Viviani, Vincenzo
,
De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei
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rum ex eodem diametri puncto F: </
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<
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xml:space
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tremis punctis communium applicatarum ad vtraſque diametri partes: </
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<
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re huiuſmodi ſectiones erunt in totum congruentes: </
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<
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minis; </
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<
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xml:space
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">quoniam cum regula Parabolæ æquidiſtet diametro; </
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<
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xml:space
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">Hyperbolæ au-
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tem conueniat cum diametro extra ſectionem; </
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<
s
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xml:space
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">Ellipſis verò eidem diametro
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intra ſectionem occurrat, hoc eſt ad extremum tranſuerſi lateris, cumque
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harum ſectionum diametri ſimul congruant (nam ſectiones ſunt ſimul adſcri-
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ptæ) ſi diuerſi nominis eſſent ipſarum regulæ nunquam congruerent, quod
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eſt contra hypoteſim. </
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<
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">Sunt ergo tales ſectiones congruentes ſimul, & </
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<
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dem nominis. </
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<
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">Quod primò, &</
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<
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</
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<
s
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xml:space
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">Si verò regulæ GOI, HPL infra contingentem BGH nunquam conueniũt,
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diſiunctim ſimul procedentes, vt in 26. </
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<
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xml:space
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">proximè ſubſequentibus figuris ap-
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paret, in quarum primis quatuor, regulæ ſunt parallelæ, in reliquis autem à
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contingente BGH ad partes ſectionum ſunt ſemper inter ſe recedentes, eſtq;
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</
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<
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">regula GOI propinquior diametro quàm HPL; </
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<
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">facta eadem conſtructione,
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vt ſupra; </
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">quoniam latitudo FO minor eſt latitudine FP, & </
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">altitudo BF eſt ea-
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dem, erit rectangulum BFO ſiue quadratum applicatæ NF in
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prop. I. h.</
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DBE, maius rectangulo BFP ſiue quadrato applicatæ MF in ſectione
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prop. I. h.</
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C, hoc eſt applicata NF erit minor ipſa MF: </
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<
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xml:space
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">quare punctum m ſectionis AB
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C cadit extra ſectionem DBE: </
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<
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ABC ad vtranque diametri partem. </
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ctionem DBE; </
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<
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">ideoq; </
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<
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">tales ſectiones ſunt in totum diſiunctæ (eò quod ſem-
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per diſiunctim procedant ipſarum regulæ) & </
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<
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ſe mutuò contingunt. </
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<
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xml:space
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">Quod ſecundò, &</
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<
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<
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</
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<
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xml:space
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">Sitandem ſectionum regulę GOI, HPL infra contingentem BGH ad par-
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tes ſectionum ſe mutuò ſecant in P, vt videre eſt in 9. </
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<
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">vltimis figuris; </
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<
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tur ex P communis ſectionum applicata PFNM ſecans diametrum in F, ſe-
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ctionem ABC in M, & </
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<
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<
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xml:space
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">Iam cum in ſectione ABC quadratum
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applicatæ MF æquale ſit rectangulo BFP, & </
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<
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">quadratum applicatæ NF
<
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xml:space
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">Coroll.
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prop. I. h.</
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ſectione DBE æquale ſit eidem rectangulo BFP, erunt quadrata MF, NF in-
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ter ſe æqualia, hoc eſt ipſæ applicatæ æquales; </
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<
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">quare huiuſmodi ſectiones
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conueniunt ſimul in puncto M. </
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<
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">Eadem omnino ratione oſtendetur has ſe-
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ctiones ad alteram quoque diametri partem ſimul conuenire in extremo pũ-
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cto R reliquæ ad vnam ſectionum applicatæ ex eodem diametri puncto F:
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</
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<
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">ergo in duobus punctis M & </
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<
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">R, præter in communi vertice B, ſimul conue-
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niunt, in quibus patet has ſectiones ſe mutuò ſecare; </
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<
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xml:space
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">nam regulæ HL, GI
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conueniunt ſimul in vnico puncto P, in quo ſe mutuò ſecantes, hinc inde di-
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ſiunctim procedunt, cadens PH ſegmentum regulæ LPH remotius à diame-
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tro BF, quàm PG ſegmentum regulæ GOI; </
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<
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<
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ABC ſupra applicatam MR totum cadet extra ſegmentum ſectionis DBE
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ſupra eandem applicatam; </
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<
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">è contra verò reliquum portionis ABC infra ap-
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plicatam MR cadet totum intra reliquum portionis DBE infra eandem ap-
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plicatam, cum ſegmentũ PL propriæ regulæ HPL diſiunctum ſit, & </
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<
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diametro BF quàm ſegmentum PI propriæ regulæ GOI: </
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<
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ditur eadem penitus ratione, ac in ſecunda parte huius Theorematis demõ-
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ſtrauimus: </
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<
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">quare huiuſmodi coni-ſectiones per vertices ſimul adſcriptæ, & </
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quarũ regulæ ſe mutuò ſecant infra contingentem ex vertice, in ipſis </
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