GENIOMETRY, a method of measur ing angles with a pair of compasses, and that without any scale whatever, except an undivided semicircle. Thus, having any angle drawn upon paper to be mea sured; produce one of the sides of the angle backwards behind the angular point; then with a pair of fine compasses de scribe a pretty large semicircle from the angular point as a centre, cutting the sides of the proposed angle, which will intercept a part of the semicircle. Take then this intercepted part very exactly between the points of the compasies, and turn them successively over upon the arc of the semicircle, to find how of ten it is contained in it, after which there is commonly some remainder : then take this remainder in the compasses, and in like manner find how often it is contained in the last of the integral parts of the first arc, with again some remainder : find in like manner how often this last remainder is contained in the former ; and so on continually, till the remainder become too small to be taken and applied as a measure. By this means is obtained a series of quotients, or fractional carts, one of another, which, being properly reduced into one fraction, give the ratio of the first arc to the semicircle, or of the proposed angle to two right angles, or 180 degrees, and consequently that angle itself in degrees and minutes. Thus,
suppose the angle B A C (plate VI. Mis cellany, fig. 4.) be proposed to be mea sured. Produce B A out towards f, and from the centre, A, describe the semicir cle a b e f, in which a b is the measure of the proposed angle. Take a b in the compasses, and apply it four times on the semicircle, as at b, c, tr, and e; then take the remainderf e, and apply it back upon e d, which is but once, viz. at g; again, take the remainder g d, and ap ply it five times on g e, as at h, i, k, 4 and ; lastly, take the remainder m e, and it is contained just two times in m 1. Hence the series of quotients is 4, I, 5, 2 ; consequently the fourth, or last arc, e m, is the third, m 1 of 8. d ; and therefore 1 , t the third arc g d, is 5i or of the second arc, e f; and therefore, again, this 1 second arc, e f, is --, or 11 of the first TT arc, a b ; and consequently this first arc, a b, is 1 — , or of the whole semi circle a f. But *Ids of 180° are or 37° 8' 34,r, which therefore is the mea sure of the angle sought.