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Concessis figurarum curvilinearum Quadraturis, invenire vires qui∣bus corpora in datis curvis lineis Oscillationes semper Isochronas peragent.
Oscilletur cor∣pus
Nam si vis, qua corpus trahitur de T versus C, exponatur per rectam TZ captam ipsi proportionalem, resolvetur haec in vires
TY, YZ; quarum YZ trahendo corpus secundum longitudinem fili PT, motum ejus nil mutat, vis autem altera TY motum ejus in curva STRQ directe accelerat vel directe retardat. Proinde cum haec sit ut via describenda TR, accelerationes corporis vel retardationes in Oscillationum duarum (majoris & minoris) par∣tibus proportionalibus describendis, erunt semper ut partes illae, & propterea facient ut partes illae simul describantur. Corpora autem quae partes totis semper proportionales simul describunt, simul describent totas. Q.E.D.
Corol. 1. Hinc si corpus T filo rectilineo AT a centro A pen∣dens, describat arcum circularem STRQ, & interea urgeatur secundum lineas parallelas deorsum a vi aliqua, quae sit ad vim u∣niformem gravitatis, ut arcus TR ad ejus sinum TN: aequalia e∣run Oscillationum singularum tempora. Etenim ob parallelas TZ, AR, similia erunt triangula ANT, TYZ; & propterea TZ erit ad AT ut TY ad TN; hoc est, si gravitatis vis unifor∣mis exponatur per longitudinem datam AT, vis TZ, qua Oscil∣lationes evadent Isochronae, erit ad vim gravitatis AT, ut arcus TR ipsi TY aequalis ad arcus illius sinum TN.
Corol. 2. Igitur in Horologiis, si vires a Machina in Pendulum ad motum conservandum impressae ita cum vi gravitatis compo∣ni possint, ut vis tota deorsum semper sit ut linea quae oritur ap∣plicando rectangulum sub arcu TR & radio AR, ad sinum TN, Oscillationes omnes erunt Isochronae.