iv MODERN SCIENCE 353 



with a model. Now, what do the laws of Kepler say r 

 They lay down a relation between the areas described 

 by the heliocentric radius-vector of a planet and the 

 time employed in describing them, a relation between 

 the longer axis of the orbit and the time taken up by 

 the course. And what was the principle discovered by 

 Galileo ? A law which connected the space traversed 

 by a falling body with the time occupied by the fall. 

 Furthermore, in what did the first of the great 

 transformations of geometry in modern times consist, 

 if not in introducing in a veiled form, it is true time 

 and movement even in the consideration of figures ? 

 For the ancients, geometry was a purely static science. 

 Figures were given to it at once, completely finished, 

 like the Platonic Ideas. But the essence of the 

 Cartesian geometry (although Descartes did not give 

 it this form) was to regard every plane curve as de 

 scribed by the movement of a point on a movable 

 straight line which is displaced, parallel to itself, along 

 the axis of the abscissae, the displacement of the 

 movable straight line being supposed to be uniform and 

 the abscissa thus becoming representative of the time. 

 The curve is then defined if we can state the relation 

 connecting the space traversed on the movable straight 

 line to the time employed in traversing it, that is, if 

 we are able to indicate the position of the movable 

 point, on the straight line which it traverses, at any 

 moment whatever of its course. This relation is just 

 what we call the equation of the curve. To substitute 

 an equation for a figure consists, therefore, in seeing 

 the actual position of the moving points in the tracing 

 of the curve af &quot;any moment whatever, instead of re 

 garding this tracing all at once, gathered up in the unique 

 moment when the curve has reached its finished state. 



