6 EOYAL SOCIETY OF CANADA 



But we who view the matter fi'om a higher point, in a larger field, 

 know that the plane of the lilUputian, is a minute portion of a great 

 spherical surface. That his supposed straight lines, although the shortest 

 from point to point upon his sujîposed plane, are only geodetic lines on 

 the sjshere, and therefore not straight in the Euclidean sense ; that his 

 parallels will meet in no less than two j^oints if produced far enough ; 

 and consequently that two of his straight lines can include a space ; and 

 that the sum of the internal angles of a triangle drawn on his plane 

 would be greater than two right angles. 



On the other hand if we suppose our lilliputian to be living upon the 

 surface of a hyperboloid of one sheet, or a hyperbolic paraboloid whose 

 radius of curvature at any point is exceedingly great as compared with 

 the limits of his migrations, then, although he would draw the same con- 

 clusions in regard to his plane geometry as befoi'e, we know that under 

 his conditions of existence he could have two straight lines, and no more, 

 passing through any one point in his plane, and that all other lines 

 through this point, although appearing to be straight, would be slightly 

 curved, being geodetic lines upon the surface, that bis parallels would 

 never meet, but would gradually separate when continued sulficiently 

 outwards, and that his supposed plane triangle might have the sum of its 

 internal angles less than two right angles. 



Now the new geometry holds that the properties and relations of 

 space may be such as to compel a state of things something such as here 

 illustrated. That although we, on account of our limitations, might be 

 never able to detect any deviation from the strictest Euclidean conclu- 

 sions, yet our so-called parallels might either apjjroximate when very 

 greatly extended, when our space would be called elliptical ; or they 

 might separate under like conditions, in which case our space would be 

 hyperbolic, and thus, of course, all the other consequences would follow. 



Prof. Clili'ord, whose versatility and ingenuity made him somewhat 

 of a vandal amongst mathematical usages, gave great attention to illus- 

 trating these transcendental speculations, and I shall make use of some of 

 his illustrations. 



Clifford begins by assuming an intellectual worm of infinitesimal 

 thickness, part of a geometrical line, in fact, moving in a tube of infini- 

 tesimal bore. Such a worm, he claims, could have knowledge of space 

 of one dimension only. If his tube were straight it might be infinite in 

 length, and thus the one dimensional space of the worm would be infinite 

 in extent, and same; f.e., identically the same in every part. But if the 

 tube were bent into a circle^ so that the bore would form a continuous 

 circumference, the space would be equally endless, and as the worm could 

 still pass along the tube without any knowledge of change or limitation 

 to motion, it would naturally infer, as before, that the one dimensional 

 space which it occupied was infinite and same. If, however, the tube 



