312 Mr. Youne on an Extension of a Theorem of Euler, 
This quotation is from the Philosophical Magazine for April, 1845. No 
demonstration, however, of the extended case of Euler’s theorem having appeared, 
the author of the present communication was led to examine into the practicabi- 
lity of this extension; and, early in May last, arrived at the conclusion that, as 
previously affirmed by Mr. J. T. Graves, the theorem really had place for sums 
of eight squares, as well as for sums of two and of four. He communicated his 
investigation to the Royal Irish Academy in the beginning of June; and the 
formula for eight squares, to which this investigation conducted, was printed in 
the «‘ Proceedings” for the then current month. At the same time Mr. Graves’s 
results were also presented : both formule appear on the same page of the “ Pro- 
ceedings ;” and, as they are equally correct, it is needless to add that they are 
mutually convertible ito each other. 
In drawing up this more detailed communication for the Transactions, the 
writer should not have thought it worth while, in a matter of such comparative 
insignificance, to have adverted to these preliminary particulars, but for his 
anxiety to avoid all appearance of preferring claims, however unimportant, to 
which he has no title. He may, perhaps, be permitted to add, that, even up to 
this date, no investigation of the principles upon which the eight-square theorem 
depends has been made public ; nor has any intimation been as yet given, as to 
whether the proposition terminates here, or may be extended indefinitely. It was 
the author’s original impression that such an indefinite extension was really prac- 
ticable ; and such, he has reason to believe, was the prevailing opinion with those 
to whom Mr. Graves’s results had been made known. It is probable, therefore, 
that the principal point of interest—perhaps the only point of interest—which the 
present paper offers, is that which sets all conjecture on this head at rest, by 
showing that the supposed extension is impossible. 
To these introductory remarks it may not be superfluous to add, that the 
demonstration hereafter given, of the impossibility of the sixteen-square form, 
except under special restrictions, adds one more to the very few instances hitherto 
furnished of proving a negative : that is, of showing that a proposition, the 
affirmative of which would fully harmonize with pre-established truths, and actu- 
ally involve them in its expression, is, nevertheless, impossible. In modern 
times, the most remarkable specimen of a proof of this kind is that which has 
been offered by Abel and Hamilton, in their researches respecting general for- 
