TRACT 21.

LOGARITHMS.

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as ab : 2ae

or ab : 2be

now because of the equiangular triangles abe, TLfl, km b,alt, AMV, &c, it will be ab -• AE :: KT : s a (=alv + £Ls) ::lt : t a (=amv+aks,) and ab : be :: lt : La (= 4 ls—alv)

:: kt (=4km) : 4m b (=4mv — 4ks.) Hence are given thesines and cosines ks, mv, ls, lv. And the method of conti-nuing the progressions is evident. Namely,

i LV : MT -f MX : ; MX : NV 4- NY &C,

( MV : NX + LT : : NX : OY + MV StC,

( Lv : nx — lt :: MX : oy — mv &c,

■ ■ ( MV : MT — MX : : NX : NV — NY &c.

And, on the other hand,*AB : 2ae :: ls : KT + KR &c.Therefore put ab^I, and make be x LT = Lfl, aex kt = Sti,sa — La = lv, 2ae x lv — tm = mx, &c.

The sense of these general theorems is this, that if p be anyone among a series of angles in arithmetical progression, theangle d being their common difference, then as radius or

1 : 2 COS, d : : ' C0S ' P : C0S ‘ P + d + cos * p ~

1 sin. p : sin. p + d 4- sin. p — d,

1 : 2 sin. d:: $ COS ' P : sin> p + d ~ sin - p ~ d \

( sin. p : cos. p + d — cos. p — d;

where the 4th terms of these proportions are the sums or dif-ferences of the sines or cosines of the two angles next less andgreater than any angle p in the series ; and therefore, sub-tracting the less extreme from the sum, or adding it to thedifference, the result will be the greater extreme, or the nextsine or cosine beyond that of the term p. And in the samemanner are all the rest to be found. This method, it is evi-dent, is equally applicable, rvhether the common differenced, or angle A, be equal to one term of the series or not: whenit is one of the terms, then the whole series of sines and co-sines becomes thus, viz, as 1 : 2 cos. d ::

sin. d : sin. 2 d :: sin. 2 d: sin r + sin. 3 d : : sin. 3 d : sin. 2rf+ sin* 4 d &c.cos. d : 1 + cos- 2 d :: cos. 2 d: cos. d + cos r 3 d :: cos. 3d : cos. 2 d + cos. 4 d 8fc.which is the very method contained in the directions given byAbraham Sharp , for constructing the canon of sines.

Sir I. Newton remarks, that it only remains to fiud the sineand cosine of a first angle a, by some other method; and for