ANCIENT SCIENCE. 13 



also had his answer to the question concerning the universal principle 

 of things, and his doctrine appears at first sight much stranger than 

 those of his predecessors. For him, this universal principle was order, 

 harmony, proportion. His disciples held that number was the essence 

 of all things, and that its power pervaded all the works of art and of 

 nature. We cannot suppose that they believed that numbers were 

 the causes of things ; the meaning probably was that all things must 

 be considered under the condition of number, quantity, or proportion. 

 Possibly, therefore, the Pythagorean doctrine contains a recognition of 

 the necessity of studying the laws of nature as expressible by quanti- 

 tative relations. The Pythagoreans, however, attached extraordinary 

 importance to some numbers, merely as numbers. Thus, 10 was held 

 to be above all others the perfect number, because it is formed by 

 the addition of the first four numbers (that is, 1 + 2+3+4=10), and 

 probably also because it was connected with a mysterious symbol in 

 the religion of the Chaldeans and Egyptians. Great significance was 

 also attached to the number 3, which was called "the Number of the 

 Universe," because everything, it was said, must have three parts, 

 namely the beginning, the middle, and the end. Reasons were 

 found for also appropriating to the numbers 5, 7, and 9, certain sym- 

 bolical or mystical meanings. It is curious to remark how general 

 has been the tendency to attach significance to certain numbers, and 

 the observant reader will not fail to notice the existence of this ten- 

 dency even at the present day. 



Pythagoras was perhaps the first of the Greeks who taught arithmetic 

 and geometry on a scientific plan. He invented the multiplication 

 table, and the abacus which is still used in schools to impart the 

 elements of arithmetic. He is reputed to have been the first to in- 

 troduce among the Greeks a system of exact weights and measures. 

 Some important theorems of geometry are considered to be his dis- 

 coveries, and among these is the famous 

 proposition about the squares described 

 on the sides of a right-angled triangle. 

 As one of the most simple, elegant, and 

 useful of geometrical propositions, we 

 may express it here for the benefit of the 

 non-mathematical reader, leaving him to 

 test its truth. 



Let ABC (Fig. 4) be any right-angled 

 triangle, having its right angle at B. If 

 squares be formed on each side of the 

 triangle, it will be found that the area of 

 the square on the side opposite the right FIG. 4 . 



angle is exactly equal to the sum of the 



areas of the other two squares. It was said that Pythagoras was so de- 

 lighted by the discovery of this truth that he offered up a sacrifice ; 



