Newton, Isaac - Philosophia naturalis principia mathematica

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LIBER
PRIMUS.

PROPOSITIO XX. PROBLEMA XII.

Circa datum umbilicum Trajectoriam quamvis ſpecie datam deſcribe
re, quæ per data puncta tranſibit & rectas tanget pofitione datas.

Cas.1. Dato umbilico S,deſcribenda ſit Trajectoria ABCper
puncta duo B, C.Quoniam Trajectoria datur ſpecie, dabitur ra
tio axis principalis ad diſtantiam

31[Figure 31]
umbilieorum. In ea ratione cape
KBad BS,& LCad CS.Cen
tris B, C,intervallis BK, CL,de
ſcribe circulos duos, & ad rectam
KL,quæ tangat eoſdem in K&
L,demitte perpendiculum SG,idemque ſeca in A& a,ita ut ſit
GAad AS& Gaad aS,ut eſt KBad BS,& axe &c. Aa,verticibus
A, a,deſcribatur Trajectoria. Dico factum. Sit enim Humbilicus
alter Figuræ deſcriptæ, & cum ſit GAad ASut Gaad aS,erit di
viſim Ga-GAſeu Aaad aS-ASſeu SHin eadem &c. ratione,
adeoQ.E.I. ratione quam habet axis principalis Figuræ deſcribendæ
ad diſtantiam umbilieorum ejus; & propterea Figura deſcripta eſt
ejuſdem ſpeciei cum deſcribenda. Cumque ſint KBad BS& LC
ad CSin eadem ratione, tranſibit hæc Figura per puncta B, C,ut
ex Conicis manifeſtum eſt.

Cas.2. Dato umbilico S,deſcribenda ſit Trajectoria quæ rectas
duas TR, tralicubi contingat. Ab umbilico in tangentes demitte
perpendicula ST, St& produc ea

32[Figure 32]
dem ad V, v,ut ſint TV, tvæ
quales TS, tS.Biſeca Vvin O,
& erige perpendiculum infinitum
OH,rectamque VSinfinite pro
ductam ſeca in K& kita, ut ſit
VKad KS& Vkad kSut eſt
Trajectoriæ deſcribendæ axis prin
cipalis ad umbilieorum diſtantiam.
Super diametro Kkdeſcribatur
circulus ſecans OHin H; & umbilicis S, H,axe principali ipſam
VHæquante, deſcribatur Trajectoria. Dico factum. Nam biſeca
Kkin X,& junge HX, HS, HV, Hv.Quoniam eſt VKad KS
ut Vkad kS; & compofite ut VK+Vkad KS+kS; diviſimque

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Scan	Original
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