TRACT 19. TRIGONOMETRICAL TABLES, &C. 301

gives 2909 for the sine of l'; which may be doubled, tripled,&c , for the sines of 2', 3', &c, up to 45'.

Then, from all the foregoing primary sines, by the theoremsfor halving, doubling, or tripling, and by those for the sumsand differences, the rest of the sines are deduced, to completethe quadrant.

But having thus determined the sines and cosines of thefirst 30° of the quadrant, that is, the sines of the first and last30°, those of the intermediate 30° are, by theor. 4, found byone single subtraction for each sine.

The sines of the whole quadrant being thus completed, thetangents are found by theor. 18,19,22, namely, for one halfof the quadrant by the 18th and 19th, and the other half, byone single addition or subtraction for each, by the 22d theorem.And lastly, by theor. 24 and 25, the secants are deduced fromthe tangents, by addition and subtraction only.

Among the various means used for constructing the canonof sines, tangents, and secants, the writers above enumeratedseem not to have been possessed of the method of differences,so profitably used since, and first of all I believe by Briggs,in computing his trigonometrical canon and his logarithms,as we shall see hereafter, when we come to describe thoseworks. They took however the successive differences of thenumbers, after they we-e computed, to verify or prove thetruth of them; and if found erroneous, by any irregularityin the last differences, from thence they had a method of cor-recting the original numbers themselves. At least, this me-thod is used by Pitiscus , Trig. lib. 2, where the differencesare extended to the third order.—In page 44 of the samebook also is described, for the first time that I know of, thecommon notation of decimal fractions, as now used. And thissame notation was afterwards described and used by baronNapier, in positio 4 and 5 of bis posthumous works, on theconstruction of logarithms, published by bis son in the year1619. But the decimal fractions themselves may be consi-dered as having been introduced by Regiomontanus , by bisdecimal division of the radius, &c, of the circle; and from