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Sit ABC Parabola umbilicum habens S. Chordâ AC bisectâ in I abscindatur segmentum ABCI, cujus diameter sit I μ & vertex μ. In I μ productâ capiatur μ O aequalis dimidio ipsius I μ. Iungatur OS, &producatur ea ad ξ, ut sit S ξ aequalis 2 SO. Et si Cometa B moveatur in arcu CBA, & agatur ξ B secans AC in E: dico quod punctum E ab∣scindet[illustration]
de chorda AC segmentum AE tempori proportionale quam∣proxime.
Jungatur enim EO secans arcum Parabolicum ABC in Y, & erit area curvilinea AEY ad aream curvilineam ACY ut AE ad AC quamproximè. Ideoque cum triangulum ASE sie ad triangulum ASC in eadem ratione, erit area tota ASEY ad aream totam ASCY ut AE ad AC quamproximè. Cum autem ξ O sit ad SO ut 3 ad 1 & EO ad YO prope in eadem ratione, erit SY ipsi EB parallela quamproximè, & propterea triangulum SEB, triangulo YEB quamproximè aequale. Unde si ad aream ASEY addatur triangulum EYB, & de summa auferatur triangulum SEB, ma∣nebit area ASBY areae ASEY aequalis quamproximè, atque adeo ad aream ASCY ut AE ad AC. Sed area ASBY est ad aream ASCY ut tempus descripti arcus AB ad tempus descripti arcus to∣tius. Ideoque AE est ad AC in ratione temporum quamproximè. Q.E.D.