or Pair of COMPASSES, a mathematical instrument for describing circles, measuring figures, &c.
The common compasses consist of two sharp-pointed branches or legs of iron, steel, brass, or other metal, joined together at the top by a rivet, whereon they move as on a centre. Those compasses are of the best sort in which the pin or axle on which the joint turns, and also half the joint itself, is made of steel, as the opposite metals wear more equable. The perfection of them may be known by the easy and uniform opening and shutting of their legs; one of which is sometimes made to take in and out, in order to make room for two other points to describe with ink, black-lead, or other materials.
There are now used compasses of various kinds and contrivances, accommodated to the various uses they are intended for; as,
COMPASSES of three Legs, or Triangular COMPASSES, are, setting aside the excess of a leg, of the same structure with the common ones: their use being to take three points at once, and so to form triangles; to lay down three positions of a map, to be copied at once, &c.
Beam COMPASSES consist of a long branch, or beam, made of brass or wood, carrying two brass cursors, the one fixed at one end, the other sliding along the beam, with a screw to fasten it on occasion. To the cursors may be screwed points of any kind, whether steel for pencils, or the like. It is used to draw large circles, to take great extents, &c. To the fixed cursor is sometimes applied an adjusting or micrometer screw, by which an extent is obtained to extreme nicety. Mr Jones of Holborn has made beam compasses to adjust to the 1/100th of an inch.
Caliber COMPASSES. See Caliber.
Clockmaker's COMPASSES are joined like the common compasses, with a quadrant, or bow, like the spring compasses; only of different use, serving here to keep the instrument firm at any opening. They are made very strong, with the points of their legs of well tempered steel, as being used to draw lines on paste-board COMPASSES, or copper.
Cylindrical and Spherical COMPASSES, consist 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, thickness, or caliber of round or cylindric bodies; such as cannon, pipes, &c.
Elliptic COMPASSES. Their use is to draw ellipses, or ovals of any kind: they consist of a beam A B about a foot long, bearing three cursors; to one of which may be screwed points of any kind: to the bottom of the other two are riveted two sliding dovetails, adjusted in grooves made in the cross branches of the beam. The dovetails having a motion every way, by turning about the long branch, go backwards and forwards along the cross; so 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 same dovetail has got back the whole length of the branch. Understand the same of the other dovetail.
Note, the distance between the two sliding dovetails is the distance between the two foci of the ellipse; so that by changing that distance, the ellipse will be rounder or flenderer. Under the ends of the branches of the cross are placed four steel points to keep it fast.
The use of this compass is easy; by turning round the long branch, the ink, pencil, or other point, will draw the ellipse required. Its figure shows both its use and construction.
German COMPASSES have their legs a little bent outwards, towards the top; so that when shut, the points only meet.
Hair COMPASSES are so contrived within side by a small 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 shaft of a plane, cleft at top, as far as half its length; with this they measure the angles, &c. of jewels and precious stones, as they cut them. There is in the cleft a little brass rule, fastened there at one end by a pin; but so that it may be moved in the manner of a brass level: with this kind of square they take the angles of the stones, laying them on the shaft as they cut them.
Proportional COMPASSES are those whose joint lies between the points terminating each leg: they are either simple or compound. In the former sort the centre is fixed, so that one pair of these serves only for one proportion.
Compound proportional COMPASSES consist of two parts or sides of brass, which lie upon each other so nicely as to appear but one when they are shut. These sides easily open, and move about a centre, which is itself moveable in a hollow canal cut through the greatest part of their length. To this centre on each side is affixed a sliding piece A of a small length, with a fine line drawn on it serving as an index, to be set against other lines or divisions placed upon the compasses on both sides. These lines are, 1. A line of lines. 2. A line of superficies, areas, or planes. 3. A line of solids. 4. A line of circles, or rather of polygons to be inscribed in circles. These lines are all unequally divided; the three first from 1 to 25, the last from 6 to 20. Their uses are as follow:
N n 2
By By the line of lines you divide a given line into any number of equal parts; for by placing the index against 1, and screwing it fast, if you open the compasses, then the distance between the points at each end will be equal. If you place the index against 2, and open the compasses, the distance between the points of the longer legs B B, will be twice the distance between the shorter ones C C; and thus a line is bisected, or divided into two equal parts. If the index be placed against 3, and the compasses opened, the distances between the points will be as 3 to 1, and so a line is divided into three equal parts; and so you proceed for any other number of parts under 10.
The numbers of the line of planes answer to the squares of those in the line of lines; for because surfaces or planes are to each other as the squares of their like sides; therefore, if the index be placed against 2 in the line of planes, then the distance between the small points will be the side of a plane whose area is one; but the distance of the larger points will be the like side of a plane whose area is two; or twice as large. If the index be placed at 3, and the compasses opened, the distances between the points at each end will be the like side of planes whose area are as 1 to 3; and so of others.
The numbers of the line or solids answer to the cubes of those in the line of lines; because all solids are to each other as the cubes of their sides or diameters: therefore, if the index be placed to number 2, 3, 4, &c. in the line of solids, the distance between the lesser and larger points will be the like sides of solids, which are to each other as 1 to 2, 1 to 3, 1 to 4, &c. For example: If the index be placed at 10, and the compasses be opened so that the small points may take the diameter of a bullet whose 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.
Lastly, The numbers in the line or circles are the sides of polygons to be inscribed 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 compasses at either end, when opened to the radius of a given circle, will contain the side of a hexagon, or divide the circle into six equal parts. If the index be placed against 7, and the compasses opened so that the larger points may take in the radius of the circle, then the shorter points will divide the circle into seven equal parts for inscribing a heptagon. Again, placing the index to 8, and opening the compasses, the larger points will contain the radius, and the lesser points divide the circle into eight equal parts for inscribing an octagon or square. And thus you may proceed for others.
Proportional Compasses with the Sector Lines. The structure of these is so like that of the common proportional compasses, only a little nicer, that it needs no particular description. 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 sines to 90°, and a line of tangents to 45°. On one side are the tangents from 45° to 71° 34′; on the other, secants from 0° to 70° 30′.
For the use of these compasses: 1. To divide a line into any number of equal parts less than 100: divide 100 by the number of parts required; flip the cursor till the line on the sliding dovetail be against the quotient on the line of lines: then, the whole line being taken between the points of the compasses most remote from the centre, the aperture of the other will show the division required. 2. A right line given, supposed to be divided into 100 parts, to take any number of those parts; flip the line on the sliding dovetail to the number of parts required: the whole line being taken between the points farthest from the centre, the aperture of the other two will include the number of divisions required. 3. The radius being given, to find the chord of any arch under 60°; flip the line on the sliding dovetail to the degrees required on the line of chords: the radius being taken between the points farthest from the centre of the cursor; the aperture of the other line will be the chord required, provided the number of degrees be greater than 29°; if it be less, 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 dovetail to the degrees given on the line of chords: the given chord being taken between the two points next the cursor, the aperture of the other will be the radius required. 5. The radius being given, to find the sine of any number of degrees; flip the line on the dovetail to the degree on the line of sines whose fine is required: the radius taken between the points furthest from the cursor, the aperture of the other will give the sine of the angle required. But if the sine sought be less than 30°, the difference of the apertures of the opposite points will be the sine 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 dovetail to the degree proposed on the tangent line; the radius taken between the points farthest from the cursor, the aperture of the others will be the tangent of the degrees required: if the tangent required be above 26° 30′, but under 45°, the line on the cursor must be flipped to the degrees given on the tangent line: then the radius being taken between the points furthest from the cursor, the aperture of the others will be the tangent. If the tangent required be greater than 45°, but less than 56° 20′, flip the notch on the tangent side of the turned cheek to the degree o in the tangent line on the side of the compass; the radius taken between the points farthest from the cursor; the difference between the aperture of the other and these, added together, will be the tangent required. Thus, for the tangents of other degrees under 71°. After the like manner may the secant of any number of degrees under 71° be found.
Mr Heath, a mathematical instrument-maker in London, constructed a pair of proportional compasses, in 1746, with a curious and useful contrivance for preventing the shorter legs from changing their position, when these compasses were used. It consisted of a small beam foldered to a screw, and running parallel to the leg of the compasses, nearly of the length of the groove; in this beam a slit was made, which admitted of a sliding-nut, the other end of which fell into a hole in the bottom of the screw, belonging to the great nut. of the compasses. The screw-pin of the beam passed through an adjuster, by means of which the mark on the slider might be brought exactly to any division. But the proportional compasses have been much out of use since the invention of the sector.
Spring Compasses, or dividers; those with an arched head, which by its spring opens the legs; the opening being directed by a circular screw fastened to one of the legs, and let through the other, worked with a nut. These compasses are made of hardened steel.
Trifling Compasses consist of two central rules, and an arch of a circle of 120 degrees, immovable, with its radius; which is fastened with one of the central rules like the two legs of a sector, that the central rule may be carried through all the points of the circumference of the arch. The radius and rule should be as thin as possible; and the rule fastened to the radius should be hammered cold, to attain the greater elasticity; and the breadth of the central rule should be triple that of the radius: there must also be a groove in this rule, with a dove-tail fastened on it for its motion, and a hole in the centre of each rule. The use of this instrument is to facilitate the trification of angles geometrically; and it is said to have been invented by M. Tarragon for that purpose.
Turn-up Compasses. The body of this instrument is like the common compasses: but towards the bottom of the legs, without-side, are added two other points besides the usual ones; the one whereof carries a drawing pen point, and the other a port-crayon, both adjusted so as to turn round, and be in the way of use, or out of it, as occasion requires. These compasses have been contrived to save the trouble of changing the points.