224 Mr. Woodhouse on the Integration 
properties of the sines, and tangents, of circular arcs. Again, 
Fagnani's theorem, by which a right line is assigned equal to 
the difference of two elliptic arcs, affords a method of arith- 
metically computing the length of the ellipse; and, conversely, 
the analytical artifice by which the integral of dx (~~~?) 
is computed; translated into geometrical language, becomes 
Fagnani’s theorem. And again, the analytical resolution of 
f dx J ( ‘~_ 5 ) int ° Au ' + Ba " + v s du ' J [ '-‘S ) + 
QJdu" J ( x ~ e (where the integrals, on account of the 
2 / — (contained between the values of v, o and-- l -r=), which latter series, 
J a/(i-^) V 5 ] 
from the smallness of v, v', converge with considerable rapidity ; or the latter part 
thus, putu — — — ,v‘ 
V(i + *') 
_ rdz rdz' 
-J 1 +Z*' + J l +Z ,z ’ 
Now, if u' — 1 _ = , y — i , 
A/ 2 
if v — , z — i 
V (»+*'*)’ t/(i +:v z ) ; 
then 
z-\-z 
and /r 
dy 
-t-y 
lfv Vio’ Z 3 * 
Consequently, the integral of— ^ — — (between the values of y, o and i) r: / — 
C' dz f 
(between the values of z, o and f) -f J (between the values of z', o and i), 
and consequently,^^^ — ~ (between o and i) ~ z f -f ~~r — j* 
j. 2 ( - — — 1 1 &c. ‘1 which is, in fact, Euler’s method of determining 
l 3 3 - 3 3 5 3 s i 
the periphery of a circle. Now, from this analytical artifice of putting the integral of 
— — f rr + ~zr ; , by which means its arithmetical value is 
VO—**) 7 VO-* 4 ) VVO-«"*) y 
computed, may be deduced th,ose theorems which relate to the sines, and tangents, of 
the sum and difference of arcs. See. by translating the formula k'VO — + 
zt" V' ( i — u! z ) —x into geometrical language. 
