282 Proceedings of Royal Society of PJdinliwgh. [sess. 
adequately to the world his new conception of the Quaternion. I 
got it from him by correspondence, and in conversation. AVhen 
he was pressed to answer a definite question, and could he kept to 
it, he replied in ready and effective terms, and no man could express 
vivd voce his opinions on such subjects more clearly and concisely 
than he could : — but he perpetually planed and repolished his 
printed work at the risk of attenuating the substance : and he 
fatigued and often irritated his readers by constant excursions 
into metaphysics. One of his many letters to me gave, in a few 
dazzling lines, the whole substance of what afterwards became a 
Chapter of the Elements ; and some of his shorter papers in the 
Proc. R. I. A. are veritable gems. But these were dashed off at a 
sitting, and were not planed and repolished. 
Should I be called upon, in the future, to produce a fourth edition 
of my book, the Chapter which Prof. Cayley so kindly furnished 
for the third edition will probably preserve by far the greater part 
of the allusions to i, j, k (except, of course, the necessary intro- 
ductory and historical ones) which it will contain.] 
In the sense above explained, I consider Prof. Cayley’s remarks to 
be so far warranted, hard to bear though some of them undoubtedly 
are. But the Quaternion, when it is regarded from the true 
point of view^, is seen to be untouched, in fact unassailable, by any 
criticism based upon such grounds as reference to co-ordinates. It 
occupies a region altogether apart. To compare it to a pocket map 
is to regard it as a mere artificial mode of wrapping up and conceal- 
ing the ^, j, h or the x, y, z which are supposed to be its ultimate 
constituents. To be of any use it must be unfolded, and its neatly 
hidden contents turned out. But, from my point of view’’, this 
comparison is entirely misleading. The quaternion exists, as a 
space-reality, altogether independent of and antecedent to i, j, k or 
X, y, z. It is the natural, they the altogether artificial, weapon. 
And I venture further to assert (1) that if Descartes, or some of 
his brilliant contemporaries, had recognized the quaternion, (and it 
is quite conceivable that they might have done so), science would 
have then advanced with even more tremendous strides than those 
which it has recently taken ; and (2) that the wretch who, under 
such conditions, had ventured to introduce i, j, k, would have been 
justly regarded as a miscreant of the very basest and most depraved 
