Fragmenta ex Adversarits depromta. 237 



i. Si desiderenlur duo tales numeri, ut sit ab{a-i-b) = l, ponatur a = ab eritque a6'(a-i- 1) = 1, «ive 

 6'=— rr) sicque a(a-i-l) debet eaae cubus, quod fieri nequit. 



2. Si tres de«iderentur numeri «6« (a -i- 6 -4- c) = 1 , ponatur a = a(6-t-c), ideoque 



oH-6-*-c = (l -i-a) (6-f-c), ergo a (an- 1) 6e (6 -♦-«)'= 1. 

 Nunc ponatur 6 = jffc eritque 



(6-l-c)*=cr(i-i-/?)^ ideoque a/? (a-H 1) (/?-♦- 1)* c*=i, sive -V = a/?(a-*- 1 )(/?-•- i)*. 

 Sumatur a = /3/?-f-2^, et debet esse 



-1- = /?/?(/? -f- 2) (5 -Hi)*, sive -,^^=/?/?(^-i-2). 



Sit ;5=j)p-2, unde c«(^h-1)4 = ^ ^^""^^^^ ^^^^ cc(p-*-l)' "^P^PP"^^' 



quod fit quadratum I. si sumatur p = 2; tum enim erit -— — — = 2: deinde /? = 2, a = 8, c=-, ft=-r-j 



C(/)-4-l) b w 



..11 9 2 . 



a = 4. Consequenter tres numeri quaesiti 4, — ? — , quorum summa est — et productum -— . Deinde 



9 9 49 1 21 49 



p{pp—2) fit quadratum sumendo p = — . Erit enim p(pp— 2) = — .— , hinc -y^ — — =— , porro /? = — > 



49.81 8.16 ^ 7.S , . 49.81.8 7.27 ^ . ... 7.27 , 7.8 



« = 16^' '^ =21:65' *=3:65' ^^™^"^ «=16^=1672' ^^« ^'^^* ""'"^" ^"^^"^^^ « = -32"' ^^Os' 



8.16 65^ A , 21.32 



* = 2r65' ^"^^"™ summa ^^-^, productum vero -^^- 



Alia Solutio. Sumatur /?=pp — i, ut fiat -^-4 =« (a-l- i) (pp — i), jam sumatur a=p — 2, unde 



c p 



--i-=(p — 2) (p-Hi)(p— i)*, staluatur (p — 2) (p-+- i) = (i)-i- i)*??, unde p — 2 = (p-i- i) 99 et j) = |l^, 

 c* p* 1 — 99 



p -t- 1 = , p — l = i ^i ideoque —^ ^sr- Superest ergo reddi quadratum 3q{i~t-2qq), quod 



11 1 3 



manifesto fit sumto 0=77; ^^inc — = 2, p = 3, a=i, /? = 8, c = — , ergo tres numeri sunt a= — , 

 ■* 2 cp b 2 



41 1 1 



fc=— , c= — , quorum summa est = 3 et productum = — . Deinde solutionem praebet positio q = —. 

 3 6 «> 24 



Aliter, sumto statim a=i, fit -^^=2/?(/?-*-i)^; sumatur p = 2pp, fiet 



c p 



\ 



— = 2»(2»«-Hi), cui satisfacit p = 2. 

 ecpp i -^ J.L t 



3. Si desiderentur quatuor numeri, ut sit a6crf(a-H JH-c-Hd) = i, ponatur 



a = a(6-i-c-4-rf), b = ^{c-+-d) et c = yd, 



unde b = p{y-*-\)d, a = a (/?-f- i) (y-f- i) d et summa omnium (a-t-i) (j8-i-i) ^-f- i) rf, productum vero 

 a^y {^ -i- i) {y -^- i)^ d* . Debet ergo esse 



a/?y (a-H 1) (/?-4- i)My-*- l)'<i'= i- 

 Sumatur y = /? eritque a(3(3{a-^i) {fi-^i)^d^=i, ideoque (/?-i-i)^rf'= ■ ^^ » 



Sumaturporro (/3^i)rf=-^ fietque t-^'^^^^ /? = "-^^', 



ubi a el fc pro arbitrio sumi possunt; tum autem habebitur /3= tj ; hinc .y jji',i>nJ 



