40 Colorado College Studies. 



but never that they be exclusively integral. The distinction 

 between the modern Theory of Numbers and Diophantine 

 Analysis has not been observed by American and many 

 European writers; and we find it therefore most convenient, in 

 this article, not to insist upon this distinction between the two. 

 The leading characteristics of Diophantus's solutions of 

 indeterminate problems is the startling ingenuity and dexterity 

 displayed, the lack of general methods, and the failure to detect 

 multiple values in the answers. 



II. Work in the United States. 



The study of Diophantine Analysis was introduced into, the 

 United States by Robert Adrain, a mathematician whose work 

 in one department of mathematics has not been sufficiently 

 appreciated. He contributed to the Matliematical Corre- 

 spondent, the earliest American -journal of mathematics, started 

 in 1804 in New York city, an essay, entitled, " A View of the 

 Diophantine Algebra," in which he gives a general discussion 

 of principles. He contributed later on the same subject to the 

 Anahjst, of which he himself was editor. 



One of Professor Adrain's pupils, William Lenhart (died in 

 1840 at Frederick, Maryland), devoted much of his time, year 

 after year, to the study of Diophantine Analysis. He was a 

 cripple,, and pursued mathematics for pastime. Frequent con- 

 tributions were made by him to the Matliematical Diarij and to 

 the Mathematical Miscellany. In the latter were published 

 " Useful Tables Relating to Cube Numbers." The editor states 

 that besides these tables, the manuscript compiled by Mr. Len- 

 hart with so much labor and care included a table containing a 

 variety of numbers between 1 and 100,000 and the roots, not 

 exceeding two places of figures, of two cubes, to whose differ- 

 ence the numbers are respectively equal; together with another 

 table, no less curious, exhibiting the roots of three cubes to 

 satisfy the indeterminate equation, x^ + y^ -\- z^ ~ A, for all 

 values of A from 1 to 50 inclusive. These enabled him to solve 



