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.