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savage who counted only by tlie help of outward objects, 

 to whom 15 was " half the hands and both the feet," and 

 Newton or Laplace ? The answer is the history of Ma- 

 thematics and its successive developments, arithmetic, 

 geometry, algebra, &c. The first and greatest step in all 

 this was the transition from number in the concrete to 

 number in the abstract. This was the beginning not 

 only of Mathematics but of all abstract thought. The 

 reason and mode of it was the same as in the individual. 

 There was the same general influx of evidence, the 

 same unsought for experimental proof, the same re- 

 I cognition of general laws running through all manner of 

 I purposes and relations of life. No wonder then if, 

 1 under such circunistances. Mathematics, like some other 

 subjects, and perhaps with better excuse, came after a 

 [ time to be clothed with mysticism ; nor that, even in 

 I modern times, they should have been placed upon an 

 : a ^priori basis, as in the philosophy of Kant. Number Their sur 

 \was so soon found to be a principle common to many ' 



1 branches of knowledge that it was readily assumed to 

 I be the key to all. It gave distinctness of expression, if 

 !not clearness of thought, to ideas which were float- 

 ing in the untutored mind, and even suggested to 

 it new conceptions. In " the one," " the all," " the 

 many in one," (terms of purely arithmetic origin,) it 

 gave the earliest utterance to men's first crude notions 

 labout God and the world. In " the equal," " the solid," 

 "the straight," and " the crooked," which still survive 

 |as figures of speech among ourselves, it supplied a 

 vocabulary for the moral notions of mankind, and 

 quickened them by giving them the power of expres- 

 sion. In this lies the great and enduring interest in 

 the fragments which remain to us of the Pythagorean 

 philosophy. 



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