PROGRESS OF MATHEMATICS AND MECHANICS 247 



41 nature abhors a vacuum" as an explanation. He shows experi- 

 mentally that a body descends an inclined plane with uniformly 

 accelerated motion. 



In a board 12 ells in length a groove half an inch wide was made. 

 It was drawn straight and lined with very smooth parchment. The 

 board was then raised at one end, first one ell, then two. Then Galileo 

 let a polished brass ball roll through the groove and determined the 

 time of descent for the whole length of the groove. If on the other hand 

 he let the ball roll through only one quarter of the length, this required 

 just half the time. . . . The distances were to each other as the 

 squares of the times, 



a law verified by hundredfold repetitions for all sorts of distances 

 and slopes. The time was still determined by weighing water 

 escaping through a small orifice. He shows by ingenious experi- 

 ments the dependence of velocity on height alone, and that a 

 freely falling body has the necessary energy to reach its original 

 level. The whole theory of the falling body is now easily deduced. 



When, therefore, I observe a stone initially at rest falling from an 

 elevated position and continually acquiring new increments of speed, 

 why should I not believe that such increases take place in a manner 

 which is exceedingly simple and rather obvious to everybody? If 

 now we examine the matter carefully we find no addition or increment 

 more simple than that which repeats itself always in the same manner. 

 This we readily understand when we consider the intimate relation- 

 ship between time and motion ; for just as uniformity of motion is de- 

 fined by and conceived through equal times and equal spaces (thus we 

 call a motion uniform when equal distances are traversed during equal 

 time-intervals), so also we may, in a similar manner, through equal 

 time-intervals, conceive additions of speed as taking place without 

 complication ; thus we may picture to our mind a motion as uniformly 

 and continuously accelerated when, during any equal intervals of time 

 whatever, equal increments of speed are given to it. ... 



Hence the definition of motion which we are about to discuss may 



column of water. And really is not that thing which is attracted in the pump a 

 column of water attached at the upper end and stretched more and more until 

 finally a point is reached where it breaks, like a rope, on account of its excessive 

 weight? . . .' 



