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]

4O~ Tke Ltfr of LenA art Ike MaMematician. UULY
I have done, by any species of guessing he might invent!
Why, sud~ a problem, ten years ago, would I am sure have
been considered quite inN~ossible: but we see that it is now
an easy matter. Besides, I am clearly of the opinion, that
the method of a known case as practised in the resolution
of Diophantine Problems, is more like, and comes nearer to
a species of guessing than any method I know: but it is all
right in those who fancy its use, to employ it when necessary." To this Prof. Gill replied in a letter from which we
niake il~ese extracts.`I never construed the matter extracted from one of your letters, as in the least respect as boastiug
or tending to dictation, (~1r. L. had intimated in his letter
that, perhaps Prof. Gill supposed that he wished to dictate
to him:) but had I so construed it I should have considered
you fully justified in doing it. You are well qualified to dictate to any man on that subject, much more to me." Prof.
Gill then, after stating that while every other branch of science had improved, the Diophantine Analysis had remained
almost stationary, remarks, " Diophantus left it in as complete a state as a science, as it is at the present day. Do not
misnnderstand me again. I do not mean that Diophantus
could have done what Euler has done, what Barlow or you
have done. While I admire and wonder at the almost magical works of geometrical analysis, I cannot but regret that
the theory of abstract numbers, has not joined in the onward march. 1 have it is true, glimpses uncertain and transient, of points where the march should begin, and one of
these points I have, in so bungling a manner tried to point
out to you. In order to do so, I took a specimen of the analysis in an article of your own: but I might have gone, and
with far greaterj ustice to the most finished works of the analysts
who have preceded you. Scarcely a problem of Euler's but
is liable to the same objection-and the argument I wished
to urge was, the usual mode of numerical interpretation of
Diophantine analysis is`loose and unsatisfactory,' it is liable to the charge of`guessing.' I beg you sir to believe,
that if I have sinned it is the sin of execution alone not that
of design. My purpose was good, and my aim single. No
one will now deny that you have done more with the Diophantine analysis than any man who ever lived: then who
better qualified than yourself to detect the deficiencies of the
analysis and to amend them. To urge you to this was my
aim." To this Mr. L. rejoined in a letter from which we
make the following extract: "When I so anxiously solici