130 PHILOSOPHICAL TRANSACTIONS. [aNNO 16q6-7 . 



Prob. 1. — The first problem is a mechanico-geometrical one, concerning the 

 line of swiftest descent, and is in these words : " To determine the curve-line 

 connecting two given points, which are at different distances from the horizon, 

 and not in the same vertical line, along which a body passing by its own gravity, 

 and beginning to move at the upper point, shall descend to the lower point in 

 the shortest time possible." 



" The sense of the problem is this: of the infinite number of lines that may 

 be drawn between those points, from one to the other, to make choice of that 

 according to which if a plate be bent, having the form of a tube or a canal, so 

 that a ball being laid upon it, and suffered to descend freely, it may perform its 

 passage from one point to the other in the shortest time possible." 



Solution. — From the one given point a, (fig. 4, pi. .3,) let there be drawn an 

 indefinite right line apcz parallel to the horizon, and on the same right line let 

 there be described any cycloid aqb, meeting in q with the right line ab, drawn 

 and produced if necessary, through the other given point b; as also another 

 cycloid ABC, whose base and altitude may be to the base and altitude of the 

 former respectively, as ab to aq. Then this last cycloid will pass tiirough the 

 point B, and will be that curve line in which a body, falling by the force of 

 gravity, will pass soonest from the point a to the point b. q. e. i. 



Prob. 1. — " To find a curve line of this property, that the two segments (of 

 a right line drawn from a given point through the curve), being raised to any 

 given power, and taken together, may make everywhere the same sum." 



This problem, if I rightly understand its meaning, (for I have not yet seen 

 the part of the Leipsic Acts concerning it, cited by the author), may be thus 

 proposed: required a curve-line kil (fig. 5, pi. 3) under this condition, that if 

 a right-line pkl be any how drawn from some given point or pole p, and meet- 

 ing that curve in two points k and l, the powers of its two segments pk and 

 PL, intercepted between the given point p and the two points of intersection, 

 if they be raised to an equal height (that is, both squares, or cubes, or biqua- 

 drats, &c.), in every position of the right-line, may make the same sum, 



PK^ + PL'^ or PK^ -f PL% &c. 



to the celebrated Newton, who solved them as soon as he received the paper containing their pro- 

 posal; although for the soUilion of tlie former of them, viz. to tind the curve of swiftest descent, 

 Bernouilli had proposed, in the Leipsic Acts, to allow all mathematicians the term of 6 months to 

 give in their solutions; and even at the request of M. Leibnitz, exteiuled tiie term for that purpose 

 to the end of 12 months. The history of this celebrated problem, and of various solutions that 

 were given to it, may be seen in Montuda's History of Mathematics, vol. ii. p. 437, 2d edition. 

 Although Newton's solution was anonymous, the masterly manner of it readily detected the author, 

 and Bernouilli said he perceived the lion thryugli hii disgui':e. 



