316 Mr. Younc on an Extension of a Theorem of Euler, 
sw! — ws! 4>at) — ty’ 
at) — ta’ + uy! — yu 
uy — yu + zv' — v2, 
all conform to the previously established model for four, and harmonize with the 
other binomials in which the same quantities occur. At present, this arrange- 
ment may be regarded simply as matter of observation : there is, indeed, a confor- 
mity to the model still more general than this, and which we shall find to be equally 
essential to the accuracy of the eight-square formula; it will be more distinctly 
adverted to presently: but as the additional conformity here hinted at is a spon- 
taneous result of that stipulated for above, no cautionary precepts, beyond those 
now mentioned, are necessary for the actual construction of the eight-square for- 
mula. 
It was at this stage of the investigation that the writer of the present com- 
munication, abandoning all further attempts at practically applying the principles 
that had thus led him to correct constructions of the four-square and eight-square 
formule, resorted to theoretical considerations and abstract general reasonings, 
to prove that the laws of construction, thus far found to be successful, must 
necessarily prevail in the sixteen-square and subsequent forms. He conceived 
that when the sixteen rows of combinations,—each dashed quantity being com- 
bined with an undashed one, as in the preceding cases,—were written down 
without any intervening signs, and which would involve little or no trouble :—he 
conceived that it was always possible afterwards to introduce such an arrangement 
of signs as to bring about the conformity here alluded to. And it was not till 
after the expenditure of much time, and the trial of every arrangement that gave 
any promise of success in the case of sixteen squares, that he was led to suspect 
the applicability of the assumed principle beyond the eight-square form already 
attained; and thence to the demonstration of its utter failure beyond that 
limit. 
Before entering upon this demonstration, it may be well to verify, by actual 
development, the expression for the product of eight squares already given; not 
that such development is absolutely necessary for the purpose of verification, 
for, as will shortly be noticed, abundant proof of the accuracy of the formula 
may be derived from a simple inspection of its component terms; but it will be 
found convenient, in showing the impossibility of the sixteen-square form, to 
