NEWTON'S THEORY OF PLANETARY MOTIONS 93 



it be constantly directed towards the sun, the radius-vector of the 

 planet will sweep over equal areas in equal times. And, con- 

 versely, it cannot sweep over equal areas in equal times if the 

 force acts in any other direction than that of the sun. Hence it 

 follows, from Kepler's second law, that the force is directed to- 

 wards the sun itself. 



In transcribing the geometrical figure given by New- 

 comb, I have taken the liberty of adding the dotted lines 

 and using final letters of the alphabet to designate new 

 points of reference. Let Z, then, represent the center of 

 the earth, Ze a radius vector of the moon (the projectile 

 body), and let the line exv be drawn parallel with CAZ. 



In this demonstration (for which, by the way, New- 

 comb is not primarily responsible) we have a concrete 

 example of the difficulties that beset the path of those who 

 persist in the attempt to compress into a scheme of but 

 two dynamical dimensions the phenomena actually taking 

 place in one of three. Observe, on the one hand, (1) that 

 in Fig. 1 the circulating body (earth) is pictured cor- 

 rectly as forever falling toward the, central attraction 

 S, and (2) that in the last paragraph of Newcomb's own 

 statement he accepts this point, when he says: " Hence 

 it follows, from Kepler's second law, that the force is 

 directed toward the sun itself." On the other hand, how- 

 ever, when we turn to his diagram and the descriptive 

 text accompanying it, we discover to our surprise thai 

 the path of the body, DB, instead of being pointed rad- 

 ially toward Z as it honestly should, is cunningly drawn 

 parallel with CAZ, and points to nothing. Why does he 

 assert that the earth is always falling sunward, but 

 imply that the point of attraction races out along the 

 line ZQ'Q pari passu with the projectile body and as if it 

 will be at the end of the quadrant to greet the earth when 

 she shall have descended to that point ? Does he suppose 

 the sun to possess an astral body which he can thus pro- 

 ject out of his physical corpus and thereby prompt the 

 planets to fall to places where he himself is not? 



Now, for the sake of convenience, let us drop the ex- 

 ample of the sun and the earth and substitute the more 

 familiar one of the earth and the moon : 



