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The children, therefore, were but little noticed, especially as they were at quite a distance from where the multitude of spectators was assembled, being considerably higher up and near the flat-land, bearing the undignified name which only historical accuracy compels us to introduce. After a time a cake, on which one of the boys was standing, began slowly to slip away from the shore.
The gossiping circle in flat-land about Earl's Court was bad enough, but the Redcliffe Square set, being slightly higher in the social scale, was infinitely worse. "Oh! all the ill-natured people are commenting upon your apparent coolness. Once, not long ago, you used to be seen everywhere with Ethelwynn, and now no one ever sees you.
This is perfectly simple to me; it is impossible to you solely because your conceptions do not admit of more than three dimensions." Supposing the inhabitants of "flat-land" to be intellectual beings as we are, it would be interesting to them to be told what dwellers of space in three dimensions could do. Let us pursue the analogy by showing what dwellers in four dimensions might do.
Now, to pursue the analogy, suppose that, after we have drawn three mutually perpendicular lines, some being from another sphere proposes to us the drawing of a fourth line through the same point, perpendicular to all three of the lines already there. We should answer him in the same way that the inhabitants of "flat-land" answered us: "The problem is impossible.
It is very curious that, in these transcendental speculations, the most rigorous mathematical methods correspond to the most mystical ideas of the Swedenborgian and other forms of religion. Right around us, but in a direction which we cannot conceive any more than the inhabitants of "flat-land" can conceive up and down, there may exist not merely another universe, but any number of universes.
A savage can be taught to use a rifle or an electric switch or even a flame-projecter, but this is no proof that he is really civilised. On the other hand, the scholarly recluse and philosopher whose works uplift and refine humanity may bungle even with the "fool-proof" lift which takes him up to his own eyrie in Flat-land, but he is none the less civilised.
Place a dweller of "flat-land" inside a circle drawn on his plane, and ask him to step outside of it without breaking through it. He would go all around, and, finding every inch of it closed, he would say it was impossible from the very nature of the conditions. "But," we would reply, "that is because of your limited conceptions. We can step over it." "Step over it!" he would exclaim.
The two lines AB and CD, starting parallel, would eventually, perhaps at distances greater than that of the fixed stars, gradually diverge from each other. This system does not admit of being shown by analogy so easily as the other, but an idea of it may be had by supposing that the surface of "flat-land," instead of being spherical, is saddle-shaped.
You cannot draw any such line in space as we understand it." If our visitor conceived of the fourth dimension, he would reply to us as we replied to the "flat-land" people: "The problem is absurd and impossible if you confine your line to space as you understand it. But for me there is a fourth dimension in space. Draw your line through that dimension, and the problem will be solved.
Euclid takes it for granted that the one triangle can be laid upon the other so that the two shall fit together. But this cannot be done unless we lift one up and turn it over. In the geometry of "flat-land" such a thing as lifting up is inconceivable; the two triangles could never be fitted together. Now let us suppose two pyramids similarly related.
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