1. The Mathematical Correspondent, Edited by G. Baron, New York, 1804. 2. The Analyst, Edited by Robert Adrain, Philadelphia, 1808. 3. The Scientific Journal, Edited by W. Marratt, New York, 1818. 4. The Ladies' and Gentlemen's Diary, Edited by M. Nash, New York, 1820. 5. The Mathematical Diary, Edited by Robert Adrain and afterwards by Mr. Ryan, New York, 1825. 6. The Mathematical Miscellany, Edited by C. Gill, New York, 1836. [pp. 394-416]

1S41.j TIte Lzfe of Len/tart I/te ~JaThernatician. 405
puzzled the Arithineticians of England and America, viz:
`To divide unity into three such positive parts, that if ead~
part be increased by unity, the sums shalt be three rational
cubes.' The common denominator of these three parts of
unity is 54427098504275016408, and the three numerators
are 57430152918127737:36, 773815889:3915488717, and
40945924:31854675:3965 As was before intimated, it would
be impossible to deduce a solution to this question, from the
investigation in the Ladies' Diary of 1825.
"The next question Mr. L. resolves, is the one previously
noticed as being the one which probably first turned his attention to this I1class of problems; and for this he gives various methods of solution, illustrated by numerous examples.
~Ve can do little more than give the enunciation of the problen~ which he has eithcr originated or improved, in these
excellent articles.`To find n cube numbers, such that if
from each or them a given number (a) be subtracted the
sum of the remainders shall be a square number.'`To
find n numbers such that, if ~ach of them be added to the
cube of their sum, the respective sums shall be cube numbers.'`To find n numbers such that each of them, being
severally subtracted from the cube of their sum, the n remainders may be cubes.'`To find m numbers, in arithmeticial progression, such that the sum of their cubes shall be
itself a cube number.' As an example of this last problem,
he finds that if the cubes of the 1000 consecutive numbers
of the natural series, beginning with 11:34, and ending with
~1~:3, are added together, their sum will be the cube of the
number, 168:30. These numbers had been previously exhibited as possessing this property, by M. Pagliani, who had
inc.'autiously boasted that no other solution, than his own,
could be obtained to this question, except by a very complicated analysis. But, perhaps, the most difficult application
of these principles, is in the solution of the question,`To
find four integers, such that the sum of every two of them
may be a cube number,' proposed by ~Mr. Lenhart, in the
second number of the Mathematical Miscellany, and resolved
by himself in the succeeding number. Mr. L. succeeds in
obtaining the following numbers,
208091:30829564551426:36,
49:37801:3475106807:32948,
726281047641001616:3052,
21497210869:3241589340948;
and if these numbers be added, two and two, the resulls will