THE STUDY OF DIOPHANTINE ANALYSIS IN THE 

 UNITED STATES. 



By Florian Cajoei, 



I. Infrodiicfion. 



The term " number " was used b}' the Greeks in a restricted 

 sense. With early mathematicians, as the Pythagoreans, it was 

 never used except for integers. AVith them even unity was no 

 number; a collection of units only was designated by that name. 

 Fractious were looked upon as merely ratios of numbers. Later 

 Oreek authors, particularly Diophantus, enlarged the use of the 

 term so as to include fractional numbers; but never were irra- 

 tionals classified as numbers by any Greek mathematician. 



The oldest extant work on algebra is the one of Diophantus. 

 He took pains to exclude from his book the methods and con- 

 ■ceptions of geometry which before his time had been extended 

 by the Greeks even to arithmetic. Since irrational quantities 

 were not looked upon by the Greeks as numbers and since, to 

 their minds, they represented only lines, surfaces, and solids of 

 definite dimensions, they could not be admitted into the algebra 

 of Diophantus. According to him, algebra could be built up 

 independently of geometry only by the exclusion of irrational 

 solutions. Hence arose the condition, imposed by him, that 

 solutions should be given in numbers. 



The term " Diophantine Analysis " is sometimes used in a 

 wrong sense. An equation, nx -\- by = c, in which only inteo-ral 

 values of x and y are to be found, is frequently called " Dio- 

 phantine." But the solution of this equation was unknown to 

 Diophantus; nor did he ever propose to himself such a problem. 

 He always imposed the condition that the results be rational, 



