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| {195} THEOREMA V, PROPOSITIO V. | {195} THEOREM V, PROPOSITION V |
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| Si duo mobilia aequabili motu ferantur, sint tamen velocitates inaequales, et inaequalia spatia peracta, ratio temporum composita erit ex ratione spatiorum et ex ratione velocitatum contrarie sumptarum. | If two particles are moved at a uniform rate, but with unequal speeds, through unequal distances, then the ratio of the time-intervals occupied will be the product of the ratio of the distances by the inverse ratio of the speeds. |
| Sint duo mobilia A, B, sitque velocitas ipsius A ad velocitatem ipsius B ut V ad T; spatia autem peracta sint ut S ad R: dico, rationem temporis quo motum est A, ad tempus quo motum est B, compositam esse ex ratione {10} velocitatis T ad velocitatem V et ex ratione spatii S ad spatium R. Sit ipsius motus A tempus C, et ut velocitas T ad velocitatem V, ita sit tempus C ad tempus E; et cum C sit tempus in quo A cum velocitate V conficit spatium S, sitque ut velocitas T (10) mobilis B ad velocitatem V, ita tempus C ad tempus E, erit tempus E illud in quo mobile B conficeret idem spatium S. Fiat modo ut spatium S ad spatium R, ita tempus E ad tempus G: constat, G esse tempus quo B conficeret spatium R.Et quia ratio C ad G componitur ex rationibus C ad E et E ad G; est autem ratio C ad E eadem cum ratione velocitatum mobilium A, B {20} contrarie sumptarum, hoc est cum ratione T ad V; ratio vero E ad G est eadem cum ratione spatiorum S, R; ergo patet propositum. | Let the two moving particles be denoted by A and B, and let the speed of A be to the speed of B in the ratio of V to T; in like manner let the distances traversed be in the ratio of S to R; then I say that the ratio of the time-interval during which the motion of A occurs to the time-interval occupied by the motion of B is the product of the ratio of the speed T to the speed V by the ratio of the distance S to the distance R. Let C be the time-interval occupied by the motion of A, and let the time-interval C bear to a time-interval E the same ratio as the speed T to the speed V. And since C is the time-interval during which A, with speed V, traverses the distance S and since T, the speed of B, is to the speed V, as the time-interval C is to the time-interval E, then E will be the time required by the particle B to traverse the distance S. If now we let the time-interval E be to the time-interval G as the distance S is to the distance R, then it follows that G is the time required by B to traverse the space R.Since the ratio of C to G is the product of the ratios C to E and E to G (while also the ratio of C to E is the inverse ratio of the speeds of A and B respectively, i. e., the ratio of T to V); and since the ratio of E to G is the same as that of the distances S and R respectively, the proposition is proved. |
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