500 L. EULERI OPERA POSTHUMA. Geometria. 



6o 1-3.5 . * 



co&^2(p = Q ■+- etc. 



cos''^© = 2 . ^' ,' ' cos 29 -*- etc. 

 ^ 2.4.6.8 '^ 



8^ 1.3.5.7 



*^^'^^ = 2:4:6:8 -^ '^'- 



Deinde notetur esse ^(/93 cos 2/19? = ^ sin 2A93 , quod casu 9) = 90° fit =0; unde patet in evolutione omnes ter- 

 minos sin 2Xcp continentes omitti posse, unde nostra formula summatoria erit 



/rf9)(lH-ftcos29r"^=/#(l-i-|;|4^'^-^-lxl^-2-il'^'-^, etc.) 



/17 o -,* I o ^—-f /-j /^ 1 1 7 1-3.5 1.3,„ 1.3.5.7.9 1.3.5,, \ 



/rf9>cos29(lH-ftcos2<p) ^=/rf<jp^-^.-/c- — .-P-^-^^^^._fcs_ etc.) 



consequenter CiV — AM = 



1 l/l ; A 1 7 . 1-3 7, 1.3.5,3 . 1.3.5.7,4 1.3.5.7.9 ,. \ 



-2 "^^yK^- 4 ^-^4-74^^-4X8^-*- 4X0^ - 4.4.8.8.12 ^-*- elc) 



casu ergo, quo « = 00, fit k = \, hic vero valor fieri debet =a, unde sequitur 



2^2 __/ 1 1.3 1.3.5 1.3.5.7 \ 



"Proposita autem vicissim hac serie, ejus valor ita investigari potest. Fiat lt = zz et ponatur 



- 1.3 . 1.3 5.7 8 , ^ 



4.4 4.4 8.8 



1 2 1.3.5 6 1.3.5.7.9 .„ 

 '=T"-^4X8" -*-4-X8Xl2^'-*- ^^'' 



ita Mi s — t praebeat nostram seriem. Hinc erit 



. -•■ ^^-u....i.k .j c , d, 1,3 1.3.5.7 . 



— = — .s^ -f- ■ ■ e ;s^ -H etc. 

 dz A 4.4.8 



• r d,tz 1.3 „ 1.3.5.7 B , d* 



— — = -j- <s* -H -T-r-^ • ;s^ H- etc. =— - 

 dz 4 4.4.8 zda 



hinc zdt -i-tdz = — ' Porro 



d.sz . 1.3.5 4 1.3.5.7.9 „ 



— =1-1- z -I z -H etc. 



dz *^ 4.4 4.4.8.8 



rf.fzz , , 1.3.5 » , z^.d.sz 1 



— — = 1 . ^^ -I — — — z' -I- etc. = — - — 

 di 4.4 dz 



hinc ;22(?< -*- 2fzrfz = ^Vs -H s^'</-z. En ergo has duas aequationes, ex quibus eliminando efo reperilur 



^_(1— 2*)d< (2 — z^) < 



unde ^^ dd^l - ^) _ ^ /1 ^ 3^ \ ^ ^ /^_ 4 ^ ^ N. 



= zzm-^tzdzi 



zdz 



*(!_«) 



unde resultat haec aequatio 



