eel eee ee ee i 
sheet of paper,” which can realise space of only ¢wo 
dimensions, so also we may conceive of beings capable 
of realising space of four dimensions, Prof. Sylvester, 
Dr. Salmon, Prof. Clifford, and others, have indicated in 
some of their profoundest mathematical demonstrations 
that they possess “an inner assurance of the reality of 
transcendental space.” We desire now to bring forward, 
with great apology to the mathematicians for our temerity, 
some ideas, which we believe may enable even the least 
mathematical amongst us, to realise,—faintly, indeed, 
and very dimly—the possibility of existence of space, 
other than that which we now occupy. This we propose 
to do, (a) by attempting to realise a condition of life in 
space of two dimensions, and (8) by adding the element 
of diverse motions, to our already known space. 
Our knowledge of the Universe involves the conception 
of space, time, and number. These are intuitive notions : 
we cannot strictly define them; in the abstract our 
notion of them is merely relative ; apart from material 
existence we cannot realise them, Extension is an 
essential property of matter, and our conception of space 
is linked with our conception of extension. Robert 
Hooke, in a series of lectures De Potentia Restitutiva, 
written nearly two hundred years ago, and too little 
known, defines a sensible body as “a determinate space, 
or extension, defended from being penetrated by another, 
by a power from within.” Now this power may be most 
readily conceived to be a vibratory motion of the particles 
across a position of rest. Let us imagine an infinitely 
thin plane vibrating between two fixed points with such 
velocity that no other matter can penetrate into the space 
limiting the vibration, then a solid bounded in one direc- 
tion by the two fixed points would be the result. For ex- 
ample, let an infinitely thin sheet of iron a metre square 
vibrate with extreme velocity in a span of one metre, and 
acubic metre ofiron would be the result. The rapid vibra- 
tion of the plate would defend the range of vibration from 
being penetrated, and impenetrable material substance 
would result. An infinitely thin line vibrating between 
two fixed points would furnish a plane, An infinitely thin 
plane vibrating between two fixed points would furnish a 
solid. Thus by the addition of motion we can convert a 
determinate space, approximately of one dimension, into 
space of two dimensions ; and by the addition of motion 
we can convert space of two dimensions into space of 
three dimensions. Can we cdnceive of any motion which 
given to space of three dimensions shall generate space of 
Jour dimensions? We do not know of such motion, but 
we can surely conceive the possibility of its existence. 
Space of four dimensions is transcendental space: it is 
beyond the limit of our experience, but not beyond the 
limit of our imagination. 
Let us now endeavour to realise the condition of 
a being living in space of two dimensions. If man pos- 
sessed the eyes and the power of flight of an eagle, super- 
added to his ordinary intellectual qualities, he would, no 
doubt, have very enlarged views of space. As it is, man 
is distinguished from the brute animals by his erect bear- 
ing, and the range of space which his vision enables him 
to scan. Our eyes are easily movable in various direc- 
tions, so also is our head ; by a slight movement of the 
head and eyes, we may take in either space bounded by 
the horizon, or by a surface a foot square. If we throw 
our head back we enlarge our view of space; if we bend 
our head forward we narrow our view of space. Now, 
imagine that a man thus endowed, and with our own 
notions of space of three dimensions, begins to stoop for- 
ward and to grow so: his eyes survey less space ; he stoops 
more forward; his body forms angles of 80°, 70°, 60°, 50° 
in succession, with a horizontal plane. Then he is obliged 
to go on all-fours, his limbs shorten and are gradually 
absorbed into the mass of his body; he crawls, he creeps; 
at length his limbs disappear altogether, and he trails 
himself along and glides like a serpent, moving in a hori- 
NATURE 
zontal plane, During these successive shrinkings in the 
direction of his thickness his head has become fixed, his 
eyes motionless, in the plane in which he moves, and his 
vision has hence become more and more limited. Now 
his body begins to diminish in thickness; he becomes 
thinner, and thinner, and thinner, and when he has be- 
come very thin indeed, let his thickness be expressed as 
the numerator of a fraction, while the denominator is an 
infinitely great number—say, if you will, as many figures 
as, written on paper, would reach ten billion miles, with 
ten figures to an inch. Now he is a mere plane, an infi- 
nitely thin surface; he occupies space approximately of 
two dimensions ; his eyes are onaline. Try to imagine 
what the ideas of space of such a being would be; com- 
pared with our own ideas of space, compared with his own 
ideas before and during his process of flattening. He 
would now contemplate only a plane surface ; he would 
see length and breadth without thickness. Compare also 
his ideas of space at each and every position between 
verticality and horizontality as his ken gets less and less, 
and at last the whole world is shut out from him. 
Again, to come nearer home, and back again to the 
world of real existences, let us compare our own ideas of 
space after concentrating our vision for awhile on a book 
a foot square, with our ideas of space acquired while we 
ascend a lofty mountain, or lie upon our back on the deck 
of a vessel in mid-ocean. Compare the views of space 
possessed by a prisoner immured for forty years in a 
dungeon eight feet square, of La Sachette in the 7yow aux 
Rats, of a being bed-ridden for half a century, with those 
of a hunter in the prairies of the West, a sailor of the 
Atlantic, even of a dweller in a flat tame country. The 
conceptions of space possessed by these different people 
will vary enormously. Contract the limits of space of 
possible contemplation ; remove the possibility of con- 
templating space of great dimensions, and the facu/ty of 
such contemplation willitself die out ; and thus, bya gradual 
process of diminution, we may arrive at our ideal being, 
living in space of two dimensions. Finally, let us imagine 
the being of two dimensions—length and breadth—to 
become narrower and narrower, and when he has become 
extremely narrow let us divide his breadth by an infinitely 
large number, and he becomes approximately of one 
dimension ; he has now only length; he lives in a line; 
his one motionless eye is a point. 
So much for space of less dimensions than our own. 
Let us now try to conceive an extension of our ordinary 
space ; and let us attempt this by the superaddition 
of motion to “known space. And let us clearly 
realise the fact that one and the same thing 
may easily possess various motions at the same time. 
For instance, when I walk across the room, talking the 
while ; my vocal chords possess five distinct motions: 
(a) their own proper motion of vibration; A/ws (8) the 
motion of translation caused by walking forward ; p/ws (y) 
the motion of rotation of the earth about its axis ; A/is (8) 
the motion of revolution of the earth about the sun ; plus 
(ec) the motion of translation of the whole solar system 
through space. Let us suppose now that our bodies, 
instead of being at apparent rest, were to vibrate in arcs, 
with an amplitude of 10,000 miles, and with an infinite 
velocity ; and let the plane of the direction of vibration 
itself vibrate between limits 10,000 miles apart; and let 
the whole vibrating system move with infinite velocity in 
a circle 1,000,000 miles diameter; and let the circle 
rotate upon its diameter; and let the sphere of revo- 
lution thus formed revolve in an infinitely great ellipse ; 
and let the ellipse rotate upon one of its axes; and 
but hold! we have surely arrived at a somewhat enlarged 
view of our own relations to space. Conceptions of this 
nature sufficiently pursued may, perchance, lead us to the 
very threshold of transcendental space ; and, once on the 
threshold, we may look wonderingly beyond. 
G, F. RoDWELI. 
