analytical and geometrical Methods of Investigation. 
generally i ==f — 4 = ~ 1} n an y number of the pro- 
gression o, 1, 2, 3, 4, &c. 
And, since mmml z=z f ~ 2ir ^— 1 ~ _ t x f 4 » ,r ^-' 1 — 
-zW-Ti x _ ls or^ 2M + I l 2 ^ z ^ = ^~(2«+02^^__ 1> 
n any number of the progression o, 1, 2, 3, 4, 5, &c. 
Hence it appears, that if xzzz ^2^/ — 1 ) 1 ^ gzV — t — — i j. ? x % 
— { 2 } -I { f * v ~ - f-*''- 7 } X — (since 
f - 4 B*V. r -i'j (2V /— )-{ ^(4«^+*)^~ (4Jwr+*)VITFJ. 
Again, since f zn+l ') 27rx/ ~ ~ : ~( zn + l ) x 
X X — ■ 1==(2s/37)” i |^\/-i_„ s -zV~ I ^ zn +i) v V-i 
= ( 2 / — — < c,(( 2 « + i)29r — 1 £— ((2«+i)z7r— z)\/in 
consequently, 
X== ( 2\/ — 1 )~“ I I M 2n + I ) 27r z) V 7 j ((2«-fi) 2 ?T— ^-/ZT J. 
or the equation * = ( 2 </- 1 )-{ ^=1 - r^=i } i s true, when 
instead of z is put (4 tv -\-z) or (87 r+2), or generally (4^-j-^) . 
and is moreover true, when instead of % is put 
(27 r— z), ( 6 n—z), or generally (src-J-i) 27r— z. 
In like manner, the equation v/ILTp I j 
is true, when instead of z is put 
47r+», 87 r-j-z, or 12 tt-1-z, or generally 4 nir-\-%; 
and is moreover true, when instead of z is put 
47 r-—z, ,87 r—*z, or 12 tt—z, or generally 4 nn—z. 
Let now x 3 —qx=r, then, by Cardan’s solution, 
r X ~/ if + $) ) + V (t -SCi~f 7 )); 
put s - a ,~— — = —b , then [a-\-b \/ — 1 )-f- 3 v / ^( — b\/H 7J. 
Let a -\~b \/—i=m r y~ .-.a~b s/Zil = m ^vzr x 
MDCCCII, Q 
