Fragmenla ex Adversariis depromta. ^87 



XXIII. 



Coiitiiiiiatio Fragiueiitoriiiii ex A d t e i* s a r i i s iii a t li e iii a t i c i s 



deproiiitoriiiii. 



(Conf. supra pagg. 157 ad 266.) 



1. SuppleiueDta Dumerorum doctriDae. 

 91. 



{LexelL) 



Phoblema. Invenire numeros p, q, r, s, ut haec formula 



/i {pp -+• ss) (q{-t- rr) 



pqrs {pp — ss) (/</ — rr) m 



Gat quadratum. 



SoLUTio. I. Primo ponalur pp -h- ss = {aa -t- bb) [xx -t- yy) et 



qq -^- rr = [cc -t- dd) [xx -h yy) 

 eritque p = ax -i- by et q = cx -t- dy 



s =: bx — ay r =. dx — cy ; 



quo facto, quadratum esse debet haec formula : 



>i (aa -¥■ bb) (cc -i- dd) 

 pqrslp-t-s){p — s){q-t-r){q — r)' 



II. Ut numerus factorum diminuatur, statuatur r = s, sive 



dx — cy = bx — ay, unde fit - = » 



^ ^ y b —d 



fiat ergo x = a — c el y = b — d, 



unde coUigitur p = aa — ac -t- bb — bd, q = ac — cc -t- bd — dd et 



s =. r = ab — bc — ab -t- ad = ad — bc, bincque 

 p -t- s =: aa — ac -t- ad -i- bb — bc — bd, q -\- r =^ ac — bc — cc -\- bd -\- ad — dd 

 p — s =: aa — ac — ad -t- bb -\- bc — bd, q — r = ac -*- 6c — tc h- 6rf — ad — dd. 



Formula ergo quadratum reddenda erit 



/i {eux -*- bb) (cc -*- dd) 

 pq(p-*-s)(p~ s){q-*-r)(q — r)' 



III. Fiat porro p = cc -\- dd, sive cc -\- dd = aa — ac -\- bb — bd, ad qiiam resolvendam statuatur d = a, 

 eritque cc = — ac -\r bb — ba, sive = — cc — ac -\- bb — ab, seu cc -\- ac - bb -\- ab = 0, quae per c -\- b 

 divisa dat a -\- c — 6 = 0, unde fit c =: b — a existente d =^ a. Habebimus ergo 



p =: cc -h- dd, q = {3a — b) {b — a), r = s = aa -t- ab — bb, 

 unde fit p-i-8 = a (3a— 6), p — s = {b — a) {2b — a), q-\- r = {2a — b) {2b — a), q — r = a{'ib — ka). 



Consequenter formula quadratum reddenda erit 



A (aa H- 66) 

 (3a - 6) (6 — a) a (3a — 6) (6 - a) (26 a) (2a 6) (26 — a) a (36 — 4a) ' 



