Discorsi Propositions 2/25-th-16 Discorsi Proposition2/25-th-16

 THEOREMA XVI, PROPOSITIO XXV. THEOREM XVI, PROPOSITION XXV Si post casum per aliquod planum inclinatum sequatur motus per planum horizontis, erit tempus casus per planum inclinatum ad tempus motus per quamlibet lineam horizontis ut dupla longitudo plani inclinati ad lineam acceptam horizontis. If descent along any inclined plane is followed by motion along a horizontal plane, the time of descent along the inclined plane bears to the time required to traverse any assigned length of the horizontal plane the same ratio which twice the length of the inclined plane bears to the given horizontal length. Sit linea honzontis CB, planum inclinatum AB, et post casum per AB sequatur motus per horizontem, in quo {30} sumatur quodlibet spatium BD: dico, tempus casus per AB ad tempus motus per BD esse ut dupla AB ad BD. Sumpta enim BC ipsius AB dupla, constat ex praedemonstratis, tempus casus per AB aequari tempori motus per BC: sed tempus motus per BC ad tempus motus per DB est ut linea CB {247} ad lineam BD: ergo tempus motus per AB ad tempus per BD est ut dupla AB ad BD: quod erat probandum. Let CB be any horizontal line and AB an inclined plane; after descent along AB let the motion continue through the assigned horizontal distance BD. Then, I say, the time of descent along AB bears to the time spent in traversing BD the same ratio which twice AB bears to BD. For, lay off BC equal to twice AB (Condition 2/23-pr-09-schol1) then it follows, from a previous proposition, that the time of descent along AB is equal to the time required to traverse BC; (Condition 1/01-th-01) but the time along BC is to the time along DB as the length CB is to the length BD. Hence the time of descent along AB {247} is to the time along BD as twice the distance AB is to the distance BD.

 Discorsi Propositions 2/25-th-16 Discorsi Proposition2/25-th-16