SCIENCE. 



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



di£Fereiit law of position among the mole- 

 cules of bodies. 



This consideration leads us to a possible 

 form of space relations distinct from those 

 of our Euclidean geometry, and from the 

 hypothesis of space of more than three di- 

 m.ensions, I refer to what is commonly 

 known as ' curved space.' The history of 

 this conception is now so well known to 

 mathematicians that I shall mention it 

 only so far as is necessary to bring it 

 to your minds. The question whether 

 Euclid's axioms of parallels is really an 

 independent axiom, underivable from the 

 other axioms of geometry, is one which 

 has occupied the attention of mathema- 

 ticians for centuries. Pei-haps the sim- 

 plest form of this axiom is that through a 

 point in a plane one straight line and no 

 more can be drawn which shall be parallel 

 to a given straight line in the plane. Here 

 we must understand that parallel lines 

 mean those which never meet. The axiom, 

 therefore, asserts that through such a point 

 we can draw one line which shall never 

 meet the other line in either direction, but 

 that if we give this one line the slightest 

 motion around the point in the plane it will 

 meet the other in one direction or the op- 

 posite. Thus stated, the proposition seems 

 to be an axiom, but it is an axiom that does 

 not grow out of any other axioms of geom- 

 etry. The question thus arising was at- 

 tacked by Lobatchevsky in this very 

 conclusive manner. If this axiom is inde- 

 pendent of the other axioms of geometry 

 then we should be able to construct a self- 

 consistent geometrical system, in conformity 

 to the other axioms, in which this axiom 

 no longer held. The axiom of parallels 

 may be deviated from in two directions. 

 In the one it is supposed that every two 

 lines in the plane must meet ; no line par- 

 allel to another can be drawn through the 

 same point in the plane. Deviating in the 

 other direction we have several lines drawn 



through the point which never meet the 

 given line ; they diverge from it as lines on 

 an hyperboloid may diverge. 



That such possibilities transcend our or- 

 dinary notions of geometrical relations is 

 beyond doubt, but the hypothesis of their 

 possibility is justified by the following^ 

 analogy. Let us suppose a class of beings- 

 whose movements and conceptions were 

 wholly confined to a space of two dimen- 

 sions as ours are to a space of three dimen- 

 sions. Let us suppose such beings to live 

 upon or in a plane and to have no concep- 

 tion of space otherwise than as plain extend- 

 ed space. These beings would then have 

 a plane geometry exactly like ours. The 

 axiom of parallels would hold for them as it 

 does for vis. But let us suppose that these 

 beings, without actually knowing it, instead 

 of being confined to a plane, were really 

 confined to the surface of a sphere, a sphere 

 such as our earth, for example. Then, 

 when they extended their motions and ob- 

 servations over regions so great as a large 

 part of the earth's surface, they would find 

 the axiom of parallels to fail them. Two 

 parallel lines would be only two parallel 

 great circles, and though each were followed 

 in a direction which would seem to be in- 

 variable they would be found to meet on 

 opposite sides of the globe. The suggestion 

 growing out of this consideration is : May 

 it not be possible that we live in a space of 

 this sort? Or, to use what seems to me to be 

 the more accurate language : May it not be 

 that two seemingly parallel straight lines 

 continued indefinitely would ultimately meet 

 or diverge ? The conceptions arising in this 

 way are certainly very interesting. If the 

 lines would meet it can easily be shown 

 that the total volume of all space is 

 a finite quantity. The sum of the three 

 angles of a triangle extending from star 

 to star would then be greater than the 

 sum of two right angles. Equally legiti- 

 mate is the hypothesis that it would 



