, and they cross each other before the rays come to the retina, consequently the two limbs on the retina will lap over; and if we diminish the refractive power of the eye, then each pencil being brought to a focus beyond the retina, and not crossing till after they have passed it, the two limbs on the retina must be separated. Microme-ter. R by the prism PR, may proceed in the direction RO; and as all the rays which were proceeding to a suffer the same refraction at the prism, they will all be refracted to Q; and therefore, instead of an image a b, which would have been formed by the lens alone, an image Q c is formed by those rays which fall on the prism PR; and for the same reason, the rays falling on the prism SR will form an image Q d: and in fig. 9, the image of the point b is brought to Q, by the prism PR, and consequently an image Q d is formed by those rays which fall on PR: and for the same reason, an image Q c is formed by the rays falling on SR. Now in both cases, as the rays EFR, WGR, coming from the two opposite limbs of the sun, and forming the point of contact of the two limbs, proceed in the same direction RO, they must thus accompany each other through the eye-glas, and also through the eye, whatever refractive power it has, and therefore to every eye the images must appear to touch. Now the angle a R b is twice the refraction of the prism, and the angle a C b is the diameter of the sun; and as these angles are very small, and have the same subtense a b, we have the angle a R b : angle a C b :: CQ : RO.—Now as CQ is constant, and also the angle a R b being twice the refraction of the prism, the angle a C b varies as RO. Hence the extent of the scale for measuring angles becomes the focal length of the object-glas, and the angle measured is in proportion to the distance of the prisms from the principal focus of the object-glas; and the micrometer can measure all angles (very small ones excepted, for the reason given in Art. 19.) which do not exceed the sum of the refractions of the prisms; for the angle a C b, the diameter of the object to be measured, is always less than the angle a R b, the sum of the refractions of the prisms, except when the prisms touch the object-glas, and then they become equal. The scale can never be out of adjustment, as the point o, where the measurement begins, answers to the focus of the object-glas, which is a fixed point for all distant objects, and we have only to find the value of the scale answering to some known angle: for instance, bring the two limbs of the sun's images into contact, and measure the distance of the prisms from the focus, and look in the nautical almanac for the sun's diameter, and you get the value of the scale.

18. In fig. 8, the limb Q of the image Q c, is illuminated by the rays falling on the object-glas between A and F, and of the image Q d by those falling between B and G; but in fig. 9, the same limbs are illuminated by the rays falling between B and F, A and G respectively, and therefore will be more illuminated than in the other case; but the difference is not considerable in achromatic telescopes, on account of the great aperture of the object-glas compared with the distance FG.

It might be convenient to have two sets of prisms, one for measuring angles not exceeding 36', and therefore fit for measuring the diameters of the sun and moon, and the lucid parts and distances of the cusps in their eclipses; and another for measuring angles not much greater than r', for the convenience of measuring the diameters of the planets. For as Q c : QR :: sum of the refractions of the prisms : angle a C b, the apparent diameter of the object, it is evident that if you diminish the third term, you must increase the second in the same ratio, in order to measure the same Microme-angle; and thus by diminishing the refractive angle of the prisms, you throw them further from Q, and consequently avoid the inconvenience of bringing them near to Q, for the reason in the next paragraph; and at the same time you will increase the illumination in a small degree. The prisms must be achromatic, each composed of two prisms of flint and crown glas, placed with their refracting angles in contrary directions, otherwise the images will be coloured.

19. In the construction here described, the angle measured becomes evanescent when the prisms come to the principal focus of the object-glas, and therefore o on the scale then begins: but if the prisms be placed in the principal focus they can have no effect, because the pencil of rays at the junction of the prisms would then vanish, and therefore it is not practicable to bring the two images together to get o on the scale. Dr Maltkeyne, therefore, thought of placing another pair of prisms within, to refract the rays before they came to the other prisms, by which means the two images would be formed into one before they came to the principal focus, and therefore o on the scale could be determined. But to avoid the error arising from the multiplication of mediums, he, instead of adding another pair of prisms, divided the object-glas through its centre, and filling the segments a little it separated the images, and then by the prisms he could form one image very distinctly, and consequently could determine o on the scale; for by separating the two segments you form two images, and you will separate the two pencils so that you may move up the two prisms, and the two pencils will fall on each respectively, and the two images may be formed into one. In the instrument which Dr Maltkeyne had made, o on the scale was chosen to be about \( \frac{1}{2} \) of the focal length of the object-glas, and each prism refracted 27'. By this means all angles are measured down to o.

20. In the Philosophical Transactions for 1779, Mr Ramfden has described two new micrometers, which he contrived with a view of remedying the defects of the object-glas micrometer.

21. 1. One of these is a catoptric micrometer, which, beside the advantage it derives from the principle of reflection, of not being disturbed by the heterogeneity of light, avoids every defect of other micrometers, and can have no aberration, nor any defect arising 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; and the separation of the images being effected by the inclination of the two specula, and not depending on the focus of any lens or mirror, any alteration in the eye of an observer cannot affect the angle measured. It has peculiar to itself the advantages of an adjustment, to make the images coincide in a direction perpendicular to that of their motion; and also of measuring the diameter of a planet on both sides of the zero, which will appear no inconsiderable advantage to observers who know how much easier it is to ascertain the contact of the external edges of two images than their perfect coincidence.

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

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

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

24. Besides the table for reducing the revolutions and parts of the screw to minutes, seconds, &c. it will require a table for correcting a small error which arises from the eccentric motion of the half-mirrors. By this motion their centres of curvature will approach a little towards the large mirror: the equation for this purpose in small angles is insensible; but when angles to be measured exceed ten minutes, it should not be neglected. Or, the angle measured may be corrected by diminishing it in the proportion the versed sine of the angle measured, supposing the eccentricity radius, bears to the focal length of the small mirror.

25. Mr Ramfden preferred Cassegrain's construction of the reflecting telescope to either the Gregorian or Newtonian; because in the former, the errors of one speculum are corrected by those of the other. From a property of the reflecting telescope, not generally known, that the apertures of the two specula are to each other very nearly in the proportion of their focal lengths, it follows, that their aberrations will be in the same proportion; and these aberrations will be in the same direction, if the two specula are concave; or in contrary directions, if one speculum is concave and the other convex. In the Gregorian telescope, both specula being concave, the aberration at the second image will be the sum of the aberrations of the two mirrors; but in the Cassegrainian telescope, one mirror being concave and the other convex, the aberration at the second image will be the difference between the two aberrations. By alluming such proportions for the foci of the specula as are generally used in the reflecting telescope, which is about as 1 to 4, the aberration in the Cassegrainian construction will be to that in the Gregorian as 3 to 5.

26. The other is a dioptric micrometer, or one suited to the principle of refraction. This micrometer is applied to the erect eye-tube of a refracting telescope, and is placed in the conjugate focus of the first eyeglas; in which position, the image being considerably magnified before it comes to the micrometer, any imperfection in its glass will be magnified only by the remaining eye-glasses, which in any telescope seldom exceeds five or fix times. By this position also the size of the micrometer glass will not be the 1/100 part of the area which would be required if it was placed in the object-glas; and, notwithstanding this great disproportion of size, which is of great moment to the practical optician, the same extent of scale is preserved, and the images are uniformly bright in every part of the field of the telescope.

27. Fig. 12. represents the glasfs of a refracting telescope; w y, the principal pencil of rays from the object-ccxxxiv. glasfs O ; t t and u u, the axis of two oblique pencils; a, the first eye-glas; m, its conjugate focus, or the place of the micrometer; b the second eye-glas; c the third; and d the fourth, or that which is nearest the eye. Let p be the diameter of the object-glas, e the diameter of a pencil at m, and f the diameter of the pencil at the eye; it is evident, that the axes of the pencils from every part of the image will cross each other at the point m; and e, the width of the micrometer-glas, is to p the diameter of the object-glas, as m a is to g o, which is the proportion of the magnifying power at the point m; and the error caused by an imperfection in the micrometer-glas placed at m will be to the error, had the micrometer been at O, as m is to p.

28. Fig. 13. represents the micrometer; A, a convex or concave lens bifected by a plane across its centre; one of these semi lenses is fixed in a frame B, and the other in the frame E; which two frames slide on a plate H, and are pressed against it by thin plates a a: the frames B and E are moved in contrary directions by turning the button D: L is a scale of equal parts on the frame B; it is numbered from each end towards the middle with 10, 20, &c. There are two verniers on the frame E, one at M and the other at N, for the convenience of measuring the diameter of a planet, &c. on both sides the zero. The first division on both these verniers coincides at the same time with the two zeros on the scale, L; and, if the frame is moved towards the right, the relative motion of the two frames is shewn on the scale L by the vernier M; but if the frame B be moved towards the left, the relative motion is shewn by the vernier N.—This micrometer has a motion round the axis of vision, for the convenience of measuring the diameter of a planet, &c. in any direction, by turning Micrometering an endiefs screw F; and the inclination of the diameter measured with the horizon is shown on the circle g by a vernier on the plate V. The telescope may be adjusted to distinct vision by a screw, which moves the whole eye-tube with the micrometer nearer to or farther from the object-glass, as telescopes are generally made; or the same effect may be produced without moving the micrometer, by sliding the part of the eye tube m on the part n, by help of a screw or pinion.

29. Notwithstanding these improvements on micrometers, they are still liable to many sources of error. The imperfections of the wire micrometer, (which was till the most correct instrument for measuring small angles) when employed to determine the distance of close double stars, have been ably pointed out by Dr Herschel.

30. When two stars are taken between the parallel wires the diameters must be included. Dr Herschel* has in vain attempted to find lines sufficiently thin to extend them across the centres of the stars so that their thickness might be neglected. The threads of the silk-worm, with such lenses as he uses, are so much magnified that their diameter is more than that of many of the stars. Besides, if they were much smaller, the deflection of light would make the attempt to measure the distance of the centres this way fruitless; for he has always found the light of the stars to play upon those lines and separate their apparent diameters into two parts. Now since the spurious diameters of the stars thus included, are continually changing with the state of the air, and the length of time we look at them, we are, in some respect, left at an uncertainty; and our measures taken at different times, and with different degrees of attention, will vary on that account. Nor can we come at the true distance of the centres of any two stars, unless we know the semidiameters of the stars themselves; for different stars have different apparent diameters, which, with a power of 227, may differ from each other as far as two seconds (d).

31. The next imperfection arises from a deflection of light upon the wires when they approach very near to each other; for if this be owing to a power of repulsion lodged at the surface, it is easy to see that such powers must interfere with each other, and give the measures larger in proportion than they would have been if the repulsive power of one wire had not been opposed by a contrary power of the other wire.

32. Another disadvantage of these micrometers is an uncertainty of the real zero. The least alteration in the situation and quantity of light will affect the zero; and a change in the position of the wires will sometimes produce a difference. To remove this difficulty Dr Herschel always found his zero while the apparatus preserved the situation which it had when his observations were made; but this introduces an additional observation.

33. The next imperfection, is that every micrometer hitherto used requires either a screw, or a divided bar and pinion, to measure the distance of the wires or the two images. Those acquainted with works of this kind are sensible how difficult it is to have screws perfectly equal in every thread or revolution of each thread; or pinions and bars that shall be so evenly divided as to be depended upon in every leaf and tooth to the two or three thousandth part of an inch: and yet, on account of the small scale of those micrometers, these quantities are of the greatest consequence; an error of a single thousandth part inducing in most instruments a mistake of several seconds.

34. The greatest imperfection of all is, that the wires require to be illuminated; and when Dr Herschel had double flars to measure, one of which was very obscure, he was obliged to be content with less light than is necessary to make the wires distinct; and several stars on this account could not be measured at all, though not too close for the micrometer.

Dr Herschel, therefore, was led to direct his attention to the improvement of these instruments; and the result of his endeavours has been a very ingenious *lamp-micrometer*, which is not only free from the imperfections above specified, but also possesses the advantages of a large scale.

35. It is represented in fig. 14, where ABGCFE is a Dr Herschel's lamp nine feet high, upon which a semicircular board q h o g p is moveable upwards or downwards, and is held in its situation by a peg p put into any one of the holes of the upright piece A.B. This board is a segment of a circle of fourteen inches radius, and is about three inches broader than a semicircle, to give room for the handles r D, e P, to work. The use of this board is to carry an arm L, thirty inches long, which is made to move upon a pivot at the centre of the circle, by means of a string, which passes in a groove upon the edge of the semicircle p g o h q; the string is fastened to a hook at o (not expressed in the figure, being at the back of the arm L), and passing along the groove from o h to q is turned over a pulley at q, and goes down to a small barrel c, within the plane of the circular board, where a double-jointed handle e P commands its motion. By this contrivance, we see, the arm L may be lifted up to any altitude from the horizontal position to the perpendicular, or be suffered to descend by its own weight below the horizontal to the reverse perpendicular situation. The weight of the handle P is sufficient to keep the arm in any given position; but if the motion should be too easy, a friction spring applied to the barrel will moderate it at pleasure.

36. In front of the arm L a small filder, about three inches long, is moveable in a rabbet from the end L towards the centre backwards and forwards. A string is fastened to the left side of the little filder, and goes towards L, where it passes round a pulley at m, and returns under the arm from m, n, towards the centre, where it is led in a groove on the edge of the arm, which is of a circular form, upwards to a barrel (raised above the plane of the circular board) at r, to which the handle r D is fastened. A second string is fastened to the filder, at the right side, and goes towards the centre, where it passes over a pulley n; and the weight w, which is suspended by the end of this string, returns the filder towards the centre, when a contrary turn of the handle permits it to act.

37. By a and b are represented two small lamps, two inches high, 1 1/2 in breadth by 1 1/4 in depth. The sides, back, and top, are made so as to permit no light to be seen, and the front consists of a thin brass sliding door. The flame in the lamp a is placed three-tenths of an inch from

(d) These imperfections are remedied in the instrument described in p. 801. from the left side, three-tenths from the front, and half an inch from the bottom. In the lamp b it is placed at the same height and distance, measuring from the right side. The wick of the flame consists only of a single very thin lamp cotton-thread; for the smallest flame being sufficient, it is easier to keep it burning in so confined a place. In the top of each lamp must be a little slit lengthways, and a small opening in one side near the upper part, to permit the air to circulate to feed the flame. To prevent every reflection of light, the side opening of the lamp a should be to the right, and that of the lamp b to the left. In the sliding door of each lamp is made a small hole with the point of a very fine needle just opposite the place where the wicks are burning, so that when the sliders are shut down, and everything dark, nothing shall be seen but two fine lucid points of the size of two stars of the third or fourth magnitude. The lamp a is placed so that its lucid point may be in the centre of the circular board where it is fixed. The lamp b is hung to the little slider which moves in the rabbet of the arm, so that its lucid point, in an horizontal position of the arm, may be on a level with the lucid point in the centre. The moveable lamp is suspended upon a piece of brafs fastened to the slider by a pin exactly behind the flame, upon which it moves as a pivot. The lamp is balanced at the bottom by a leaden weight, so as to remain upright, when the arm is either lifted above or depressed below the horizontal position. The double-jointed handles r D, e P, consist of deal rods, 10 feet long, and the lowest of them may have divisions, marked upon it near the end P, expressing exactly the distance from the central lucid point in feet, inches, and tenths.

38. Hence we see, that a person at a distance of 10 feet may govern the two lucid points, so as to bring them into any required position south or north preceding or following from 0 to 90° by using the handle P, and also to any distance from six-tenths of an inch to five or fix and twenty inches by means of the handle D. If any reflection or appearance of light should be left from the top or sides of the lamps, a temporary screen, consisting of a long piece of pasteboard, or a wire frame covered with black cloth, of the length of the whole arm, and of any required breadth, with a slit of half an inch broad in the middle, may be affixed to the arm by four bent wires projecting an inch or two before the lamps, situated so that the moveable lucid point may pass along the opening left for that purpose.

Fig. 15. represents part of the arm L, half the real size; S the slider; m the pulley, over which the cord x t y z is returned towards the centre; v the other cord going to the pulley n of fig. 14. R the brafs piece moveable upon the pin e, to keep the lamp upright. At R is a wire riveted to the brafs piece, upon which Fig. 16. 17. is held the lamp by a nut and screw. Fig. 16. 17. represent the lamps a, l, with the sliding doors open, to show the situation of the wicks. W is the leaden weight with a hole d in it, through which the wire R of fig. 15. is to be passed when the lamp is to be fastened to the slider S. Fig. 18 represents the lamp a with the sliding door shut; l the lucid point; and i k the openings at the top, and s at the sides, for the admission of air.

39. The motions of this micrometer are capable of great improvement by the application of wheels and pinions, and other mechanical resources; but as the principal object is only to be able to adjust the two lucid points to the required position and distance, and to keep them there for a few minutes, while the observer measures their distance, it will be unnecessary to say more upon the subject.

40. It is well known that we can with one eye look into a telescope, and see an object much magnified, while the other eye may see a scale upon which the magnified picture is thrown. In this manner Dr Herschel generally determined the power of his telescopes; and any one who has been accustomed to make such observations will seldom mistake so much as one in fifty in determining the power of an instrument, and that degree of exactness is fully sufficient for the purpose.

41. When Dr Herschel uses this instrument he puts it at ten feet distance from the left eye, in a line perpendicular to the tube of his Newtonian telescope, and raises the moveable board to such a height that the lucid point of the central lamp may be upon a level with the eye. The handles, lifted up, are passed through two loops fastened to the tube, just by the observer, so as to be ready for his use. The end of the tube is cut away, so as to leave the left eye entirely free to see the whole micrometer.

42. The telescope being directed to a double star, it is viewed with the right eye, and at the same time with the left it is seen projected upon the micrometer: then, by the handle P, the arm is raised or depressed so as to bring the two lucid points to a similar situation with the two stars; and, by the handle D, the moveable lucid point is brought to the same distance of the two stars, so that the two lucid points may be exactly covered by the stars.

43. With a rule, divided into inches and fortieth parts, the distance of the lucid points is thus determined with the greatest accuracy; and the measure thus obtained is the tangent of the magnified angle under which the stars are seen to a radius of ten feet; therefore, the angle being found and divided by the power of the telescope, the real angular distance of the centres of a double star is ascertained. On September 25, 1781, Dr Herschel measured a Herculis with this instrument. Having caused the two lucid points to coincide with the stars, he found the radius or distance of the central lamp from the eye 10 feet 4.15 inches; the tangent or distance of the two lucid points 50.6 fortieth parts of an inch; this gives the magnified angle 3.5', and dividing by the power 460, we obtain 4" 3.4"' for the distance of the centres of the two stars. The scale of the micrometer at this convenient distance, with the power of 460, is above a quarter of an inch to a second; and by putting on a power of 932, we obtain a scale of more than half an inch to a second, without increasing the distance of the micrometer; whereas the most perfect micrometers, with the same instrument, had a scale of less than the two thousandth part of an inch to a second.

44. Mr Brewster has lately directed his attention to the improvement of micrometers, and has invented one in particular which appears to be highly deserving of notice in this place. In this instrument a pair of fixed wires is made to subtend different angles by varying the magnifying power of the telescope, by sliding one tube within another; whereas in all other micrometers with wires this effect is produced by mechanical contrivances.

Mr Brewster's method of fluttering and opening the wires optically is therefore free from all those sources of error to which other micrometers are subject, and renders it particularly useful to the practical astronomer; while the mode of changing the magnifying power by the motion of a second object-glas affords a length of scale equal to the local distance of the principal object-glas. The same principle is peculiarly applicable to the Gregorian telescope; for the magnifying power of this instrument can be changed by merely increasing or diminishing the distance of the eye-piece from the large speculum.

45. In the common micrometer, which can manifestly, as well as Mr Cavallo's and Mr Brewster's, be used in the measurement of distances, the focal length of the telescope to which it is attached remains always the same; so that a correction computed from an optical theorem must be applied to every angle that is measured: but in Mr Brewster's telescope and micrometer, the focal length varies in the same proportion as the distance of the object; and consequently no correction of the angles can be necessary. To obviate the necessity of having a stand for the instrument, which would prevent its usefulness at sea, Mr Brewster divides the second or moveable object-glas into two, as in the divided object-glas micrometer. By this contrivance two images are formed, and these images are separated or made to form different angles at the eye, by bringing the moveable object-glas nearer to the fixed one. In determining the angle, therefore, we have only to bring the two images of the object into contact; and such contact the eye is capable of ascertaining even during the agitation of a carriage, as the two images retain the same relative position whatever be their absolute motion.

This ingenious instrument, being formed with sliding tubes, is very portable and convenient; and will be found extremely useful to military gentlemen, and others who may wish to ascertain distances without a more cumbersome apparatus. Huxley's Nat. Phil. by Gregory, v. ii. p. 427.

46. Mr Brewster, we understand, still continues to direct his attention to the subject of micrometers, keeping in view the improvement of these instruments, not only in greater accuracy of construction, but also in their more extensive application to various practical purposes. An account of those uses and improvements is to form the subject of an appropriate publication; and, if we are rightly informed, the author deems them of sufficient importance to secure to himself, by patent, the exclusive right to the advantages which he thinks will arise from using them.

47. A very simple micrometer for measuring small angles with the telescope has been invented by Mr Cavallo*. It consists of a thin and narrow slip of mother-of-pearl finely divided, and situated in the focus of the eye-glas of a telescope, just where the image of the object is formed. It is immaterial whether the telescope be a refractor or a reflector, provided the eye-glas be a convex lens.

The simplest way of fixing it is to stick it upon the diaphragm, which generally stands within the tube, in the focus of the eye-glas. When thus fixed, the divisions of the micrometrical scale will appear very distinct, unless the diaphragm is not exactly in the focus; in which case, the scale must be placed accurately in the focus of the eye-glas, either by moving the diaphragm, or by interposing any thin substance, such as paper or card between it and the scale. This construction is fully sufficient, when the telescope is always to be used by the same person; but when different persons are to use it, then the diaphragm which supports the micrometer must be constructed so as to be easily moved backwards or forwards, though that motion need not be greater than about \( \frac{1}{8} \) or \( \frac{1}{16} \) of an inch.

The scale of the micrometer is represented in fig. 19. Fig. 19, which is about four times greater than one which Mr Cavallo has adapted to a three-feet achromatic telescope that magnifies about 84 times. It is something less than the 24th part of an inch broad; its thickness is equal to that of common writing paper; and the length of it is determined by the breadth of the field of view. The divisions are 200ths of an inch, and the lines which form them reach from one edge of the scale to about the middle of it, excepting every fifth and tenth division, which are longer. Two divisions of the scale in the telescope already mentioned are very nearly equal to one minute; and as a quarter of one of those divisions may be easily distinguished by the eye, an angle of one-eighth part of a minute, or of \( \frac{\pi}{32} \), may be measured with it.

In looking through a telescope furnished with such a micrometer, the field of view appears divided by the micrometer scale, the breadth of which occupies about \( \frac{1}{4} \)th of the aperture; and as the scale is semitransparent, that part of the object which is behind it may be discerned sufficiently well to ascertain the division, with which its borders coincide. Fig. 20. shows the appearance of the field of the telescope with the micrometer, when directed to the title page of the Philosophical Transactions, in which it appears that the thickness of the letter C is equal to three-fourths of a division, the diameter of the O is equal to three divisions, and so on.

48. After having adapted this micrometer to the telescope, we must then ascertain the value of the divisions. It is hardly necessary to mention in this place, that though those divisions measure the chords of the angles, and not the angles or arches themselves, and the chords are not as the arches, yet in small angles the chords, arches, sines, and tangents, follow the same proportion so very nearly, that the difference may be safely neglected: so that if one division of this micrometer is equal to one minute, we may conclude, that two divisions are equal to two minutes, three divisions to three minutes, and so on. In order to ascertain the value of the divisions of this micrometer, the following simple and accurate method may be adopted.

Mark upon a wall the length of six inches, by making two dots or lines fix inches asunder, or by fixing a fix-inch ruler upon a stand; then place the telescope before it so that the ruler or fix-inch length may be at right angles with the direction of the telescope, and just 57 feet 3\( \frac{1}{2} \) inches distant from the object-glas of the telescope: this done, look through the telescope at the ruler or other extension of fix inches, and observe how many divisions of the micrometer are equal to it, and that same number of divisions is equal to half a degree, or 30', as may be shewn by plane trigonometry.

49. When this value has been once ascertained, any other angle measured by any other number of divisions is determined by simple proportion. Thus, if the diameter of the sun seen through the same telescope, be equal to MIC

12 divisions, say as \( \frac{1}{11} \) divisions are to 30 minutes, so are 12 divisions to \( \left( \frac{12' \times 30'}{11.5} \right) = 31'3'' \), which is the required diameter of the fun.

Notwithstanding the facility of this calculation, a scale may be made answering to the divisions of a micrometer, which will show the angle corresponding to any number of divisions by mere inspection. Thus, for the above-mentioned small telescope, the scale is represented in fig. 21. AB is a line drawn at pleasure; it is then divided into 23 equal parts, and those divisions which represent the divisions of the micrometer that are equal to one degree, are marked on one side of it. The line then is divided again into 60 equal parts, which are marked on the other side of it; and these divisions represent the minutes which correspond to the divisions of the micrometer: thus the figure shows, that fix divisions of the micrometer are equal to 15\( \frac{1}{2} \) minutes, 11\( \frac{1}{4} \) divisions are nearly equal to 29 minutes, &c. What has been said of minutes may be said of seconds also, when the scale is to be applied to a large telescope.

50. We shall therefore add some practical rules to render this micrometer useful to persons unacquainted with trigonometry and the use of logarithms.

Problem I. The angle, not exceeding one degree, which is subtended by an extension of one foot perpendicular to the axis of the telescope being given, to find its distance from the object-glass of the telescope.

Rule 1. If the angle be expressed in minutes, say, as the given angle is to 60, so is 687.55 to a fourth proportional, which gives the answer in inches.—2. If the angle be expressed in seconds, say, as the given angle is to 3600, so is 687.55 to a fourth proportional, which expresses the answer in inches.

Example. At what distance is a globe of one foot diameter when it subtends an angle of two seconds?

\[ 2'' : 3600'' :: 687.55 : \frac{3600 \times 687.55}{2} = 1237590 \]

inches, or 103132\( \frac{1}{2} \) feet, which is the answer required.

Problem II. The angle, not exceeding one degree, which is subtended by any known extension, being given, to find its distance from the object-glass of the telescope.

Rule. Proceed as if the extension were of one foot by Problem I. and call the answer B; then, if the extension in question be expressed in inches, say, as 12 inches are to that extension, so is B to a fourth proportional, which is the answer in inches; but if the extension in question be expressed in feet, then you need only multiply it by B, and the product is the answer in inches.

Example. At what distance is a man fix feet high, when he appears to subtend an angle of 32''.

By Problem I. if the man were one foot high, the distance would be 82506 inches; but as he is fix feet high, therefore multiply 82506 by 6, and the product gives the required distance, which is 495036 inches, or 41233 feet.

For greater convenience, especially in travelling, or in such circumstances in which one has not the opportunity of making even the easy calculations required in those problems, the following two tables have been computed; the first of which shows the distance answering to any angle from one minute to one degree, which is subtended by a man, the height of which has been called an extension of fix feet; because, at a mean, such is the height of a man when dressed with hat and Microme. shoes on.

Thus, if it is required to measure the extension of a street, let a foot ruler be placed at the end of the street; measure the angle it subtends, which suppose to be 36', and in the table you will have the required distance opposite 36', which is 955 feet. Thus also a man who appears to be 49' high, is at the distance of 421 feet.

Angles subtended by an extension of one Foot at different Distances.

<table> <tr> <th rowspan="2">Angles.</th> <th colspan="2">Distances in Feet.</th> <th rowspan="2">Angles.</th> <th colspan="2">Distances in Feet.</th> </tr> <tr> <th>Min. 1</th> <th>Min. 31</th> <th>Min. 1</th> <th>Min. 31</th> </tr> <tr><td>1</td><td>3437.7</td><td></td><td>32</td><td>1074</td><td></td></tr> <tr><td>2</td><td>1718.9</td><td></td><td>33</td><td>1042</td><td></td></tr> <tr><td>3</td><td>1145.9</td><td></td><td>34</td><td>1011</td><td></td></tr> <tr><td>4</td><td>859.4</td><td></td><td>35</td><td>982</td><td></td></tr> <tr><td>5</td><td>687.5</td><td></td><td>36</td><td>955</td><td></td></tr> <tr><td>6</td><td>572.9</td><td></td><td>37</td><td>929</td><td></td></tr> <tr><td>7</td><td>491.1</td><td></td><td>38</td><td>904</td><td></td></tr> <tr><td>8</td><td>429.7</td><td></td><td>39</td><td>881</td><td></td></tr> <tr><td>9</td><td>382.0</td><td></td><td>40</td><td>859</td><td></td></tr> <tr><td>10</td><td>343.7</td><td></td><td>41</td><td>838</td><td></td></tr> <tr><td>11</td><td>312.5</td><td></td><td>42</td><td>818</td><td></td></tr> <tr><td>12</td><td>286.5</td><td></td><td>43</td><td>799</td><td></td></tr> <tr><td>13</td><td>264.4</td><td></td><td>44</td><td>781</td><td></td></tr> <tr><td>14</td><td>245.5</td><td></td><td>45</td><td>764</td><td></td></tr> <tr><td>15</td><td>229.2</td><td></td><td>46</td><td>747</td><td></td></tr> <tr><td>16</td><td>214.8</td><td></td><td>47</td><td>731</td><td></td></tr> <tr><td>17</td><td>202.2</td><td></td><td>48</td><td>716</td><td></td></tr> <tr><td>18</td><td>191.0</td><td></td><td>49</td><td>701</td><td></td></tr> <tr><td>19</td><td>180.9</td><td></td><td>50</td><td>687</td><td></td></tr> <tr><td>20</td><td>171.8</td><td></td><td>51</td><td>674</td><td></td></tr> <tr><td>21</td><td>162.7</td><td></td><td>52</td><td>661</td><td></td></tr> <tr><td>22</td><td>156.2</td><td></td><td>53</td><td>648</td><td></td></tr> <tr><td>23</td><td>149.4</td><td></td><td>54</td><td>636</td><td></td></tr> <tr><td>24</td><td>143.2</td><td></td><td>55</td><td>625</td><td></td></tr> <tr><td>25</td><td>137.5</td><td></td><td>56</td><td>614</td><td></td></tr> <tr><td>26</td><td>132.2</td><td></td><td>57</td><td>603</td><td></td></tr> <tr><td>27</td><td>127.3</td><td></td><td>58</td><td>592</td><td></td></tr> <tr><td>28</td><td>122.7</td><td></td><td>59</td><td>582</td><td></td></tr> <tr><td>29</td><td>118.5</td><td></td><td>60</td><td>583</td><td></td></tr> <tr><td>30</td><td>114.6</td><td></td><td></td><td></td><td></td></tr> </table>

Angles subtended by an Extension of six Feet at different Distances.

<table> <tr> <th rowspan="2">Angles.</th> <th colspan="2">Distances in Feet.</th> <th rowspan="2">Angles.</th> <th colspan="2">Distances in Feet.</th> </tr> <tr> <th>Min. 1</th> <th>Min. 14</th> <th>Min. 1</th> <th>Min. 14</th> </tr> <tr><td>1</td><td>20626.8</td><td></td><td>14</td><td>1473.3</td><td></td></tr> <tr><td>2</td><td>19313.</td><td></td><td>15</td><td>1375.</td><td></td></tr> <tr><td>3</td><td>1875.4</td><td></td><td>16</td><td>1298.1</td><td></td></tr> <tr><td>4</td><td>1556.5</td><td></td><td>17</td><td>1213.3</td><td></td></tr> <tr><td>5</td><td>1412.5</td><td></td><td>18</td><td>1145.9</td><td></td></tr> <tr><td>6</td><td>13437.7</td><td></td><td>19</td><td>1085.6</td><td></td></tr> <tr><td>7</td><td>2046.6</td><td></td><td>20</td><td>1031.4</td><td></td></tr> <tr><td>8</td><td>2578.2</td><td></td><td>21</td><td>982.2</td><td></td></tr> <tr><td>9</td><td>2291.8</td><td></td><td>22</td><td>937.6</td><td></td></tr> <tr><td>10</td><td>2062.6</td><td></td><td>23</td><td>896.8</td><td></td></tr> <tr><td>11</td><td>1875.2</td><td></td><td>24</td><td>859.4</td><td></td></tr> <tr><td>12</td><td>1718.8</td><td></td><td>25</td><td>825.</td><td></td></tr> <tr><td>13</td><td>1586.7</td><td></td><td>26</td><td>793.3</td><td></td></tr> </table> Angles. Distances in Feet. Angles. Distances in Feet.

<table> <tr> <th>Angles.</th> <th>Distances in Feet.</th> <th>Angles.</th> <th>Distances in Feet.</th> </tr> <tr> <td>Min. 27</td> <td>763.9</td> <td>Min. 44</td> <td>4688</td> </tr> <tr> <td>28</td> <td>736.6</td> <td>45</td> <td>4584</td> </tr> <tr> <td>29</td> <td>711.3</td> <td>46</td> <td>448.4</td> </tr> <tr> <td>30</td> <td>687.5</td> <td>47</td> <td>438.9</td> </tr> <tr> <td>31</td> <td>665.4</td> <td>48</td> <td>429.7</td> </tr> <tr> <td>32</td> <td>644.5</td> <td>49</td> <td>421.</td> </tr> <tr> <td>33</td> <td>625.</td> <td>50</td> <td>412.5</td> </tr> <tr> <td>34</td> <td>606.6</td> <td>51</td> <td>404.4</td> </tr> <tr> <td>35</td> <td>589.3</td> <td>52</td> <td>396.7</td> </tr> <tr> <td>36</td> <td>572.9</td> <td>53</td> <td>389.2</td> </tr> <tr> <td>37</td> <td>557.5</td> <td>54</td> <td>381.9</td> </tr> <tr> <td>38</td> <td>542.8</td> <td>55</td> <td>375.</td> </tr> <tr> <td>39</td> <td>528.9</td> <td>56</td> <td>368.3</td> </tr> <tr> <td>40</td> <td>515.6</td> <td>57</td> <td>361.9</td> </tr> <tr> <td>41</td> <td>503.1</td> <td>58</td> <td>355.6</td> </tr> <tr> <td>42</td> <td>491.1</td> <td>59</td> <td>349.6</td> </tr> <tr> <td>43</td> <td>479.7</td> <td>60</td> <td>343.7</td> </tr> </table>

Mr Brewster's circumvented by Mr Brewster, of the circumstances which led to the invention, and of its advantages. We shall give it in his own words*.

"In the winter of 1805 (he observes), when I was employed in delineating the surface of the moon, I wished to measure the diameter of the lunar spots by applying Mr Cavallo's micrometer to a thirty-inch achromatic telescope made by Berge. But as the eye-piece was moved by a rack and pinion, and consequently could not turn round its axis, the micrometer must have remained stationary, and could only measure angles in one direction. This difficulty, indeed, might have been surmounted by a mechanical contrivance for turning the diaphragm about its centre, or more simply by giving a motion of rotation to the tube which contains the third and fourth eye-glasses. Such a change in the eye-piece, however, was both inconvenient and difficult to be made. Mr Cavallo's micrometer, therefore, has this great disadvantage, that it cannot be used in reflecting telescopes, or in any achromatic telescope where the adjustment of the eye-piece is effected by rackwork, unless the structure of these instruments is altered for the purpose. Another disadvantage of this micrometer arises from the slip of mother-of-pearl passing through the centre of the field. The picture in the focus of the eye-glass is broken into two parts, and the view is rendered still more unpleasant by the inequality of the segments into which the field is divided. In addition to these disadvantages, the different divisions of the micrometer are at unequal distances from the eye-glass which views them, and therefore can neither appear equally distinct nor subtend equal angles at the eye.

"Finding that Mr Cavallo's instrument laboured under these imperfections, I thought of a circular mother-of-pearl micrometer which is free from them all, and has likewise the advantage of a kind of diagonal scale, increasing in accuracy with the angle to be measured. This micrometer, which I got executed by Miller and Adie, optical instrument-makers in Edinburgh, and which I have often used, both in determining small angles in the heavens and such as are subtended by terrestrial objects, is represented in fig. 27, which exhibits its appearances in the focus of the fourth eye-glass.

The black ring, which forms part of the figure, is the diaphragm, and the remaining part is a ring of mother-of-pearl, having its interior circumference divided into 360 equal parts. The mother-of-pearl ring, which appears connected with the diaphragm, is completely separate from it, and is fixed at the end of a brafs 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.

52. "Let A m p n B, fig. 28, 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 C m, C p, C n. The line C p will be at right angles to m n, and therefore m n will be twice the fine of half the arch m n. Consequently, AB : m n = rad. fine of \( \frac{1}{2} m p n \); therefore \( m n \times R = \sin \frac{1}{2} m p n \times AB \), and \( m p n = \frac{\sin \frac{1}{2} m p n \times AB}{R} = \frac{\sin \frac{1}{2} m p n}{R} \times AB \); a formula by which the angle subtended by the chord of any number of degrees may be easily found. The first part of the formula, viz., \( \frac{\sin \frac{1}{2} m p n}{R} \) is constant, while AB varies with the size of the micrometer and with the magnifying power which is applied. We have therefore computed the following table, containing the value of the constant part of the formula for every degree or division of the scale.

<table> <tr> <th>Deg.</th> <th>Constant Part of the Formula</th> <th>Deg.</th> <th>Constant Part</th> <th>Deg.</th> <th>Constant Part</th> <th>Deg.</th> <th>Constant Part</th> </tr> <tr> <td>1</td> <td>.087</td> <td>21</td> <td>.1822</td> <td>41</td> <td>.3502</td> <td>61</td> <td>.5075</td> </tr> <tr> <td>2</td> <td>.174</td> <td>22</td> <td>.1908</td> <td>42</td> <td>.3844</td> <td>62</td> <td>.5150</td> </tr> <tr> <td>3</td> <td>.262</td> <td>23</td> <td>.1994</td> <td>43</td> <td>.3665</td> <td>63</td> <td>.5225</td> </tr> <tr> <td>4</td> <td>.349</td> <td>24</td> <td>.2079</td> <td>44</td> <td>.3716</td> <td>64</td> <td>.5299</td> </tr> <tr> <td>5</td> <td>.436</td> <td>25</td> <td>.2164</td> <td>45</td> <td>.3877</td> <td>65</td> <td>.5373</td> </tr> <tr> <td>6</td> <td>.523</td> <td>26</td> <td>.2250</td> <td>46</td> <td>.3927</td> <td>66</td> <td>.5446</td> </tr> <tr> <td>7</td> <td>.610</td> <td>27</td> <td>.2334</td> <td>47</td> <td>.3987</td> <td>67</td> <td>.5519</td> </tr> <tr> <td>8</td> <td>.698</td> <td>28</td> <td>.2419</td> <td>48</td> <td>.4067</td> <td>68</td> <td>.5592</td> </tr> <tr> <td>9</td> <td>.785</td> <td>29</td> <td>.2504</td> <td>49</td> <td>.4147</td> <td>69</td> <td>.5664</td> </tr> <tr> <td>10</td> <td>.872</td> <td>30</td> <td>.2588</td> <td>50</td> <td>.4226</td> <td>70</td> <td>.5735</td> </tr> <tr> <td>11</td> <td>.959</td> <td>31</td> <td>.2672</td> <td>51</td> <td>.4305</td> <td>71</td> <td>.5807</td> </tr> <tr> <td>12</td> <td>1.045</td> <td>32</td> <td>.2756</td> <td>52</td> <td>.4384</td> <td>72</td> <td>.5878</td> </tr> <tr> <td>13</td> <td>1.132</td> <td>33</td> <td>.2840</td> <td>53</td> <td>.4462</td> <td>73</td> <td>.5948</td> </tr> <tr> <td>14</td> <td>1.219</td> <td>34</td> <td>.2923</td> <td>54</td> <td>.4540</td> <td>74</td> <td>.6018</td> </tr> <tr> <td>15</td> <td>1.305</td> <td>35</td> <td>.3007</td> <td>55</td> <td>.4617</td> <td>75</td> <td>.6088</td> </tr> <tr> <td>16</td> <td>1.392</td> <td>36</td> <td>.3090</td> <td>56</td> <td>.4695</td> <td>76</td> <td>.6157</td> </tr> <tr> <td>17</td> <td>1.478</td> <td>37</td> <td>.3173</td> <td>57</td> <td>.4771</td> <td>77</td> <td>.6225</td> </tr> <tr> <td>18</td> <td>1.564</td> <td>38</td> <td>.3256</td> <td>58</td> <td>.4848</td> <td>78</td> <td>.6293</td> </tr> <tr> <td>19</td> <td>1.650</td> <td>39</td> <td>.3338</td> <td>59</td> <td>.4924</td> <td>79</td> <td>.6361</td> </tr> <tr> <td>20</td> <td>1.736</td> <td>40</td> <td>.3420</td> <td>60</td> <td>.5000</td> <td>80</td> <td>.6428</td> </tr> </table>

Deg. MIC

<table> <tr> <th rowspan="2">Micrometer.</th> <th colspan="2">Constant Part of the Formula.</th> <th colspan="2">Constant Part.</th> <th colspan="2">Constant Part.</th> <th colspan="2">Constant Part.</th> </tr> <tr> <th>Deg.</th> <th>Mr Brewster's circular micrometer.</th> <th>Deg.</th> <th>Deg.</th> <th>Deg.</th> <th>Deg.</th> <th>Deg.</th> <th>Deg.</th> </tr> <tr><td>81</td><td>.6494</td><td>106</td><td>.7986</td><td>131</td><td>.9100</td><td>156</td><td>.9781</td></tr> <tr><td>82</td><td>.6561</td><td>107</td><td>.8039</td><td>132</td><td>.9135</td><td>157</td><td>.9799</td></tr> <tr><td>83</td><td>.6626</td><td>108</td><td>.8090</td><td>133</td><td>.9171</td><td>158</td><td>.9816</td></tr> <tr><td>84</td><td>.6691</td><td>109</td><td>.8141</td><td>134</td><td>.9205</td><td>159</td><td>.9833</td></tr> <tr><td>85</td><td>.6756</td><td>110</td><td>.8192</td><td>135</td><td>.9239</td><td>160</td><td>.9848</td></tr> <tr><td>86</td><td>.6820</td><td>111</td><td>.8241</td><td>136</td><td>.9272</td><td>161</td><td>.9863</td></tr> <tr><td>87</td><td>.6884</td><td>112</td><td>.8290</td><td>137</td><td>.9304</td><td>162</td><td>.9877</td></tr> <tr><td>88</td><td>.6947</td><td>113</td><td>.8339</td><td>138</td><td>.9336</td><td>163</td><td>.9890</td></tr> <tr><td>89</td><td>.7009</td><td>114</td><td>.8387</td><td>139</td><td>.9367</td><td>164</td><td>.9903</td></tr> <tr><td>90</td><td>.7071</td><td>115</td><td>.8434</td><td>140</td><td>.9397</td><td>165</td><td>.9914</td></tr> <tr><td>91</td><td>.7133</td><td>116</td><td>.8480</td><td>141</td><td>.9426</td><td>166</td><td>.9923</td></tr> <tr><td>92</td><td>.7193</td><td>117</td><td>.8526</td><td>142</td><td>.9455</td><td>167</td><td>.9936</td></tr> <tr><td>93</td><td>.7254</td><td>118</td><td>.8572</td><td>143</td><td>.9483</td><td>168</td><td>.9945</td></tr> <tr><td>94</td><td>.7314</td><td>119</td><td>.8616</td><td>144</td><td>.9511</td><td>169</td><td>.9954</td></tr> <tr><td>95</td><td>.7373</td><td>120</td><td>.8660</td><td>145</td><td>.9537</td><td>170</td><td>.9962</td></tr> <tr><td>96</td><td>.7431</td><td>121</td><td>.8704</td><td>146</td><td>.9563</td><td>171</td><td>.9969</td></tr> <tr><td>97</td><td>.7490</td><td>122</td><td>.8746</td><td>147</td><td>.9588</td><td>172</td><td>.9976</td></tr> <tr><td>98</td><td>.7547</td><td>123</td><td>.8783</td><td>148</td><td>.9613</td><td>173</td><td>.9981</td></tr> <tr><td>99</td><td>.7604</td><td>124</td><td>.8820</td><td>149</td><td>.9636</td><td>174</td><td>.9986</td></tr> <tr><td>100</td><td>.7660</td><td>125</td><td>.8870</td><td>150</td><td>.9659</td><td>175</td><td>.9990</td></tr> <tr><td>101</td><td>.7716</td><td>126</td><td>.8910</td><td>151</td><td>.9681</td><td>176</td><td>.9994</td></tr> <tr><td>102</td><td>.7771</td><td>127</td><td>.8949</td><td>152</td><td>.9703</td><td>177</td><td>.9996</td></tr> <tr><td>103</td><td>.7826</td><td>128</td><td>.8988</td><td>153</td><td>.9724</td><td>178</td><td>.9998</td></tr> <tr><td>104</td><td>.7880</td><td>129</td><td>.9026</td><td>154</td><td>.9744</td><td>179</td><td>1.0000</td></tr> <tr><td>105</td><td>.7934</td><td>130</td><td>.9063</td><td>155</td><td>.9763</td><td>180</td><td>1.0000</td></tr> </table>

53. "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 AB, or the angle subtended by the whole field. Thus if AB is 30 minutes, as it happens to be in the micrometer which I have constructed, the angle subtended by 1 degree of the scale will be 30' × .009 = 16 1/2 seconds, and the angle subtended by 40 degrees will be 30' × 34.2 = 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 degrees, a variation of 1 degree produces a variation of about 16 seconds in the angle; whereas when the arch is between 170 and 180, the variation of a degree does not produce a change much more than one second in the angle. This is a most important advantage in the circular scale, as in Cavallo's micrometer a limit is necessarily put to the size of the divisions.

"It is obvious, from an inspection of fig. 27, that there is no occasion for turning the circular micrometer round its axis, because the divided circumference lies in every possible direction. In fig. 2, for example, if the object has the direction ab it will be measured by the arch ao b, and if it lies in the line cd it will be measured by the arch cr d.

"In the circular micrometer which I have been in the habit of using, AB, or the diameter of the field of view, is exactly half an inch, the diameter of the brass tube in which it is fixed is one inch, the length of the tube half an inch, and the degrees of the divided circumference \( \frac{1}{10} \)th of an inch."

§4. II. The micrometer has not only been applied to telescopes, and employed for astronomical purposes; but Microme there have also been various contrivances for adapting it to MICROSCOPICAL observations. Mr Leeuwenhoecck's method of estimating the size of small objects was by the mi comparing them with grains of sand, of which 100 in diameter to a line took up an inch. These grains he laid upon the fame plate with his objects, and viewed them at the same time. Dr Jurin's method was similar to this; for he found the diameter of a piece of fine silver wire, by wrapping it as close as he could about a pin, and observing how many rings made an inch; and he used this wire in the same manner as Leeuwenhoecck employed his sand. Dr Hooke looked upon the magnified object with one eye, while at the same time he viewed other objects placed at the same distance with the other eye. In this manner he was able, by the help of a ruler, divided into inches and small parts, and laid on the pedestal of the microscope, to cast as it were the magnified appearance of the object upon the ruler, and thus exactly to measure the diameter which it appeared to have through the glass; which being compared with the diameter as it appeared to the naked eye, showed the degree in which it was magnified.

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

65. The mode of actual admeasurement (Mr Adams observes+) is without doubt the most simple that can be Microfoc used; as by it we comprehend, in a manner, at one glance, the different effects of combined glasses; and as it saves the trouble, and avoids the obscurity, of the usual modes of calculation: but many persons find it exceedingly difficult to adopt this method, because they have not been accustomed to observe with both eyes at once. To obviate this inconvenience, the late Mr Adams contrived an instrument called the Needle-Micrometer, which was first described in his Micrographia Illustrata; and of which, as now constructed, we have the following description by his son Mr George Adams in the ingenious Essays above quoted.

This micrometer consists of a screw, which has 50 threads to an inch; this screw carries an index, which points to the divisions on a circular plate, which is fixed at right angles to the axis of the screw. The revolutions of the screw are counted on a scale, which is an inch divided into 50 parts; the index to these divisions is a flower-de-luce marked upon the slider, which carries the needle point across the field of the microscope. Every revolution of the micrometer screw measures \( \frac{1}{50} \)th part of an inch, which is again subdivided by means of the divisions on the circular plate, Micrometer. as this is divided into 20 equal parts, over which the index passes at every revolution of the screw; by which means we obtain with ease the measure of 1/100th part of an inch: for 50, the number of threads on the screw in one inch, being multiplied by 20, the divisions on the circular plate are equal to 1000; so that each division on the circular plate shows that the needle has either advanced or receded 1/100th part of an inch.

57. To place this micrometer on the body of the microscope, open the circular part FKH, fig. 25, by taking out the screw G, throw back the femuricle FK, which moves upon a joint at K; then turn the sliding tube of the body of the microscope, so that the small holes which are in both tubes may exactly coincide, and let the needle g of the micrometer have a free passage through them; after this, screw it fast upon the body by the forew G. The needle will now traverse the field of the microscope, and measure the length and breadth of the image of any object that is applied to it. But further affiance must be had, in order to measure the object itself, which is a subject of real importance; for though we have ascertained the power of the microscope, and know that it is so many thousand times, yet this will be of little affiance towards ascertaining an accurate idea of its real size; for our ideas of bulk being formed by the comparison of one object with another, we can only judge of that of any particular body, by comparing it with another whose size is known: the fame thing is necessary, in order to form an estimate by the microscope; therefore, to ascertain the real measure of the object, we must make the point of the needle pass over the image of a known part of an inch placed on the stage, and write down the revolutions made by the screw, while the needle passed over the image of this known measure; by which means we ascertain the number of revolutions on the screw, which are adequate to a real and known measure on the stage. As it requires an attentive eye to watch the motion of the needle point as it passes over the image of a known part of an inch on the stage, we ought not to trust to one single measurement of the image, but ought to repeat it at least fix times; then add the fix measures thus obtained together, and divide their sum by fix, or the number of trials; the quotient will be the mean of all the trials. This result is to be placed in a column of a table next to that which contains the number of the magnifiers.

58. By the affiance of the sectoral scale, we obtain with ease a small part of an inch. This scale is shown at fig. 22, 23, 24, in which the two lines c a, c b, with the side a b, form an isosceles triangle; each of the sides is two inches long, and the base fill only of one-tenth of an inch. The longer lines may be of any given length, and the base fill only one-tenth of an inch. The longer lines may be considered as the line of lines upon a sector opened to one-tenth of an inch. Hence whatever number of equal parts c a, c b are divided into, their transverse measure will be such a part of one-tenth as is expressed by their divisions. Thus if it be divided into ten equal parts, this will divide the inch into 100 equal parts; the first division next c will be equal to 1/100th part of an inch, because it is the tenth part of one-tenth of an inch. If these lines are divided into twenty equal parts, the inch will be by that means divided into 200 equal parts. Lastly, Micrometer if a b, c a, are made three inches long, and divided into 100 equal parts, we obtain with ease the 1/100th part. The scale is represented as solid at fig. 23, but as perforated at fig. 22. and 24, so that the light passes through the aperture, when the sectoral part is placed on the stage.

59. To use this scale, first fix the micrometer, fig. 25, to the body of the microscope; then fit the sectoral scale, fig. 24, in the stage, and adjust the microscope to its proper focus or distance from the scale, which is to be moved till the base appears in the middle of the field of view; then bring the needle point g, fig. 25, (by turning the screw L) to touch one of the lines c a, exactly at the point answering to 20 on the sectoral scale. The index a of the micrometer is to be set to the first division, and that on the dial plate to 20, which is both the beginning and end of its divisions; we are then prepared to find the magnifying power of every magnifier in the compound microscope which we are using.

60. Example. Every thing being prepared agreeable to the foregoing directions, suppose you are desirous of ascertaining the magnifying power of the lens marked No 4; turn the micrometer screw until the point of the needle has passed over the magnified image of the tenth part of an inch; then the division, where the two indices remain, will show how many revolutions, and parts of a revolution, the screw has made, while the needle point traversed the magnified image of the one-tenth of an inch; suppose the result to be 26 revolutions of the screw, and 14 parts of another revolution, this is equal to 26 multiplied by 20, added to 14; that is, 534,000 parts of an inch.—The 26 divisions found on the straight scale of the micrometer, while the point of the needle passed over the magnified image of one-tenth part of an inch, were multiplied by 20, because the circular plate CD, fig. 25, is divided into 20 equal parts; this produced 520; then adding the 14 parts of the next revolution, we obtain the 534,000 parts of an-inch, or five-tenths and 3400 parts of another tenth, which is the measure of the magnified image of one-tenth of an inch, at the aperture of the eye-glasses or at their foci. Now if we suppose the focus of the two eye-glasses to be one inch, the double thereof is two inches; or if we reckon on the 1000th part of an inch, we have 2000 parts for the distance of the eye from the needle point of the micrometer. Again, if we take the distance of the image from the object at the stage at 6 inches, or 6000, and add thereto 2000, double the distance of the focus of the eye-glass, we shall have 8000 parts of an inch for the distance of the eye from the object; and as the glasses double the image, we must double the number 534 found upon the micrometer, which then makes 1068: then, by the following analogy, we shall obtain the number of times the microscope magnifies the diameter of the object; say, as 240, the distance of the eye from the image of the object, is to 800, the distance of the eye from the object; so is 1068, double the measure found on the micrometer, to 3563, or the number of times the microscope magnifies the diameter of the object. By working in this manner, the magnifying power of each lens used with the compound microscope may be easily found, though the result will be different in different compound microscopes, varying according to the combination of the lenses, their distance from the object and one another, &c.

61. Having discovered the magnifying power of the microscope, with the different object-lenses that are used therewith, our next subject is to find out the real size of the objects themselves, and their different parts: this is easily effected, by finding how many revolutions of the micrometer screw answer to a known measure on the factorial scale or other object placed on the stage; from the number thus found, a table should be constructed, expressing the value of the different revolutions of the micrometer with that object lens, by which the primary number was obtained. Similar tables must be constructed for each object lens. By a set of tables of this kind, the observer may readily find the measure of any object he is examining; for he has only to make the needle point traverse over this object, and observe the number of revolutions the screw has made in its passage, and then look into his table for the real measure which corresponds to this number of revolutions, which is the measure required.

62. Mr Coventry of Southwark has favoured us with the description of a micrometer of his own invention; the scale of which, for minuteness, surpasses every instrument of the kind of which we have any knowledge, and of which, indeed, we could scarcely have formed a conception, had he not indulged us with several of these instruments, graduated as underneath.

The micrometer is composed of glass, ivory, silver, &c. on which are drawn parallel lines from the 10th to the 10,000th part of an inch. But an instrument thus divided, he observes, is more for curiosity than use: but one of those which Mr Coventry has sent us is divided into squares, so small that sixteen millions of them are contained on the surface of one square inch, each square appearing under the microscope true and distinct; and though so small, it is a fact, that animalcula are found which may be contained in one of these squares.

The use of micrometers, when applied to microscopes, is to measure the natural size of the object, and how much that object is magnified. To ascertain the real size of an object in the single microscope, nothing more is required than to lay it on the micrometer, and adjust it to the focus of the magnifier, noticing how many divisions of the micrometer it covers. Suppose the parallel lines of the micrometer to be the 1000th of an inch, and the object covers two divisions; its real size is 500ths of an inch; if five, 200ths, and so on.

But to find how much the object is magnified, is not mathematically determined so easily by the single as by the compound microscope: but the following simple method (says Mr Coventry) I have generally adopted, and think it tolerably accurate. Adjust a micrometer under the microscope o, say the 100th of an inch of divisions, with a small object on it; if square, the better: notice how many divisions one side of the object covers, suppose 10: then cut a piece of white paper something larger than the magnified appearance of the object: then fix one eye on the object through the microscope, and the other at the same time on the paper, lowering it down till the object and the paper appear level and distinct: then cut the paper till it appear exactly the size of the magnified object; the paper being then measured, suppose an inch square: Now, as the object under the magnifier, which appeared to be one inch square, was in reality only ten hundredths, or the tenth of an inch, the experiment proves that it is magnified ten times in length, one hundred times in superficies, and one thousand times in cube, which is the magnifying power of the glass; and, in the same manner, a table may be made of the power of all the other glasses.

In using the compound microscope, the real size of the object is found by the same method as in the single: but to demonstrate the magnifying power of each glass to greater certainty, adopt the following method.—Lay a two-feet rule on the stage, and a micrometer level with its surface (an inch suppose, divided into 100 parts): with one eye see how many of those parts are contained in the field of the microscope, (suppose 50); and with the other, at the same time, look for the circle of light in the field of the microscope, which with a little practice will soon appear distinct; mark how much of the rule is intercepted by the circle of light, which will be half the diameter of the field. Suppose eight inches; consequently the whole diameter will be fifteen. Now, as the real size of the field, by the micrometers, appeared to be only 50 hundredths, or half an inch, and as half an inch is only one 32d part of 16 inches, it shows the magnifying power of the glass to be 32 times in length, 1024 superficies, and 32,768 cube (E).

63. Another way of finding the magnifying power of compound microscopes, is by using two micrometers of the same divisions; one adjusted under the magnifier, the other fixed in the body of the microscope in the focus of the eye-glass. Notice how many divisions of the micrometer in the body are seen in one division of the micrometer under the magnifier, which again must be multiplied by the power of the eye-glass. Example: Ten divisions of the micrometer in the body are contained in one division under the magnifier; so far the power is increased ten times: now, if the eye-glass be one inch focus, such glass will of itself magnify about seven times in length, which, with the ten times magnified before, will be seven times ten, or 70 times in length, 4900 superficies, and 343,000 cube.

"If (says Mr Coventry) these micrometers are employed in the solar microscope, they divide the object into squares on the screen in such a manner as to render it extremely easy to make a drawing of it. And (says he) I apprehend they may be employed to great advantage with such a microscope as Mr Adams's lucernal; because this instrument may be used either by day or night, or in any place, and gives the actual magnifying power without calculation."

(e) It will be necessary, for great accuracy, as well as for comparative observations, that the two-feet rule should always be placed at a certain distance from the eye: eight inches would, in general, be a proper distance.

PLATE CCCXXXV.

Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11.

E. Mitchell sculp't

PLATE CCCXXXVI.

Fig. 12. Fig. 13. Fig. 15. Fig. 14. Fig. 16. Fig. 17. Fig. 20. Fig. 18. Fig. 19. Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26. Fig. 27. Fig. 28.

SACT OF THE ROYAL SOCIETY OF NDO

F. Mitchell sculp't The case with which we have been favoured by Mr Coventry contains fix micrometers, two on ivory and four on glass. One of those on ivory is an inch divided into one hundred parts, every fifth line longer than the intermediate ones, and every tenth longer still, for the greater ease in counting the divisions under the microscope, and is generally used in measuring the magnifying power of microscopes. The other ivory one is divided into squares of the 50th and 100th of an inch, and is commonly employed in measuring opaque objects.

The glass micrometer without any mark is also divided, the outside lines into 100th, the next into 1000th, and the inside lines into the 4000th of an inch: these are again crossed with an equal number of lines in the same manner, making squares of the 100th, 1000th, and 4000th of an inch, thus demonstrating each other's size. The middle square of the 1000th of an inch (see fig. 26.) is divided into fifteen squares; now as 1000 squares in the length of an inch, multiplied by 1000, gives one million in an inch surface; by the same rule, one of those squares divided into 16 must be the sixteen millionth part of an inch surface. See fig. 26. which is a diminished view of the apparent surface exhibited under the magnifier No 1. of Wilson's microscope. In viewing the smallest lines, Mr Coventry uses No 2. or 3.; and they are all better seen, he says, by candle than by day-light.