NEWTON S PRINCIPIA 103 



that its evolute is an equal and similar cycloid, a property 

 which it has in common with another curve, the logarithmic 

 spiral, whose tangent makes with the radius vector a con 

 stant angle. He investigated the case of hypocycloids and 

 hypercycloids, rather than the common cycloids, because 

 it is that of the earth s gravity, which above the surface de 

 creases inversely as the square of the distance from the 

 centre, but within the sphere increases as the distance 

 simply. 



It follows, from the propositions respecting the vibration 

 of pendulums, that the times of the descent of falling bodies 

 may be compared together and with the times of vibrations 

 of the pendulum : So that the time of a vibration round a 

 given centre being given, as a second, the time of the 

 falling body s descent to the centre of forces can be found, 

 or the equal time of vibration in the circular arch of 

 90 with any radius. The time is to the given time as 



1 to Y), L being the length of the pendulum, and D 



the distance from the point of suspension to the centre 

 of forces; and since D becomes infinite and the lines in 

 which the central force acts parallel, and since half the 

 length of the pendulum is to the line fallen through in the 

 time of one vibration as 1 to 9,869 nearly (the proportion 

 of the square of the diameter to that of the circumference), 

 we can easily ascertain the force of gravity at any point 

 by the length of the pendulum vibrating seconds. It is 

 found to be in these latitudes about 34 44 ; consequently 

 a body falls in a second through about 16 feet 9 inches. 



Hitherto we have only considered the motions and tra- / 

 jectories of bodies acted upon by forces directed towards 

 a fixed centre whether in the plane of their motion or out 

 of that plane, and supposing that plane either to be fixed 

 or to be moved round the centre of forces. But as action 



H 4 



