SCIENCE. 



[N. S. Vol. VII. No. 158. 



ocean experiences objects which, to him, 

 seem to come from outside his universe, 

 steamships for example. If our atmos- 

 phere had been opaque to the rays of light 

 from the sun, or even if it had been so 

 filled with clouds and vapor that we could 

 never see outside of it, we also should have 

 had a similar experience. But we may be 

 said, in a certain sense, to see through the 

 whole of our conceivable space with the aid 

 of our telescopes, and the general tendency 

 of scientific thought at the present time is 

 toward the conclusion that no natural 

 agency of which we can trace the operation 

 originates outside the space into which 

 our telescopes may penetrate. Our uni- 

 verse forms, so to speak, a closed system. 

 This is true apparently even of agencies so 

 subtle as those which give vibrations to 

 ether. If there is any agency which we 

 could imagine to connect us with an out- 

 side sphere it is certainly the luminiferous 

 ether. But should this ether enter into a 

 fourth dimension the intensity of light and 

 radiant heat would diminish as the cube of 

 the distance and not as the square. To 

 speak more accurately, radiance emanating 

 from an incandescent body would be en- 

 tirely lost — would pass completely out of 

 our universe. The fact that it is not lost, 

 and indeed the general theory of the con- 

 servation of energy, shows that there is no 

 interchange of energy between our universe 

 and any possible one lying in another di- 

 mension of space. 



We may regard the limitations of the 

 dimension of space to three as expressing 

 in a certain way a physical fact. Our con- 

 ception of space is originally based upon 

 the possibility of motion. The threefold 

 posibility of relative motion can be reduced 

 to a physical fact in this way. Let a point 

 be fixed at one end of a rod, the other end 

 of which is immovably fixed to a wall. The 

 point can then have motion over the surface 

 of a sphere whose center is at the fixed 



point and whose radius is the length of the 

 rod. Now fix one end of a second rod to 

 another point of the plane and bring the 

 two ends of the rods together, and fix the 

 point on both ends ; then the point can only 

 move in a circle. Fasten it to a third point 

 of the plane with a third rod, and it cannot 

 move at all. But if we add a fourth dimen- 

 sion it could move. 



The limits of space are for us simply the 

 limits of possible motion of a material body. 

 We can imagine a body coming from any 

 point in three dimensional space to us, but 

 cannot imagine one coming from outside of 

 such space, until we add a fourth dimen- 

 sion. 



Our conclusion is that space of four 

 dimensions, with its resulting possibility 

 of an infinite number of universes along- 

 side of our own, is a perfectly legitimate 

 mathematical hypothesis. We cannot say 

 whether this conception does or does not 

 correspond to any objective reality. What 

 we can say with confidence is that if a 

 fourth dimension exists, our universe and 

 every known agency in it is, by some 

 fundamental law of its being, absolutely 

 confined to three of the dimensions. But 

 we must not carry a conclusion of this sort 

 beyond the limits set by experience. When 

 we say that experience shows that not only 

 our material universe, but all known 

 agencies in it, are, by a law of their being, 

 incapable of motion in more than three 

 dimensions we must remember that the 

 conclusion applies only to those motions 

 which our senses can perceive, the motions 

 of masses, in fact. There is no proof that 

 the molecule may not vibrate in a fourth 

 dimension. There are facts which seem to 

 indicate at least the possibility of molecular 

 motion or change of some sort not expressi- 

 ble in terms of time and three coordinates 

 in space. If we consider those conceptions 

 of mechanics which we derive from visible 

 phenomena to afford a sufficient explana- 



