THE HISTORY OF MATHEMATICS 391 



the defeat of the Spanish Armada, and the brief interlude of a com- 

 monwealth in England, had their counterpart in the invention of print- 

 ing and the appearance of scholars advancing knowledge in almost all 

 the civilized countries of the western world. The tremendous steps 

 taken in the course of two centuries, culminating in Newton and 

 Leibnitz, finally placed scientific investigation on a plane where it 

 became largely indifferent to what was going on in the political world, 

 and where it was able to pursue its own course, hampered it is true, 

 by wars and revolutions, but never without great names which will live 

 as long as the history of scientific development is remembered. 



This rapid sketch may serve to assist in seeing how the progress of 

 scientific thought has been related to the development of civilization. 

 We are all apt to regard our own concerns as independent developments 

 and too often the history of science is treated in the same way. But 

 in looking at the subject over long periods of time, we should treat 

 scientific development as one of the phenomena which will illustrate 

 progress or decline. The attitude of mind which leads to a search for 

 the secrets of nature is simply one manifestation of a common desire 

 for progress and if so, we should see signs of this desire in many di- 

 rections. The ultimate causes which underlie the changes which have 

 taken place — which produce periods of activity and inactivity in a 

 whole people or group of peoples — are unknown and will probably 

 only be finally found in the laboratories of the biologist and the 

 physicist. All we can do now is to correlate the facts as far as possible 

 and record them for future use. 



Let us return to the Greeks and examine a little more in detail their 

 contributions. In doing so we are at once faced by the difficulty that 

 most of our knowledge comes to us second hand and in the form of 

 treatises which gathered together past achievements. But few names 

 survive and it is not always easy to apportion the credit. It sometimes 

 appears that a name represents a school rather than an individual. 

 This is certainly true to some degree of Pythagoras, to whom is usually 

 credited the theorem that the sum of the squares on the sides of a right- 

 angled triangle is equal to that on the hypotenuse. He certainly formed 

 schools, but these were conducted as secret societies, the members of 

 which might not divulge the knowledge they attained. Their continued 

 existence for a long period of years was probably much assisted by the 

 custom or rule of attributing all discoveries made by the members to 

 their founder, thus avoiding much heartburning and jealousy. Never- 

 theless Pythagoras seems to have been responsible for placing 

 geometry on a scientific basis by investigating the theorems abstractly. 

 Briefly stated, it may be said that up to the middle of the fifth century 

 B. C, shortly after the battle of Salamis, the Greeks had discovered the 

 chief properties of areas in a plane and most of the regular solids. 



