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| THEOREMA IV, PROPOSITIO IV. | THEOREM IV, PROPOSITION IV |
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| Si duo mobilia ferantur motu aequabili, inaequali tamen velocitate, spatia temporibus inaequalibus AB ipsis peracta habebunt rationem compositam ex ratione velocitatum et ex ratione temporum. | If two particles are carried with uniform motion, but each with a different speed, the distances covered by them during unequal intervals of time bear to each other the compound ratio of the speeds and time intervals. |
| Mota sint duo mobilia E, F motu aequabili, et ratio velocitatis mobilis E ad velocitatem mobilis F sit ut A ad B; temporis vero quo movetur E, ad tempus quo movetur F, ratio sit ut C ad D: dico, spatium peractum ab E cum velocitate A in tempore C, ad spatium peractum ab F cum {20} velocitate B in tempore D, habere rationem compositam ex ratione velocitatis A ad velocitatem B et ex ratione temporis C ad tempus D. Sit spatium ab E cum velocitate A in tempore C peractum G, et ut velocitas A ad velocitatem B, ita fiat G ad I; ut autem tempus C ad tempus D, ita sit I ad L: constat, I esse spatium quo movetur F in tempore eodem in quo E motum est per G, cum spatia G, I sint ut velocitates A, B.Et cum sit ut tempus C ad tempus D, ita I ad L; sit autem I spatium quod conficitur A mobili F in tempore C; erit L spatium quod conficitur ab F in tempore D cum velocitate B. Ratio autem G ad L componitur ex rationibus G ad I et I ad L, nempe ex rationibus {30} velocitatis A ad velocitatem B et temporis C ad tempus D: ergo patet propositum. | Let the two particles which are carried with uniform motion be E and F and let the ratio of the speed of the body E be to that of the body F as A is to B; but let the ratio of the time consumed by the motion of E be to the time consumed by the motion of F as C is to D. Then, I say, that the distance covered by E, with speed A in time C, bears to the space traversed by F with speed B in time D a ratio which is the product of the ratio of the speed A to the speed B by the ratio of the time C to the time D. For if G is the distance traversed by E at speed A during the time-interval C, and if G is to I as the speed A is to the speed B; and if also the time-interval C is to the time-interval D as I is to L, then it follows that I is the distance traversed by F in the same time that G is traversed by E since G is to I in the same ratio as the speed A to the speed B.And since I is to L in the same ratio as the time-intervals C and D, if I is the distance traversed by F during the interval C, then L will be the distance traversed by F during the interval D at the speed B. But the ratio of G to L is the product of the ratios G to I and I to L, that is, of the ratios of the speed A to the speed B and of the time-interval C to the time-interval D. Q. E. D. |
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