Newton, Isaac - Philosophia naturalis principia mathematica

Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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1

Cas.2. Ponamus jam Trapezii latera oppoſita AC& BDnon
eſſe parallela. Age Bdparallelam AC& occurrentem tum rectæ
ST in t,tum Conicæ ſectioni in d.Junge Cdſecantem PQin r,
& ipſi PQparallelam age DM

39[Figure 39]
ſecantem Cdin M& AB in N.
Jam ob ſimilia triangula BTt,
DBN; eſt Btſeu PQad Ttut
DNad NB.Sic & Rreſt ad
AQſeu PSut DMad AN.
Ergo, ducendo antecedentes in
antecedentes & conſequentes in
conſequentes, ut rectangulum PQ
in Rreſt ad rectangulum PSin
Tt,ita rectangulum NDMeſt
ad rectangulum ANB,& (per Caſ.1) ita rectangulum PQin Preſt
ad rectangulum PSin Pt,ac diviſim ita rectangulum PQXPR
eſt ad rectangulum PSXPT. Q.E.D.

Cas.3. Ponamus denique lineas

40[Figure 40]
quatuor PQ, PR, PS, PTnon
eſſe parallelas lateribus AC, AB,
ſed ad ea utcunQ.E.I.clinatas. Ea
rum vice age Pq, Prparallelas
ipſi AC; & Ps, Ptparallelas
ipſi AB; & propter datos angu
los triangulorum PQq, PRr,
PSs, PTt,dabuntur rationes
PQad Pq, PRad Pr, PS
ad Ps,& PTad Pt; atque adeo rationes compoſitæ PQXPR
ad PqXPr,& PSXPTad PsXPt.Sed, per ſuperius de
monſtrata, ratio PqXPrad PsXPtdata eſt: Ergo & ratio
PQXPRad PSXPT. Q.E.D.

LEMMA XVIII.

Iiſdem poſitis, ſi rectangulum ductarum ad oppoſita duo latera Tra
peziiPQXPR ſit ad rectangulum ductarum ad reliqua duo late
raPSXPT in data ratione; punctum P, a quo lineæ ducuntur,
tanget Conicam ſectionem circa Trapezium deſcriptam.

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