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^visions.need not be so minute : for instance, in one of Dollond ’s? >0 cket-telescopes, which wiien drawn out for use is about 14ln ches long, a micrometer with the hundredths of an inch is quitesufficient,' and one of its divisions is equal to little less than three^unites, so that an angle of a minute may be measured by it.

' In looking through a telescope furnished with such a microme-(says our author), the field of view appears to be divided byP>e micrometer-scale, the breadth of which occupies about one-*®venth part of the aperture ; and as the scale is semitransparent,unt part of the object which happens to be behind it may be dis-Cer ned sufficiently well to ascertain the division,. and even theQuarter of a dl vision, with which its borders coincide. Fig. 13,

8 'Mvs the appearance of the field of my telescope with the micro-meter, when directed to the title-page of the Philosophical Trans-itions, wherein one may observe that the thickness of the letter>s equal to 3-4ths of a division, the diameter of the O is equal|° three divisions, and so on. At first view, one is apt to imagine,mil it is difficult to count the divisions which may happen to cover? r to measure an object ; but upon trial it will be found, that thisreadily performed ; and even people who have never been used0 observe with the telescope, soon learn to measure very quicklv' l nd accurately with this micrometer; for since every fifth ande, 'th division is longer than the rest, one soon acquires the habitsaying five, fen, fifteen; and then, by adding the other divi-* lQ ns less than five, completes the reckoning, Evenwithalcde-' c °pe which has no stand, if the object-end of it be rested againsta steady place, and the other end beheld by the hand near the eyem Hie observer, an object may be measured with accuracy suiii-munt for several purposes, as lor the estimation of small distances,° r determining the height of a house, Ac. After having con-ducted and adapted this micrometer to the telescope, it is ncces-( Sar y to ascertain the value of the divisions. It is hardly necessaryi Mention in this place, that though those divisions measure the^•'ords of the angles, and not the angles or arches themselves, and; ! e chords are not as the arches, yet it has been shewn by all thefSonomctrical writers, that in small angles the chords, arches,and tangents, follow the same proportion so very nearly,■J’ut the very minute difference may be safely neglected ; so thatll °‘ie division of this micrometer is equal to one minute, we mays , a ffily conclude, that two divisions are equal to two minutes, threec 'visions to three minutes, and so on. There are various methodsm ascertaining the value of the divisions of such a micrometer,! e y beingthe very same that are used for ascertaining the value°*ffie divisions in other micrometers. Such are, the passage of ane ffiiatoreal star over a certain number of divisions in a certain time ;(, r for the measuring the diameter of the sun, by computation from'e focal distance of the object and other lenses of the telescope ;le last of which, however, is subject to several inaccuracies; but8 Ihey are well known to astronomical persons, and have been de-j?ffied in many books, they need not be farther noticed here,towever, for the sake of workmen, and other persons not conver-a, 'tastronomy, I shall describe an easy and accurate method1 as certaining the value of the divisions of the micrometer. Markjt J °" a wall or other place the length of six inches, which may be°Ue by making two dots or lines six inches asunder, or by fixings ‘X-inch ruler upon a stand; then place the telescope before it,J! W'at the ruler or six-inch length may be at right angles with theT r ection of the telescope, and just 57 feet 3$ inches distant from°k)ect-glass of the telescope : this done, look through the te-. Sc °pe at the ruler or other extension of six inches, and observeMany divisions of the micrometer are equal to it, and that•I ! ne number of divisions is equal.to half a degree, or 30 ; andr ' ,Sls all that needs be done for the required determination ; theason of which is, because an extension of six inches subtends anpl’S'e of 30' at the distance of 57 feet 3-1 inches, as may be easily|‘ Mdated by the rules of plane trigonometry. In one of Dol-lK I s fourteen-inch pocket-telescopes, if the divisions of. the mi-n'"Meter be the hundredths of an inch, 114 ot those divisions willbe ;° Ul 'd equal to 30', or 23 to a degree. When this value hasen once .ascertained, any other angle measured by any otherMber of divisions is determined by the rule of three. 'I hus‘Ppose that the diameter of the sun seen through the same tele-°l )e > be found equal to 12 divisions, say as 11-J divisions are to 30ini , /12' X 30'\

"utes, so are 12 divisions to ( - ) 31'.3, which is the re-

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V ° L - iii,— xo. 143.

quired diameter of the sun. Notwithstanding the facility of thiscalculation, a scale may lie made answering to the divisions of amicrometer, which will shew the angle corresponding to any num-ber of divisions to mere inspection. Thus, for the above men-tioned small telescope, the scale is represented in fig. 14. AB isa line drawn at pleasure ; it is then divided into 23 equal parts,and those divisions which represent the divisions of the microme-ter that are equal to one degree, are marked cn one side of it.The line then is divided again into 60 equal pails, which aremarked on the other side of it; and these divisions represent theminutes which correspond to the divisions of the micrometer: thusthe figure shews, that six divisions of the micrometer are equal to15-| minutes, 11-'- divisions are nearly equal to 29 minutes, &c.What has been said of minutes may be said of seconds also, whenthe scale is to be applied to a large telescope. Thus far this mi-crometer and its general use have been sufficiently described ; andmathematicians may easily apply it to the various purposes towhich micrometers have been found subservient. But as the sim-plicity, cheapness, and accuracy, of this contrivance, may renderthe use of it much more general than that of any other microme-ter ; and I may venture to say, that it will be found very useful'in the army, and amongst sea-faring people, for the determinationof distances, heights, &c.; I shall therefore join some practicalrules to render this micrometer useful to persons unacquaintedwith trigonometry and die use of logarithms.

Problem I. The angle, not exceeding one degree, which issubtended by an extension of one foot, being given, to find itsdistance from the place of observation. N. B. 'I his extension ofone foot, or any oilier which may be mentioned hereafter, mustbo perpendicular to the direction of the telescope through whichit is observed. The distances are reckoned from the object-glassof tile telescope; and the answers obtained by the rules of thisproblem, though not exactly true, are however so little differentfrom the truth, that the difference seldom amounts to more thantwo or three inches, which may be safely neglected.

Rule 1. If the angle be expressed in minutes, say, asthe givenangle is to 60, so is 687.55 to a fourth proportional, which givestlie 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 pro-portional, which expresses the answer in inches. 3. If the anglebe expressed in minutes and seconds, turn it all into seconds, andproceed as above.

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

„ 3600 x 687-55

2 : 3600 : : 687-55 :- - -= 1237590

inches, or 103132-J feet, which is the answer required. This cal-culation may be shortened; for since two of the three propor-tionals are fixed, their product in the first case is 41253, and inthe other two cases is 2475180; so that in the first case, viz. whenthe angle is expressed in minutes, you need only divide 41253 bythe given angle; and in the other two cases, &c. when the angleis expressed in seconds, divide 2475180 by the given angle, andthe quotient in either case is the answer in inches,

Prob. II. The angle, not exceeding one degree, which is sub-tended by any known extension, being given, to find its distancefrom tlie place of observation.

Rule. Proceed as if the extension were of one foot by ProblemI, and call tlie answer B; then, if the extension in question beexpressed in inches, say, as 12 inches arc to that extension, so is15 to a fourth proportional, which is the answer in indies; but ifthe extension in question be expressed in feet, then you need onlymultiply it by B, and the product is the answer in inches.

Example. At what distance is a man six feet high, when lieappears to subtend an angle ot 30". By problem 1. if the manwere one foot high, the distance would be 82506 indies ; but as lieis six feet high, therefore multiply 82506 by 6, and the productgives tlie required distance, which is 495036 inches. For greaterconveniency, especially in travelling, or in such circumstances inwhich one lias not tlie opportunity ot making even tlie easy calcu-lations required in those problems, 1 lias (' calculated the followingtwo tables ; the first of which shews the distance answering to "anyangle from oik- minute to one degree, which is subtended by anextension of one foot; and the second tabic; shews tlie distance an-swering to any angle rom one minute to one degree, which is6 Y subtended