Discorsi Propositions 2/04-th-04 Discorsi Proposition2/04-th-04

 THEOREMA IV, PROPOSITIO IV. THEOREM IV, PROPOSITION IV Tempora lationum super planis aequalibus, sed inaequaliter inclinatis, sunt inter se in subdupla ratione elevationum eorumdem planorum permutatim accepta. The times of descent along planes of the same length but of different inclinations are to each other in the inverse ratio of the square roots of their heights Sint ex eodem termino B plana aequalia, sed inaequaliter inclinata, BA, BC; et ductis AE, CD, lineis horizontalibus, ad perpendiculum usque BD, {220} esto plani BA elevatio BE, plani vero BC elevatio sit BD; et ipsarum elevationum DB, BE media proportionalis sit BI: constat, rationem DB ad BI esse subduplam rationis DB ad BE. Dico iam, rationem temporum descensuum seu lationum super planis BA, BC esse eamdem cum ratione DB ad BI permutatim assumpta, ut scilicet temporis per BA homologa sit elevatio alterius plani BC, nempe BD, temporis vero per BC homologa sit BI. Demonstrandum proinde est, tempus per BA ad tempus per BC esse ut DB ad BI. Ducatur IS, {10} ipsi DC aequidistans: et quia iam demonstratum est, tempus descensus per BA ad tempus casus per perpendiculum BE esse ut ipsa BA ad BE, tempus vero per BE ad tempus per BD ut BE ad BI, tempus vero per BD ad tempus per BC ut BD ad BC, seu BI ad BS, ergo, ex aequali, tempus per BA ad tempus per BC erit ut BA ad BS, seu CB ad BS; est autem CB ad BS ut DB ad BI; ergo patet propositum. From a single point B draw the planes BA and BC, having the same length but different inclinations; let AE and CD be horizontal lines drawn to meet the perpendicular BD; and {220} let BE represent the height of the plane AB, and BD the height of BC; also let BI be a mean proportional to BD and BE; then the ratio of BD to BI is equal to the square root of the ratio of BD to BE. Now, I say, the ratio of the times of descent along BA and BC is the ratio of BD to BI; so that the time of descent along BA is related to the height of the other plane BC, namely BD as the time along BC is related to the height BI. Now it must be proved that the time of descent along BA is to that along BC as the length BD is to the length BI. Draw IS parallel to DC; (Condition 2/03-th-03-cor) and since it has been shown that the time of fall along BA is to that along the vertical BE as BA is to BE; (Condition 2/02-th-02-cor2) and also that the time along BE is to that along BD as BE is to BI; (Condition 2/03-th-03-cor) and likewise that the time along BD is to that along BC as BD is to BC, or as BI to BS; it follows, ex aequali, that the time along BA is to that along BC as BA to BS, or BC to BS. However, BC is to BS as BD is to BI; hence follows our proposition.

 Discorsi Propositions 2/04-th-04 Discorsi Proposition2/04-th-04