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# Title: [Astronomiae Physicae & Geometricae Elementa] Astronomiæ Physicæ & Geometricæ Elementa (shelfmark Leibn. Marg. 124)
# Folio 431, page id 1040780814:0449
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11D. 1.
um sire
B, (II
node
IM PaI.
CH. Fil
dictat
distat
5V,
R
Ecribali
IPOIC
Datiu
Pot.
VM;
ipsiuo
qud 1
atabo
IuICI
Chot
uncta
Bect
15 iM
etet,
ee
S1
+a0
D;
10C
Lab. V.
aE GEOMEFELOE L. LLMF AIA.
5.
hoc cA, ut S+ ad 5D: Ideoque ABad Y2 r Yr in Cubduplicata ratione
SF adl 5D. O. F. D.
PROPOSTTIO XXIII. LEMMA.
T communi foco s describantur duae quaevis Parabolae ACD.
 D ECF, in quibus ductae chordae AB C~ FF abscindtant sermenta
quorum vertices ce ea lee ut ductis rectis se C se praedictas
chordas in DS  secantibus, sit cD aequalis CH; dico a B esse ad P. F
in sub duplicata ratione rectae Sc ad rectam Sc.
Per C ducatur em segmenti a cB diameter, e per C recta CK
Termenti L CF diameter, chordis respe.
etiris in m C x occurrentes. Ilarum
diametrorum parametri aequantur qua-
druplis Sc, Sc, ut superius octensum;
Critque (per Prop. XI IX. Lib. L. Conic.)
AMg = 4 SCKCD, _ F KG = qSGLCF:
Inde AMg eA adFKg SICuEq SCrCD ad
4SGLCH, hoc est (Ob aequales cD e
GH) SICUE SC ad SE; _ isIEUE AM est ad
F. F in Subduplicata ratione sc ad Sc. Vnde elt AB (iptuS aM du
PLa) ad F. F. (ipsiuS P. K duplam) in subduplicata ratione sc ad SC.
PROPOSITTOXXIV.
Trea datum focum Parabolam describere; quae per data duo
puncta transeat.
Sit datus focus s, puncta duo data Pet qua Latavra MMCCerC
dehet a _ B. Centris a _ B, distantiis aS _ BS ducantur duo cir-
culi, e ducatur (per Constr. PrOp. CxVIIT. Lib. VII. P appi C ollect. Math.)
recta eo utrumque continsens, in quam ex s puncto demittatur
perpendicularis sF, quae bi-
secetur in V. Verticer, axc
VSx recta e latere recto x
quali quadrupla rectae rs
(per PrOP. LII. Lib. I. ConiC.
describatur Parabola: Prit
haeC qualIta.
Ouod ssit descriptae Par.
abola focus constat exinde,
quod in axe Parabola di-
stat ab illius vertice quartd parte lateris recti, qua est notistima
foci Parabola proprictas. Ouod vero haec per puncta a e Btranseat,
sic patet: Parabola axis rx ultra verticem productus c ad F, ita
UE YE. Ct aequalis rs distantix foci à rertice; et per P ducta est
cFD recta ad axem Parabola perpendicularis, et punctum a tale
est uE aS ejus distantia a foco aequalis sIt A C, gux (Per PrOP. XVIII.
Fl. II.) perpendicularis est ex a ad cD rectam: Unde (per C Orol. 2.
PFOD. XIV.) Parabola Supradescripta per a trantit, et propter s
miles rationes trantic exiam per B. G. L. F.
Pocu