angle. Then I do not know that the difference of the sum of the three angles of this triangle from two right angles would be less than ten degrees, or the ninth part of a right angle." This says that it is within the power of our astronomers to discover that our space is not flat. And already spiritualists claim to have experimentally demonstrated that our space has more than three dimensions. As for myself, I admit I am prejudiced just as you are. I do not think it probable that astronomers will prove that we are living in a curved space, and everything connected with spiritualism seems to me disgusting bosh. But it is not the probability that I want. I am simply illustrating the possibility, and this is enough to bring the matter into the domain of simple external reality.
You have the meaning of a fourth dimension strikingly put before you every time you look into a mirror. There you see yourself so turned around that your right hand has become your left. If you were to step straight out of the looking-glass every one would think you left-handed. Such a change could be accomplished by revolving you in the fourth dimension, and in no other way. Therefore a mirror will show you at any moment exactly the effect of a fourth dimension. Then why is this not a proof of the actual existence of a fourth dimension? I answer that here, as in the case of the spiritualists, there is deception.
It would be proof if there were no deception. The straight rays of light break against the mirror and are turned back. Our eyes give us no account of this break and turn, and so deceive us, putting before us, like the spiritualists, the effect of a fourth dimension. These are not questions which can be decided by reference to our space intuitions, for our intuitions are confined to Euclidean space, and even there are insufficient, approximative. For instance, you suppose you can imagine a curve on a plane, and so in physics curves are taken to represent functions. In reality you can not get any closer to it than what the Germans call a stripe. The analytical copy of the curve is not the function but the stripe.
But you may say, How can we ever go better and deeper than our intuitions? If I answer, "Logic," you are apt to feel soothed. It is wonderful what a strong though often unconscious distinction exists in the general English-speaking mind between logic and metaphysics. Metaphysics is always scorned and scouted; but if you say logic, ah! that is a very different matter. Again, I must acknowledge for myself sympathy with the general feeling. I think most metaphysics ought to be scorned; and I am glad that in English logic means formal logic, a pure science, and is rarely mixed up with a metaphysical Erkentnisslehre or ken-lore. To be sure, formal logic was for ages the most fixed of all things, and so fell into some disrepute, since to be stationary and unprogressive is to be so far unscientific. But at last came the awakening. In 1847 two mathematicians, Boole and De Morgan, published
