The Athenaeum No. 2980 (December 6, 1884), p. 733.

Flatland, the State Prison, Nov. 28, 1884.

I write from a world that has been truly and literally described as ``weary, stale, flat, and unprofitable,'' --- from the land of Two Dimensions, some of the characteristics of which I have recently endeavoured to describe in a little treatise entitled Flatland.

Into the dimness of my dull existence in this region there has penetrated a notice of my work which appeared in a recent number of the Athenaeum, and which raises a neat question --- shall I say metaphysical or psychological? --- which may possibly interest your readers.

Your not unfriendly, but, as I venture to think, to hasty critic, while complimenting me on the `ingenuity' of my simple description of my native land, and while admitting that the incidents recorded in my history, though `funny,' are nevertheless `strictly according to facts,' has, nevertheless, cast an implied censure on my intelligence, and on that of my countrymen, by declaring that, though we think we are of Two Dimensions, we are really of Three, and ought to know it. The narrative is spoilt, he says, `for mathematical minds,' because any visible line must really have thickness as well as length; and therefore all our so-called plane figures, besides having length and breadth, must really have some degree of thickness, or height --- in other words a Third Dimension; and of this, he implies, we ought not to be ignorant.

I admit your critic's facts, but I deny his conclusions. It is true, no doubt, that we really have a Third Dimension, just as it is also true that you have a Fourth. But just as you are not aware that you belong to the Fourth Dimension, so neither are we aware, nor can we be made logically aware, that we belong to the Third.

A moment's reflection will make this obvious. Dimension implies measurement. Now, our lines are so thin that they cannot be measured. Measurement implies degrees, the more and the less; but all our lines are equally and infinitesimally thin, or thick, whichever you please to call it; so that we in Flatland can neither measure their thinness, nor even take cognizance of it. Where you speak of a line as being long and thick (or thin), we speak of it as being long and bright; `thickness' (or `thinness') never enters our heads, and we do not know what you mean by it. I knew what it meant once, during the few hours I spent in Spaceland; but I cannot realize it now. I take it on trust; but I cannot now make a mental image of it even to myself, much less to my countrymen.

Does this puzzle you? Then put yourself in my place. Suppose a being of the Fourth Dimension, condescending to visit you, were to address you thus: ``You creatures of Three Dimensions see a plane (which is of Two Dimensions) and you infer a solid (which is of Three); but in reality what you call a plane has another Dimension of a kind not know to you;'' what would you reply? Would you not call for a policeman to see your visitor safely locked up in some asylum?

Well, precisely this has been my reception when I have attempted to demonstrate the facts insisted on by your critic. Only yesterday, when the Chief Circle (in other words the Chief Priest) paid his annual visit to my prison, I endeavoured to prove to him that the Figures which we saw around us had a Third non-recognized Dimension, being not only long and broad, but also what you in Spaceland call `high.' What was his reply? Simply this: ``Dimension implies measurement. You say I am `high'; measure my `high-ness' and I will believe you.'' I was crushed, and he left the room in triumph.

Sir, I am a humble Square, and I do not deny the superiority of your critic, who is doubtless a Cube; I impugn neither the exactness of his mathematics nor the regularity of his proportions; in the language of Spaceland, I am ready to admit that he is ``a regular Cube and no mistake.'' But I submit that his knowledge of human nature is not equal to his knowledge of mathematics. He has forgotten that we are all alike --- Points, Lines, Squares, Cubes, Extra-Cubes, whether of no Dimensions or of many dimensions --- liable to the prejudices of our several Dimensions, brothers in error; as one of your own poets also has said, ``One touch of nature makes the world akin,'' meaning thereby not one world only, but all worlds, and not excepting the favoured world of Three Dimensions. And I must say I take it ill that I should be, however gently, censured for appearing to be ignorant of a truth which I firmly apprehend by faith, and which I daily endeavour to inculcate upon others. --- A Square

*** If we understand the Square rightly, all that is wanting to make the Flatlanders realize a third dimension, and to settle circularism once for all, is a delicate micrometer. For he seems to admit that the edges of himself and his countrymen really are extended surfaces --- as, indeed, appears from the fact which he elsewhere mentions, that they were capable of receiving colour. He is not, therefore, in the same position with regard to the third dimension as we of this world with regard to a fourth. The truth is, it may be suspected that our Square, having once in some measure grasped the conception of three-dimensioned space, cannot now wholly divest himself of it. He thinks, so to speak, in three dimensions. For instance, he talks in one place of hearing the sound of his wife's retreating footsteps, a bold metaphor indeed to apply the motion of a line on a plane. But, with a degree of intellectual insincerity probably unconscious, certainly pardonable in a person situated as he is, he thinks it necessary to persist in saying that he apprehends by faith a truth which he has really learnt from the evidence of eyesight; thus making a serious confusion between the functions of faith and sense. The Square does his reviewer too much honour in supposing him to be a regular cube. The best he can claim to be is a rectangular parallelepiped; and he finds it hard enough to live up to that configuration in space of the kind he knows, so that he is content to do without speculations as to the ways of beings in worlds of more or fewer dimensions.