
Aug. 3, 1871] 
assigned direction, the condensation and elementary rotation 
produced by given displacements of the parts of a system, &c. 
For instance, a very elementary application of Vv to the theory of 
attraction enables us to put one of its fundamental principles in 
the following extremely suggestive form :—If the displacement 
or velocity of each particle of a medium represent in magnitude 
and direction the electric force at that particle, the corresponding 
statical distribution of electricity is proportional everywhere to 
the condensation produced. Again, Green’s celebrated theorem 
is at once seen to be merely the well-known equation of continuity 
expressed for a heterogeneous fluid, whose density at every point 
is proportional to one electric potential, and its displacement or 
velocity proportional to and in the direction of the eleciric force 
due to another potential. Butthis is not the time to pursue such 
an inquiry, for it would lead me at once to discussions as to the 
possible nature of electric phenomena and of gravitation. I 
believe myself to be fully justified in saying that, were the theory 
of this operator thoroughly developed and generally known, the 
whole mathematical treatment of such physical questions as those 
just mentioned would undergo an immediate and enormous 
simplification ; and this, in its turn, would be at once followed 
by a proportionately large extension of our knowledge. * 
And this is but ove of the claims of Quaternions to the attention 
of physicists. When we come to the important questions of 
stress and strain in an elastic solid, we find again that all the 
elaborate and puzzling machinery of coordinates commonly em- 
ployed can be at once comprehended and kept out of sight in a 
mere single symbol—a linear and vector function, which is self- 
conjugate if the strain be pure. This is simply, it appears to me, 
a proof either that the elaborate machinery ought never to have 
been introduced, or that its use was an indication of a compara- 
tively savage state of mathematical civilisation. In the motion 
of a rigid solid about a fixed point, a Quaternion, represented by 
a single symbol which is a function of the time, gives us the 
operator which could bring the body by a single rotation from its 
initial position to its position at any assigned instant. In short, 
whenever with our usual means a result can be obtained in, or 
after much labour reduced to, a single form, Quaternions will 
give it at once in that form ; so that nothing is ever /os¢ in point 
of simplicity. On the other hand, in numberless cases the Qua- 
ternion result is immeasurably simpler and more intelligible than 
any which can be obtained or even exfressed by the usual methods. 
And it is not to be supposed that the modern Higher Algebra, 
which has done so much to simplify and extend the ordinary 
Cartesian methods, would be ignored by the general employment 
of Quaternions ; on the contrary, Determinants, Invariants, &c., 
present themselves in almost every Quaternion solution, and in 
forms which have received the full benefit of that simplification 
which Quaternions generally produce. Comparing a Quaternion 
investigation, no matter in what department, with the equivalent 
Cartesian one, even when the latter has availed itself to the ut- 
most of the improvements suggested by Higher Algebra, one 
can hardly help making the remark that they contrast even more 
strongly than the decimal notation with the binary scale, or with 
the old Greek Arithmetic—or than the well-ordered subdivisions 
of the metrical system with the preposterous no-systems of Great 
Britain, a mere fragment of which (in the form of Table of 
Weights and Measures) form, perhaps the most effective, if not 
* The following extracts from letters of Sir W. R. Hamilton have a per- 
fectly general application, so that 1 do not hesitate to publish them :—‘‘ De 
Morgan was the very /z7s¢ person to zo/ice the Quaternions zz Aint ; namely 
in a paper on Triple Algebra, in the Camb. Phil. Trans. of 18.4. It was, 
1 thiuk, about that time, or not very long afterwards, that he wrote to me, 
nearly as follows:—‘I suspect, Hamilton, that you have caught the right 
sow by the ear!’ Between us, dear Mr. Tait, 1 think we sha'l degin the 
SHEARING of it!” ‘You might without offence to me, co sider that I 
abused the licence of Zofe, which may be indulged to an inventor, if 1 were 
toconfess that I expect the Quacernions to supply, hereafter, not merely 
mathematical methods, but also physical suggestions. And, in p:tticular, 
you are quite welcome to smile, it 1 say that it does not seem extravagant to 
me tO suppose that a fd? possession of those a priori principles oi mine, 
about the mxtiplication of vectors :—including the 1 aw of the Four Scales, 
and the conception of the Extra-spatial Unit—which have as yet been not 
much more than ited to the public— MIGHT have led (I do not at all mean 
that zx my hands they ever would have done so) to an ANTICIPATION of the 
great discovery of OERSTED.” 
‘It appears to me that one, and not the least, of the services which 
Quaternions may be expected to do to mathematical analysis generally, is 
that their introduction will compel those who adopt them,—or even who 
admit that they say be reasonably acopted by other persons—to consider, 
or to admit, thar others may uesfully iuquire, what common grounds can be 
established, for conclusions common to Quaternions and 10 older branches of 
mathematics.” 
“Could anything be simpler, or more satisfactory? Don’t you /e/, as well 
as think, that we are on a 7vight track, and shall be ¢hanked hereafter? 
Never mind when.” 
NATURE 


271 

the most ingenious, of the many instruments of torture employed 
in our elementary teaching. 
It is true that, in the eyes of the pure mathematician, Quater- 
nions have one grand and fatal defect. They cannot be applied 
to space of 2 dimensions, they are contented to deal with those 
poor three dimensions in which mere mortals are doomed to dwell, 
but which cannot bound the limitless aspirations of a Cayley or a 
Sylvester. From the physical point of view this, instead of a 
defect, is to be regarded as the greatest possible recommendation. 
It shows, in fact, Quaternions to be a special instrument so con< 
structed for application to the Actwal as to have thrown over- 
board everything which is not absolutely necessary, without the 
slightest consideration whether or no it was thereby being rendered 
useless for applications to the Zizconcetvable. 
The late Sir John Herschel was one of the first to perceive 
the value of Quaternions ; and there may be present some who 
remember him, at a British Association meeting not long alter 
their invention, characterising them as a ‘‘Cornucopia from which, 
turn it how you will, something valuable is sure to fall.” Is it 
not strange, to use no harsher word, that such a harvest has 
hitherto been left almost entirely to Hamilton himself? If but 
half a dozen tolerably good mathematicians, such as exist in scores 
in this country, were seriously to work at it, instead of spending 
(or rather wasting) their time, as so many who have the requisite 
leisure now do, in going over again what has been already done, 
or in working out mere details where a grand theory has beea 
sketched, a very great immediate advance would be certain. 
From the majority of the papers in our few mathematical journals, 
one would almost be led to fancy that British mathematicians 
have too much pride to use a simple method while an unnecessa- 
rily complex one can be had. No more telling example of this 
could be wished for than the insane delusion under which they 
permit Euclid to be employed in our elementary teaching. They 
seem voluntarily to weight alike themselves and their pupils for 
the race ; and a cynic might, perhaps without much injustice, say 
they do so that they may have mere self-imposed and avoidable 
difficulties to face instead of the new, real, and dreaded ones (be- 
longing to regions hitherto unpenetrated) with which Quaternions 
would too soon enable them to come into contact. But this 
game will certainly end in disaster. As surely as mathematics 
came to a relative stand-still in this country for nearly a century 
after Newton, so surely will it do so again if we leave our eager 
and watchful rivals abroad to take the initiative in developing the 
grand method of Hamilton. And it is not alone French and 
Germans whom we have now to dread, Russia, America, regene- 
rated Italy, and other nations, are all fairly entered for the contest. 
The flights of the imagination which occur to the pure mathe- 
matician are in general so much better described in his formule 
than in words, that it is not remarkable to find the subject treated 
by outsiders as something essentially cold and uninteresting— 
while even the most abstruse branches of physics, as yet totally 
incapable of being popularised, attract the attention of the un- 
initiated. The reason may perhaps be sought in the fact that, 
while perhaps the only successful attempt to invest mathematical 
reasoning with a halo of glory—that made in this section by 
Prof. Sylvester—- is known to a comparative few, several of the 
highest problems of physics are connected with those observations 
which are possible to the many. The smell of lightning has 
been observed for thousands of years, it required the sagacity of 
Schonbein to trace it to the formation of Ozone. Not to speak 
of the (probably fabulous) apple of Newton—what enormous 
consequences did he obtain by passing light through a mere 
wedge of glass, and by simply laying a lens on a flat plate! The 
patching of a trumpery model led Watt to his magnificent in- 
ventions. As children at the sea-shore playing with a “‘ roaring 
buckie,” or in later life lazily puffing out rings of tobacco-smoke, 
we are illustrating two of the splendid researches of Helmholtz. 
And our President, by the bold, because simple, use of reaction, 
has eclipsed even his former services to the Submarine Telegraph, 
and given it powers which but a few years ago would have been 
deemed unattainable. 
In Experimental Physics our case is not hopeless, perlaps not 
as yet even alarming. Still something of the same kind may te 
said in this as in pure Mathematics. If Thomson’s Theory of 
Dissipation, for instance, be not speedily developed in this 
country, we shall soon learn its consequences from abroad. The 
grand test of our science, the proof of its being a reality and not 
a mere inventing of new terms and squabbling as to what they 
shall mean, is that it is ever advancing. There is no standing 
still ; there is no running round and round as in a beaten donkey- 
track, coming back at the end of a century or so into the old posi- 
