$$ i 7 i, 172] Motion in a Circle 215 



constantly to draw the body towards o, or counteracting 

 the tendency which it has, in virtu'e of the First Law of 

 Motion, to get farther and farther away from o. To 

 express either of these tendencies numerically we want a 

 more complex idea than that of velocity or rate of motion, 

 namely acceleration or rate of change of velocity, an idea 

 which Galilei added to science in his discussion of the 

 law of falling bodies (chapter vi., 116, 133). A falling 

 body, for example, is moving after one second with the 

 velocity of about 32 feet per second, after two seconds 

 with 'the velocity of 64, after three seconds with the velocity 

 of 96, and so on ; thus in every second it gains a downward 

 velocity of 32 feet per second; and this may be expressed 

 otherwise by saying that the body has a downward accele- 

 ration of 32 feet per second per second. A further in- 

 vestigation of the motion in a circle shews that the motion 

 is completely explained if the moving body has, in addition 

 to its original velocity, an acceleration of a certain magnitude 

 directed towards the centre of the circle. It can be shewn 

 further that the acceleration 'may be numerically expressed 

 by taking the square of the velocity of the moving body 

 (expressed, say, in feet per second), and dividing this by 

 the radius of the circle in feet. If, for example, the body 

 is moving in a circle having a radius of four feet, at the 

 rate of ten feet a second, then the acceleration towards 



the centre is ( = J 25 feet per second per second. 



These results, with others of a similar character, were 

 first published by Huygens not of course precisely in this 

 form in his book on the Pendulum Clock (chapter VIIL, 

 158) ; and discovered independently by Newton in 1666. 



If then a body is seen to move in a circle, its motion 

 becomes intelligible if some other body can be discovered 

 which produces this acceleration. In a common case, such 

 as when a stone is tied to a string and whirled round, 

 this acceleration is produced by the string which pulls 

 the stone ; in a spinning-top the acceleration of the outer 

 parts is produced by the forces binding them on to the 

 inner part, and so on. 



172. In the most important cases of this kind which 

 occur in astronomy, a planet is known to revolve round 



