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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 66, page id 1041163592:0100
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PTTILOSODHILA NATORALIS
SE CTTC
Inventio Orbium ubi umbilicus neuter datur.
L E M NI A XVII.
Si a datae Conicae Sectionis puncto quovis P, ad Trapezii alicujus
ABDC, in Conica illa sectione inscripri, latera quatuor infinite
producta AB, CD, AC, DB, totidem recta PO, PR, PS, P Y
in datis anculis ducantur, sinculae ad singula: rectanculum due-
tarum ad opposita duo latera P O Z PR, erit ad rectanculum duc-
tarum ad alia duo latera opposita PS X P P in data ratione.
Cas. F. Fonamus primo lineas ad e
opposita latera ductas parallelaser-
se alcerutri reliquorum lacerum,
DuEa F Q_ F R lateri AC, ex FS
2c F Flateri AB. Sintque insuper
latera duo ex opposcis, puta IC
e B D. Abi invicem paealle.
la. FC recta quae bisecat paralle-
Ia illa lacera erit una ex diametris
Conicae sectionis, et bisecabit eti
am RD. Sit O punctum in quo
R Q bileCaTuT, eE Crit P O ordinatim applicata ad diamerrutillam.
Produc PO ad X UE SIC OK aequalis PO, e crit OK ordinatim
applicata ad contrarias partes diametri. Cum isitur puncta , B,
P _ K Enc ad Conieam sectionem, ec F _ sccet AB in daco an-
sulo, erie (per Prop. 17 ec 18L ib. III Conicorum Apollonii) rectangu-
lum P QR ad rectangulum QB in data rationc. Scd ) R e YR
aequales TunE, uepote aequalium OK, O P, e O  O R disterentiae,
ec inde etiam rectanoula F OK eE F OX F R aequalia TUNE; 2C-
que adeo rectangulum P Q X F R est ad rectangulum 1 B, hoc
est ad rectangulum P SX P F in data ratione. Q. F. D.
DE MIOTO
POMPOROM
C450