Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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dio) ut
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BW
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ad
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BV,
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ſeu
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AO+OR
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ad
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AO,
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id eſt (cum ſint
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CA
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ad
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CO, CO
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ad
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CR
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& diviſim
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AO
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ad
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OR
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proportionales,) ut
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CA+CO
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ad
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CA
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vel, ſi biſecetur
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BV
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in
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E,
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ut 2
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CE
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ad
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CB.
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Proinde, per Corol. </
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>1. Prop. </
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>XLIX, longitudo partis rectæ Fili
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PT
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æquatur ſemper Cycloidis arcui
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PS,
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& Filum totum
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APT
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æquatur
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ſemper Cycloidis arcui dimidio
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APS,
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hoc eſt (per Corol. </
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XLIX) longitudini
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AR.
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Et propterea viciſſim ſi Filum manet ſem
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per æquale longitudini
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AR
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movebitur punctum
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T
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in Cycloide
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data
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QRS.
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E. D.
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LIBER
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PRIMUS.</
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Corol.
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Filum
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AR
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æquatur Semicycloidi
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AS,
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adeoque ad ſemi
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diametrum
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AC
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eandem habet rationem quam ſimilis illi Semicy
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clois
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SR
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habet ad ſemidiametrum
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CO.
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PROPOSITIO LI. THEOREMA XVIII.
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Si Vis centripeta tendens undique ad Globi centrum
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C
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ſit in locis
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ſingulis ut diſtantia loci cujuſque a centro, & hac ſola Vi a
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gente corpus
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T
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oſcilletur (modo jam deſcripto) in perimetro Cy
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cloidis
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QRS:
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dico quod oſcillationum utcunQ.E.I.æqualium
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æqualia erunt Tempora.
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>Nam in Cycloidis tangentem
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TW
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infinite productam cadat per
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pendiculum
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CX
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& jungatur
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CT.
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Quoniam vis centripeta qua cor
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pus
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T
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impellitur verſus
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C
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eſt ut diſtantia
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CT,
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atque hæc (per Legum
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Corol. </
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>2.) reſolvitur in partes
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CX, TX,
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quarum
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CX
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impellen
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do corpus directe a
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P
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diſtendit filum
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PT
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& per ejus reſiſtentiam
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tota ceſſat, nullum alium edens effectum; pars autem altera
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TX,
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urgendo corpus tranſverſim ſeu verſus
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X,
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directe accelerat motum
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ejus in Cycloide; manifeſtum eſt quod corporis acceleratio, huic
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vi acceleratrici proportionalis, ſit ſingulis momentis ut longitudo
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TX,
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id eſt, (ob datas
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CV, WV
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iiſque proportionales
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TX, TW,
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)
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ut longitudo
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TW,
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hoc eſt (per Corol. </
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>1. Prop. </
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<
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>XLIX,) ut longitudo
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arcus Cycloidis
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TR.
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Pendulis igitur duobus
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APT, Apt
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de per
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pendiculo
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AR
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inæqualiter deductis & ſimul dimiſſis, acceleratio
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nes eorum ſemper erunt ut arcus deſcribendi
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TR, tR.
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Sunt au
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tem partes ſub initio deſcriptæ ut accelerationes, hoc eſt, ut totæ
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ſub initio deſcribendæ, & propterea partes quæ manent deſcriben-</
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