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logic, or psychology, these definitions would have to be reduced to 

 forms far more precise ; but I purposely refrain from an attempt at 

 exact definition, because I wish to remain on ground common to all who 

 have made the matter in hand the subject of their investigation. For 

 my purpose, it is of little consequence whether or not the distinction 

 here indicated between representations and concepts is accurate and 

 clear ; nor is it necessary to determine the exact nature of the relations 

 established in conception between the constituents of a concept, or 

 between the various concepts themselves ; it is sufficient to know that 

 both in the representation and in the concept we have in some form 

 a complex of attributes which are ultimately, in the case of material 

 objects at least, traceable to sensible experience, and that the elabora- 

 tion of representations into concepts involves the establishment of 

 some sort of mental relations between their elements, as well as be- 

 tween the several concepts themselves. 



At this point, it is important to guard against a confusion which 

 naturally arises from the fact that logicians and psychologists habit- 

 ually illustrate the evolution of concepts by examples taken from the 

 abstract sciences. There is a very wide distinction between the rela- 

 tion of a concept to the object of thought in mathematics, for instance, 

 and the corresponding relation between a representation, or concept 

 of a material object, to that object itself. In mathematics, as in all the 

 sciences which are conversant with single relations or groups of rela- 

 tions established (and, within the limits of the constitutive laws of the 

 mind, arbitrarily established) by the mind itself, all concepts are ex- 

 haustive in the sense that they imply, if they do not explicitly state, all 

 the properties belonging to the object of thought. Not only the con- 

 stituents of such an object, but also the laws of their interdependence, 

 being determined by the intellect, they may be strictly deduced, each 

 from the other. 1 Thus, a parabola is a line, every point in which is 

 equidistant from a fixed point and a given straight line : that is one 



1 The truth of the proposition that the system of forms and relations, whose discus- 

 sion constitutes the science of mathematics, is of purely subjective determination, does 

 not involve the assumption (erroneously attributed to Kant, who, on the very first page 

 of his " Critique of Pure Reason," expressly draws the distinction between the " begin- 

 ning of all knowledge with experience," and " the derivation of all knowledge from expe- 

 rience "), that the mind is furnished a priori with ready-made ideas or concepts ; nor is 

 it affected by the circumstance that these forms and relations are ultimately referable to 

 the facts of sensible experience. Mill's refusal to recognize this has betrayed him into 

 writing the extraordinary fifth chapter of the second book of his " System of Logic," 

 in which he questions — albeit falteringly — the necessary truth of the propositions of 

 geometry. The inevitable outcome of this is seen in the writings of Mr. Buckle, who not 

 only boldly asserts that there are no lines without breadth (he strangely forgets the thick- 

 ness), but also that the neglect of this breadth by the geometrician vitiates his conclu- 

 sions. His comfort is that the error, after all, is not very considerable. " Since, how- 

 ever," is his language (" History of Civilization in England," ii., 342, Appletons' edition), 

 " the breadth of the faintest line is so slight as to be incapable of measurement, except 

 by an instrument used under the microscope, it follows that the assumption, that there 



