[ 7 6 4 ] 
terms, wherein thefe two quantities enter, will 
ftand thus; — a — — i| x Log. i — qx 
— a -j- v/ ace. — ^ X 1 ^ •*' 
But, if ^ be afiumed to exprefs the co-fine of an 
arch (£>J, m times as great as that whofe co- 
fine is here denoted by ct ; then will A — \/ A A — i 
= * cc — \/aa — il ”, and A + \/ A A — i = 
* Becaufe 
and 
- are known to exprefs the 
•y 1 ■ — • xx *y i — xx 
fluxions of the circular arcs whofe co-fines are * and X, it is evi- 
dent, if thofe arcs be fuppofed in any conftant ratio of i to n, that 
n x 
X • 
n x 
V 1 
(= 
X X 
•y — iXy/l 
XX 
and confequently that 
</—[ x yy^Acxj ' y xx — i- 
4/ I — xx 
»*.' x A 
— jzzri x s / 1 — xx) 
V X X I 
X 
From whence, by taking the fluents, n x Log. x -f- y xx ■ — i (or Log. 
x V'fr - il") = Log. X + y XX — i ; and confequently 
x _p s/ xx - w — X-\- y XX - i : whence alfo, feeing x-*/xx- i 
is the reciprocal of at -f- V xx — i, and X — s/ XX — i of X -f- 
\/ XX- i, it is likewife evident, that x - V xx - 1 1 = X- \/ XX- i . 
Hence, not only the truth of the above aflumption, but what has been 
advanced in relation to the roots of the equation z ’ — i =: o, will ap- 
pear manifeft. For if *• + ^ xx — i be put = s, then will z" ( = 
x + s/ XX — tl”) =r X Hr y XX — I : where, afluming X = i = 
co-f. o r= co-f. 360° co-f. 2 X 360° =r co-f. 3 x 360°, &c. the equa- 
tion will become z'’— 1 , or z n — 1 — o ; and the different values of x, 
in the expreffion {x + y xx — 1 ) for the root z, will confequently be 
&c. thefe arcs being 
„ r 1 ° 36 ° 
the co-fines of the arcs, — • 
n 
n 
11 
