306 DISTANCE VERSUS VELOCITY. 



its eccentricity. The extremity of the major axis, the nearest 10 

 the focus, is \hzperihelion, and its farthest extremity the aphelion. 

 The difference of the angles formed by the vector radii in- 

 dicate the inequality of the movements : to the greatest angle, 

 the perihelion, corresponds the maximum of velocity (a a'd'\ 

 just as to the smallest, or aphelion, corresponds the minimum 

 (m m') ; the other angles mark the velocities intermediary 

 between these two extremes. We have thus before us a series 

 of triangles with their apices at the focus of the ellipse, and their 

 bases on the contour of the curve. 



But these latter are not sufficient for the mind, whose prin- 

 cipal function lies in seeking unity among the variety of 

 phenomena. 



In what way are the variations of distance connected with 

 the variations of velocity ? What is the simplest expression of 

 their relationship ? These are questions which naturally pre- 

 sented themselves to Kepler's inquiring intellect. By dint of 

 immeasurable patience, and recommencing more than once 

 the same toil, this great astronomer discovered that the vari- 

 able arc traversed by the earth (or, in appearance, the sun), in 

 four-and-twenty hours, multiplied by one half the correspond- 

 ing vector radius, is a constant quantity: is the product which, 

 as elementary geometry teaches, gives the surface of a triangle. 

 And, in fact, look at the matter carefully: the vector radii 

 form triangles whose base is the arc traversed in the same 

 interval of time, and whose apices rest upon the centre of 

 the sun (or, in appearance, the observer, or the centre of the 

 earth). 



