JOHN STUART MILL'S PHILOSOPHY TESTED. 



283 



which they rest ? I answer, they are experimen- 

 tal truths ; generalizations from ohservation. The 

 proposition, two straight lines cannot inclose a 

 space — or, in other words, two straight lines which 

 have once met, do not meet again, but continue to 

 diverge — is an induction from the evidence of our 

 senses." 



This opinion, as Mill goes on to remark, runs 

 counter to a scientific prejudice of long standing 

 and great force, and there is probably no propo- 

 sition enunciated in the whole treatise for which 

 a more unfavorable reception was to be expected. 

 I think that the "scientific prejudice" still pre- 

 vails, but I am perfectly willing to agree with 

 Mill's demand that the opinion is entitled to be 

 judged, not by its novelty, but by the strength of 

 the arguments which are adduced in support of 

 it. These arguments are the subject of our in- 

 quiry. Mill proceeds to point out that the prop- 

 erties of parallel or intersecting straight lines are 

 apparent to us in almost every instant of our 

 lives. " We cannot look at any two straight lines 

 which intersect one another, without seeing that 

 from that point they continue to diverge more 

 and more." 1 Even Whewell, the chief opponent 

 of Mill's views, allowed that observation suggests 

 'be properties of geometrical figures ; but Mill is 

 not satisfied with this, and proceeds to controvert 

 the arguments by which Whewell and others have 

 attempted to show that experience cannot prove 

 the axiom. 



The chief difficulty is this : before we can as- 

 sure ourselves that two straight lines do not in- 

 close space, we must follow them to infinity. Mill 

 faces the difficulty with boldness and candor : 



"What says the axiom? That two straight 

 lines cannot inclose a space ; that after having once 

 intersected, if they are prolonged to infinity they 

 do not meet, but continue to diverge from one an- 

 other. How can this, in any single case, be proved 

 by actual observation ? We may follow the lines 

 to any distance we please ; but we cannot follow 

 them to infinity ; for aught our senses can testify, 

 they may, immediately beyond the farthest point 

 to which we have traced them, begin to approach, 

 and at last meet. Unless, therefore, we had some 

 other proof of the impossibility than observation 

 affords us, we should have no ground for believing 

 the axiom at all. 



" To these arguments, which I trust I cannot 

 be accused of understating, a satisfactory answer 

 will, I conceive, be found, if we advert to one of 

 the characteristic properties of geometrical forms 

 — their capacity of being painted in the imagina- 

 tion with a distinctness equal to reality : in other 



1 Same section, near the beginning of fourth para- 

 graph. 



words, the exact resemblance of our ideas of form 

 to the sensations which suggest them. This, in 

 the first place, enables us to make (at least with a 

 little practice) mental pictures of all possible com- 

 binations of lines and angles, which resemble the 

 realities quite as well as any which we could make 

 on paper ; and in the next place, make those pict- 

 ures just as fit subjects of geometrical experimenta- 

 tion as the realities themselves ; inasmuch as pict- 

 ures, if sufficiently accurate, exhibit of course all 

 the properties which would be manifested by the 

 realities at one given instant, and on simple inspec- 

 tion ; and in geometry we are concerned only with 

 such properties, and not with that which pictures 

 could not exhibit, the mutual action of bodies one 

 upon another. The foundations of geometry would 

 therefore be laid in direct experience, even if the 

 experiments (which in this case consist merely in 

 attentive contemplation) were practised solely upon 

 what we call our ideas, that is, upon the diagrams 

 in our minds, and not upon outward objects. For 

 in all systems of experimentation we take some 

 objects to serve as representatives of all which re- 

 semble them ; and in the present case the condi- 

 tions which qualify a real object to be the repre- 

 sentative of its class, are completely fulfilled by an 

 object existing only in our fancy. Without deny- 

 ing, therefore, the possibility of satisfying our- 

 selves that two straight lines cannot inclose a 

 space, by merely thinking of straight lines with- 

 out actually looking at them — I contend, that we 

 do not believe this truth on the ground of the 

 imaginary intuition simply, but because we know 

 that the imaginary lines exactly resemble real ones, 

 and that we may conclude from them to real ones 

 with quite as much certainty as we could conclude 

 from one real line to another. The conclusion, 

 therefore, is still an induction from observation." i 



I have been obliged to give this long extract 

 in full, because, unless the reader has it all fresh- 

 ly before him, he will scarcely accept my analy- 

 sis. In the first place, what are we to make of 

 Mill's previous statement that the axioms are 

 mductions from the evidence of our senses ? Mill 

 admits that, for aught our senses can testify, two 

 straight lines, although they have once met, may 

 again approach and intersect beyond the range 

 of our vision. " Unless, therefore, we had some 

 other proof of the impossibility than observa- 

 tion affords us, we should have no ground for be- 

 lieving the axiom at all." 2 Probably it would 

 not occur to most readers to inquire whether such 

 a statement is consistent with that made two or 

 three pages before, but on examination we find it 

 entirely inconsistent. Before, the axioms were 



1 Book II., chapter v., section 5. The passage oc- 

 curs in the second and third paragraphs. 

 8 End of the second paragraph. 



