such epoch during the entire interval. This was when 

 abstract geometrical reasoning commenced, and astro- 

 nomical observations aiming at precision were recorded, 

 compared and discussed. Closely associated with it 

 must have been the construction of the forms of logic. 

 The radical difference between the demonstration of a 

 theorem of geometry and the reasoning of everyday 

 life which the masses of men must have practiced from 

 the beginning, and which few even today ever get 

 beyond, is so evident at a glance that I need not dwell 

 upon it. The principal feature of this advance is that, 

 by one of those antinomies of the human intellect of 

 which examples are not wanting even in our own time, 

 the development of abstract ideas preceded the concrete 

 knowledge of natural phenomena. When we reflect 

 that in the geometry of Euclid the science of space was 

 brought to such logical perfection that even today its 

 teachers are not agreed as to the practicability of any 

 great improvement upon it, we cannot avoid the feeling 

 that a very slight change in the direction of the intel- 

 lectual activity of the Greeks would have led to the 

 beginning of natural science. But it would seem that 

 the very purity and perfection which was aimed at in 

 their system of geometry stood in the way of any 

 extension or application of its methods and spirit to 

 the field of nature. One example of this is worthy of 

 attention. In modern teaching the idea of magnitude 

 as generated by motion is freely introduced. A line is 

 described by a moving point ; a plane by a moving 

 line ; a solid by a moving plane. It may, at first sight, 

 seem singular that this conception finds no place in the 

 Euclidian system. But we may regard the omission 



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