GONIOMETRY

, a method of measuring angles, so called by M. de Lagny, who gave several papers, on this method, in the Memoirs of the Royal Acad. an. 1724, 1725, 1729. M. de Lagny's method of Goniometry consists in measuring the 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 measured; produce one of the sides of the angle backwards behind the angular point; then with a pair of fine compasses describe 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 compasses, and turn them successively over upon the arc of the semicircle, to find how often 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 1st 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 he obtains a series of quotients, or fractional parts, 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 BAC be proposed to be measured. Produce BA out towards f; and from the centre A describe the semicircle abcf, in which ab is the measure of the proposed angle. Take ab in the compasses, and apply it 4 times on the semicircle, as at b, c, d, and e; then take the remainder fe, and apply it back upon ed, which is but once, viz at g; again take the remainder gd, and apply it 5 times on ge, as at h, i, k, l, and m; lastly, take the remainder me, and it is contained just 2 times in ml. Hence the series of quotients is 4, 1, 5, 2; consequently the 4th or last arc em is 1/2 the third ml or gd, and therefore the 3d arc gd is 1/(5 1/2) or 2/11 of the 2d arc ef; and therefore again this 2d arc ef is 1/(1 2/11) or 11/1<*> of the 1st arc ab; and consequently this 1st arc ab is 1/(4 11/13) or 13/63 of the whole semicircle af. But 13/63 of 180° are 37 1/7 degrees, or 37° 8′ 34″ 2/7, which therefore is the measure of the angle sought. When the operation is nicely performed, this angle may be within 2 or 3 minutes of the truth; though M. de Lagny pretends to measure muchnearer than that.

It may be added, that the series of fractions forms what is called a continued fraction. Thus, in the example above, the continued fraction, and its reduction, will be as follow: ; the quotients being the successive denominators, and 1 always for each numerator.