279 
1893 - 94 .] Prof. Tait on the Quaternion Method. 
conception; but it is nothing more than the full development of 
tlie system of imaginaries y, h ; defined by the equations 
= - 1 , 
with the associative, but not the commutative, law for the factors. 
The novel and splendid points in it were the treatment of all direc- 
tions in space as essentially alike in character, and the recognition 
of the unit vector’s claim to rank also as a quadrantal versor. 
These were indeed inventions of the first magnitude, and of vast 
importance. And here I thoroughly agree with Prof. Cayley 
in his admiration. Considered as an analytical system, based 
throughout on pure imaginaries, the Quaternion method is elegant 
in the extreme. But, unless it had been also something more, 
something very different and much higher in the scale of develop- 
ment, I should have been content to admire it : — and to pass it by. 
It has always appeared to me that, magnificent as are Hamilton’s 
many contributions to mathematical science : — his Fluctuating Func- 
tions, and his Varying Action, for instance: — nothing that he (or 
indeed any other man) ever did in such matters can be regarded^, as 
a higher step in pure reasoning than that which he took when he 
raised Quaternions from the comparatively low estate of a mere 
system of Imaginaries to the proud position of an Organ of Expres- 
sion ; giving simple, comprehensive, and (above all) transparently 
intelligible, embodiment to the most complicated of Real geometrical 
and physical relations. From the most intensely artificial of systems 
arose, as if hy magic, an absolutely nahiral one ! 
Most unfortunately, alike for himself and for his grand concep- 
tion, Hamilton’s nerve failed him in the composition of his first 
great Volume. Had he then renounced, for ever, all dealings with 
i, j, h, his triumph would have been complete. He spared Agag, 
and the best of the sheep, and did not utterly destroy them ! He 
had a paternal fondness for i, j, k ; perhaps also a (not unnatural) 
liking for a meretricious title such as the mysterious word Quatern- 
ion ; and, above all, he had an earnest desire to make the utmost 
return in his power for the liberality shown him by the authorities 
of Trinity College, Dublin. He had fully recognized, and proved to 
others, that his i, j, k were mere excrescences and blots on his 
improved method : — but he unfortunately considered that their 
