Prop. LI. Theor. XVIII.
Si vis centripeta tendens undi{que} ad Globi centrum C sit in locis singu∣lis ut distantia loci cujus{que} a centro, & hac sola vi agente Corpus T
Philosophiæ naturalis principia mathematica autore Js. Newton ...
About this Item
- Title
- Philosophiæ naturalis principia mathematica autore Js. Newton ...
- Author
- Newton, Isaac, Sir, 1642-1727.
- Publication
- Londini :: Jussu Societatis Regiae ac Typis Josephi Streater ...,
- 1687.
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- Subject terms
- Mechanics -- Early works to 1800.
- Celestial mechanics -- Early works to 1800.
- Cite this Item
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"Philosophiæ naturalis principia mathematica autore Js. Newton ..." In the digital collection Early English Books Online. https://name.umdl.umich.edu/A52251.0001.001. University of Michigan Library Digital Collections. Accessed May 29, 2024.
Pages
Page 152
oscilletur (modo jam descripto) in perimetro Cycloidis QRS: dico quod oscillationum utcun{que} inaequalium aequalia erunt Tempora
Nam in Cycloidis tangentem TW infinite productam cadat perpendiculum CX & jungatur CT. Quoniam vis centripeta qua corpus T impellitur versus C est ut distantia CT, (per Legum Corol. 2.) resolvitur in partes CX, TX, quarum CX impellen∣do corpus directe a P distendit filum PT & per cujus resistenti∣am tota cessat, nullum alium edens effectum; pars autem altera TX urgendo corpus transversim seu versus X, directe accelerat motum ejus in Cycloide; manifestum est quod corporis acceleratio huic vi acceleratrici proportionalis sit singulis momentis ut longitudo TX, id est, ob datas CV, WV iis{que} proportionales TX, TW, ut longitudo TW, hoc est (per Corol. 1. Prop. XLIX.) ut longi∣tudo arcus Cycloidis TR. Pendulis igitur duabus APT, Apt de perpendiculo AR inaequaliter deductis & simul dimissis, accele∣rationes eorum semper erunt ut arcus describendi TR, tR. Sunt autem partes sub initio descriptae ut accelerationes, hoc est ut totae sub initio describendae, & propterea partes quae manent describendae & accelerationes subsequentes his partibus pro∣portionales sunt etiam ut totae; & sic deinceps. Sunt igitur ac∣celerationes at{que} adeo velocitates genitae & partes his velocitati∣bus descriptae partes{que} describendae, semper ut totae; & propterea partes describendae datam servantes rationem ad invicem simul e∣vanescent, id est corpora duo oscillantia simul pervenient ad per∣pendiculum AR. Cum{que} vicissim ascensus perpendiculorum de loco infimo R, per eosdem arcus Trochoidales motu retrogrado facti, retardentur in locis singulis a viribus iisdem a quibus descen∣sus accelerabantur, patet velocitates ascensuum ac descensuum per eosdem arcus factorum aequales esse, at{que} adeo temporibus ae∣qualibus fieri; & propterea cum Cycloidis partes duae RS & RQ ad utrum{que} perpendiculi latus jacentes sint similes & aequales, pen∣dula duo oscillationes suas tam totas quam dimidias iisdem tem∣poribus semper peragant. Q.E.D.