50 A SHORT HISTORY OF SCIENCE 



the custom to credit its founder with all sorts of knowledge which 

 he could not possibly have possessed. 



Pythagoras makes the classification, arithmetic (numbers 

 absolute), music (numbers applied), geometry (magnitudes at 

 rest), astronomy (magnitudes in motion), this fourfold division or 

 " quadrivium " continuing in vogue for some two thousand years. 

 The distinction between abstract and concrete arithmetic had been 

 emphasized among the Greeks in comparatively early times. 

 Arithmetic and geometry were distinguished on one side from 

 mechanics, astronomy, optics, surveying, music, and computation 

 on the other. The aim of Greek arithmetic "was entirely differ- 

 ent from that of the ordinary calculator, and it was natural that 

 the philosopher who sought in numbers to find the plan on which 

 the Creator worked, should begin to regard with contempt the 

 merchant who wanted only to know how many sardines, at 10 for 

 an obol, he could buy for a talent." 



The limited mathematics of the practical Egyptians had con- 

 sisted of numerical cases. It was an easy step for Pythagoras to 

 make number in a somewhat mystical sense the central element 

 in his philosophy. 



PYTHAGOREAN ARITHMETIC. In pure arithmetic or number 

 theory as we should call it, the Pythagoreans enunciated such 

 dicta as, for example, "Unity is the origin and beginning of all 

 numbers but not itself a number." Prime and composite num- 

 bers were also distinguished, and theorems of considerable alge- 

 braic complexity discovered. There is naturally no algebraic 

 symbolism, but "unknown" and "given" quantities are employed 

 in the modern sense. Odd and even numbers received special 

 names, and besides the series of squares and cubes and the arith- 

 . metic and geometric progressions previously known, 



other series were derived from these, for example, 



the triangular numbers : 1, 3, 6, 10, 15, etc., by 

 successive addition of the natural numbers. The 

 reason for the name triangular will be clear if one 

 counts the dots in the triangle formed by taking one, two, three or 

 more rows beginning at the top of the figure. 



