iO SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



meet with considered from the point of view that I shall indicate. 

 I do not need to say that the exact and general knowledge of 

 this law will be important in many questions, for example, in the 

 problem of ballistics. But for the object which I have in view in 

 this present memoir I can admit the ordinary law of the resistance 

 proportional to the square of the velocity as being sufficiently 

 accurate. 



It is Newton, also, who has given the first example of the deter- 

 mination of the motion of a heavy body in a resisting medium. 

 He solved the problem when the motion is vertical by assuming 

 the resistance proportional either to the velocity or to its square, 

 but when the projectile is projected into the atmosphere in any 

 direction whatever he confined himself to considering the case 

 of a resisting force proportional to the simple velocity, observing 

 nevertheless that this case is not that of nature. The two equa- 

 tions that Newton was obliged to integrate in order to determine 

 the horizontal and vertical components of the velocity at any 

 instant, are linear of the first order and with constant coefficients; 

 and the two unknown quantities are so separated in them that 

 these two equations are solved independently of each other, and 

 their solution really implies only a simple direct integration. This 

 is no longer true in the case of a resistance proportional to the 

 square of the velocity; the two unknown quantities enter at the 

 same time into each of the equations of motion, which are no 

 longer linear, and it is only by a special combination that we 

 succeed in separating the variables therein and in reducing them 

 to quadratures, which we consider as the complete solution of 

 the problem. 



This was done by John Bernoulli, who published it in the Acta 

 Eruditorum, Leipzig, May 1719, pp. 216-226, more than thirty 

 years after the solution by Newton, and at an epoch when the 

 integral calculus had already made great progress. However, Euler, 

 at the beginning of his memoir on this subject, 5 expresses his sur- 

 prise at seeing that Newton, "who has well solved other problems 

 more difficult," should stop with the case of the resistance pro- 

 portional to the simple velocity, and not consider the case of 

 nature. We know, however, that the question of the trajectory 

 in a medium resisting in proportion to the square of the velocity 

 was proposed as a challenge to the geometers of the continent 

 by an Englishman named Keil, who believed the problem insol- 



1 Memoirs of the Academy of Berlin; year 1753. 



