160 THE POPULAR SCIENCE MONTHLY. 



or from the abstract concept " quantity," using this term as compre- 

 hending both algebraic " quantities " and geometrical magnitudes. As 

 subsidiary to these questions I also discuss certain minor questions, 

 such as that of the representability of non-homaloidal forms of space ; 

 but upon the proof that there is actually no such thing as non-homa- 

 loidal or four-dimensional space I do not waste a syllable. In other 

 words (which Professor Newcomb may find more intelligible, perhaps) : 

 my first inquiry is, not whether any one has ever discovered a fourth 

 dimension or an inherent spatial crook by looking through a telescope, 

 but whether there would be any use or sense in trying to make such a 

 discovery by looking through a telescope, even if we could get a base- 

 line large enough to meet the requirements of Professor Helmholtz ; 

 and my second inquiry is, whether or not there is any world-producing 

 potency in an algebraic formula or an " abstract noun." 



Professor Newcomb claims that investigations respecting geometry 

 of more than three dimensions are at least harmless, and even useful, 

 inasmuch as " they have thrown a flood of light on the origin and 

 meaning of geometrical axioms." My answer to this is, that specula- 

 tions of this sort are harmless only so long as it is not pretended that 

 they can teach us anything respecting either empirical reality or em- 

 pirical possibility. And they can throw light on the origin and mean- 

 ing of geometrical axioms only by giving us an insight into the nature 

 of the forms or modes in which the world of objective reality is or 

 may be reproduced in the intellect. But what shall we say, then, 

 about the grin at speculation in science which stares at us from the 

 very title of Professor Newcomb's article ? If he may throw a flood 

 of light on the foundations of geometry, by speculating about space 

 of four dimensions, am I to be jeered at when I endeavor to direct a 

 feeble ray from the general theory of cognition on the same subject 

 when I try to do methodically what he is doing at random, and with- 

 out the least suspicion that anything more is necessary for the accom- 

 plishment of his purpose than skill in the handling of an analytical 

 formula ? It may be that my undertaking has not been very success- 

 ful ; but in magnis voluisse sat est. And this leads me to say a few 

 words in answer to the intimation of Professor Newcomb and the 

 direct charge of my reviewer in " The Critic," that inquiries into the 

 forms and laws of thought are sheer impertinence, and of no conse- 

 quence to the physicist. 



In the introductory part of his article Professor Newcomb flings 

 at me the case of De Morgan's paradoxer Smith, who fancied that he 

 could prove the ratio of the circumference of a circle to its diameter 

 to be exactly 3^, by getting somebody to admit that the ratio of the 

 circumference to the diameter is the same for all circles, and then tell- 

 ing him to draw one circle with the diameter 1 and circumference 3^. 

 ISTow, the intellectual plight of this paradoxer, who, besides assuming 

 the very thing to be proved, failed to see that his argument would 



