analytical and geometrical Methods of Investigation. 101 
mentions it as a remarkable paradox, that the series for the arc 
in terms of the sine represents only one arc, viz. the arc less 
than go degrees ; whereas the series for the sine, produced by 
reversion from the former series, exhibits all possible arcs that 
have the same sine. I shall endeavour to solve this paradox, 
which, I think, originated partly from the introduction of geo- 
metrical considerations into an analytical investigation, by which 
the true derivation of certain expressions was concealed. 
It has appeared that the equation — i ) —1 __ ^— z V~x j 
is true, when instead of %■ is put, 8 +z, or 20+ z, .... or n 9 -\-z, 
0 30 2«-fi 
or z, or — % or 9 — 2, 
2 7 2 2 
Now, if the fluxions of these equations are taken, and the equa- 
tions cleared of exponential quantities, there results from each 
the same equation, to wit, z- = Hence, if the symbol 
J denotes the operation by which 
ginal equations from which z‘ = 
strict consequence from fz'=f- 
we are to ascend to the ori- 
is derived, the only 
V 1 — — * X 
X' 
V I—* 1 " 
is that x — (,/— i) '{g V- lV ‘ },or = -(as/— 1)- ‘ 
e — (0-+^:) VZT 7 j 
or generally =3(2 s/ — 1 )~ l { 1 
or 
= ( 2 V /-1 + 
in -}- 1 
• qV — 1 
£ — e 
(2«~J-l) 
seul arc de 90 degres au plus, l’expression du sinus par 1’arc, qu’on peut deduire (par 
la rnechode du retour de suites) de l’expression de l’arc par le sinus, represents 
exactement, etant poussee a 1’infini, le sinus de tous les arcs possibles, plus petits 
45 ou plus grands que go 0 , et meme que la circonference ou demi circonference, prise 
tant de fois qu’on voudra. Je laisse a d’autres geometres, le sola d’eclaircir ce 
rnystere, ainsi que plusieurs autres,” &c. p. 183. 
