374 NEWTON S PRINCIPIA. 



where a is the radius and h the vertical height through 

 which the pendulum oscillates. The time is therefore 

 longer than in a cycloid. The time also depends on the 

 length of the arc described. When a is very great and 

 h small, we may often neglect this term and say, 



T = , 1. 

 9 



We may learn many lessons from this ^important &quot;re 

 sult. 



I. It furnishes us with a method of comparing bodies 

 as to the quantity of matter in each. For we see that 

 for pendulums of the same length 



varies as T 2 ; 



60 



if then we take pendulums of the same weight, we can, 

 by observations on r, determine the masses or ^quantities 

 of matter in them. By experiments made with the 

 greatest accuracy, Newton and Bessel always found this 

 ratio constant; so that the weight of a body varies in 

 exact proportion to the quantity of matter in it. This 

 ratio we call y. And hence, 



g 



By observing the value of r for any value of /, we can 

 deduce that of y, giving, when the unit of time is a se 

 cond, 



g = 32-18 feet. 



II. The time of oscillation is independent of the arc. 

 By the use, therefore, of a cycloid, we are enabled to pre 

 vent any variations of the arc described produced by any 

 irregularities from affecting the rate of going of a clock. 



III. Since the time of an oscillation varies as the square 



