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XI.—On an Extension of a Theorem of Euler, with a Determination of the 
Limit beyond which it fails. By Joun Raprorp Youne, Professor of 
Mathematics in Belfast College. 
Read November 30, 1847. 
THE theorem, into the extension and necessary limitation of which it is here 
proposed to examine, was first demonstrated by Euler, under the following enun- 
ciation :—The sum of four squares multiplied by the sum of four squares gives 
a product which is also the sum of four squares. 
This theorem has been lately called in request by Sir William R. Hamilton, 
in connexion with his researches in reference to the calculus of quaternions ; in 
the establishment of the fundamental principles of which he was independently 
led to the theorem in question, as a consequence and a confirmation of the con- 
sistency of his original conventions, with respect to the new imaginary quantities 
which that calculus involves. 
Attention thus came to be directed to the inquiry, as to whether or not 
Euler’s theorem admits of extension ; for it was naturally enough suspected that 
if such extension should be found to be practicable, a corresponding extension of 
the new theory of imaginaries would suggest itself. 
Mr. John T. Graves, who, as well as Professors Charles Graves and De Morgan, 
has paid much attention to these researches, was, I believe, the first to announce 
that the theorem in question was equally applicable to sums of eight squares ; 
and he remarks that, “as Euler’s theorem is connected with Hamilton’s quater- 
nions, so my theorem concerning sums of eight squares may be made the basis of 
sets of eight, and was actually so applied by me, about Christmas, 1843. But,” 
he adds, “ the full statement and proof of the theorem concerning sums of eight 
squares, and of several other new theorems connected with the doctrine of num- 
bers, must be reserved for another time.” 
VOL. XxI. 27 
