  Discorsi Propositions 1/06-th-06 Discorsi Proposition1/06-th-06     {196} THEOREMA VI, PROPOSITIO VI. {196} THEOREM VI, PROPOSITION VI Si duo mobilia aequabili motu ferantur, ratio velocitatum ipsorum composita erit ex ratione spatiorum peractorum et ex ratione temporum contrarie sumptorum. If two particles are carried at a uniform rate, the ratio of their speeds will be the product of the ratio of the distances traversed by the inverse ratio of the time-intervals occupied. Sint duo mobilia A, B, aequabili motu lata; sint autem spatia ab illis peracta in ratione V ad T, tempora vero sint ut S ad R: dico, velocitatem mobilis A ad velocitatem ipsius B habere rationem compositam ex ratione spatii V {10} ad spatium T et temporis R ad tempus S.Sit velocitas C ea cum qua mobile A conficit spatium V in tempore S, et quam rationem habet spatium V ad spatium T, hanc habeat velocitas C ad aliam E; erit E velocitas cum qua mobile B conficit spatium T in tempore eodem S: quod si fiat, ut tempus R ad tempus S, ita velocitas E ad aliam G, erit velocitas G illa secundum quam mobile B conficit spatium T in tempore R. Habemus itaque velocitatem C, cum qua mobile A conficit spatium V in tempore S, et velocitatem G, cum qua mobile B conficit spatium T in tempore R, et est ratio C ad G composita ex rationibus C ad E et E ad G; {20} ratio autem C ad E posita est eadem cum ratione spatii V ad spatium T; ratio vero E ad G est eadem cum ratione R ad S: ergo patet propositum. Let A and B be the two particles which move at a uniform rate; and let the the respective distances traversed by them have the ratio of V to T, but let the time-intervals be as S to R. Then I say the speed of A will bear to the speed of B a ratio which is the product of the ratio of the distance V to the distance T and the time-interval R to the time-interval S. Let C be the speed at which A traverses the distance V during the time-interval S; and let the speed C bear the same ratio to another speed E as V bears to T; then E will be the speed at which B traverses the distance T during the time-interval S. If now the speed E is to another speed G as the time-interval R is to the time-interval S, then G will be the speed at which the particle B traverses the distance T during the time-interval R. Thus we have the speed C at which the particle A covers the distance V during the time S and also the speed G at which the particle B traverses the distance T during the time R. The ratio of C to G is the product of the ratio C to E and E to G; the ratio of C to E is by definition the same as the ratio of the distance V to distance T; and the ratio of E to G is the same as the ratio of R to S. Hence follows the proposition.  Discorsi Propositions 1/06-th-06 Discorsi Proposition1/06-th-06     