HoGBEN. — Transcendental Geometry. 511 



I shall not enter into all the questions raised between Mr. 

 Frankland and his critic, Mr. Skey, but content myself with 

 noticing the three most important propositions laid down in the 

 two papers contributed by the former. These are : — 



(1.) That the axioms of geometry may be only approxi- 

 mately true : 



(2.) That the actual properties of space may be somewhat 

 different from those which we are in the habit of 

 ascribing to it : 



(3.) That the extent of space may be a finite number of 

 cubic miles. 



If these propositions are sound, the transcendental geometers 

 may be right ; if not, the position of the Euclidian geometers, 

 who maintain that space has three dimensions, and three only, 

 remains unassailed. 



The subject, of course, has often been discussed, and the 

 argument on the orthodox side is well represented by Stallo, 

 ("Concepts of Modern Physics,'") and Lotze (" Metaphysic "'). 

 While acknowledging my obligations to these great writers, 

 each of whom, however, gives only part of the argument, I shall 

 endeavour to state the case in a somewhat different form : — 



(1.) "The axioms of geometry may be only approximately 

 true " ; or, again, as Mr. Frankland says in another place, 

 " geometry is a physical and experimental science." 



This idea of geometry, though countenanced by John Stuart 

 Mill, is founded upon a serious misconception as to what the 

 subjects are of which geometry treats. The line of reasoning 

 pursued is shortly as follows : — ' Geometry treats, among other 

 things, ol straight lines ; but straight lines cannot be conceived 

 apart from objects, and nowhere are we acquainted with Imes 

 that are more than approximately straight. Therefore geometry 

 is only an approximate science.' The argument, as Stallo and 

 others have shown, contains its own answer. How do you 

 know that any given line must be only approximately straight, 

 except by reference to some standard ? The very phrase " only 

 approximately straight " implies the existence of such a standard 

 in the mind of the person who makes it. When Mr. Frankland 

 speaks of a line on his supposed manifold as having such feeble 

 curvature as hardly to be distinguishable from a Euclidian 

 straight line, he is really implying this standard. In a similar 

 manner it could be shown that we must admit the concepts of 

 a line, a surface, a plane surface, a right angle, a solid, and 

 so on. 



In fact, geometry is the science of such standards as this, or 

 rather of such concepts as this. It has been, I think, rightly 

 defined as the science of the concepts of the limits of the modes 

 of extension. It starts with a limited number of concepts, and 



