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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
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1III
PFTTLOSOPMLAE NATORALIS
PROPOSITIO XV. THEOREMA XII
Si Medii densitas in locis sinculis sit reciproce ut distantia locorum
a centro immobili, sitque vis centripeta in duplicata ratione den-
sitatis: dico quod corpus errari potest in Spirali, quae radios
omnes a centro illo ductos inter secat in anculo dato.
Ponantur quae in superiore Lemmate, ee producatur 59 ad V,
uE sic SV aequalis SP. Tempore quoris, in Medio resistente, de
seribac corpus arcum quam minimum P G ec tempore duplo ar-
cum quam minimum P R; et decremenca horum areuum ex reli-
sentia oriunda, sive dete-
ctus ab arcubus qui in MC.
dio non resistente iisdem
temporibus describeren-
tur, crune ad invicem ue
quadrata temporum in
quibus generantur: LA
itaque decrementum arcus
P D ParS quarta decre.
menti arcus F R. Vndc
etiam, AI AFCX PS Xe
qualis capiatur arca dSr,
erit decrementum arcus
P9 aequale dimidiolincolae Rr; adeoque vis resistentix ec vis cen-
trixeca une ad invicem ue lineolae : Re ec F  quas simul generant.
Duoniam vis centripeta, qua corpus urvetur in P, Cs reciproce ut
SF9, _ (per L em. Z. Lib. T,) lineola F Q, quae vi illa generatur, cA
in ratione composita ex ratione hujus vis ex ratione duplicata tem-
Poris quo arcus F  describitur. (Nam resictentiam in hoc casus
uc infinice minorem quam ris centripeta, nesliso) erit T Q XS Pq
id est (per Lemma novissimum) =P QgZSP, in ratione duplicata
temporis, adeoque tempus est ut P QX VSP, ec corporis reloci-
EaS, qua arcus P Q illo tempore deseribitur, ut ∞
et
Q
QXYS
Ceu
 hoc cA, in subduplicata ratione ipsUS S P reCIPFOCC. PF S-
mili argumento, relocitas qua arcus  R describitur, eA in sub-
duplicata
DE MOTD
EORPORVM