220 PHILOSOPHICAL TRANSACTIONS. [aNNO 1746. 



of 2 miles in 2 hours is the double of walking 1 mile in 1 hour, it will follow 

 that the walking of 2 miles in 1 hour is the quadruple of walking 1 mile in 

 1 hour." 



These disturbing words, virtually and formally, being now removed, which 

 had hitherto fouled this clear fountain of truth, Leibnitz not only took off 

 Bernoulli's objection, but brought him over entirely to his side. " Your answer, 

 says he in his next letter, quite satisfies me; for I perceive what you mean by 

 those two terms : and your argumentation appears to me very elegant ; so that it 

 ought no longer to be detained from the public ; and it will give great weight to 

 the argument a posteriori." 



Perhaps Bernoulli would not urge the matter fiirther, as Leibnitz seemed to be 

 in a more than ordinary commotion : " I dare not, says he, promise any thing 

 great ; but I hope to be not guilty of an obvious paralogism, in an argumentation 

 which did not escape from me suddenly, but had been considered for several years, 

 and I had boasted of it as a thing of some moment." And yet, that Leibnitz was 

 guilty of an obvious paralogism, I believe will soon be shown. 



We need not dwell on Leibnitz's examination of both Bernoulli's demonstra- 

 tions, because they depend on the sense of the words virtually and formally, 

 understood differently from the meaning of Leibnitz, " I took the terms, says 

 Bernoulli, in a sense different from that in which you now explain them." But 

 Leibnitz, being still in doubt what weight his first demonstration would have with 

 Bernoulli, adds the following to it. " I add another, says he, which, if you 

 examine it to the bottom, comes to the same as the former, and yet it has its 

 own proper weight. Moving actions, I mean equable ones, of the same move- 

 able, are in a ratio compounded of the immediate effects, viz. the spaces run 

 through and the velocities. Now the lengths equably run through, are in a ratio 

 compounded of the times and velocities. Therefore moving actions are in a ratio 

 compounded of a simple ratio of the times, and a duplicate one of the velocities. 

 So that, in the same times, or elements of times, the moving actions of the same 

 moveable are in the duplicate ratio of the velocities ; or, if the moveables are 

 different, in a ratio compounded of the simple ratio of the moveables, and the 

 duplicate one of the velocities." 



As Bernoulli had said, in his letter of April, that he acquiesced in the former 

 demonstration, without mentioning the latter, Leibnitz asked him in May, 

 " what he thought of the other demonstration, which says he is a little more in 

 the manner of the received form, though they both agree in the root. Bernoulli 

 therefore, when he could no longer avoid opening his mind, in his letter of June 

 thus expresses himself: 



" Your other demonstration of the proposition concerning the ratio of moving 

 actions, seems contrived no less ingeniously than the former, and as you express 



