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]

1841.] The LJe of Len hart the Mathematician. 409
ted you to point out what you might consider defects in my
system, (Mr. L. had solicited Prof. Gill to do so after that
gentleman had intimated that he would do so, if he were
better acquainted with Mr. L.) I took it for granted that
they would be of a nature to effect the substance of it, in such
a manner as to render another edition or possibly a total
change in its construction necessary. Hence my anxiety.
But such it seems has not been the case; for you have not
offered a suggestion or asked a single question that I had
not in the course of my study on the subject anticipated, and
iii consequence had under flill coiisideration. Indeed, I cannot well imagine after having furnished you with my paper,
and thus made you conscious of the study necessary to produce it, how yoii could for one moment suppose that I had
not observed all these things. Among many which you have
not yourself noticed, because you did not view the subject in
the same light, I shall only mention briefly, the nature and
form of the remainders 4 r3-s3, 2 r'~s% r and s being prime
to each other; a knowledge of which in my search after
three cubes, each greater or less than unity and equal to four
or three, was highly important, and saved me much time and
unnecessary labour. I cannot stop to say more about them at
present, knowing that you will understand my meaning, and
their value. In order to show you that I was liot as unmindflil of the nature of the formulas in Rule II, and their
applications in a more scientific manner, as you may have
supposed, I will briefly sketch a few considerations relative
to them which ought to have been attached to Rule II, as a
remark, like that to Rule I, or else have been embodied in
Article 2, page 124. That the formulas in Rule II, or the
values of ax, bx, and cx in the solution to Prob. I, may be
each greater or less than unity, we must have s (2r'A+s3),
6 (r3A-s3) and a (r3A-s3) each greater or less than r (r3A
+ 2s'); or by a reduction of these inequalities, $3greater or
less than 2r+a 2r+6 and 2r-s
A(a-r)' A(6-r) A(20-~) Now we readily
perceive that a, 6 ~nd 20 must each be greater than r, and
that the nearer a, 6 and 20 approach an equality the greater
the latitude for r, and the greater the certainty of effecting the
required division. Hence when we have the equation r3A
+s'-a'+ 6% as in Art. II, we may easily ascertain from
what has been here shown whether the numbers resulting
from a substitution of the terms which compose it, in the for