VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 1^0 



Each of these may be expressed differently in a series, either by the sine 

 and cosine conjointly, or by either of them separately. 



Thus Y the sine of the multiple arch n X a, may be in either of these two 



forms, viz. 



z»-' . . n— 1. n-2 y' , «— 3. n— 4 y* « „ 



= — rV into n — ■ — -— A. •^ H r— —-- B.~ — OfC. 



r—i^ 2 3 z' ' 4 5 z* 



nn—l 3 nn— 9 . nn— 25 , . 



o"- = "i/ - T3;7 *^ - T^ "^y - tttt ""y - *^"- 



Where the letters a, b, c, &c. stand, as usual, for the co-efficients of the 

 preceding terms. 



The first of these theorems terminates when n is any integer number; the 

 other, which is Sir Isaac Newton's rule, and is derived from the former by sub- 

 stituting Vrr — yy for z, terminates when n is any odd number. 



The cosine z may, in like manner, be in either of these two forms, viz. 



2» • , ".'»— 1. y' 1 n. n— 1. n— 2, n— 3, y* » 



nn 3 «»— 4 4 n»— 16 g . 



2rr " 3.4rr " 5.6rr " 



The latter of which terminates when the number n is even, and the other 

 as before, when it is any integer. 



Corol. 2. Hence the sine, cosine, and tangent of any submultiple part of 



an arch, suppose - a, may be determined thus: 



_i^ j^ 



The tangentof i- a will be r+TK^IZlhf . 



r+TJ" 4- r— t| 



I 



The sine of ^a will be i+ifcinlCj. 



2r" 



For these equations will arise from the transposition and reduction of the 

 former, for the tangent and sine of the multiple arch, on the substitution of 

 /, y, z and A; for t, y, z and n X a. 



Corol. 3. Hence regular polygons of any given number of sides may be 

 inscribed within, or circumscribed without, a given arch of a circle. For if 

 the number n express the double of the number of sides to be inscribed within, 

 or circumscribed about, the given arch A; then one of the sides inscribed will 

 be the double of the sine, and one of the sides circumscribed the double of 

 the tangent of the sub-multiple part of the arch, viz. -a. 



Lemma Il.—-ToJind the Length of the Arch of a Circle within certain Limits, 

 by means of the Tangent and Sine of the Arch. — Let t be the tangent, y the 

 sine, and 2 the cosine of the arch a, whose length is to be determined; and 



A A 2 



