NEWTON S PRINCIPIA. 201 



and the time of describing this space is to be found 

 from 



or, 



In a preceding section Newton had determined the path 

 of a projectile when the resistance varied as the velocity, 

 and here was the place to give the solution of the cor 

 responding problem, when the resistance varied as the 

 square of the velocity. But this is a far harder question ; 

 ~we~ are even now unable to find quite accurately the path 

 described. Newton considered the problem in an indirect 

 manner. He determined the law of density that a given 

 curve may be described, but he could not thence deduce 

 the curve that gave the density uniform. He even made 

 several mistakes, which were corrected at the suggestion 

 of John Bernoulli, in the edition of 1713.* In 1718 

 Keill, in the course of the quarrel between the supporters 

 of Newton and Leibnitz, dared the foreigners to attempt 

 this question. Bernoulli was the first who gave a solution, 

 and challenged the proposer to furnish his own solution 

 within a certain time. This, however, Keill was unable 

 to do. Meantime Nicholas Bernoulli, of Padua, supplied 

 a solution ; and seventeen days after the time fixed had 

 elapsed, Taylor vindicated the honour of England by a 

 tardy solution. The problem we shall now consider is 

 somewhat more general than that enunciated by Newton, 

 and it is as follows : 



* Montucla, Part IV. Liv. VII. 6. 



