SPACE 647 



Euclid's postulate from the several axioms, it is evident that by reject- 

 ing the postulate and retaining the other axioms we should be led to 

 contradictory consequences. It would be, therefore, impossible to 

 found on those premisses a coherent geometry. Now, this is pre- 

 cisely what Lobatschewsky has done. He assumes at the outset that 

 several parallels may be drawn through a point to a given straight 

 line, and he retains all the other axioms of Euclid. From these 

 hypotheses he deduces a series of theorems between which it is im- 

 possible to find any contradiction, and he constructs a geometry as 

 impeccable in its logic as Euclidean geometry. The theorems are very 

 different, however, from those to which we are accustomed, and at first 

 will be found a little disconcerting. For instance, the sum of the 

 angles of a triangle is always less than two right angles, and the differ- 

 ence between that sum and two right angles is proportional to the 

 area of the triangle. It is impossible to construct a figure similar to a 

 given figure but of different dimensions. If the circumference of a 

 circle be divided into n equal parts, and tangents be drawn at the 

 points of intersection, the n tangents will form a polygon if the 

 radius of the circle is small enough, but if the radius is large enough 

 they will never meet. We need not multiply these examples. Lobat- 

 schewsky's propositions have no relation to those of Euclid, but they 

 are none the less logically interconnected. 



Riemann's Geometry. Let us imagine to ourselves a world only 

 peopled with beings of no thickness, and suppose these " infinitely 

 flat " animals are all in one and the same plane, from which they 

 cannot emerge. Let us further admit that this world is sufficiently 

 distant from other worlds to be withdrawn from their influence, and 

 while we are making these hypotheses it will not cost us much to en- 

 dow these beings with reasoning power, and to believe them capable 

 of making a geometry. In that case they will certainly attribute to 

 space only two dimensions. But now suppose that these imaginary 

 animals, while remaining without thickness, have the form of a spher- 

 ical, and not of a plane figure, and are all on the same sphere, from 

 which they cannot escape. What kind of a geometry will they con- 

 struct? In the first place, it is clear that they will attribute to 

 space only two dimensions. The straight line to them will be the 

 shortest distance from one point on the sphere to another that is 

 to say, an arc of a great circle. In a word, their geometry will be 

 spherical geometry. What they will call space will be the sphere on 

 which they are confined, and on which take place all the phenomena 

 with which they are acquainted. Their space will therefore be 

 unbounded, since on a sphere one may always walk forward without 

 ever being brought to a stop, and yet it will be finite; the end will 

 never be found, but the complete tour can be made. Well, Kiemamrs 

 geometry is spherical geometry extended to three dimensions. To 



