January 7, 1898.] 



SCIENCE. 



back it would be obverted, or appear as in 

 a looking glass. A man capable of such a 

 motion would come back into our sight 

 similarly obverted, his left side would now 

 be his right, without any change having 

 taken place in the relative positions of the 

 particles of his body. The somerset he 

 would have turned would have completely 

 obverted every atom and molecule of his 

 body without introducing any disturbance 

 into its operations. 



This possibility of obversion brings in a 

 curious question concerning the rigor of 

 one of the fundamental propositions in ele- 

 mentary geometry. Euclid proves by super- 

 position that the two triangles in a plane 

 having two angles and the included side 

 equal are equal to each other. In the dem- 

 onstration it is assumed that the triangles 

 can be made congruent by simply placing 

 one upon the other without taking it out of 

 the plane. From this the conclusion is 

 drawn that the same conclusion holds true 

 if one of the triangles be obverted. But in 

 this case they cannot be brought into con- 

 gruence without taking one of them out of 

 the plane and turning it over. The third- 

 dimension is thus assumed in geometry in- 

 volving only two dimensions. 



Now consider the analogous case in 

 space. Two pyramids upon congruent 

 bases may be proved equal by bringing 

 them into congruence with each other. 

 But suppose that they differ only in that 

 one is the obverse of the other, so that they 

 could be brought into congruence only by 

 looking at one of them in a mirror and then 

 placing the other into congruence with the 

 image of the first as seen in the mirror. 

 Would we detract from the rigor of the 

 demonstration by assuming the possibility 

 of such an obversion without changing the 

 volume of the pyramid? With a fourth 

 dimension we should have no detraction 

 from rigor. We would simply obvert the 

 pyramid as we would turn over the triangle. 



The question of the fourth dimension as 

 a reality may be considered from two points 

 of view, its conceivabilitj' and its possible 

 objective reality. If by conceivability we 

 mean the power of being imaged in the 

 mind it must be admitted that it is ab- 

 solutely inconceivable. We have no diffi- 

 culty in forming a visual conception of 

 three lines passing through the same point, 

 each of which is at right angles to the other 

 two. Such is the familiar system of coordi- 

 nate axes in space. But he who would con- 

 ceive a fourth dimension must be able to 

 imagine a fourth axis perpendicular to all 

 three of the others. This clearly transcends 

 all possibility even of imagination. The 

 fourth dimension in this sense is certainly 

 inconceivable. 



The question of the objective possibility 

 of the fourth dimension is quite a distinct 

 one from that of its conceivabilitj'. The 

 latter limitation upon our faculties grows 

 out of the objective fact that we and our 

 ancestors have had no experience of a 

 fourth dimension ; that we have always 

 lived in a universe of three dimensions 

 only. But we should not too readily con- 

 clude that all being is necessarily confined 

 to these three dimensions. Those who 

 speculate on the possible have taken great 

 pleasure in imagining another universe 

 alongside of our own and yet distinct from 

 it. The mathematician has shown that 

 there is nothing absurd or contradictory in 

 such a supposition. But when we come to 

 the question of physical fact we must admit 

 that there appears to be no evidence of 

 such a universe. If it exists, none of its 

 agencies intrude into our own universe, 

 at least in the opinion of sober think- 

 ers. The intrusion of spirits from without 

 into our world is a favorite idea among 

 primitive men, but tends to die out with 

 enlightenment and civilization. Yet there 

 is nothing self-contradictory or illogical in 

 the supposition. The fish that swims the 



