328 HISTORY OF MECHANICS. 



equally increasing the velocity of a descending body, — will become 

 clear by a little attention ; but it undoubtedly presents difficulty at 

 first. Accordingly, we find that Descartes did not accept it. " It is 

 certain," he says, "that a stone is not equally disposed to receive anew 

 motion or increase of velocity when it is already moving very quickly, 

 and when it is moving slowly." 



Descartes showed, by other expressions, that he had not caught hold 

 of the true notion of accelerating force. Thus, he says in a letter to 

 Mersenne, "I am astonished at what you tell me, of having found, by 

 experiment, that bodies thrown up in the air take neither more nor 

 less time to rise than to fall again ; and you will excuse me if I say 

 that I look upon the experiment as a very difficult one to make ac- 

 curately." Yet it is clear from the Notion of a Constant Force that 

 (omitting the resistance of the air) this equality must take place ; for 

 the Force which will gradually destroy the whole velocity in a certain 

 time in ascending, will, in the same time, generate again the same ve- 

 locity by the same gradations inverted ; and therefore the same space 

 will be passed over in the same time in the descent and in the ascent. 



Another difficulty arose from a necessary consequence of the Laws 

 of Falling Bodies thus established ; — the proposition, namely, that in 

 acquiring its motion, a body passes through every intermediate degree 

 of velocity, from the smallest conceivable, up to that which it at last 

 acquires. When a body falls from rest, it begins to fall with no velo- 

 city ; the velocity increases with the time ; and in one-thousandth par: 

 of a second, the body has only acquired one-thousandth part of the 

 velocity which it has at the end of one second. 



This is certain, and manifest on consideration ; yet there was at first 

 much difficulty raised on the subject of this assertion ; and disputes 

 took place concerning the velocity w T ith which a body begins to fall. 

 On this subject also Descartes did not form clear notions. He writes 

 to a correspondent, " I have been revising my notes on Galileo, in 

 which I have not said expressly that falling bodies do not pass through 

 every degree of slowness, but I said that this cannot be known without 

 knowing what Weight is, which comes to the same thing ; as to your 

 example, I grant that it proves that every degree of velocity is infi- 

 nitely divisible, but not that a falling body actually passes through all 

 these divisions." 



The Principles of the Motion of Falling Bodies being thus establish- 

 ed by Galileo, the Deduction of the principal mathematical conse- 

 quences was, as is usual, effected with great rapidity, and is to be found 



