Newton, Isaac - Philosophia naturalis principia mathematica

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LIBER
PRIMUS.

Nam in recta MNdetur punctum N,& ubi punctum mobile
Mincidit in immotum N,incidat punctum mobile Din immo
tum P,Junge CN, BN,

47[Figure 47]
CP, BP,& a puncto
Page rectas PT, PR
occurrentes ipſis BD,
CDin T& R,& fa
cientes angulum BPT
æqualem angulo dato
BNM,& angulum
CPRæqualem angu
gulo dato CNM.Cum
ergo (ex Hypotheſi)
æquales ſint anguli
MBD, NBP,ut &
anguli MCD, NCP;
aufer communes NBD
& NCD,& reſtabunt
æquales NBM& PBT,
NCM& PCR:adeoque triangula NBM, PBTſimilia ſunt, ut
& triangula NCM, PCR.Quare PTeſt ad NMut PBad
NB,& PRad NMut PCad NC.Sunt autem puncta B, C, N, P
immobilia. Ergo PT& PRdatam habent rationem ad NM,pro
indeQ.E.D.tam rationem inter ſe; atque adeo, per Lemma xx,
punctum D(perpetuus rectarum mobilium BT& CRconcurſus)
contingit ſectionem Conicam, per puncta B, C, Ptranſeuntem.
Q.E.D.

Et contra, ſi punctum mobile Dcontingat ſectionem Conicam
tranſeuntem per data puncta B, C, A,& ſit angulus DBMſemper
æqualis angulo dato ABC,& angulus DCMſemper æqualis angu
lo dato ACB,& ubi punctum Dincidit ſucceſſive in duo quævis ſe
ctionis puncta immobilia p, P,punctum mobile Mincidat ſucceſſive
in puncta duo immobilia n, N:per eadem n, Nagatur Recta n N,
& hæc erit Locus perpetuus puncti illius mobilis M.Nam, ſi fieri
poteſt, verſetur punctum Min linea aliqua Curva. Tanget ergo
punctum Dſectionem Conicam per puncta quinque B, CA, p, P,
tranſeuntem, ubi punctum Mperpetuo tangit lineam Curvam. Sed
& ex jam demonſtratis tanget etiam punctum Dſectionem CoNI
cam per eadem quinque puncta B, C, A, p, Ptranſeuntem, ubi pun-

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