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]

404 The LJe of Lenhart the Mathematician. [JULY
lege, L.I., Nov. 183&.) In a subsequent number of that periodical, the table calculated from these formulas,`exhibiting a variety of numbers, between 1 and 100,000, and the
roots of the two cubes of which they are composed.' He
continued, from time to time, to extend this table, which will
ever be the most striking monument of his untiring ind~istry, and indomitable perseverance, among circm~stances the
most disheartening, and in a state of alternate bodily debility and acute pain, that renders his success astonishing.
The applications which Mr. Lenhart has made of these
tables are no less interestin~ than are the tables themselves,
and the process of deducing them; most of these are exhibited iii a series of papers published in the Mathematical
Miscellany, under the title of`Diophantine Speculations.'
The first of these is the problem`To divide a given number
(A) into three cubes.' Two methods of solution are given,
the first and most simple of which, is the fbllowing:-Rule
I. Multiply the given number (A) by any cnbe, (r',) and
from the product deduct a series of cubes, prime to (r3,)
~mtil you find a remainder (t) that shall be equal to one of
the tabular numbers composed of two cubes. Substitute the
two cubes in place of the remainder, transpose the deducting cube that made the remainder, divide by the multiple
cube, (r3), and the result will be three cubes, equal to the
given number, (A.)'
"The evident defect in this process, as well as in its accompanying one, is its tentative character; for as it nowhere
appears that the given number (A) is necessarily capable of
being decomposed into three cubes, in the man~er proposed,
there is nothing to ins~ire us that by successive trials we
shall at last arrive at a n~m~ber which is the sum of two
cubes, much less that we shall arrive at one of those already
tabulated, since it can scarcely be expected that ail the numbers within reasonable limitr, composed of two cube num
bers, will ever be tabulated.
"Whatever may be the theoretical defects of his methods,
however, the application of them afiords the most triumphant vindication of their practical utility. For instance, he
finds that the number 4 is composed of three cube numbers,
whose roots are fractions, each greater than unity, their commoil denominator being 3789702, and their numerators
~918564, 3961405, and 4568867; and by means of these
numbers he arrives at far less numbers than had ever before
been attained, to answer the question which had so long