ALGEBRAS, SPACES, LOGICS. 521 



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 as- 

 tronomers will prove that we are living in a curved space, and every- 

 thing connected with spiritualism seems to me disgusting bosh. But 

 it is not the probability that I want. I am simply illustrating the pos- 

 sibility, and this is enough to bring the matter into the domain of sim- 

 ple 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 dimen- 

 sion ? 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 im- 

 agine a curve on a plane, and so in physics curves are taken to repre- 

 sent 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 Erkentniss- 

 lehre or hen-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 awaken- 

 ing. In 1847 two mathematicians, Boole and De Morgan, published 



