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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 190, page id 1041163592:0224
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190
PHTLOSOPMILL NATDRALIS
Dy. MOrO Cem uE SP quad ad S A quad: Si in quadruplicata, ut SF cub ad
Corronvi A cub. Onde cum accractio in P, in hoc ultimo casu, inventa
fuiE reciprOCC UE F Scub X P F actractio in I crit reciproce ut
S Acub Z P I, id eA (ob datum S A cub) reciproce ut PY. EF
similis est progressus in infinitum. Theorema vero sic demon-
AratUr.
Scancibus jam ante constructis, e existente corpore in loco
quoris P, Ordinatim appliCaTA D N inVeNEA FUIE UE  F. 9X FS
PFX
FrGO A agatuF PF, ordinata illa ad alium quemris locum B, mu-
ratis mutandis, evadet ut  FgX Id
FXV. PONC VirES CentrIDCTAS, C
Sphaerae puncto quoris P manances, ese ad invicem in distantiis
TF P L UE PP ad YF, (ubi numerus n designet indicem
DF 9XES
porestatum PF ec PF.) ec ordinatae illae AEnE UE P F Z P F n de
DFgX7S
TF X TFn quarum ratio ad inricem es UE PSXTF XIF  ad
TSXPF. XP Fn. Ouoniam ob similia triangula SF F, SF. I, AE
TF ad F F. uE IS ad SF vel S A; PrO rationC PF ad P F. IcribC
rationem IS ad S A; ex ordinatarum ratio eradet PSXIFn 2d
SAXF Fn. Sed FS ad SA subduplicata est ratio distantiarum
DS, SF; ec TFn ad F Fn Subduplicata est ratio virium in distan-
tiis FS, YS. Eroo ordinatae, et propterca arcae quas ordinata
describune, hisque proportionales actractiones, lune in rarione com-
Posica ex subduplicatis illis rationibus. Q. F. D.
PROPOSITIOLXZXIII. PROBLEMA XLII.
Invenire vim qua corpusculum in centro Sphaerae locatum ad ejus
Sermentum quodcunque attrahitur.
Sit P COrpUS in CCNErO SpHrX, _E RBSD Sermentum ejus
plano R DS e CaperEcie SpHXrICa RBS CONTENTUM. SUPCrE-
eie Sphaerica P F G centro P descripta seCCtUE D B in F, ac di-
stinguatur Seomentum in partes BR F F CS, F F D G. Sit
aurem superticies illa non pure Machematica, Sed Phrsica, PrO-
funditarem habens quam minimam. Nominetur icta protundi-
TaSO