514 Transactions. — Miscellaneous, 



universe of space. Absolute space, in short, is the sum of all 

 structure-spaces and all potential structure-spaces. 



In order to be finite, it must, as the new school admit, 

 return into itself, as the circumference of a circle or the surface 

 of a sphere or spheroid returns into itself. Now, since by our 

 three-dimension geometry (the sufficiency of which has, I trust, 

 been made clear) we can always obtain a figure of small extent 

 similar to any corresponding figure of finite extent ; we could, 

 if space were finite, obtain a similar small structure-space which 

 likewise returns into itself. But we cannot — the phrase is 

 meaningless ; therefore the universe of space cannot be finite. 



The only possible meaning that could logically be given to 

 the statement that the Universe is finite, is that the structure- 

 space occupied by all the bodies subject to the physical con- 

 ditions known to us is a finite number of cubic miles.''' But 

 outside this structure- space, again, there must be space, just as 

 there is space outside my piece of chalk. 



It is possible that I may be accused of neglecting the argu- 

 ment upon which Mr. Frankland relies — that, namely, which is 

 based upon the assumption that all the axioms of Euclid are 

 true, except the twelfth ; and that the twelfth is not true. That 

 axiom is easily shown to be identical with the modern substi- 

 tute! ; the advantage of the latter being, to my mind, the fact 

 that it is at once seen to flow directly from the concept of 

 parallel straight lines ; whereas Euclid's 12th needs the 28th 

 proposition before its force can be properly appreciated. I do 

 not know whether it would not be just as easy to approach the 

 subject by assuming as the axiom the second part of Euclid I., 

 29 : — "If a straight line fall upon two parallel straight lines, it 

 shall make the exterior angle equal to the interior and opposite 

 angle on the same side." This is an immediate consequence, 

 hardly more than a re-statement, of the concept of parallel 

 straight lines (which may be roughly described as straight lines 

 drawn in the same direction). 



What Mr. Frankland seems to lose sight of is this : That 

 the notion of parallel straight lines is as truly a concept as is 

 that of a straight line ; that the definitions are not and cannot 

 be equivalents for the concepts ; they are merely indexes to the 

 nature of the several concepts ; and, in like manner, the axioms 

 are indexes of certain concepts so closely related to those pointed 

 to in the definitions as to need no detailed proof. 



The inclusion of the twelfth axiom does not make geometry 

 an experimental science. The very question brought as an 



* In this, of course, there must be included not merely the space these 

 bodies occupy in a literal sense, but the whole space within the range of 

 which all phenomena connected with them take jjlace. 



t Through the same point there cannot be two straight lines, each of 

 which is parallel to a third straight line, 



