282 



A SHORT HISTORY OF SCIENCE 



are so related that lines joining corresponding vertices meet in a 

 point O, then the intersections of corresponding sides will lie 



in a straight line A"B"C". It 

 remained for Monge, the inventor of 

 descriptive geometry (p. 335) and 

 others more than a century later 

 to carry this development forward. 

 Desargues's work was indeed prac- 

 tically lost until Poncelet in 1822 

 proclaimed him the Monge of his 

 century. 



THEORY OF NUMBERS AND PROBABILITY : FERMAT, PASCAL. 

 But little younger than Descartes and Cavalieri was Pierre de 

 Fermat (1601-1665) a man of quite exceptional position in mathe- 

 matical history. Devoting to mathematics such leisure as his 

 public duties afforded, he nevertheless published almost noth- 

 ing, many of his results being known to us only in the form of 

 brief marginal notes without proof. In editing Diophantus he 

 enunciated numerous theorems on integers, for example, 



An odd prime can be expressed as the difference of two square 

 integers in one and only one way. 



No integral values of x, y, z can be found to satisfy the equation 

 x* + y n = 2 "> if n be an integer greater than 2. 



This seemingly simple theorem has been verified for so wide a 

 range of values of n, that its truth can hardly be doubted, but no 

 general proof has yet been given in spite of a prize of 100,000 marks 

 awaiting him who either proves or disproves it. Some writers even 

 credit Fermat with a substantial share in the invention of the new 

 analytic geometry, in which he had certainly done independent 

 work for some years before Descartes's publication. Laplace in- 

 deed calls Fermat "the true inventor of the differential calculus." 

 He discusses problems of maxima and minima, and passing to 

 concrete phenomena, enunciates the interesting theorem : that 

 Nature, the great workman which has no need of our instru- 

 ments and machines, lets everything happen with a minimum of 



