4o6 THE POPULAR SCIENCE MONTHLY 



his solution by an actual cut, and then brought me written out the 

 process of reasoning whereby he had solved the problem. 



The process of reasoning in question, or any process of reasoning 

 by which the problem could be solved in advance of actually cutting the 

 strip, is in part a genuine instance of deductive reasoning. Just because 

 of the intimate mingling of empirical data and of exactly definable 

 relationships, the case in question forms an admirable instance of the 

 study of the genuine psychology of the deductive process; but I con- 

 fess that no psychologist would make much of the study who was not 

 fairly well acquainted with deductive processes of a certain complexity, 

 — processes which in their more exact forms you will find anywhere in 

 pure mathematics, where a symbolic language with an exact definition 

 is used, as the only means of presenting data to the imagination. 



Let me mention another instance of a deductive process of some 

 complexity, but of great ingenuity and of interesting psychological 

 relationships. We know that about 500 b. c. a member of the Pythago- 

 rean school discovered that granting the ordinary principles of metrical 

 geometry as they were then and ever since have been used, the diagonal 

 of a square could not be commensurate with the side of the square. The 

 strictly deductive portion of this proof can be with fair ease distin- 

 guished from that portion of the proof in question which is indeed em- 

 pirical and not deductive. That figures resembling squares exist is a 

 matter of experience. That if you make a square exactly enough and 

 large enough, and measure carefully enough you will discover that by no 

 rule you seem to be certain of stating the ratio of diagonal to side 

 exactly in terms of whole numbers : this again is so far empirical. And 

 the ordinary so-called axioms of metrical geometr}', considered as prin- 

 ciples about the constitution of the physical world, are of course gen- 

 eralizations from physical experience. On the other hand, the purely 

 mathematical portion of geometry, that is, the purely deductive portion, 

 consists in the discovery, not that the geometrical axioms are self- 

 evident or otherwise certain, and not that the physical world has any 

 properties whatever, but that certain assumed geometrical principles 

 which can be stated wholly in symbols, actually imply certain geomet- 

 rical conclusions. Xow the early Greek geometer who discovered that 

 the diagonal and the side of the square are from the point of view of geo- 

 metrical theory incommensurable, was no doubt guided by the empirical 

 diflSculty of discovering any rule whereby a common measure for the 

 diagonal and the side could be stated. Furthermore, he was not clear in 

 his own mind as to the precise distinction between the deductive and 

 the inductive part of his geometrical science. But he was possessed of 

 the power to draw an exact deductive conclusion. And what lie found 

 out was that if certain principles of measurement and certain purely 

 mathematical properties of whole numbers be admitted, the diagonal and 



