340 Mr. Youne on an Extension of a Theorem of Euler, 
which satisfies the law of the moduli, namely, 
(w? + a? + y? + 2) (w? + a? + 7? + 2”) 
= w+ oe? + y? + 27; (g) 
so my friend Graves pointed out to me, in return, that the product of two octaves is an 
octave which satisfies the same modular law ; or that he could write, consistently with 
his extended definitions, 
(a+ ib +jc + kd + le + mf + ng + oh) | 
x(a + ib 4+ jc + hd’ + le + mf + ng + oh’) (h) 
= a+ ib’ + je’+ kd’ + le’ + mf" + ng’ + oh’; | 
where 
(W@+P +04 PVP +64 fP +9 +h’) | 
X(2 +B? + +d? + e+ Hf? + 9 +h’) b 
= a?4+ 074 674 d7 4+ 674 f7 4+ 97? +h”. j 
(i) 
«‘ And thus he succeeded in connecting his eight-square formula with a theory of 
octaves, as I had already been led from quaternions to a four-square formula, which latter 
formula appears, however, to have been previously discovered by Euler. 
<< It was natural that, when Graves had gone so far, he should entertain the hope of 
extending similar principles to systems of sixteen, and generally of 2™ squares; and, 
accordingly, in his letter of December 26, 1843, he spoke of what he proposed to call 
2m_jons. But he soon afterwards told me that he had met with what he called ‘an unex- 
pected hitch,’ in seeking to extend the law of the moduli to systems of sixteen numbers ; 
and, in a letter of February 3, 1844, he said: ‘I cannot help harping on the strange- 
ness of not being able to arrange the product of two sums of sixteen squares as a sum of 
sixteen rational squares.’ He then proceeded to point out certain cases in which this 
arrangement could be effected, and enclosed me two schemes with that view; and, after 
offering some suggestions respecting the effects of signs and substitutions, he said, ‘it 
ought to be capable of d priori proof that the problem is impossible, ¢fit be so.’ He 
also expressed a wish that I should attempt to furnish a proof of its impossibility. Being 
engaged at the time on other matters, I forbore to make that attempt; and as I believe 
that my friend Graves did not pursue the inquiry, the honour of the a priori investiga- 
tion respecting the products of sums of sixteen squares has been reserved for you. 
«I regret that you did not apply to me to furnish you sooner with a sketch of 
those early researches of my friend, John Graves, which contained other things that 
would have interested you; for instance, a mode of introducing certain arbitrary coeff- 
cients into the eight-square formula, and certain extensions of results from squares to 
