Viviani, Vincenzo - De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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6339 rum ex eodem diametri puncto F: idemque oſtendetur de omnibus alijs ex-
tremis punctis communium applicatarum ad vtraſque diametri partes: qua-
re huiuſmodi ſectiones erunt in totum congruentes: eruntque eiuſdem no-
minis; quoniam cum regula Parabolæ æquidiſtet diametro; Hyperbolæ au-
tem conueniat cum diametro extra ſectionem; Ellipſis verò eidem diametro
intra ſectionem occurrat, hoc eſt ad extremum tranſuerſi lateris, cumque
harum ſectionum diametri ſimul congruant (nam ſectiones ſunt ſimul adſcri-
ptæ) ſi diuerſi nominis eſſent ipſarum regulæ nunquam congruerent, quod
eſt contra hypoteſim. Sunt ergo tales ſectiones congruentes ſimul, & eiuſ-
dem nominis. Quod primò, & c.

Si verò regulæ GOI, HPL infra contingentem BGH nunquam conueniũt,
diſiunctim ſimul procedentes, vt in 26. proximè ſubſequentibus figuris ap-
paret, in quarum primis quatuor, regulæ ſunt parallelæ, in reliquis autem à
contingente BGH ad partes ſectionum ſunt ſemper inter ſe recedentes, eſtq;
regula GOI propinquior diametro quàm HPL; facta eadem conſtructione,
vt ſupra; quoniam latitudo FO minor eſt latitudine FP, & altitudo BF eſt ea-
dem, erit rectangulum BFO ſiue quadratum applicatæ NF in 11Coroll.
prop. I. h. DBE, maius rectangulo BFP ſiue quadrato applicatæ MF in ſectione 22Coroll.
prop. I. h. C, hoc eſt applicata NF erit minor ipſa MF: quare punctum m ſectionis AB
C cadit extra ſectionem DBE: idemque de omnibus alijs punctis ſectionis
ABC ad vtranque diametri partem. Vnde tota ſectio ABC cadit extra ſe-
ctionem DBE; ideoq; tales ſectiones ſunt in totum diſiunctæ (eò quod ſem-
per diſiunctim procedant ipſarum regulæ) & in communi tantùm vertice B
ſe mutuò contingunt. Quod ſecundò, & c.

Sitandem ſectionum regulę GOI, HPL infra contingentem BGH ad par-
tes ſectionum ſe mutuò ſecant in P, vt videre eſt in 9. vltimis figuris; duca-
tur ex P communis ſectionum applicata PFNM ſecans diametrum in F, ſe-
ctionem ABC in M, & DBE in N. Iam cum in ſectione ABC quadratum
applicatæ MF æquale ſit rectangulo BFP, & quadratum applicatæ NF 33Coroll.
prop. I. h. ſectione DBE æquale ſit eidem rectangulo BFP, erunt quadrata MF, NF in-
ter ſe æqualia, hoc eſt ipſæ applicatæ æquales; quare huiuſmodi ſectiones
conueniunt ſimul in puncto M. Eadem omnino ratione oſtendetur has ſe-
ctiones ad alteram quoque diametri partem ſimul conuenire in extremo pũ-
cto R reliquæ ad vnam ſectionum applicatæ ex eodem diametri puncto F:
ergo in duobus punctis M & R, præter in communi vertice B, ſimul conue-
niunt, in quibus patet has ſectiones ſe mutuò ſecare; nam regulæ HL, GI
conueniunt ſimul in vnico puncto P, in quo ſe mutuò ſecantes, hinc inde di-
ſiunctim procedunt, cadens PH ſegmentum regulæ LPH remotius à diame-
tro BF, quàm PG ſegmentum regulæ GOI; ideoque & ſegmentum ſectionis
ABC ſupra applicatam MR totum cadet extra ſegmentum ſectionis DBE
ſupra eandem applicatam; è contra verò reliquum portionis ABC infra ap-
plicatam MR cadet totum intra reliquum portionis DBE infra eandem ap-
plicatam, cum ſegmentũ PL propriæ regulæ HPL diſiunctum ſit, & propius
diametro BF quàm ſegmentum PI propriæ regulæ GOI: omneque id oſten-
ditur eadem penitus ratione, ac in ſecunda parte huius Theorematis demõ-
ſtrauimus: quare huiuſmodi coni-ſectiones per vertices ſimul adſcriptæ, &
quarũ regulæ ſe mutuò ſecant infra contingentem ex vertice, in ipſis

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