[ 568 ] 
made above when we took the fluents of the given 
fluxionary equation by logarithms. Therefore if A 
be put for the leaft arc whofe cofine is y, and C for the 
whole circumference,, radius being i ; y being the co- 
line of A, A -f- C, A + 2 C, A + 3 C, &c. x will 
be the cofine of — , Sec. . 
n n n 
A 4- »• — i * C 
to ! . 
n 
l 
Confequently, exprefling the lafl-mention'd co- 
fines, or the feverai values of a*, by p , y, r, s, &c. 
z zn — 2 y z n + i will be =.3* — 2 /> 3 + 1 x 
A — 2 y 3 + 1 x 3* — 2 r 3 + ij &c. (»), when ?i 
is a pofitive integer (as we fliall always iuppofe it to 
be), let 3 be what it will. 
Hence may be ealily deduc’d a demonftration of 
that remarkable property of the circle flffl: difeover’d 
by Mr. Cotes : But as that property has already been 
demonftrated by feverai mathematicians, I fliall omit 
taking any farther notice of it, and proceed in the in- 
veftigation of fome other ufeful theorems which I do 
not find have ever yet been publilh’d. 
II. 
If y be = 1 ; then, A being = o ; />, y, r, &c. 
will be the cofines of , &c. ( n ) refpedl- 
n n n 
ively : Therefore p will be = 1 and, if n be an even 
number, one of the colines y, r, s , &c. will be — — 1, 
one of the arcs , occ. being then = - . 
n 71 
III. 
