Philosophiae naturalis principia mathematica . p;a MN detur pun&um N, & ubi pun&um mobileM incidit in immotum N, incidat pun&um mobile D in immo-tum P. Junge C N,BN, CP, BP, &apunclo P age reftasP T, P K occurren-tes ipfis BD, CDin T & R, & faci-cntes angulum B P Tsequalem angulo B-N M & angulumC?K ^qualem an-<mlo CNM. Cumergo ( ex Hypo-thefi ) sequales fintanguliMBD,NBP,ut &anguli MCD,NCP: aufer com- munes NBDScMC P,& reftabunt sequales NB M 8c P B T, NC-M&PCR: adeoq^ triangula NBM, PBT fimilia funf, ut &triangula NCM, ¥CK. Quare P T eft ad N M ut P B ad NB,& P R ad NM ut P C ad NC. Ergo P
Philosophiae naturalis principia mathematica . p;a MN detur pun&um N, & ubi pun&um mobileM incidit in immotum N, incidat pun&um mobile D in immo-tum P. Junge C N,BN, CP, BP, &apunclo P age reftasP T, P K occurren-tes ipfis BD, CDin T & R, & faci-cntes angulum B P Tsequalem angulo B-N M & angulumC?K ^qualem an-<mlo CNM. Cumergo ( ex Hypo-thefi ) sequales fintanguliMBD,NBP,ut &anguli MCD,NCP: aufer com- munes NBDScMC P,& reftabunt sequales NB M 8c P B T, NC-M&PCR: adeoq^ triangula NBM, PBT fimilia funf, ut &triangula NCM, ¥CK. Quare P T eft ad N M ut P B ad NB,& P R ad NM ut P C ad NC. Ergo P T & P K datam babentrationem ad NM^ proindeq; datam rationem inter fe, atq^ adeo3per Lcmma XX, puncTum P ( perpetuus re£tarum mobilum B T& C R concurfus ) contingit feclionem Conicam. Q^_E. D. Et contra, fi punclum D contingit fe&ionein Conicam tranfe-untem per B, C, A, & ubi recla; BM, CMcoinciduntcumrecla B C, punctum illud D incidit in aliquod feclionis ptinclum. Ah ubi vero punclum D incidit iucccffive in alia duo qnsevis fec-tionis puncla p, P, puncTum mobile M incidit fucceffive in puiic-ta immobilia //, N: per eadem #, N agatur recTa n N, Sc ha?c e-rit Locus perpetuus pun&i illius mobilis M. Nam, fi fieri potefl,verfetur puncTum M in Iinea aliqua curva. Tangct ergo punc-tum D fecTionem Conicam per puncTa quinq; C, p> P, £, ^tran-feuntem, ubi pundTum M perpetuo tangit lineam curvam. Sed&: ex jam tanget etiam puncTum D fecTionem Co-nicam per eadem quinq; puncTa C, p, P, B, A tranfeuntem, ubipuncTuni Mperpetuo tangitlineam recTam. Ergo dux fecTionesConicae tranfibunt per eadem quinqj puncTa, contra Corol. XX. Igitur puncTum M verfari in linea curva abfurdumeiT. Q^E. D. TrajeSioriam per data qninq^ punSla pun&a quinq; A7 B, C, D, P. Ab eorum aliquo A2Aalia duo quaevis f>,C, quae poli nominentur, age redTas AByAChifq ; parallelas .TPS,P R £2_per puncTumquartum P. D
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