SCIENCE. 



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



three dimensions. A hypothesis which is 

 simpler in its fundamental basis, and yet 

 seems absurd enough iu itself, is that of 

 what is sometimes, improperly I think, 

 called curved space. This also we may call 

 hyper- space, defining the latter in general 

 as space in which the axioms of the Euclid- 

 ean geometry are not true and complete. 

 Curved space and space of four or more 

 dimensions are completely distinct in their 

 characteristics, and must, therefore, be 

 treated separately. 



The hypothesis of a fourth dimension can 

 be introduced in so simple a way that it 

 should give rise to no question or diificulty 

 whatever. Indeed, the whole conception 

 is so simple that I should hardlj;^ deem it 

 necessary to explain the matter to a pro- 

 fessional mathematical student. But as we 

 all have to come in contact with educated 

 men who have not had the time to com- 

 pletely master mathematical conceptions, 

 and yet are interested in the fundamental 

 philosophy of our subject, I have deemed it 

 appropriate to present the question in what 

 seems to me the simplest light. 



The student of geometry begins his study 

 with the theory of figures in a plane. In 

 this field he reaches certain conclusions, 

 among them that only one perpendicular 

 can be drawn to a line at a given point, 

 and that only one triangle can be erected 

 with given sides on a ^iven base in a given 

 order. Having constructed this plane 

 geometry, he passes to geomej^ of three 

 dimensions. Here he enters a region in 

 which some of the propositions of plane 

 geometry cease to be true. An infinity of 

 perpendiculars can now be drawn to a given 

 line at a given point, and an infinity of tri- 

 angles can be constructed on a given base 

 with given sides. He has thus considered 

 in succession geometry of two dimensions, 

 and then passed to geometry of three di- 

 mensions. Why should he stop there? 

 You reply, perhaps, because there are only 



three dimensions in actual space. But in 

 making hypotheses we need not limit our- 

 selves to actualities ; we can improve our 

 methods of research, and gain clearer con- 

 ceptions of the actual by passing outside 

 and considering the possible. 



For logical purposes there is no limit to 

 the admissibility of hypotheses, provided 

 we consider them purely as hypotheses, and 

 do not teach that they are actual facts of 

 the universe. It is, therefore, perfectly 

 legitimate to inquire what our geometry 

 would be if, instead of being confined to 

 three dimensions, we introduced a fourth. 

 Many curious conclusions follow. When 

 we are confined to a plane a circle com- 

 pletely bounds a region within the plane, 

 so that we cannot pass from the inside to the 

 outside of the circle without intersecting it. 

 Beings conscious only of two dimensions 

 and moving only in two dimensions, and 

 placed inside such a material circle, would 

 find themselves completely imprisoned, 

 with no possibility of getting outside. 

 But give them a third dimension, with the 

 power to move into it, and they simply 

 step over the circle without breaking it. 

 They do not have to even touch it. Liv- 

 ing, as we do, in space of three dimen- 

 sions, the four walls, pavement and ceiling 

 of a dungeon, confine a person so com- 

 pletely that there is no possibility of escap- 

 ing without making an opening through 

 the bounding surface. But give us a fourth 

 dimension, with the faculty of moving into 

 it, and we pass completely outside of our 

 three dimensional universe, by a single 

 step, and get outside the dungeon as easily 

 as a man steps over a line drawn on the 

 ground. Were motion in the fourth dimen- 

 sion possible, an object moving in that 

 dimension by the smallest amount would 

 be completely outside of what we recognize 

 as the universe, and would, therefore, be- 

 come invisible. It could then be turned 

 around in such a way that on being brought 



