C ]' 
Lemma I. 
The Tangent of an Arch being given, to find the 
Tangent of its Multiple . 
Let r be the Radius of the Circle, t the Tangent of 
a given MdiA, and n a given Number. And let T 
be the Tangent of the Multiple Arch nxA to be 
found. 
Then if gg be put for— rr, and tt for — tt; 
-j — |* - — ,» 
The Tangent T will be — - — — - ? : 
r+T I +r — rl 
Which Binomials being raifed according to Sir- 
Ifaac Newton’s Rule, the fictitious Quantities t and p 
will difappear, and the Tangent T’will become equal to 
71*71 1 # »— - 2 . 
71 1 I 2 3 Y 2 *" I 
n n — i % n — m — 3 . 72 — 4 . t* 
2 
3 
4 
— - &c. 
71 71 - 
I 
tt , 
rr 1 i 
71 71 1 71 - 
i 2 rr i i 3 4 
This Theorem (which I formerly found for the 
Quadrature of the Circle, at a time when it was not 
known here to have been invented before) has now 
been common for many Years 5 for which Reafon I 
fhall premife it, at prefent, without any Proof 5 only- 
for the fake of fome Ufes that have not yet been made 
of it. 
Corollary 1. Erom this Theorem for the Tangent, 
the Sine (fuppofe) T> and Cofine Z of the Multiple. 
Arch n may be readily found. 
For if y be the Sine, and z, the Cofine of the given 
Arch Ai then putting v v for — yy> and fubftituting ~ 
rT 
x 
rv 
for .t, and — for t, and 
Vrr-\-TT 
D d 
forT 
The 
