Enodatio instgnis cujusdam paradoxi ctrca multiphcationem anguiorum. 307 



21. Evolvamus etiam casum n = —, ac reperiraus , 



^^^ i Vx /. i -2 1.8 — * 1. 11.14 -6 1.14.17.20 — « \ 



COS-q.= -(1--X -^^X -ia^^e^ - 12.24.36.48 ^ - ^^') 



1 /. 1 — » 1.10 — « 1.1.3.16 — « 1.16.19.22 — « \ 



-*-^l*-*-T2^ -*-i2724^ -*^12T4.T6^ -^ 12.24.36.48 ^ -^^^^•)' 



quae bioae series ita conjungaulur : 





„^„1 1 T 1 — r 1 — T 1 — "T 1.8 — T 1.10 — T 



cos— y = — (E H- -^ — jc j-cc H ^ — X ~x -t — X — etc. 



V4 Vl6 12/4 I2V16 12.24^4 12.24V^16 



Jam ad irrationaiitatem tollendam statuatur x^ = yyh-, seu x = ky^, ac prodibit 



j_ _ 1 1 i_ 1.8 1.10 Liii*i_ _ t~^ 



COSg^C—y-H^^ 12.42y5"*~12.4'y'' 12.24.4*y"-*-12.24.45y»' 12.24.36.4«y*' "*" 



Sit porro y=—} erit 



Jl^ \^Oi«#fi|^ j^ ^^ I g 1.10 _ii**_i£_ 1.13.16 



COS g^p— 2 -*-2z 6«5-^6^7 2.3.62^1"*- 2.3.6«^' 2.3.6T97»' ~^2.3.6.9«»' """ 



o 1 1 1 1 1.8 1.10 1.11.14 1.13.16 



seu 2cos-9P=2-i--~3^-4-~,- 3^, -h ^^, - 3-^-^ -1- 3-^,-, -etc. 



22. In genere autem casus a? = ^> unde fit ^ = 60**, seu ^ = —71;, denotante n semicir- 

 cumferentiam circuli, cuius radius = 1 , omni attentione diffnus videtur. Nam ob 2a3=l, fit . 



cos -^;r = 1 fl --^H-"l^-^>-~-^-'^J"-i:^^ 



3 2 V 1 1.2 1.2.3 1.2.3.4 V 



1 / n n(n-H3) n(nH-4) (n-f-5) n(n-H5) (nH-6)(n-+-7) \ 



"*" 2"^*-*" i"*"-T:2^~*~ rn ' 17273^ ^-eic.j, 



ubi notari convenit utriusque seriei, summam seorslm sumtam esse imaginariam, et quia utraque est 

 divergons, minime licet eas pro lubitu combinare. Veluti si termini ordinate conjungerentur, prodiret 



n . nn 9wi nn (nn -t- 107) 



cos 3 ;t = 1 -+- — -H j-^ -+— j-^^3^ -^- etc. 

 unde sequeretur fore cos— tt^ 1, quod tamen est absurdum. Intcrim tamen binarum illarum 



l/n ^ .n\ . 1/n /j'**A 



prions summa est ^ ( cos ^ tt h- y — 1 .sm— rr j, poslerions vero « ( cos —n — y — 1 • ^10-^:» j> 

 sicque nullum est dubium, quin ambae conjunetim praebcaut cos — /r , etiam si non pateat, quemad- 

 modum hic valor ex conjunctione facta clici possit. Hinc ergo deuuo insignc paradoxon resultat, 

 cujus explicatio haud parum ardua videtur; sine dubio autem ex serierum divergentia est petenda, 

 et series signis alternantibus ita scribenda , terminorum numero neque pari neque impari reputato: 



» n » n n n(3 — n) n(3H-n) n(4 — n)(5 — n) n (4H-n)(5-*-n) 



ita ut sit r,u\i 



