sirr. i.] DISSERTATION SECOND. 17 



proposed by Bernoulli in 1690; and had been resolved by 

 Huygens, Leibnitz, and himself. 



A question had been proposed, also, concerning the line 

 of swiftest descent in 1697, or the line along which a 

 body must descend, in order to go from one point to an- 

 other not perpendicularly under it, in the least time pos- 

 sible. Though a straight line be the shortest distance 

 between two points, it does not necessarily follow, that 

 the descent in that line will be most speedily performed, 

 for, by falling in a curve that has at first a very rapid de- 

 clivity, the body may acquire in the beginning of its motion 

 so great a velocity, as shall carry it over a long line in 

 less time than it would describe a short one, w r ith a ve- 

 locity more slowly acquired. This, however, is a problem 

 that belongs to a class of questions of peculiar difficulty; 

 and accordingly it was resolved only by a few of the most 

 distinguished mathematicians. The solutions which ap- 

 peared within the time prescribed were from Leibnitz, 

 Newton, the two Bernoullis, and M. de l'Hopital. New- 

 ton's appeared in the Philosophical Transactions without a 

 name ; but the author was easily recognised. John Ber- 

 noulli, on seeing it, is said to have exclaimed, Ex ungw 

 teonem ! 



The curve that has the property required is the cycloid ; 

 Newton has given the construction, but has not accompa- 

 nied it with the analysis. He added afterwards 'the de- 

 monstration of a very curious theorem for determining the 

 time of the actual descent. Leibnitz resolved the problem 

 the same day that he received the programme in which it 

 was proposed. 



The problem of orthogonal trajectories, as it is called, 

 had been long ago proposed in the Acta Eruditorum, with 

 an invitation to all who were skilled in the new analysis 

 to attempt the solution. The problem had not, at first 



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