BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 283 



outlay, an idea not indeed strange to some of the Greeks. 

 The law of refraction of a ray of light he deals with correctly as 

 a particular case of the principle of economy, a principle which 

 exerted a potent influence in the scientific philosophy of the 

 following century. Thus for example Euler says in 1744 : 



Since the organization of the world is the most excellent, nothing is 

 found in it, out of which some sort of a maximum or minimum 

 property does not shine forth. Therefore no doubt can exist, that all 

 action in the world can be derived by the method of maxima and 

 minima as well as from the actual operating causes. 



Fermat's work in the theory of probability is fundamental. He 

 discusses the case of two players, A and B, where A wants two 

 points to win and B three points. Then the game will certainly 

 be decided in the course of four trials. Take the letters a and b, 

 and write down all the combinations that can be formed of four 

 letters. These combinations are 16 in number, namely aaaa, 

 aaab, aaba, aabb, abaa, abab, abba, abbb, baaa, baab, baba, babb, 

 bbaa, bbab, bbba, bbbb. Now every combination in which a oc- 

 curs twice or oftener represents a case favorable to A, and every 

 combination in which b occurs three times or oftener represents a 

 case favorable to B. Thus, on counting them, it will be found 

 that there are 11 cases favorable to A, and 5 cases favorable to 

 B ; and, since these cases are all equally likely, A's chance of win- 

 ning the game is to B's chance as 11 is to 5. 



Like Descartes, Pascal (1623-1662) devoted but a fraction of 

 his great talent to mathematical science. 



I have spent much time in the study of the abstract sciences, 

 but the paucity of persons with whom you can communicate on such 

 subjects gave me a distaste for them. When I began to study man, 

 I saw that these abstract studies were not suited to him, and that in 

 diving into them, I wandered farther from my real track than those 

 who were ignorant of them, and I forgave men for not having at- 

 tended to these things. But I thought at least I should find many 

 companions in the study of mankind, which is the true and proper study 

 of man. Again I was mistaken. There are yet fewer students of Man 

 than of Geometry. 



