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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
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PRIHOTPTA MATTIEMATICA
PROPOSITIO ZZ. PROBLEMA XII.
Circa datum umbilicum Trajectoriam quamvis specie datam de scribe-
re, quae per data puneta transibit e rectas tans et positione datas.
Cas. T. Dato umbilico S, describenda sit Frajectoria ABC per
puncta duo B, C. Ouoniam Frajectoria datur specie, dabitur ra-
cio axis principalis ad dictantiam:t
umbilicorum. In ca ratione cape
RB ad BS, _E L C ad CS. C CN- :
eris B, C, interrallis BR, CL, dc.
Tcribe circulos duos, ex ad rectam
R I, quae tangat cordem in  e
L demitre perpendiculum SC, idemque C
SA 2d AG _ 5 a 2d a G, ut CA SB ad BR, _ aXc
A, a delcribatur Trajectoria. Dico Factum. Sic Cum E umbilicus
alter Figurae descripex, ec cum sIC S Aad AC ue dd ad a Cr, crie di
visim S a =S A sCu SFT ad a G - AC Ceu La III Cadlem ratione,
adeoque in ratione quam habet axis principalis lsguta dlescribendae
ad distanciam umbilicorum ejus; ee proprerca FIgura Clcripta cA
ejusdem speciei cum describenda. Cumque SiInE R B dd ∞ _ L. C
ad CS in eadem ratione, cransibit haec Figura per puncta B, C ut
ex Conicis manifestum cst.
Cas. 2. Dato umbilico S, describenda sit Trajectoria quae rectas
duas T R, er alicubi concingac. Ab umbilico in cansentes demicte
Derpendicula ST, S+ e produc ca-
dem ad V, V, uc SInE TV, +V ∞-
quales F5,+5. Bilcca V V in O.
de erise perpendiculum infinicum
DFI, rectamque VS intinice pro
ductam seca in H ec R ita, UE (IL
K ad IFS _ VK ad RS uE CA
Prajectorix describendae axis prin-
cipalis ad umbilicorum distantiam.
Super diametro RR describatur
circulus sCCanS OFY in FL; de umbilicis S, FZ, axe principali ipsam
Il aequance describatur Prajectoria. Dico Factum. Nam biscca
(Ck in X, ee junge IYX, TTS, LTV, IIV. Ouoniam eAt V X 2d 5
AE V H ad LS; ee COmPOHICUEY L + Y h ad KS + kS: divisimque
b il
bus
a
F. IBFH
P RI N BS