688 THE POPULAR SCIEXCE MONTHLY. 



form of the earth, but the elephant and the tortoise would be more 

 than ever necessary to suj^port it. There would be no science of as- 

 tronomy, no knowledge of the law of gravitation, none of physics, so 

 far as it was dependent upon astronomy. The sciences so act and re- 

 act on one another that it is difficult to say just how far the absence 

 of one would affect the others, but we know it would greatly. Such 

 suppositions as we have made may be more or less fully realized in 

 other worlds, and we can thus see that their science may differ very 

 widely from ours and still be no less correct. Science is everywhere 

 relative to the facts with which it has to deal. The difficulty of con- 

 ceiving physical and spatial relations different from those we are famil- 

 iar with does not prove them non-existent ; nor does the ease of such 

 conceptions prove their existence. 



Helmholtz has very ably shown that our geometrical axioms have 

 a truth relative only to the space they are applied to. He supposes 

 beings, of the same mental capacity as ourselves, but of two dimen- 

 sions only, to inhabit a plane surface. They would possess a plane 

 geometry like ours, but would have no solid geometry whatsoever. 

 Thickness would be as inconceivable to them as a fourth dimension in 

 space is to us. Transplant these beings to the surface of a sphere, and 

 their planimetry would change to a spherical geometry. Defining a 

 straight line as the shortest distance between two points, all their 

 straight lines would be arcs of great circles, and every straight line, when 

 sufficiently extended, would return to itself. Between two points, 

 half the circumference of a great circle apart, an infinite number of 

 straight lines, of equal length, could be drawn ; and, as two points 

 would always cut the great circle on which they were situated into 

 two arcs of unequal length, there would always be, besides the shortest 

 straight line connecting the points, a longer straight line (i. e., a line 

 made up of shorter lines, each of which is the shortest distance between 

 its extremities) also connecting them. There could be no parallel 

 straight lines and no similar triangles. The sum of the angles of a 

 triangle would always be more than two right angles, and the amount 

 of the excess would depend upon the length of the sides. 



Helmholtz again supposes these beings of two dimensions to be 

 placed upon (what has been called by Beltrami) a pseudo-spherical sur- 

 face a surface shaped somewhat like the sides of an hour-glass. Here 

 our axiom of parallels does not hold good. Through a given point, 

 a whole pencil of straight lines may be drawn, none of which shall 

 cut a given line, though infinitely produced, and limited, at the two 

 extremes, by lines that cut the given line at infinite distances in the 

 opposite directions. Helmholtz makes other suppositions which it is 

 unnecessary to follow, as we have already gone far enough to show 

 that even the fundamental axioms of geometry are quite as dependent 

 upon the conditions under which they are used as upon any intuitive 

 necessity we may think belongs to them. 



