VOL. XLIV.] PHILOSOPHICAL TRANSACTIONS. 219 



be dazzled by the too great splendour of the light, if a doubt from Bernoulli 

 himself had not relieved me. This ingenious man was so far from acquiescing in 

 this clear light of truth, that he not only made an objection, but even produced 

 a double demonstration. 



" I do not see, says he, what can be said by an adversary to the contrary ; un- 

 less perhaps that the virtual seems to be confounded with the formal action ; 

 denying the consequence, that a is quadruple of c, because a is virtually double 

 of B, and B formally double of c." 



Having proposed this objection, he adds his demonstrations. 



" 1 . An action performing double in a single time, is virtually double of an 

 action performing the same double in a double time." — " 2. An action performing 

 double in a double time, is virtually single of an action performing single in a 

 single time." — " 3. Therefore an action performing double in a single time, is 

 double of an action performing single in a single time." Or, 



" 1 . An action performing double in a single time, is formally single of an 

 action performing the same double in a double time." — " 2. An action perform- 

 ing double in a double time, is formally double of an action performing single in 

 a single time." — " 3. Therefore an action performing double in a single time, is 

 double of an action performing single in a single time." 



" You see then the two arguments, which plainly conclude the same thing, 

 but are quite contrary to your conclusion, and depend on that common axiom, 

 that those things which are equal to the same are equal to each other; which 

 however holds only in homogeneous quantities, as here in comparing a virtual 

 action with a virtual, and a formal with a formal, but not one with the other." 



Thus Bernoulli, with no less acuteness than modesty. But Leibnitz first 

 endeavours to take off Bernoulli's objection; thus, " I do not well understand, 

 says he, what you mean when you say, a virtual action is confounded with a 

 formal one : for I do not here treat of an action as being either virtual or formal ; 

 but one action is double of another, either virtually or formally : virtually, when 

 it is double in estimation, though it be not double in bulk, or congruence; as a 

 ducat is the double of a dollar: but formally, as a dollar is the double of a half dollar. 

 And you must know that what is double formally, is double also in virtue or esti- 

 mation. Therefore, as the inquiry here is only concerning virtue or estimation, 

 there is no confusion of the different kind of quantities or estimation; for by 

 virtually double, I understand that which is so only virtually ; but I call that for- 

 mally double, which is double both formally and virtually." 



He then proceeds, " I might abstain from words used only for the sake of a 

 certain harmony ; for as a ducat is the double of a dollar, and a dollar of a half 

 dollar, I conclude that a ducat is the quadruple of a half dollar; so because the 

 walking of 2 miles in 1 hour is the double of 2 miles in 2 hours, and the walking 



F F 2 



