8o 
Space and its Dimensions. 
[February, 
Here both our authors have recourse to a perfectly legiti- 
mate artifice. Both suppose a set of beings, who are merely 
two-dimensional, having length and breadth, but not thick- 
ness, and to whom space — in fadl the universe — appears two- 
dimensional also. To these hypothetical beings the third 
dimension, depth or thickness, appears as inconceivable and 
impossible as the fourth dimensional is apt to do to us. Mr. 
Hinton examines formally external phenomena as they must 
present themselves to such beings. The anonymous author 
of “ Flatland ” represents himself as a dweller in two-dimen- 
sional space. He describes to us his world with all its 
strange peculiarities, interweaving with his story much 
satire of perhaps questionable relevance. Especially is it to 
be noted that research in certain directions is in “ Flatland ” 
discountenanced, not merely by social penalties, but — as in 
modern Britain— by positive law. Researches into the origin 
of light are strictly forbidden, and the author himself is ulti- 
mately doomed to perpetual imprisonment for having indis- 
creetly expressed a belief in the possibility of three dimen- 
sions. 
Whether to a polydimensional intelligence our beings and 
doings will appear as absurd as those of the Flatlanders, — 
here described — do to us, is, from the present human point 
of view, an unapproachable question. 
Previous to the supposed author’s conversion to a recog- 
nition of three dimensions he has a remarkable dream, after 
an evening spent in geometrical studies. He sees in imagi- 
nation a one-dimensional world, Line-land. Its inmates 
are aware of only one dimension in space, and can no more 
conceive of two dimensions than the Flatlanders can of three, 
or than we worthy Spacelanders, — as the author calls us — 
can of 4, 5, or n dimensions. He has a long discussion with 
the king of Line-land, and vainly tries to convince him of the 
limits of his own existence and of the possibility and reality 
of two-dimensional space with its necessary results. To the 
king of Line-land a voice or any other manifestation coming 
from any point save exactly before or behind him seemed to 
originate in his own inside. This vision made our Flatlander 
thoughtful. There dawned on his mind a vague, dim pre- 
sentiment that there might exist worlds or regions of being 
of more than two dimension, — as inexplicable to him as a 
fourth dimension is to us. 
A further incident now occurred to disturb his traditional 
notions of the nature of space. He has been giving his little 
grandson, a boy of “unusual brilliancy and perfect angu- 
larity,” a lesson in arithmetic as applied to geometry. He 
