PLANETARY MOTIONS 43 



earth should not be construed as a falling body with 

 reference to the moon ; hence the Newtonian tidal theory 

 is erroneous ! 



The figure on page 34, to which attention has once 

 before been directed, is that usually employed by writers 

 and teachers to illustrate Newton's doctrine. The " ex- 

 planation" runs thus: 



Suppose that the moon originally started from the 

 point with a velocity capable of carrying her along the 

 tangent to the point A. Instead of arriving at A, how- 

 ever, on account of the earth's attraction she arrives at P, 

 having followed the arc in place of the tangent, and so 

 on." That phrase, " and so on", is part of the explan- 

 ation, and is as far as it ever goes, as far as the theorists 

 dare go. They stop precisely at the point where the 

 trouble begins! Every tyro in mathematics knows that 

 an arc is shorter than its tangent, that is, that OP is 

 shorter than OA; hence, logically, the moon's velocity at 

 P cam-not possibly be as great as it was at 0, and by the 

 same token she would not have sufficient momentum left 

 to carry her in the succeeding second as far, either along 

 the tangent or along the arc, as it did in the first. Un- 

 less, therefore, nature has some secret way of restoring 

 to the moon this lost energy of motion so that her velocity 

 is constantly whipt up and kept up, she must sink ever 

 lower and lower, as any ordinary projectile would, and 

 soon strike the earth. When, therefore, astronomers as- 

 sert that the moon's "momentum" sustains her in her or- 

 bit, that her momentum is thereby nevertheless not im- 

 paired and that there is no way of renewing that momen- 

 tum, they assert in effect that the whole minus many times 

 its parts is still equal to the whole that children can eat 

 their pudding and have it too. So, indeed, they, can pro- 

 vided, however, that new pudding be baked as fast as the 

 supply on hand disappears ; which js not shown to be the 

 case here. 



By way of an attempt to parry this difficulty, New- 

 tonians have succeeded in persuading themselves that 

 mathematical exactness between the velocity of the tan- 

 gential motion and the force of gravity is not vital to the 



