108 Transactions. — Miscellaneous. 



idea that the Universe is of finite extent, because, (as Mr. Frankland in 

 effect puts it), the properties of any small area of a sphere do not " sensibly 

 differ" from those of a plane, he argues that " in this case the Universe is 

 again* a valid conception," (by the way a curious sort of equivalence this) 

 " for the extent of space is a finite number of cubic miles." 



Observe here the very important qualifying term sensibly, which forms a 

 part of the proposition but is omitted in the deductions. To make the 

 conclusion agree with the premises, it should have gone no further than to 

 affirm that the Universe may not sensibly differ from an infinite one. Un- 

 scientific and illogical conclusions of a very startling character are easily 

 got by suppressions of this kind — suppressions which lead us all uncon- 

 sciously to mistake appearances for reahties. 



Proceeding, however, with this quotation, we observe further that the 

 Professor, having perchance, after all, some doubts as to the validity of this 

 deduction, or possibly forgetting he has proved it, essays to prove it again ; 

 be says, "and this (finiteness of the Universe) comes about in a very 

 curious way. If you were to start in any direction whatever, and 

 move in that direction in a perfect straight line, according to the definition 



of Leibnitz, you would arrive at this place. Only if you had 



started upwards you would appear fr-om below." 



Mark, now, the qualification put upon straight lines, ^^ straight according 

 to Leibnitz,'" — put, no doubt, all in good faith, as explanative of straight 

 lines, it does still, I feel assured, confer upon them properties which 

 straight Unes have not, and in such a way as has not manifested itself 

 to him, able as he undoubtedly is. To those geometricians whose grosser 

 ideas forbid their translation to that high realm of thought where this 

 new geometry is analytically conceivable, it does seem that a definition 

 of straight hues which allows of the idea being held that a man can get 

 back on his tracks by going straight away from them, is a definition that is 

 just a little wrong. 



Our idea of what straight lines are is a fixed and definite one, whether 

 or not we can get a diagrammatical or verbal definition of them,f and it is 



* Eeferring, I suppose, to that happy time when the firmament was held to be a soHd, 

 studded with sparks, and the earth a plane supported on pillars. Delusions once started 

 seem to be ever perennial, except indeed that they are modified to suit the times. I dare- 

 say in a few centuries this geometry wUl give way in its turn to something if possible, still 

 more transcendental, and so ad infinitum. 



f This so-termed definition is on all sides acknowledged to be no definition at all in 

 a strict sense. Euclid's meaning is clear, although the terms used are ambiguous, and 

 do not exactly fit it. But our conceptions of what is the necessary property of parallel 

 hues should not be affected thereby. The fact that the definition when "worked out" 

 lets in Unes which were not contemplated by Euclid does not make these parallel, but 

 merely shows the faultiness of this definition. 



