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the true amplitude be fix degrees north, and the mag- Compaís. netical amplitude five degrees fouth; the variation, d which in this cafe is north-weft, will be equal to the fum of the magnetical and true amplitudes : understand the fame for weft amplitudes.

Compafs. by the motions of the veffel. Thus the centres of motion, gravity, and of magnetifm, are brought almoft all to the fame point; the advantages of which will be readily perceived by any perfon acquainted with mechanical principles. Experience therefore will afcertain the utility of this improvement." M'Culloch's Account.

Of Dr Knight's Azimuth Compafs, as improved by Mr Smeaton, a defcription was given under the article AZIMUTH, and a figure in Plate LXXVII. The ufe of the azimuth compafs is for finding the fun's magnetical azimuth, or amplitude; and thence the variation of the compais. If the observation be for an amplitude at fun-rifing, or for an azimuth before noon, apply the centre of the index on the weft point of the card, within the box; fo that the four lines on the edge of the card, and thofe on the infide of the box, may meet. If the observation be for the fun's amplitude fetting, or an azimuth in the afternoon, turn the centre of the index right against the east point of the card, and make the lines within the box concur with those on the card: the inftrument thus fitted for obfervation, turn the index be towards the fun, till the fhadow of the thread ae fall directly on the flit of the fight, and on the line that is along the middle of the index: then will the inner edge of the index cut the degree and minute of the fun's magnetical azimuth from the north or fouth. But note, that if, when the compafs is thus placed, the azimuth is lefs than 45° from the fouth, and the index turned towards the fun, it will pafs off the divifions of the limb: the inftrument therefore in this cafe must be turned just a quarter of the compafs, i. e. the centre of the index must be placed on the north or fouth point of the card, according as the fun is from you; and then the edge will cut the degree of the magnetic azimuth, or the fun's azimuth from the north, as before.

The fun's magnetical amplitude thus found, the variation of the needle is thus determined. Being out at fea the 15th of May 1717, in 45° north latitude, the tables give me the fun's latitude 19° north, and his eaft amplitude 27 25' north: by the azimuth compafs, I find the fun's magnetical amplitude at his rifing and fetting; and find he rifes, v. gr. between the 62d and 63d degree, reckoning from the north towards the caft point of the compafs, i. e. between the 27th and 28th degree, reckoning from the eaft. The magnetical amplitude, therefore, being here equal to the true one, the needle has no variation; but if the fun at his rifing fhould have appeared between the 52d and 534 degree from the north towards the caft; his magnetical amplitude would then have been between 37 and 38 degrees, i. e. about 10 degrees greater than the true amplitude: therefore, the needle would vary about 10 degrees north-eafterly. If the magnetical raft amplitude found by the inftrument fhould be lefs than the true amplitude, their difference would show the variation of the needle easterly. If the true eaft amplitude be fouthward, as alfo the magnetical amplitude, and this last be the greater; the variation of the needle will be north-weft; and vice verfa.

What has been faid of north-eaft amplitudes holds alfo of fouth-weft; and what of fouth-eaft amplitudes holds of north-weft amplitudes. Lastly, if amplitudes be found of different denominations, v. gr. if

The variation may likewife be found from the azi. muth but in that cafe, the fun's declina ion, latitude of the place, and his altitude, must be given, that his true azimuth may be found.

This inftrument is alfo ufeful in fettling the ship's wake, in order to find the lee-way; and alfo to find the bearings of head-lands, and other objects.

COMPASS is alfo an inftrument of confiderable use in furveying land, dialing, &c.

Its ftructure, in the main, is the fame with that of the mariner's compafs; confifting, like that, of a box and needle: the principal difference confiits in this, that inftead of the needle's being fitted into the card, and playing with it on a pivot, it here plays alone; the card being drawn on the bottom of the box, and a circle divided in 360 degrees on the limb. See fig. 5. Plate This inftrument is of obvious use to travellers, to di- CXLIV. rect them in their road; and to miners, to show them what way to dig, with other confiderable uses.

1. To take the declination of a wall by the Compass. Apply that fide of the compafs whereon the north is marked along the fide of the wall; the number of degrees over which the north end of the needle fixes will be the declination of the wall, and on that fide; v. gr. if the north point of the needle tends towards the north, that wall may be fhone on by the fun at noon; if it fix over fifty degrees, counting from the north towards the eaft, the declination is fo many degrees from north towards caft.

But fince the needle itself declines from the north towards the weft, with us, 13°; it must be noted, that to retrieve the irregularity, 130 are always to be added to the degrees fhown by the needle, when the declination of the wall is towards the eaft; on the contrary, when the declination is towards the west, the declination of the needle is to be fubtracted.

2. To take an angle with the Compafs. Suppofe the angle required be DAE, fig. 4. apply that fide of the compafs whereon the north is marked to one of the lines AD; when the needle rests, obferve the degrees at which its north point ftands, which fuppofe 80: so many degrees does the line decline from the meridian. In the fame manner take the declination of the line AE, which fuppofe 215°; fubtract 80° from 215, the remainder is 135; which fubtracted from 180, there will remain 45°; the quantity of the angle required. But if the difference between the declination of the two lines exceed 180°; in that cafe, 180° must be fubtracted from that difference: the remainder then is the angle required.

In meafuring angles by the compafs, there needs not any regard be had to the variation; that being sup pofed the fame in all the lines of the angles.

3. To take a plot of a field by the Compass. Suppofe the field A, B, C, D, E, fig. 10. for the greater accuracy let there be two fights fitted to the meridian line of the compafs, place it horizontal, and through the fights look along the fide AB, or a line parallel to it; applying the eye to the fight at the fouth point of the compafs. Draw a rough sketch of the field by

the

Compafs, the eye, and on the correfponding line enter down the Compaffes degree to which the needle points, which fuppofe 90; measure the length of the fide, and enter that too, which fuppofe 10 chains.

In this manner proceed with all the rest of the fides and angles of the field; the fides, which fuppofe 70, 65, 70, 44, 50 fathom; and the angles, which fuppole 30, 100, 130, 240, 300, degrees. To protract the field, fet down the feveral angles obferved, one after another, and fubtract the leffer from the next greater thus will you have the quantity of the feveral angles, and the length of the lines that include them. For the reft, see GEOMETRY.

Note, All the angles of the figure taken together, must make twice as many right angles; abating two if no mistake has been committed.

Azimuth COMPASS. See AZIMUTH. COMPASS-Dials, are fmall horizontal dials, fitted in brafs or filver boxes, for the pocket, to fhow the hour of the day, by the direction of a needle that indicates how to place them right, by turning the dial about till the cock or ftyle ftand directly over the needle; but these can never be very exact, because of the variation .of the needle itfelf. See COMPASS, and DIALING. . COMPASSES, or Pair of COMPASSES, a mathematical inftrument for defcribing circles, measuring figures, &c..

The common compaffes confit of two fharp-pointed branches or legs of iron, fteel, brafs, or other metal, joined together at the top by a rivet, whereon they move as on a centre. Thofe compaffes are of the beft fort in which the pin or axle on which the joint turns, and also half the joint itself, is made of fteel, as the oppofite metals wear more equable. The perfection of them may be known by the eafy and uniform opening and fhutting of their legs; one of which is fometimes made to take in and out, in order to make room for two other points to describe with ink, blacklead, or other materials.

There are now used compaffes of various kinds and contrivances, accommodated to the various ufes they are intended for; as,

COMPASSES of three Legs, or Triangular Compaffes, are, fetting afide the excefs of a leg, of the fame ftructure with the common ones: their ufe being to take three points at once, and fo to form triangles; to lay down three pofitions of a map, to be copied at once, &c.

Beam COMPASSES confift of a long branch, or beam, made of brafs or wood, carrying two brafs curfors, the one fixed at one end, the other fliding along the beam, with a screw to faften it on occafion. To the curfors may be fcrewed points of any kind, whether fteel for pencils, or the like. It is ufed to draw large circles, to take great extents, &c. To the fixed curfor is fometimes applied an adjutting or micrometer fcrew, by which an extent is obtained to extreme nicety. Mr Jones of Holborn has made beam compaffes to adjust to the dth of an inch.

30

Caliber COMPASSES. See CALIBER. Clockmaker's COMPASSES are joined like the common compaffes, with a quadrant, or bow, like the fpring compaffes; only of different use, serving here to keep the inftrument firm at any opening. They are made very ftrong, with the points of their legs of well tem

pered steel, as being ufed to draw lines on pafte-board Compaffes. or copper.

Cylindrical and Spherical COMPASSES, confift of four branches, joined in a centre, two of which are circular, and two flat, a little bent on the ends: their use is to take the diameter, thicknefs, or caliber of round or cylindric bodies; fuch as cannons, pipes, &c.

Elliptic COMPASSES. Their ufe is to draw ellipfes, Plate or ovals of any kind: they confift of a beam A B CXLIV about a foot long, bearing three curfors; to one of 8 y. fig, which may be fcrewed points of any kind: to the bottom of the other two are rivited two sliding dovetails, adjusted in grooves made in the cross branches of the beam. The dove-tails having a motion every way, by turning about the long branch, go backwards and forwards along the crofs; fo that when the beam has gone half-way about, one of these will have moved the whole length of one of the branches; and when the beam has got quite round, the fame dove-tail has got back the whole length of the branch. Understand the fame of the other dove-tail.

Note, the diftance between the two fliding dove-tails is the distance between the two foci of the ellipfis; fo that by changing that distance, the ellipfis will be rounder or flenderer. Under the ends of the branches of the crofs are placed four fteel points to keep it faft.

The ufe of this compafs is eafy; by turning round the long branch, the ink, pencil, or other point, will draw the ellipfis required. Its figure shows both its use and conftruction.

German COMPASSES have their legs a little bent outwards, towards the top; fo that when shut, the points only meet.

Hair COMPASSES are fo contrived within fide by a fmall adjusting screw to one of the legs, as to take an extent to a hair's breadth.

Lapidary's COMPASSES are a piece of wood, in form of the fhaft of a plane, cleft at top, as far as half its length; with this they measure the angles, &c. of jewels and precious ftones, as they cut them. There is in the cleft a little brafs rule, faftened there at one end by a pin; but fo that it may be moved in the manner of a brass level: with this kind of fquare they take the angles of the tones, laying them on the fhaft as they cut them.

Proportional COMPASSES are thofe whofe joint lies between the points terminating each leg they are either fimple or compound. In the former fort the centre is fixed, fo that one pair of these ferves only for one proportion.

Compound proportional COMPASSES Confist of two parts or fides of brafs, which lie upon each other so nicely as to appear but one when they are fhut. Thefe lides eafily open, and move about a centre, which is itfelf moveable in a hollow canal cut through the greatest part of their length. To this centre on each fide is affixed a fliding piece A of a fmall length, with a fine line drawn on it ferving as an index, to be fet against other lines or divifions placed upon the compaffes on both fides. Thefe lines are, 1. A line of lines. line of fuperficies, areas, or planes. 3. A line of folids. 4. A line of circles, or rather of polygons to be infcribed in circles. Thefe lines are all unequally divided; the three first from 1 to 20, the laft from 6 to 20. Their uses are as follow: Nn 2 By

2. A

Plate

CXLV.

fig. 6.

Compafs. By the line of lines you divide a given line into any number of equal parts; for by placing the index A against 1, and fcrewing it faft, if you open the com. paffes, then the diftance between the points at each end will be equal. If you place the index against 2, and open the compaffes, the diftance between the points of the longer legs BB, will be twice the distance between the shorter ones CC; and thus a line is bifected, or divided into two equal parts. If the index be placed against 3, and the compaffes opened, the distances between the points will be as 3 to 1, and fo a line is divided into three equal parts; and fo you proceed for any other number of parts under to.

The numbers of the line of planes anfwer to the fquares of thefe in the line of lines; for because fuperficies or planes are to each other as the fquares of their like fides; therefore, if the index be placed gainft z in the line of planes, then the distance between the fmall points will be the fide of a plane whofe area is one; but the diflance of the larger points will be the like fide of a plane whofe area is two; or twice as large. If the index be placed at 3, and the compaffes opened, the distances between the points at each end will be the like fide of planes whose area are as I to 3; and fo of others.

The numbers of the line or folids answer to the cubes of thofe in the line of lines; because all folids arg to each other as the cubes of their fides or diameters: therefore, if the index be placed to number 2,3,4, &c. in the line of folids, the distance between the leffer and larger points will be the like fides of folids, which are to each other as 1 to 2, I to 3, I to 4, &c. For example: If the index be placed at 10, and the compaffes be opened fo that the fmall points may take the diameter of a bullet whofe weight is one ounce, the distance between the large points will be the diameter of a bullet or globe of 10 ounces, or which is 10 times as large.

Laftly, The numbers in the line or circles are the fides of polygons to be infcribed in a given circle, or by which a circle may be divided into the equal parts, from 6 to 20. Thus, if the index be placed at 6, the points of the compaffes at either end, when opened to the radius of a given circle, will contain the fide of a hexagon, or divide the circle into fix equal parts. If the index be placed against 7, and the compaffes opened fo that the larger points may take in the radius of the circle, then the fhorter points will divide the circle into feven equal parts for infcribing a heptagon. Again, placing the index to 8, and opening the compaffes, the larger points will contain the radius, and the leffer points divide the circle into eight equal parts for infcribing an octagon or square. And thus you may proceed for others.

Proportional CoMMPASSES with the fector lines. The Aructure of theie is fo like that of the common proportional compaffes, only a little nicer, that it needs no particular defcription. The lines on the first face are the line of lines, marked lines; it is divided into 100 equal parts, every tenth numbered: and the line of chords, which goes to 60°, is marked chords. On the other face are a line of fines to 90°, and a line of tangents to 45°. On one fide are the tangents from 45 to 71° 34'; on the other, fecants from oto 70°30',

For the use of these compaffes: 1. To divide a line Compafs. into any number of equal parts lefs than soo: divide 100 by the number of parts required; flip the curfor till the line on the fliding dove-tail be againft the quotient on the line of lines: then, the whole line being taken between the points of the compaffes most remote from the centre, the aperture of the other will show the divifion required. 2. A right line given, fuppofed to be divided into 100 parts, to take any number of these parts; flip the line on the fliding dove-tail to the number of parts required the whole line being taken between the points fattheft from the centre, the aperture of the other two will include the number of divifions required. 3. The radius being given, to find the chord of any arch under 60°; flip the line on the fliding dove-tail to the degrees required on the line of chords: the radius being taken between the points fartheft from the centre of the curfor; the aperture of the other line will be the chord required, provided the number of degrees be greater than 29: if it be lefs, the aperture taken from the radius will leave the chord required. 4. If the chord of an arch under 60° be given, and the radius required; flip the line on the dove-tail to the degrees given on the line of chords: the given chord being taken between the two points next the curfor, the aperture of the other will be the radius required. 5. The radius being given, to find the fine of any number of degrees; flip the line on the dove-tail to the degree on the line of fines whose fine is required: the radius taken between the points furtheft from the curfor, the aperture of the other will give the fine of the angle required. But if the fine fought be lefs than 30°, the difference of the apertures of the oppofite points will be the fine required. 6. The radius being given, to find the tangent of any number of degrees under 71: if the tangent required be under 26 30, then flip the line on the dove-tail to the degree proposed on the tangent line; the radius taken betw-en the points fartheft from the curfor, the aperture of the others will be the tangent of the degrees required: if the tangent required be above 260 30', but under 45°, the line on the curfor must be flipped to the degrees given on the tangent line: then the radius being taken between the points furtheft from the curfor, the aperture of the others will be the tangent. If the tangent required be greater than 45°, but lefa than 56° 20', flip the notch on the tangent side of the turned cheek to the degree o in the tangent line on the fide of the compafs; the radius taken between the points fartheft from the curfor; the difference between the aperture of the other and thefe, added together, will be the tangent required. Thus, for the tangents of other degrees under 71. After the like manner may the fecant of any number of degrees under 71 be found.

Mr Heath, a mathematical inftrument-maker in London, conftructed a pair of proportional compaffes, in 1746, with a curious and ufeful contrivance for preventing the fhorter legs from changing their pofition, when thefe compaffes were used. It confifted of a fmall beam foldered to a ferew, and running parallel to the leg of the compaffes, nearly of the length of the groove; in this beam a flit was made, which admitted of a fliding-nut, the other end of which fell into a hole in the bottom of the fcrew, belonging to the great nut

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