210 L. EULERl OPERA POSTHUMA. AnthmettaL 



Ponatur porro oc = nqq et y = nrr, fietque 



n^ qqrrp{qq-t-rr) . ss np (qq-t-rr) 

 gg — z 1 Sive ___liz i. 



nnqqrr—pp ' nnqqrr nnqqrr — pp' 



Ponatur nunc nqr — p =:k{qq-\- rr) eritque p = nqr — k{qq-t- rr) , ergo 



ss nnqr — nk (qq -t-rr) n{k {qq-t-rr) — nqr) 





nnqqrr kCinqr — k{qq-t-rr)) k{k{qq-i-rr) — ^nqr)' 



M- r. T '^•^ rti -^ ** qq-t-rr — ^iqr (? — »*) 

 «r- Casus I. Sit n = 2k, erit = — — -- — J- = —-^ i . 



nnqqrr qq-t-rr — Aqr qq-v-rr — Aqr 

 Sicque quadratum essQ debet gg-i-rr — 4^r, cujus radix ponatur q-^—r, ita ut fiat 



rr^\qr=-j-qr-\--^rr, vel ggr — h-ggq = 2fgq-^ffr. 



vel {gg — ff)r = {h-gg-^2fg)q, sive 4 



Sumatur ergo q = gg — ff et r = h-gg -t- 2fg , eritque 



5, _ {q-r)^ ^ ideoaue " ^ ^-^ ^ ^og-^ff-^^fg ^^ ^ ^ ^Hgg-ff) {^gg-^m{^gg-*-ff-^m 



nnqqrr r f m ^ nqr f gg-*-ff-^4fg - gg-^ff^^fg 



{q~\ r)^ q-\-—r 



^^ g ^ ^ g 



Quocirca ev'\i f^^kqr — k{qq-\-rr) = — k{q — r)"^ , ideoque x = 2kqq, y = 2krr, z=- — — — 



9? OO I ' TT QV j /* 



Casus II. Sumatur n = k, erit — =—, to^- Sit Vfoo-i-rr — or) =o-*- — r, erit 



kkqqrr (? — »*) fif 



rr — qr = —qr-\-—rr, vel qgr — gqq = 2fgq-^ffr, hinc JL —^ ^^ n 

 g gg ^^ ^^^ '^ " r gg-¥-^fg 



Sumatur ergo q = gg — ff ei r = gg-i-2fg, ita ut sit 



f 

 q-i — r 



s g gg-*-fg-*-ff , . .. 

 hmcque ent 



(a;-t-y)pp 



p = kqr — k {qq -t- rr) , vel p = k{qr — qq — rr), indeque y = krr, x = kqq, z = 



xy—pp 



s$ 



4 ^ ., ocyp{x-*-y) ,. ., . ., i' (^ -+-!/) c . ^irj/ ^ ^ 



AnTER. Cum sit ss=-^^ —, dividetur per xv, eritque ss = • Sumatur p= — —, fietque 



xy—pp f y^ ^ l__?^ «-*-y 



xy 

 = -^j— — r^— ■ Superest ergo ut 2xy fiat quadratum. Sumatur x = 2qq et y = rr fietque 



^y^rr^ays^H-rr)* .. 2ar(2w-4-rr) ,. 4ggrr 



''=^(^-7r7^' ^deoque . = ---_-^, hmcque 1> = 2^^, 



indeque a.=2y?, q = rr, ^=,^i!?5^), vel , = S^ir^^C^^ -> . 



^ 'iqqrr — pp {'iqq — rr)* 



ExEMPLUM CASus PRiMi. Sit g = 2 et /*=!, erit 8= k, ^=3, r = 20, p = 17*ft, x=2.9 k, 



„ q^2, 818.174*3 2.174* ^. 



y=2.20^/., ^ = ^5^:j3^,, vel ^= -^, hmc 



_ _« _ 3.13 . _ X _ 43 . 20 _ ^ _ _17^ 

 '^"~ « ~ 17.20' T-~3TT7"' ^— T"~3. 13.20' 



quorum productum est ^^7 et summa .... falsa! 



p. 122 - 127. 



