MERITS AND DEFECTS 281 



Let US try to trace this curve by the motion of a 

 point starting from the origin of co-ordinates. In 

 which direction must the point move from the 

 origin ? To answer this question we differentiate, 

 and find dy j dx=s\n {i I x) — {i / x) cos ( i / x). At 

 the origin we have x = o and 7 = 0. No value can be 

 assigned to dy / dx, because i / x has no meaning 

 when^ = o ; moreover,the equationjr = ^' sin (i / x) ìs 

 expressly stated above to apply only when x is not 

 zero. There is, therefore, no way of ascertaining 

 the direction in which the point must depart from 

 the origin. Perhaps we can do better if the moving 

 point is started at another part of the curve. An 

 attempt to plot the curve reveals the fact that it 

 lies between two right Hnes, of which one makes 

 with the ;i'-axis an angle of 45°, the other an angle 

 of —45°. As the point moves along the curve 

 toward the origin, the curve is found to oscillate 

 with ever-increasing rapidity. When we try to 

 determine the direction by which it jumps into the 

 origin, we encounter the same difficulty as before. 

 As long as x is finite, the direction of motion is 

 determinable. But as soon as we try x=o, the 

 determination is impossible. This conclusion must 

 be accepted, in spite of the fact that the curve is 

 contiiiuous in ali its parts, including the origin. 

 This example illustrates the inadequacy of motion 

 as a fundamental concept. 



239. Difficulties are encountered in the notion of 

 velocity. Is variable velocity an objective reality ? 

 Take a body falling from rest. We say that its 



