INACCURACY OF SOME LOGARITHMIC FORMULAS. 
173 
Postulates. 
To obviate the necessity of interrupting the course of the argument here- 
after, it may be satisfactory to enumerate the principal truths immediately 
connected with our subject and not immediately evident, which will be taken 
for granted in this paper. 
For their support, the authority of Dr. Lardner’s Trigonometry, Part III. 
Sections 1 and 2, may be referred to. 
Euler’s development of f 0, or 
f 0 = l + v'— 10... 
+ 
(A/^10) n 
1 . 2 ...» 
De Moivre’s theorem, or 
f (x 0) =s a value of (f 0) 1 
f _1 f0 = 2 zV + 0 
De Moivre’s theorem as extended by M. Poinsot, or 
f{j?(2 in + 0)} = (ffl)* 
f(0 + h) = U.fh 
Subsidiary division. 
1st, To possess a development of f 0. 
The development of Euler [7] is accurate and sufficient. 
2nd, It remains to obtain a development of f _1 0. 
Differentiating [6] we obtain 
= V — 1 (cos 0 + */ — 1 sin 0) or — 1 f 0 
[7] 
[ 8 ] 
W 
[ 10 ] 
[ 11 ] 
L12] 
Substituting in [12] f 6 for 6, we obtain 
dff ~ 1 0 . — -„-i. . ff -l. 
= \/— iff 0;or since f f 0 = 
df _1 0 
Hence we find 
df _1 0 
d0 — (V— 10) 
-l 
d 0 
df _1 0 
V - 1 0 . 
[13] 
It is evident by [13], that when 6 becomes = 0, 
df _1 0 
d0 
becomes infinite, and 
