Caklile. — On Neeessary Tricth. 651 



formed by extension of the sides, reversing one of them, and 

 placing it on top of the other.* Why not at once take up the 

 isosceles triangle itself, reverse it, and superimpose it on its 

 former self '? We shall then plainly have two triangles to coin- 

 pare, possessing all the characteristics of those in the Fourth 

 Proposition. 



Mr. Mill seems always to speak of the Imes and circles of 

 geometry as if they were specimens which we had picked up 

 in our rambles. He would have avoided much confusion of 

 thought if he had contemplated them as what they really are 

 — the lines and circles which we suppose ourselves to have just 

 drawn or to have just constructed. 



In the light of the above, let us glance again at what Mr. 

 Mill has to say in regai'd to the axiom that two straight lines 

 cannot enclose a space. The upholders of the necessity of 

 this proposition, he says accurately enough, uphold it on the 

 ground that we can see its truth by merely thinking of the 

 lines. The answer to this, he thinks, is to be found in the 

 capacity of geometrical forms for being painted in the imagina- 

 tion with a distinctness equal to reality. " Thus," he says, 

 •' although we cannot follow two diverging lines by the eye to 

 infinity, yet we know that if they begin to converge it must 

 be at a finite distance ; thither we can follow them in our 

 imagination, and satisfy ourselves that if they approach they 

 will not be straight, but curved." That is an accurate de- 

 scription of the process by which we satisfy ourselves that 

 two straight lines cannot enclose a space ; but we cannot help 

 asking, Is it a description of the process by which truths of 

 experience are learned? It is one of Mr. Mill's most cha- 

 racteristic doctrines, and one on which he repeatedly and 

 emphatically insists, that f " Whenever we form a new judg- 

 ment — judge a truth new to us — the judgment is not a recog- 

 nition of a relation between concepts, but of a succession, a 

 coexistence, or a similitude between facts." Whether we 

 admit that this applies to all new truths or not, we may cer- 

 tainly admit that it applies to all new truths of experience. 

 It is quite plain that a new truth of experience cannot be 

 learned by the process of comparing one of our concepts with 

 another. Yet it is precisely by doing this — by comparing our 

 concept of straight lines with our concept of lines that enclose 

 a space, and finding them incongruous — -that Mr. Mill describes 

 us as arriving at the truth that they never enclose a space. 

 His description, in fact, is not a description of the mental 

 process by which truths of experience are learned, but of that 



* That is, of course, taking the proof of Props. IV. and V. together, 

 as Mr. Mill does. 



t Ex. Ham. Phil., 4th ed., p. 426. 



