Pythagoras studied science in Egypt and first became familiar with Egyptian and Babylonian mathematics and geometry. He also studied the Milesian cosmological philosophy. On his return to Greece from his foreign studies he sought to discover a principle of homogeneity in the universe more acceptable than any suggested by the earlier philosophers. He had noticed numerous relationships between numbers and natural phenomena, and believed that the true basis of philosophy was to be found in numbers. In seeking data to sustain this thesis, he discovered harmonic progression. His experiments showed that when harp strings of equal length were stretched by weights having the proportion of ½:⅔:¾, they produced harmonic intervals of an octave, a fifth and a fourth apart. Since he saw that harmony of sounds depended upon proportion he concluded that order and beauty in the world originate in numbers. There are seven intervals in a musical scale, and seven planets sweeping the heavens. Seven must, therefore, be a basic number. This suggested to him his ideas regarding the harmony of the spheres. Pythagoras and his students found that the sum of a series of odd numbers from 1 to 2n+1 was always a complete square. When even numbers are added to the above series we get 2, 6, 12, 20, etc., in which every member can be broken into two factors differing from each other by unity. Thus 6 = 2.3, 12 = 3.4, 20 = 4.5, etc. Such numbers were called heteromecic. Numbers like n(n+1)⁄₂ were called triangular. A large number of other arithmetical relations were found and given distinctive names. The Pythagoreans were also familiar with the principles of arithmetical, geometrical, harmonic, and musical proportion. [Illustration: DE WITT CLINTON TRAIN OF 1831 BESIDE A MODERN LOCOMOTIVE] [Illustration: LOCOMOTIVE OF THE 1870 PERIOD STILL IN USE IN THE OZARKS] [Illustration: "JOHN BULL," A LOCOMOTIVE BROUGHT FROM ENGLAND AND PUT INTO SERVICE IN AMERICA IN 1831] Pythagoras made similar advances in geometry. He believed that each arithmetical fact had an analogue in geometry, and each geometrical fact a counterpart in arithmetic. He devised a rule by which integral numbers could be found so that the sum of the squares of two of them equaled the square of the third. He also developed the theory of irrational quantities. The first incommensurable ratio discovered is said to have been that of the side of a square to its diagonal which is 1:√2̅. Euclid (300 B. C.) developed this theory in the tenth book of his geometry as still used. Pythagoras not only placed mathematics on a solid scientific basis, he also established the fact that the physical phenomena of the world are governed by mathematical laws.