Discorsi Propositions 2/08-th-08 Discorsi Proposition2/08-th-08

 THEOREMA VIII, PROPOSITIO VIII. THEOREM VIII, PROPOSITION VIII In planis ab eodem sectis circulo ad horizontem erecto, in iis quae cum termino diametri erecti conveniunt, sive imo sive sublimi, lationum tempora sunt aequalia tempori casus in diametro; in illis vero quae ad diametrum non pertingunt, tempora sunt breviora; in eis tandem quae {30} diametrum secant, sunt longiora. The times of descent along all inclined planes which intersect one and the same vertical circle, either at its highest or lowest point, are equal to the time of fall along the vertical diameter; for those planes which fall short of this diameter the times are shorter; for planes which cut this diameter, the times are longer. Circuli ad horizontem erecti esto diameter perpendicularis AB.De planis ex terminis A, B ad circumferentiam usque productis, quod tempora lationum {227} super eis sint aequalia, iam demonstratum est. De plano DF ad diametrum non pertingente, quod tempus descensus in eo sit brevius, demonstratur ducto plano DG, quod et longius erit et minus declive quam DF; ergo tempus per DF brevius quam per DG, hoc est per AB. De plano vero diametrum secante, ut CO, quod tempus descensus in eo sit longius, itidem constat; est enim et longius et minus declive quam CB. Ergo patet propositum. Let AB be the vertical diameter of a circle which touches the horizontal plane.(Condition 2/06-th-06) It has already been proven that the times of descent along planes drawn from either end, A or B, to the circumference are equal. In order to show that the time of descent {227} along the plane DF which falls short of the diameter is shorter we may draw the plane DB which is both longer and less steeply inclined than DF; whence it follows that the time along DF is less than that along DB and consequently along AB. In like manner, it is shown that the time of descent along CO which cuts the diameter is greater: for it is both longer and less steeply inclined than CB. Hence follows the theorem.

 Discorsi Propositions 2/08-th-08 Discorsi Proposition2/08-th-08