The Eight Squares Identity and Graves-Cayley Numbers
The eight squares identity was described in a letter by John Thomas Graves to William Rowan Hamilton on 26 December 1843. There were some errors of signs in the version Graves described in that letter which he corrected in a second letter to Hamilton on 18 January 1844. It was three years later before Hamilton made this result public in volume 3 of the Proceedings of the Royal Irish Academy (1836-1869) in the form which we now give:
(s′2+t′2+u′2+v′2+w′2+x′2+y′2+z′2)×(s2+t2+u2+v2+w2+x2+y2+z2)=(ss′+tt′+uu′+vv′+ww′+xx′+yy′+zz′)2+(st′−ts′+uv′+vu′+wx′−xw′+yz′−zy′)2+(su′−us′+vt′−tv′+yw′−wy′+xz′−zx′)2+(sv′−vs′+tu′−ut′+wz′−zw′+xy′−yx′)2+(sw′−ws′+xt′−tx′+uy′−yu′+zv′−vz′)2+(sx′−xs′+tw′−wt′+yv′−vy′+zu′−uz′)2+(sy′−ys′+zt′−tz′+vx′−xv′+wu′−uw′)2+(sz′−zs′+ty′−yt′+vw′−wv′+ux′−xu′)2.
In the same note, Hamilton writes:-
In a letter dated January 18th, 1844, Mr. Graves communicated to Sir William Hamilton his theorem respecting sums of eight squares under the form:
(a2+b2+c2+d2+e2+f2+g2+h2)×(a′2+b′2+c′2+d′2+e′2+f′2+g′2+h′2)=a′′2+b′′2+c′′2+d′′2+e′′2+f′′2+g′′2+h′′2
where a", ..., h" denoted the following expressions:
a′′=aa′−bb′−cc′−dd′−ee′−ff′−gg′−hh′;
b′′=ba′+ab′−dc′+cd′−fe′+ef′+hg′−gh′;
c′′=ca′+db′+ac′−bd′−ge′−hf′+eg′+fh′;
d′′=da′−cb′+bc′+ad′−he′+gf′−fg′+eh′;
e′′=ea′+fb′+gc′+kd′+ae′−bf′−cg′−dh′;
f′′=fa′−eb′+hc′−gd′+be′+af′+dg′−ch′;
g′′=ga′−hb′−ec′+fd′+ce′−df′+ag′+bh′;
h′′=ha′+gb′−fc′−ed′+de′+cf′−bg′+ah′.
In a letter of somewhat earlier date, but evidently written in haste, upon a journey, and dated December 26th, 1843, analogous expressions had been given, containing, however, some errors in the signs, which were soon afterwards corrected as above. That earlier letter also indicated an expectation that a theory of octaves, including a new and extended system of imaginaries, which had thus been suggested to the writer (J T Graves, Esq.) by Sir William R Hamilton's theory of quaternions, might itself be extended so as to form a theory of what Mr Graves at the time proposed to call 2n -ions: but in a letter written shortly afterwards, doubts were expressed respecting the possibility of this additional extension, from octaves to sets of sixteen.
Graves used the eight squares identity to construct what today is known as the octonians or Cayley numbers or sometimes the Graves-Cayley numbers. To see how this works let us look at multiplication of two complex numbers
(a+bi)(c+di)=(ac−bd)+(ad+bc)i.
Taking the modulus (or norm), we get
(a2+b2)(c2+d2)=(ac−bd)2+(ad+bc)2.
If we call this the two squares identity, then Euler's four squares identity would correspond in the same way to the product of two quaternions while the eight squares identity corresponds to the multiplication of the Graves-Cayley Numbers. Perhaps the eight squares identity should be named the Degen-Graves-Cayley Eight-Square Identity since these three mathematicians all discovered it independently.
Last Updated February 2016
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