Discorsi Propositions
2/03-th-03
Discorsi Proposition
2/03-th-03

{215} THEOREMA III, PROPOSITIO III.{215} THEOREM III, PROPOSITION III
Si super plano inclinato atque in perpendiculo, quorum eadem sit altitudo, feratur ex quiete idem mobile, tempora lationum erunt inter se ut plani ipsius et perpendiculi longitudines.If one and the same body, starting from rest, falls along an inclined plane and also along a vertical, each having the same height, the times of descent will be to each other as the lengths of the inclined plane and the vertical.
Sit planum inclinatum AC, et perpendiculum AB, quorum eadem sit altitudo supra horizontem CB, nempe ipsamet linea BA: dico, tempus descensus {216} eiusdem mobilis super plano AC, ad tempus casus in perpendiculo AB, eam habere rationem, quam habet longitudo plani AC ad ipsius perpendiculi ab longitudinem.Intelligantur enim quotlibet lineae DG, EI, FL, horizonti CB parallelae: constat ex assumpto, gradus velocitatis mobilis ex A, primo motus initio, in punctis G, D acquisitos, esse aequales, cum accessus ad horizontem aequales sint; similiter, gradus in punctis I, E iidem erunt, nec non gradus in L et F. Quod si non hae tantum parallelae, sed ex punctis omnibus lineae AB usque ad lineam AC protractae {10} intelligantur, momenta seu gradus velocitatum in terminis singularum parallelarum semper erunt inter se paria. Conficiuntur itaque spatia duo AC, AB iisdem gradibus velocitatis. Sed demonstratum {217} est, quod si duo spatia conficiantur a mobili quod iisdem velocitatis gradibus feratur, quam rationem habent ipsa spatia, eamdem habent tempora lationum; ergo tempus lationis per AC ad tempus per AB est ut longitudo plani AC ad longitudinem perpendiculi AB: quod erat demonstrandum.Let AC be the inclined plane and AB the perpendicular, each having the same vertical height above the horizontal, namely, BA; then I say, the time of descent of one and the same body {216} along the plane AC bears a ratio to the time of fall along the perpendicular AB, which is the same as the ratio of the length AC to the length AB. Let DG, EI and LF be any lines parallel to the horizontal CB; (Condition 2/02-th-02-schol-dialog4) then it follows from what has preceded that a body starting from A will acquire the same speed at the point G as at D, since in each case the vertical fall is the same; (Condition 2/02-th-02-schol-dialog4) in like manner the speeds at I and E will be the same; so also those at L and F. (Condition 2/02-th-02-schol-dialog4) And in general the speeds at the two extremities of any parallel drawn from any point on AB to the corresponding point on AC will be equal Thus the two distances AC and AB are traversed at the same speed. (Condition 1/01-th-01) But it has already been proved {217} that if two distances are traversed by a body moving with equal speeds, then the ratio of the times of descent will be the ratio of the distances themselves; therefore, the time of descent along AC is to that along AB as the length of the plane AC is to the vertical distance AB. Q. E. D.

Discorsi Propositions
2/03-th-03
Discorsi Proposition
2/03-th-03