NEWTON S PRINCIPIA. 303 



city of the Sun, yet still it is impossible to explain the 

 existing phenomena by a vortex. The third law of Kepler 

 declares that the squares of the periodic times of the 

 planets are as the cubes of their mean distances. But his 

 preceding propositions have shown that this is not the case 

 in a vortex. It is true that his reasoning is founded on 

 two hypotheses ; first, that the resistance arising from 

 friction varies as the velocity; and, secondly, that the 

 degree of fluidity is the same throughout the vortex. Let 

 us, however, suppose that the friction varies as the ?rcth 

 power of the velocity, and the frictional power of the 

 several strata as the ttth power of the distance. Then, 

 according to Newton, we have 



r ) . r n = constant 

 d rJ 



r 



Hence, that Kepler s law may be true, we must have 



3 



n + 2 = - m\ 



either, then, m is greater than unity, or n is negative. But 

 Newton considered that if the resistance did not vary as the 

 velocity, it would vary in a less ratio, that is, m less than 

 unity, and that if the parts of the fluid had various degrees 

 of fluidity, those parts that had least fluidity would be 

 heaviest, and would, therefore, be furthest from the centre, 

 that is, n positive ; hence he thinks that the theory of 

 vortices cannot be made to explain the third law of Kepler. 

 The motion of the vortex cannot be made to agree with 

 the two first laws of. Kepler. The planets move in ellipses 

 which are so placed that their major axes are not parallel. 

 To account for this Descartes supposed the vortices them- 



