238 
NATURE 
eye, and in conformity with this view always paid the 
most punctilious attention to preserve his text free from 
typographical errors : the ever to be lamented Riemann 
has written a thesis to show that the basis of-our concep- 
tion of space is purely empirical, and our knowledge of 
its laws the result of observation; that other kinds of space 
might be conceived to exist, subject to laws different from 
those which govern the actual space in which we are 
immersed ; and that there is no evidence of these laws 
extending to the ultimate infinitesimal elements of which 
space is composed. Like his master Gauss, Riemann 
refuses to accept ‘Kant’s doctrine of space being a form 
of intuition,* and regards it as possessed of physical 
and objective reality. I may mention that Baron Sar- 
torius von Waltershausen (a member of this Associa- 
tion), in his biography of Gauss (“Gauss zu gedachtniss”), 
published shortly after his death, relates that this great 
man was used to say that he had laid aside several ques- 
tions which he had treated analytically, and hoped to 
apply to them geometrical methods in a future state of 
existence, when his conceptions of space should have 
become amplified and extended ; for as we can conceive 
beings (like infinitely attenuated book-worms in an 
infinitely thin sheet of paper) which possess only the notion 
of space of two dimensions, so we may imagine beings 
capable of realising space of four or a greater number 
of dimensions.| Our Cayley, the central luminary, the 
Darwin of the English school of mathematicians, started 
and elaborated at an early age, and with happy conse- 
quences, the same bold hypothesis. 
Most, if not all, of the great ideas of modern mathematics 
have had their origin in observation, Take, for instance, the 
arithmetical theory of forms, of which the foundation was 
laid in the diophantine theorems of Fermat, left without 
proof by their author, which resisted all the efforts of the 
myriad-minded Euler to reduce to demonstration, and only 
yielded up their cause of being when turned over in the 
blowpipe flame of Gauss’s transcendent genius ; or the 
doctrine of double periodicity, which resulted from the 
* It is very common, not to say universal, with English writers, even such 
authorised onesas Whewell, Lewes, or Herbert Spencer, to refer to Kant’s 
doctrine as affirming space ‘‘to be a form of thought,” or “ of the understand- 
ing.” This is putting into Kant’s mouth (as pointed out to me by Dr. C. M. 
Ingleby), words which he would have been the first to disclaim, and is as 
inaccurate a form of expression as to speak: of “the plane of a sphere,” meaning 
its surface or a superficial layer, as not long ago [heard a famous naturalist do 
at a meeting of the Royal Society. Whoever wishes to gain a notion of Kant's 
leading doctrines in a succinct form, weighty with thought, and free from all 
impertinent comment, should study Schwegler’s Handbook of Philosophy, 
translated by Stirling. He will find in the same book a most lucid account 
of Aristotle’s doctrine of matter and form, showing how matter passes un- 
ceasingly upwards into form, and form downwards into matter; which will 
remind many of the readers of NATURE of the chain of depolarisations and 
repolarisations which are supposed to explain the decomposition of water 
under galvanic action, eventuating in oxygen being thrown off at one pole 
and hydrogen at the other (it recalls also the high algebraical theories in 
which the same symbols play the part of operands to their antecedents and 
operators to their consequents): at one end of the Aristotelian chain comes 
out mpdm7n UAn, at the other, tpa@tov eidos. We have, then, only to accept and 
apply the familiar mathematical principle of the two ends of infinity being one 
and the same point, and thé otherwise immoveable stumblingblock of duality is 
done away with, and the universe reintegrated in the wished-for unity. For 
this corollary, which to many will appear fanciful, neither Aristotle nor 
Schwegler is responsible. We perfectly understand how in perspective the 
latent polarities of any point in a closed curve (taken as the object) may be 
developed into and displayed in the form ofa duad of gvasé points at an infinite 
distance from each other in the picture. In like manner we conceive how 
actuality and potentiality which exist indistiuguishably as one in the absolute 
may be projected into seemingly separate elements or moments on the plane 
of the human understanding. Whatever may be the merits of the theory in 
itself, this view seems to me to give it a completeness which its author could . 
not have anticipated, and to accomplish what Aristotle attempted but 
avowedly failed to effect, viz. the complete subversion of the ‘Platonic 
Duality,” and the reintegration of matter aud mind into one. 
+ It is well known to those who have gone into these views, that the laws 
of motion accepted as a fact suffice to prove in a general way that the space 
we live in is a flat or level space (a ‘‘ homaloid’’), our existence therein being 
assimilable to the life of the bookworm in a flat page; but what if the page 
should be undergoing a process of gradual bending into a curved form? 
Mr. W. K.. Clifford has indulged in some remarkable speculations as 
to the possibility of our being able to infer, from certain unexplained pheno- 
mena of light and magnetism, the fact of our level space of three dimensions 
being in the act of undergoing in space of four dimensions (space as incon- 
ceivable to us as our space to the supposititious bookworm) a distortion analo- 
pous to the rumpling of the page. I know there are many, who, like my 
1onoured and deeply lamented friend the late eminent Prof. Donkin, regard 
observation by Jacobi of a purely analytical fact of trans- 
formation ; or Legendre’s. law of reciprocity ; or Sturm’s 
theorem about the roots of equations, which, as he informed 
me with his own lips, stared him in the face in the midst 
of some mechanical investigations connected -(if my 
memory serves me right) with the motion of compound 
pendulums; or Huyghen’s method of continued frac- 
tions, characterised by Lagrange as one of the principal 
discoveries of “that great mathematician, and to which 
he appears to have been led by the construction of his 
“ Planetary Automaton ;” or the new algebra, speaking of 
which one of my predecessors (Mr. Spottiswoode) has 
said, not without just reason and authority, from this chair, 
“that it reaches out and indissolubly connects itself each 
year with fresh branches of mathematics, that the theory 
of equations has almost become new through it, algebraic 
[ Dee. 30, 1869 
geometry transfigured in its light, that the calculus of © 
variations, molecular physics, and mechanics” (he might, 
if speaking at the present moment, go on to add the theory 
of elasticity and the developments of the integral calculus) 
“have all felt its influence.” 
Now this gigantic outcome of modern analytical thought, 
itself, too, only the precursor and progenitor of a future 
still more heaven-reaching theory, which will comprise a 
complete study of the interoperation, the actions and 
reactions, of algebraic forms (Analytical Morphology in 
its absolute sense), how did this originate? In. the 
accidental observation by Eisenstein, some twenty or more 
years ago, of a single invariant (the Quadrinvariant of a 
Binary Quartic) which he met with in the course of certain 
researches just as accidentally and unexpectedly as M. 
Du Chaillu might meet a Gorilla in the country of the 
Fantees, or any one of us in London a White Polar Bear 
escaped from the Zoological Gardens. Fortunately, he 
pounced down upon his prey and preserved it for the 
contemplation and study of future mathematicians. It 
occupies only part of a page in his collected posthumous 
works. This single result of observation (as well entitled 
to be so called as the discovery of Globigerinze in chalk _ 
the alleged notion of generalised space as only a disguised form of algebraical 
formulisation ; but the same might be said with equal truth of our notion 
of infinity in algebra, or of impossible lines, or lines making a zero angle 
in geometry, the utility of dealing with which as positive substantiated 
notions no one will be found to dispute. _ Dr. Salmon, in his extensions of 
Chasles’ theory of characteristics to surfaces, Mr. Clifford, in a question 
of probability (published in the Zducational Tintes), and myself in my theory 
of partitions, and also in my paper on Burycentric Projection in the Pilo- 
sophical Magazine, have all felt and given evidence of the practical utility.of 
handling space of four dimensions, as if it were conceivable space. More- 
over, it should be borne in mind that every perspective representation of 
figured space of four dimensions is a figure in real space, and that the pro- 
perties of figures admit of being studied to a great extent, if not completely, 
in their perspective representations. In philosophy, as in esthetic, the highest 
knowledge comes by faith. I know (from personal experience of the fact) 
that Mr. Linnell can distinguish purple tints in clouds where my untutored 
eye and unpurged vision can perceive only confused grey. If an Aristotle, 
or Descartes, or Kant assures me that he recognises God in the conscience, I 
accuse my own blindness if I fail to see with him. If Gauss, Cayley, Riemann, 
“Schalfli, Salmon, Clifford, Krénecker, have an inner assurance of the reality 
of transcendental space, [ strive to bring my faculties of mental vision into 
accordance with theirs. ‘The positive evidence in such cases is more worthy 
than the negative, and actuality is not cancelled or balanced by priyation, as 
matter plus space is none the less matter. I acknowledge two separate 
sources of authority—the collective sense of mankind, and the illumination 
of privileged intellects. As a parallel case, I would ask whether it is by 
demonstrative processes that the doctrine of limits and of infinitely greats ~ 
and smalls, has found its way to the ready acceptance of the multitude; or 
whether, after deducting whatever may be due to modified hereditary 
cerebral organisation, it is not a consequence rather of the insensible 
moulding of the ideas under the influence of language which has become 
permeated with the notions originating in the minds of a few great thinkers? 
1am assured that Germans even of the non-literary classes, such as ladies 
of fashion and novel readers, are often appalled by the habitude of their 
English friends in muddling up together, as if they were nearly or quite 
the same thing, the reason and the understanding in doing into English the 
words Vernunft and Verstand, thereby confounding distinctions now become 
familiar (such is the force of language) to the very milkmaids of Fatherland. 
Asa public teacher of mere striplings, I am often amazed by the facility and 
absence of resistance with which the principles of the infinitesimal calculus _ 
are accepted and assimilated by the present race of learners. When I was 
young, a boy of sixteen or seventeen who knew his infinitesimal calculus « _ 
would have been almost pointed at in the streets as a prodigy, like Dante, 
as aman who had seen hell. Now-a-days, our Woolwich cadets at the same 
age talk with glee of asymptotes and points of contrary flexure, and discuss 
questions of double maxima and minima, or ballistic pendulums, or miotion 
in a resisting meditm, under the familiar and ignoble name of swzrs. 
