[ioZ ] 
The Sine T will be z ~r' IJ l 
"l/ 1 
2 r n 
TheCoiinci^ will be z 4*^] ”4 - * — *4 ” 
2 ^ «-i 
Each of thefe may be expreffed differently in a 
Series, either by the Sine and Coiine conjointly, or 
by either of them feparately. 
Thus T the Sine of the multiple Arch A, may 
be in either of thefe two Forms, viz. 
^' h i l + ^=3.!n^_ 
Z. A. C ~4 
# I 
»— I 72 2 
nn— i 
- A ^ s 
3 
» » — 9 
4- 5 r r 
i it f 
B s_JL£21.Cj7-+c. 
or— ny __ — , 
^ .2. 3?*r y 4-5 rr y 6 .jrr 
t it i 
Wherein the Letters A, B C, . Hand, as ufual, 
for the Coefficients of the preceding Terms. 
The firft of thefe Theorems terminates when n is 
any integer Number, the other (which is Sir Ifaac 
Newtons Rule, and is derived from the former by 
fubftituting / r r—yy for z) terminates when n is any 
odd Number. 
The Cojine Z may, in like manner, be in either 
of thefe two Forms, viz. 
mi 
n — i 
nn ’ 
or = r— • — A/ 
irr J 
». »— x 9 y 
1 2 
nn 
•t + *• n JZl- SZ3 in 
2 a A a. 4 & ■ 
3 3 
nn~\6 
C^6 — 
3-4rr " 5 . 6 rr 
The latter of which terminates when the Num- 
ber n is even, and the other as before, when it is 
any Integer. 
Corollary i. Hence the Sine, Cofine, and Tangent 
of any Submultiple Part of an Arch (fuppofe) *A> 
may be determined thus : 
The 
