Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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1
Per puncta A, B, C, D& aliquod infinitorum punctorum P,pu­
ta p,concipe Conicam ſectionem deſcribi: dico punctum Phanc
ſemper tangere. Si negas,
41[Figure 41]
junge APſecantem hanc
Conicam ſectionem alibi
quam in P,ſi fieri poteſt,
puta in b.Ergo ſi ab his
punctis p& bducantur in
datis angulis ad latera Tra­
pezii rectæ pq, pr, ps, pt
& bk, br, bſ, bd; erit
ut bkXbr ad bſXbdita
(per Lem. XVII) pqXpr
ad psXpt,& ita (per
Hypoth.) PQXPRad
PSXPT.Eſt & prop­
ter ſimilitudinem Trapeziorum bkAſ, PQAS,ut bkad bſita
PQad PS.Quare, applicando terminos prioris proportionis ad
terminos correſpondentes hujus, erit br ad bdut PRad PT.Er­
go Trapezia æquiangula Dr bd, DRPTſimilia ſunt, & eorum
diagonales Db, DPpropterea coincidunt. Incidit itaque bin
interſectionem rectarum AP, DPadeoque coincidit cum puncto
P.Quare punctum P,ubicunque ſumatur, incidit in aſſignatam
Conicam ſectionem. Q.E.D.
DE MOTU
CORPORUM
Corol.Hinc ſi rectæ tres PQ, PR, PSa puncto communi P
ad alias totidem poſitione datas rectas AB, CD, AC,ſingulæ ad
ſingulas, in datis angulis ducantur, ſitque rectangulum ſub duabus
ductis PQXPRad quadratum tertiæ PS quad.in data ratione:
punctum P,a quibus rectæ ducuntur, locabitur in ſectione Conica
quæ tangit lineas AB, CDin A& C; & contra. Nam coeat linea
BDcum linea ACmanente poſitione trium AB, CD, AC; de­
in coeat etiam linea PTcum linea PS:& rectangulum PSXPT
evadet PS quad.rectæque AB, CDquæ curvam in punctis A& B,
C& Dſecabant, jam Curvam in punctis illis coeuntibus non am­
plius ſecare poſſunt ſed tantum tangent.
Scholium.
Nomen Conicæ ſectionis in hoc Lemmate late ſumitur, ita ut
ſectio tam Rectilinea per verticem Coni tranſiens, quam Circularis
baſi parallela includatur. Nam ſi punctum pincidit in rectam, qua
quævis ex punctis quatuor A, B, C, Djunguntur, Conica ſectio

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