1
Corol.3. Et concurſus tangentis cujuſvis PMcum recta SN,
quæ ab umbilico in ipſam perpendicularis eſt, incidit in rectam AN,
quæ Parabolam tangit in vertice principali.
quæ ab umbilico in ipſam perpendicularis eſt, incidit in rectam AN,
quæ Parabolam tangit in vertice principali.
PROPOSITIO. XIII. PROBLEMA VIII.
Moveatur corpus in perimetro Parabolæ: requiritur Lex vis centri
petæ tendentis ad umbilicum hujus figuræ.
petæ tendentis ad umbilicum hujus figuræ.
Maneat conſtructio Lemmatis, ſitque Pcorpus in perimetro Pa
rabolæ, & a loco Qin quem corpus proxime movetur, age ipſi SP
parallelam QR& perpendicularem QT,necnon Qvtangenti pa
rallelam & occurrentem tum diametro YPGin v,tum diſtantiæ
SPin x.Jam ob ſimilia triangula Pxv, SPM& æqualia unius
latera SM, SP,æqualia ſunt alterius latera Pxſeu QR& Pv.
Sed, ex Conicis, quadratum ordinatæ Qvæquale eſt rectangulo ſub
latere recto & ſegmento diametri Pv,id eſt (per Lem. XIII.) rectangu
lo 4 PSXPv,ſeu 4 PSXQR; & punctis P& Qcoeuntibus, ra
tio Qvad Qxper (per Corol. 2 Lem. VII.) fit ratio æqualitatis. Er
go Qxquad.eo
24[Figure 24]
in caſu, æquale
eſt rectangu
lo 4 PSXQR.
Eſt autem (ob
ſimilia trian
gula QxT,
SPN) Qxque
ad QTqueut
PSquead SNque
hoc eſt (per
Corol. 1. Lem. XIV.) ut PSad SA,id eſt ut 4 PSXQR
ad 4SAXQR,& inde (per Prop. IX. Lib. v. Elem.) QTque&
4SAXQRæquantur. Ducantur hæc æqualia in (SPq./QR), & fiet
(SPq.XQTq./QR) æquale SPq.X4SA:& propterea (per Corol. 1 & 5
Prop. VI.) vis centripeta eſt reciproce ut SPq.X4SA,id eſt, ob da
tam 4SA,reciproce in duplicata ratione diſtantiæ SP. Q.E.I.
rabolæ, & a loco Qin quem corpus proxime movetur, age ipſi SP
parallelam QR& perpendicularem QT,necnon Qvtangenti pa
rallelam & occurrentem tum diametro YPGin v,tum diſtantiæ
SPin x.Jam ob ſimilia triangula Pxv, SPM& æqualia unius
latera SM, SP,æqualia ſunt alterius latera Pxſeu QR& Pv.
Sed, ex Conicis, quadratum ordinatæ Qvæquale eſt rectangulo ſub
latere recto & ſegmento diametri Pv,id eſt (per Lem. XIII.) rectangu
lo 4 PSXPv,ſeu 4 PSXQR; & punctis P& Qcoeuntibus, ra
tio Qvad Qxper (per Corol. 2 Lem. VII.) fit ratio æqualitatis. Er
go Qxquad.eo
24[Figure 24]in caſu, æquale
eſt rectangu
lo 4 PSXQR.
Eſt autem (ob
ſimilia trian
gula QxT,
SPN) Qxque
ad QTqueut
PSquead SNque
hoc eſt (per
Corol. 1. Lem. XIV.) ut PSad SA,id eſt ut 4 PSXQR
ad 4SAXQR,& inde (per Prop. IX. Lib. v. Elem.) QTque&
4SAXQRæquantur. Ducantur hæc æqualia in (SPq./QR), & fiet
(SPq.XQTq./QR) æquale SPq.X4SA:& propterea (per Corol. 1 & 5
Prop. VI.) vis centripeta eſt reciproce ut SPq.X4SA,id eſt, ob da
tam 4SA,reciproce in duplicata ratione diſtantiæ SP. Q.E.I.