The Athenaeum No. 2977 (November 15, 1884), p. 622
That whimsical book Flatland by a Square
(Seeley & Co.), seems to have a purpose, but what that
may be it is hard to discover. At first it read as if it
were intended to teach young people the elementary
principles of geometry. Next it seemed to have been written
in support of the more transcendental branches of the same
science. Lastly we fancied we could see indications that it
was meant to enforce spiritualistic doctrines, with perhaps
an admixture of covert satire on various social and
political theories. The general purport of it is to show
how being shaped like a square, born and bred in a world in
which everything took place on a plane surface, and where
consequently only two dimensions were conceived, obtained by
a sort of revelation knowledge of a third dimension. He has
previously in a dream studied the conditions of existence in
a world of one dimension, where everything is a line or
point, and nobody can pass any one else. There is some
ingenuity in the way in which these conceptions are worked
out, but it is rather spoilt to the mathematical mind by the
conception (which, indeed, was unavoidable) of lines and
points as objects which can be seen. Of course, if our
friend the Square and his polygonal relations could see each
other edgewise, they must have had some thickness, and need
not, therefore, have been so distressed at the doctrine of a
third dimension. There is something rather funny in the
idea that a being of n dimensions, when addressed by a being of n+1, fancies the voice which he hears to proceed from his own inside; but no doubt it is in strict harmony with facts, and probably represents what we should all feel if we got into a region where it was possible to tie a knot in a closed loop of string, as it is in the world of four dimensions. When we saw the feat performed we should doubtless be as much surprised as our Square was when the Sphere told him the contents of his house without opening the door or taking off the roof. If we came back and told about it, we should, equally without doubt, fare much as the unlucky narrator of this history did.
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