128 FROM NEBULA TO NEBULA 



lev er but, mark you, not of a stationary lever, but of one 

 whose pendent weights are in the act of falling. 



To begin with, what is the principle of the simple 

 lever or balance arm? It is this: Suppose a bar to be 

 supported on a pivot so as to rotate in a horizontal plane, 

 then, in order that its arms, if unequal in length, shall 

 balance, the weights at the ends must be inversely pro- 

 portional to those lengths. That is to say, if one of the 

 arms be half the length of the other, the weight on the 

 shorter end must be doubled, if one-third the length, 

 trebled, and so on. 



And what are the laws of falling bodies? These are 

 given by Ganot (Art. 49) in these words: 



1. The velocities are proportional to the times dur- 

 ing which the motion has lasted. 



2. The spaces described are proportional to the 

 squares of the times employed in their description. 



3. The spaces described are proportional to the 

 squares of the velocities acquired during their descrip- 

 tion. 



4. The spaces described in equal successive periods 

 of time increase by a constant quantity. 



Suppose a bar, whose longer arm we shall call R and 

 its shorter r, be rotated horizontally around a stationary 

 pivot, it will then descibe two circles^ and, by geometry, 

 we get the equation, 



r :R ::2*r :2-xR (1) 



By our hypothesis, however, the pivot is not station- 

 ary, but is falling, and the ends of the bar are not describ- 

 ing closed circles in fact, but coils of open spirals. In 

 still other words, the last two terms of our equation are, 

 properly construed, "heights fallen through." Under 

 the second rule given above, then, our last two terms, in 

 order to express the element of time instead of space, 

 must be amended to read 



Vy^F; V171T (2) 



