Anaximenes contended that the basic element was not boundless, but determinate. Innumerable substances are derivable from it and, just as our soul, like an atmosphere, holds us together, so do breath and air encompass the whole world. Air is always in motion, otherwise so many changes could not be made by it. It differs in various substances in virtue of its rarefaction and condensation.
The perpetual changes taking place in the world owing to the instability of matter were emphasized by Heraclitus. He taught that there is nothing immutable in the world process excepting the law or principle which governs it.
Cosmological speculations were not the only ones attracting the attention of the Greek scientists. Pythagoras, for example, founded a philosophical college devoted to mathematical studies which resulted in the development of arithmetic to points beyond the requirements of commerce. He made arithmetic the basis of a profound philosophical system.
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.
DE WITT CLINTON TRAIN OF 1831 BESIDE A MODERN LOCOMOTIVE
