266 POPULAR SCIENCE MONTHLY. 



actual measurement departs from two right angles, the fact of the non- 

 Euclidean character of our space would be established at once. But no 

 such triangle has been discovered. Even the largest, which are con- 

 cerned in the measurement of stellar parallax, do not help us, and it 

 does not seem possible to get larger ones. Nevertheless Clifford and 

 others have shown that some physical phenomena, which require the 

 conception of elaborate and complex machinery for their explanation, 

 are capable of very simple explanation upon the hypothesis of a fourth 

 dimension. Then, too, in the domain of pure mathematics several 

 phenomena find a ready explanation upon the basis of such an assump- 

 tion. In the theory of curves we constantly make use of the assump- 

 tion that a curve may return into itself after passing through infinity, 

 which is only another aspect of the same hypothesis. In fact, with- 

 out this aid our processes of generalization, so important to the develop- 

 ment of modern geometry, would be sadly hampered. Professor New- 

 comb has carried this matter to its logical conclusion and has deduced 

 the actual dimensions of the visible universe in terms of the measure- 

 ment of curvature in the fourth dimension. In such a space it becomes 

 actually possible for a curve with infinite branches to pass through in- 

 finity (so-called) and return into itself. Upon this hypothesis our uni- 

 verse is unbounded in the sense that however far we travel we can never 

 reach its limits, for it has none, but it is not infinite. Just as we can 

 travel forever on the surface of the earth without reaching any limits,, 

 but that surface is not infinite. But even supposing that all this is- 

 true, the question still presses home: What is beyond? 



