PLANETARY MOTIONS 33 



difference in the world when this distance is taken in con- 

 nection with the velocity of the moon in her orbit. For 

 the feasibility of the Newtonian hypothesis presupposes 

 the precisest, undeviating correspondence between the 

 length of the space fallen through by the moon in one sec- 

 ond of time and the rate of her tangential velocity per 

 second, else must she fall to, or escape from, the earth. 

 This point may be made clearer by a reference to Figure 

 1, copied from Sir Oliver Lodge's book, Pioneers of 

 Science (p. 171), with the text accompanying it: 



Now consider circular motion in the same way, say a ball 

 whirled round by a string. 



Attending to the body at O, it is for an instant moving to- 

 wards A, and if no force acted it would get to A in a time which, 

 for brevity, we may call a second. But a force, the pull of the 

 string, is continually drawing it towards S, and so it really finds 

 itself at P, having described the circular arc OP, which may be 

 considered to be compounded of, and analyzable into the rectilin- 

 ear motion OA and the drop AP. At P it is for an instant mov- 

 ing towards B, and the same process therefore carries it to Q ; in 

 the third second it gets to R; and so on: always falling, so to 

 speak, from its natural rectilinear path, towards the centre, but 

 never getting any nearer to the centre. 



The force with which it has thus to be constantly pulled in 

 towards the centre, or, which is the same thing, the force with 

 which it is tugging at whatever constraint it is that holds it in, is 



; where m is the mass of the particle, v its velocity, and r the 



radius of its circle of movement. This is the formula first given 

 by Huyghens for centrifugal force. 



But suppose that, for any reason whatsoever, as by 

 etheric or meteoric resistance, the moon's momentum 

 (which the reader should never forget is, according to 

 Newtonian theory, altogether unexplained) should not 

 avail to carry her clear to A, but only to some point by 

 ever so little short of it, then the moon would inevitably 

 sink at P within the line of her orbit, that is, nearer to the 

 earth, where the latter 's attraction would become even 

 greater, absolutely and relatively, and consequently, dur- 

 ing the succeeding second, overmatch the tangential ve- 

 locity still more ; a process which could not be stayed and 



