Page:A History of Mathematics (1893).djvu/87

 so of all things. Four is the most perfect number, and was in some mystic way conceived to correspond to the human soul. Philolaus believed that 5 is the cause of color, 6 of cold, 7 of mind and health and light, 8 of love and friendship.[6] In Plato's works are evidences of a similar belief in religious relations of numbers. Even Aristotle referred the virtues to numbers.

Enough has been said about these mystic speculations to show what lively interest in mathematics they must have created and maintained. Avenues of mathematical inquiry were opened up by them which otherwise would probably have remained closed at that time.

The Pythagoreans classified numbers into odd and even. They observed that the sum of the series of odd numbers from 1 to $$\scriptstyle{2n+1}$$ was always a complete square, and that by addition of the even numbers arises the series 2, 6, 12, 20, in which every number can be decomposed into two factors differing from each other by unity. Thus, $$\scriptstyle{6=2 \cdot 3}$$, $$\scriptstyle{12=3 \cdot 4}$$, etc. These latter numbers were considered of sufficient importance to receive the separate name of heteromecic (not equilateral).[7] Numbers of the form $$\scriptstyle{{n(n+1)} \over 2}$$ were called triangular, because they could always be arranged thus, $$\scriptstyle{\overset{\overset{\overset{\bullet}{\bullet \bullet}}{\bullet \bullet \bullet}}{\bullet \bullet \bullet \bullet}}$$. Numbers which were equal to the sum of all their possible factors, such as 6, 28, 496, were called perfect; those exceeding that sum, excessive; and those which were less, defective. Amicable numbers were those of which each was the sum of the factors in the other. Much attention was paid by the Pythagoreans to the subject of proportion. The quantities $$\scriptstyle{a,~b,~c,~d}$$ were said to be in arithmetical proportion when $$\scriptstyle{a-b=c-d}$$; in geometrical proportion, when $$\scriptstyle {a:b=c:d}$$; in harmonic proportion, when $$\scriptstyle{a-b:b-c=a:c}$$. It is probable that the Pythagoreans were also familiar with the musical