290

HISTORY OF

TRACT 19.

are given of three different sections, namely, for 3, 5, and7 equal parts, the forms of which are respectively these,

3c — c 3 ~ g

5c - 5C 3 + C 5 . . = g7c - 14c 3 + c 5 - c 7 — g

where g is the chord of the given arc or angle, and c the re-quired chord of the 3d, 5th, or 7th part of it. And it isshown, geometrically, that the first of these equations has 2real positive roots, the second 3, and the last 4; also, fromthe same principles, the relations of these roots arc pointedout.

The method then annexed for constructing the canon ofsines, from the foregoing theorems is thus : By dividing theradius in extreme-and-mean ratio, is obtained the sine of 18degrees; this quinquisoeted, gives the sine of 3° 36'. Again,by trisecting the arc of 60°, there is obtained the sine of 20°;this again trisected gives that of 6° 40'; and this bisected givesthat of 3° 20': Then, by the theorem for the difference of twoarcs, there, will be found the sine of 16', the difference be-tween 3° 36' and 3° 20': Lastly, by four successive bisections,will at length be found the sines of 8', 4', 2', and l\ Thislast being found, the sines of its multiples, and again of themultiples of these multiples, &c, throughout the quadrant,are to be taken by the proper theorems before laid down.—, And the same subject is still further pursued and explained,in the tract containing the. answer given by Vieta, to the.problem proposed to the whole world by Adrianus Romanus .In the same collection of Vieta’s works, from page 400 to 432,is given a complete treatiso on practical trigonometry, con-taining rules for resolving all the cases of plane and sphericaltriangles, by the Canon Mathematicus, or table of sines, tan-gents and secants.

The next authors whose labours in this way have beenprinted, arc Rheticus, Otho, and Pitiscus : to all of whom weowe very great improvements in trigonometry.—GeorgeJoachim Rheticus, professor of mathematics in the univer-sity of Wittemberg, and sometime pupil to Copernicus , died