NECESSARY TRUTHS— EFFECTS OF EXPERIENCE. 659 



turn them mentally, and find that parts of them will super- 

 pose. We add imaginary lines which subdivide or enlarge 

 them, and find that the new figures resemble each other in 

 ways which show us that the old ones are equivalent too. 

 We thus end by expressing all sorts of forms in terms of 

 other forms, enlarging our knowledge of the kinds of things 

 which certain other kinds of things are, or to which they 

 are equivalent. 



The result is a new system of mental objects which can 

 be treated as identical for certain purposes, a new series of 

 ts's almost indefinitely prolonged, just like the series of 

 equivalencies among numbers, part of wdiich the multipli- 

 cation-table expresses. And all this is in the first instance 

 regardless of the coexistences and sequences of nature, 

 and regardless of whether the figures we speak of have ever 

 been outwardly experienced or not. 



CONSCIOUSNESS OF SEKIES IS THE BASIS OF RATIONALITY. 



Classification, logic, and mathematics all result, then, 

 from the mere play of the mind comparing its conceptions, 

 no matter whence the latter may have come. The essential 

 condition for the formation of all these sciences is that we 

 should have grown capable of apprehending series as such, 

 and of distinguishing them as homogeneous or hetero- 

 geneous, and as possessing definite directions of what I have 

 called 'increase.' This consciousness of series is a human 

 perfection which has been gradually evolved, and which 

 varies greatly from man to man. There is no accounting 

 for it as a result of habitual associations among outward 

 impressions, so we must simply ascribe it to the factors, 

 whatever they be, of inward cerebral growth. Once this 

 consciousness attained to, however, mediate thought be- 

 comes possible ; with our very awareness of a series may 

 go an awareness that dropping terms out of it will leave 

 identical relations between the terms that remain ; and 

 thus arises a perception of relations between things so 

 naturally separate that we should otherwdse never have 

 compared them together at all. 



The axiom of skipped intermediaries applies, however, 

 only to certain particular series, and among them to those 



