C 307 ] 
prefled hy e put ^ for | of the periphery of the circle 
whofe radius is r ; and let the whole fuents of 
T i 
and 
r z 
, — . =, generated whilft z from 0 be- 
V f — Vi — z’’ 
comes I, be denoted by F and G refpecliveiy. 
Then, by what is faid above, F G will be ^ ; 
and, by my theorem for comparing curvilineal areas, 
or fluents, publifhed in the Philof. TranfaSl. for the 
year 1768, it appears that F x G is ~ \c. From 
which equations we find F =: i e — — 2c, 
and Pjz=.\e^\'^e^ — 2 <;• 
Bat m and n being each — i, L is =r F ; there- 
fore I -P 2 — 2 AE, the value of L, from Art. 5. 
is, in this cafe, ■=.\e — i '^e'' ■ — 2c. Confequently, 
in the equilateral hyper boluy the arc AE, whofe abfeifl'a 
BC is=V I w'ill be = |-P-^— + — 2c, 
V 2 V 2 
by what is faid in the article lafl; mentioned. Hence 
the r edification of that arc may be effedted by means 
of the circle and ellipjis ! 
The application of thefe Improvements will be 
eafily made by the intelligent Reader, who is ac- 
quainted with what has been before written on the 
iubjedt. But there is a theorem (demonftrable by 
what is proved in Art. 8.) fo remarkable, that 1 
cannot conclude this difquihtion without taking no- 
tice of it. 
1 2. 
R r 2 
