216 L. EULERI OPERA POSTHUMA. Arithmem. 



t .-.,.r.,*.. 'i' ..<.r.;^<t mJT. iii.r';! ««H-ww-t-l . . yzz-t-y^-t-y ... 



«== ^ ? hmc qq = - ^ — 1. 



Statuatur nunc y = nz fietque qq = nzz~t-n^zz~i~n — l. Capiatur ergo n=2, ut sit y=2z, erit 9^=10jz'^-f-l, 



cui «atisfit : 



' ' .0 ^Q y ^ 29 7 



1) Si z=— ; tum erit qq=--, ergo g=-r» Porro i/=— , unde a; = — et p=-7' Tum igitur formula 



/49 A* /49 A2 .. - , ,. . 841 . 16 1097 



f — -f- 1 j -i-( — -+-1 j , erit quadratum radicis xX'+-yy = ~-^ — =^^' 



rtv c. . 2 ., 121 , . 11 / 4 , 52z-f-l 101 



2) humatur ^=q^> «"1 qq = --^, hmc 9 = -9 ; *"»" vero j/ = — et ar = — ^— = — , ergo 



. _101_^ 83 



-<ri^ 'TfMj au)nfiT»«ju|» »h«i»b :iji«|«4r— ^V 81 9 81 MiiBtinh 



181 109 



1092 A^ , /181=^ ..A^ 



i ' 181 109 



* 3) Sumatur 5 = 6, erit ^^=361 el </=19, porro ^=12 et a: = -j— et /)=-—, ergo 



/^1092 A^ ,_, ,,„ /181 =* ...N^* 

 CW-^*) --(3bl-f-l)^=(-^-»-lu) 



A. m. T. I. p. 279. 



tiN' 9^^ immnii iu|iil'M ^"* 



-«ffljniuzs boup .mlitbu. Problema difficillimum, 



quo quatuor biquadrata quaeruntur, quorum summa itidem sit biquadratum, 



^^._-^^j;.,,. ^hmmm; m'. : 1 v. ^^ ^^ ^.^ : ^^_^ g,^ ^*^ D'= E\ ' ''^'*-'" •a;j(iMru.. 



K-iu.M ■,(i4fii *^' - • 11. t'> I " . '. .• ; .'.,.1 -! f 



siquidem fieri potest, sequenti modo tractandum videtur: 



Statuatur scilicet ^z^PP-*-99-*-rr-ss ^ B'=^^, C^=^ et D'=^; tum enim fiet 



1^2 pp-i-qq-t-rr-*-ss 



E= , 



ita, ut haec quinque formulae quadrata reddi debeant. Incipiamus a prima et ultima, ita, ut reddi debeat 



.;.. » pp-t-qq-t-rr _^ ss 



n n ' — '* 



Cum igitur sit aa-i-bh±2ab = O , statuatur 



~ — ^i— _ —aa-\-bb et — = 2ab, 



n n , , 



fit ergo $i = 2nab = n. Sit 2n = a/? et statuatur a = aff et b = (3gg eritque 

 bv ., >:,.>).. .V. bv ,.', ,m, ,.,iJi=T««/^/^/'/"^^' «^g» s = apfg = 2nfg', 

 tiiih 'vefo'-^]se'deb6t' '''^*— *»> »ii«"^^^ -Pllt^lltH^^aaf*-^- /Sfig*. 



Pro reliquis conditionibus faciamus -^ = h-pfg = !i.xx , unde j) = — , porro —=Kqfg = Kyy, hinc q = --, pro 

 quarta — = k-rfg = h-zz, habemus r = — . Hinc superest ista aequatio 



'^"*"^^"*"'' = «ar-4-/?/?gS «vc pp -4- g<3r-f-rr=ia/?(aar-t- /?/?/), 

 et lo(;o p, 9, r valores inventos substituendo 



■^z-^-f!*g2~^^i = ^cc^{(^ccr-^PM^ "ve x*-*-y*-*-z* = \-a^ffgg{aaf*^(ipg*). 



