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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 140, page id 1041163592:0174
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NIIIIHIII
T40
PHITLOSODMLLE NATORALIS
De Moro de ex accelerationes subsequentes, his partibus proxortionales, sune
COn CFIaM UE FOEX; ec sic deincepS. Sune isitur accelerationes atque
2deo relocitates senitae et partes his relocitatibus descriprae par-
te(que describendaes Semper ut totae; e propterea partes describen-
de datam serrantes rationem ad inricem simul cranelcent, id cA,
corpora duo oscillancia simul pervenient ad perpendiculum AR.
Cunquo vicissim ascentus perpendiculorum de loco infimo R, per
eoldem arcus Creloidales moru rerrosrado Factis recardentur in
locis singulis a viribus iisdem a quibus descensus accelerabantur,
patet relocitates alcentuum ac descensuum per coldem arcus Fa-
ctorum aequales esse, atque 2deo temporibus aequalibus Heri; ec
propterea, cum Creloidis partes duxe RSe R  ad ucrum que per-
pendiculi lacus jacentes sine similes ec aequales, pendula duo oscil-
lationes luas tam totas quam dimidias iildem cemporibus semper
PeraSeNE. D. L. D.
Corol. Vis qua corpus T in loco quoris T acceleratur vel retar-
tur in Creloide, est ad totum corporis ejuldem Pondus in loco
altissimo srel Q ut Crcloidis arcus P R ad ejusdem arcum S R
vel QR.
PROPOSITIO LII PROBLEMA XAXIV.
Desmire ex Velocitates Pendulorum in locis sineulis, ex Tempora
quibus tum oscillationes totae, tum sinsulae oscillationum partes
peraruntur.
Centro quoris G, interrallo G H Cycloidis areum R S equantes
describe semicirculum IVR MC Semidiameero CR bisectum. RF
si vis ceneripeca, distantiis locorum a cenero proportionalis, tendat
2d centrum C, ltque ea in perimetro P aqualis vi centripete
in Perimetro Cilobi OS (Vide Fis. Prop. L..) 2d ipsius cen-
crum cendenti; et eodem tempore quo pendulum L dimitcitur c
loco supremo S, cadat corpus aliquod L. ab Y ad C: Quoniam
vires quibus corpora urgentur lunt aequales sub initio ex spariis
describendis TR, L. C Temper proportionales, aeque adeo, (i x-
QUanEur T R _c L. C, aequales in lOcis Y ec L; Datet corpora illa
describere spatia SZ, FYL. aequalia sub inirio, adcoque subinde per-
serexequaliter urgeri, ex aequalia spacia deleribere. Ouare, per Prop.
XREVIIE cemPuS quo Corpus deleribit arcum SY est ad cempus
Olcil-