124 SCIENCE AND THE HUMAN MIND 



consciousness. As he found in human consciousness 

 an apprehension of God, he passed neither to material- 

 ism nor to that complete divorce between faith and 

 reason which, necessary for the well-being of both 

 at a certain stage of development, leads eventually to 

 the decay of faith. Descartes indeed remained a good 

 Catholic. But his belief in the possibility of proving 

 the existence of God by philosophic evidence savoured 

 of heresy to the theologians both of Utrecht and of 

 Leyden, and, more than once, Descartes had to appeal 

 to the protection of the Prince of Orange. 



In mathematics, Descartes' greatest achievement was 

 the application of algebra to the problems of geometry, 

 in which hitherto each problem had to be solved by 

 special treatment and by a fresh display of ingenuity. 

 For the first time, Descartes introduced a general 

 method by which this isolation was broken down. 

 The primary idea of co-ordinate geometry is easily 

 stated. If two straight lines or axes start from a 

 point or origin and set out at right angles to each 

 other, it is possible to specify the position of any point 

 in their plane by stating its distance x from one axis 

 and its distance y from the other ; x and y are called 

 the co-ordinates of the point, and different relations 

 between x and y correspond to different lines or curves 

 in the plane of the diagram. Thus if y be proportional 

 to x, or y = x x const ant, we get a straight line; if 

 y= x 2 x constant, a parabola; and so on. The pro- 

 blems of geometry are reduced to the operations of 

 algebra. 



Descartes' physical speculations were less successful 

 than his mathematical discoveries. We hear an echo 

 of the old ideas of contrasted words which deceived 



