222 L. EULERI OPERA POSTHUMA. Anthmetica. 



Analysis, qua haec reductio est inventa. Posito 2x*—y*=zz^ debet esse 



%^' ^». t-f-t^^^a;x=pp-*-qq, yy=pp-t-^pq — qq, tum enim fiet z = qq-h-2pq--pp. 



Hic ergo p et q ita definiri debent, ut scx et yy fiant quadrata, quod sequenti modo praeslari potest: Gum sit 



2a 6 



yy — XX =2q{p — q)= {y-*-cc){y — x), iAm statuatur y-i-x = y.q et y — x=—{p — q), Sic enim fiet 



yy-^xx^ 2q {p — q)- Addantur jam quadrata , fiet 



• ,,_,,, ^,2yy-i-2xx=—qq-i--pp--pq-^-^qq. 



At vero ex primis formulis fiet ^yy-h-^xx^k^pp-t-k^pq, qui valor illi aequatus et multiplicatione £acta per aabb dat 



(6*— 4aa66) pp — 2pq {h^-^2,aabb) -^^ {fy^-i-ki*) qq = Q. 



P 



Hinc radicem extrahendo fit —=...,. , 



Theorema. Si fuerit wa* — n6*=c<?, inde assignari potest talis forma x* — mny*=zz. 



Demonstratio. Posito enim ma*-t-nb*=A, erit 44 = c*-+-4mna*6^. At in altera formula si ponatur 

 xx=pp-i-mnqq et yy = 2pq, fiet z=pp — mnqq. Jam statuatur p = rr et q = 2ss, ut fiat y = 2rs, hinc fiet 

 xx=:r*-t-k^mns*. Facta ergo comparatione erit x = A, r = c, s = ab, unde fit y=^2ahc, z = c* — h^mna*h*. 

 Hinc ergo necesse est ut fiat x* — mny*=:zz. 



0- '■ -•*' 



A. m. T. III. p. 129. 131. 



38. 



Observatio. Ut formula ^^^ — ?*- fiat quadratum, sumatur 



rs{rr — ss) ^ 



p= aa -*-bb ' r=aa-t-hb 



q = 2aa — hb 8 = 2hb — aa 



M ^ m ,t M ./ jjjj^^, p~i-q=3aa r-\-s = m 



p — q = 2bb — aa r — s = 2aa — bb 



substituendo fit formula PiiPP-n) ^ aa 



rs{rr — 5«) 66 



Aliter, sumi etiam potest 



p=zbb — 2aa r = 2hb — km 



q == Qaa s = hh -^ k^aa 



hinc p-\-q=zhh-i- kaa r h- s = 366 



, . p — q = bb — Saa r — s = bh — Saa 



u 44 j P9{PP — 99) oa 



ac substituendo: '««>"t<»«^ " ■ , — ^ = — . 



rs {rr — ss) 66 



Ita sumi potest p = T, S = 6, r=14 et s=13, eritque pq {pp — qq) = 5h-& , rs (rr — ss) = 4914 



hinc foronula iuti »««= 77771 



' 4914 



~"9— (t)""* """ 



EvoLUTio GENERALiOR formulac • , ~ , = D = n. 



rs (rr — ss) 



Hic ponatur q = at et s = ^t, tum verojp = t^r, unde reperitur 



A. m. T. I. p. 295. 



