Studjj of Dlophantine Analysis hi the United States. 41 



many difficult problems relating to cube numbers. Thus, it 

 can be found that 4 is composed of three cubes whose roots 

 are greater than unity. This gives an answer at once to the old 

 puzzle, "to divide unity into three such positive parts that, 

 if each part be increased by unity, the sums shall be three 

 rational cubes." " And if the division of unity," says David 

 Engel in Our Schoolday Visitor, 1871, p. 36, "into three such 

 parts as above stated has beaten the mathematicians of two 

 continents, what would be the effect of dividing unity into 12 

 such parts, which (by means of these rules) can be done with, 

 about as much ease and facility in one case as in the other?" 

 Mr. C. Gill, the editor of the JSIailiemaiical MisceUanij, who 

 furnished an abstract of Lenhart's speculations, points out the 

 great defect in them, which robs them of any scientific value. 

 It lies in their " tentative character." As it nowhere appears 

 that a given number A, is necessarily capable of being decom- 

 posed into three cubes, in the manner proposed, " there is 

 nothing to insure us that, by successive trials, we shall at last 

 arrive at a number which is the sum of two cubes, much less 

 that we shall arrive at one already tabulated." 



Some attention to our subject was paid by John D. Williams, 

 the author of w^orks on elementary mathematics. In 1828 he 

 started the Mcdhematical Companion as a rival, it would seem^ 

 to the Matliematiccd 'Diary. Williams had many opponents, 

 and a bitter contest was carried on between the two parties. 

 He finally issued his 14 famous "challenge problems,'' 

 directed against all the mathematicians in America, excepting 

 three.* The fact that all 14 problems were on Diophantine 

 Analysis tends to show to what great extent that subject was 



engrossing the attention of a certain class of mathematicians 



I 



♦ To illustrate the bitterness with which the contest was carried on, we quote from a 

 communication tea newspaper made by Williams in 1832 a passage which has reference 



to one of his rivals. " I beg leave to state that I received from this gentleman correct 



solutions to questions 1, 2, 3, 4, 5, 10, 13 and 14. This, I suppose, is about liis »ie plws ultra 

 —beyond which I defy him to advance ' till riper age shall with raaturer force burnish his 

 mind.' " 



4 



