or from the abstract concept "quantity," using this term as comprehending 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-homaloidal 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 speculations of this sort are harmless only so long as it is not pretended that they can teach us anything respecting either empirical reality or empirical possibility. And they can throw light on the origin and meaning 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 without the least suspicion that anything more is necessary for the accomplishment of his purpose than skill in the handling of an analytical formula? It may be that my undertaking has not been very successful; 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 consequence 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 318, by getting somebody to admit that the ratio of the circumference to the diameter is the same for all circles, and then telling him to draw one circle with the diameter 1 and circumference 318. Now, the intellectual plight of this paradoxer, who, besides assuming the very thing to be proved, failed to see that his argument would
