TRACT 29.

DIVISION OF THIS QUADRANT.

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viz, as radius : 2 cos. a :: sin. na : sin. (re — 1) x « + s ’ n *(re -f- l) X a ; taking a = 1, radius 10000, the sine of a willbe 1 — '0000000000^, and the cosine of a will be 100000"—'000005 ; then the above proportion will become 100000:200000 — -00001, or 1:2- 0000000001 :: sin. v : sin. (re- l)+ sin. (re -j- l); consequently sin. (n — l) + sin. (re + l) is =2 sin. 7i— '0000000001 sin. re, and the sines are in arithme-tical progression except only for the small difference of'0000000001 sin. 7 1 ; hence sin. (re + 1) is = (2— '0000000001)x sin. 7i — sin. (re — ll; and therefore, taking re succes-sively equal to 1, 2, 3, 4, &c, the series of sines wdl be asfollows :

sin. 1 = 1 --OOOOOOOOOOA ;

sin. 2 = (2 —'0000000001) x sin. 1 ;

sin. 3 = (2 -'0000000001) x sin. 2 - sin. 1 ;

sin. 4 = (2 —'0000000001) x sin. 3—sin. 2;

sin. 5 = (2 -'0000000001) x sin. 4-sin. 3;

&c.

And by this theorem, viz, sin. (re + 1) = (2 —'0000000001)X sin. re — sin. (re — 1), may easily be filled up the intervalsbetween those primary numbers mentioned in former arti-cles.

16. In like manner, as radius : 2 cos. a :: cos. 7ia : cos.(re— 1). a + cos. (re+ 1) . a; and hence this theorem, cos.(re + 1) = (2 — '0000000001) x cos. re —cos. (re— 1), by whichthe cosines will be all easily filled up. And these twotheorems, for the sines and cosines, are so easy and accu-rate, that we need not have recourse to any other, but onlyto check and verify these at certain intervals, as at every100th number, by a proportion from Rheticus’s canon, asmentioned at Art. 11, or by any other way.

17. The sines and cosines being completed, the differencebetween the radius and cosine will be the versed sine; thedifference between radius and sine will be the co-versedsine; and the sum of the radius and cosine will be the sup-plement versed sine.

K 2