282 



THE POPULAR SCIENCE MONTHLY.— SUPPLEMENT. 



2. We experiment in our minds upon imagi- 

 nary straight lines. 



3. These imaginary straight lines exactly re- 

 semble the real ones. 



4. If these imaginary straight lines are not 

 perfectly straight, they will not enable us to 

 prove the truths of geometry. 



5. If they are perfectly straight, then the real 

 ones, which exactly resemble them, must be per- 

 fectly straight : ergo, perfectly straight lines do 

 exist. 



It would not be right to attribute such rea- 

 soning to Mill without fully substantiating the 

 statements. I must, therefore, ask the reader to 

 bear with me while I give somewhat full extracts 

 from the fifth chapter of the second book of the 

 " System of Logic." 



Previous to the publication of this " system," 

 it had been generally thought that the certainty 

 of geometrical and other mathematical truths 

 was a property not exclusively confined to these 

 truths, but nevertheless existent. Mill, however, 

 at the commencement of the chapter, altogether 

 calls in question this supposed certainty, and de- 

 scribes it as an illusion, in order to sustain which 

 it is necessary to suppose that those truths re- 

 late to, and express the properties of, purely im- 

 aginary objects. He proceeds : ' 



"It is acknowledged that the conclusions of 

 geometry are deduced, partly at least, from the so- 

 called definitions, and that those definitions are 

 assumed to he correct descriptions, as far as they 

 go, of the objects with which geometry is conver- 

 sant. Now, we have pointed out that, from a defi- 

 nition as such, no proposition, unless it be one 

 concerning the meaning of a word, can ever fol- 

 low, and that what apparently follows from a defi- 

 nition, follows in reality from an implied assump- 

 tion that there exists a real thing conformable 

 thereto. This assumption, in the case of the defi- 

 nitions of geometry, is false : 2 there exist no real 

 things exactly conformable to the definitions. 

 There exist no points without magnitude ; no 

 lines without breadth, nor perfectly straight ; no 

 circles with all their radii exactly equal, nor 

 squares with all their angles perfectly right. It 

 will, perhaps, be said that the assumption does 

 not extend to the actual, but only to the possible, 

 existence of such things. I answer that, accord- 

 ing to any test we have of possibility, they are 

 not even possible. Their existence, as far as we 

 can form any judgment, would seem to be incon- 



1 Book II., chapter v.. section 1, near the com- 

 mencement of the second paragraph. 



2 The word false occurs in the editions up to at 

 least the fifth edition. In the latest, or ninth edition, 

 I find the words, not strictly trite, substituted for false. 



sistent with the physical constitution of our planet 

 at least, if not of the universe." 



About the meaning of this statement no doubt 

 can arise. In the clearest possible language Mill 

 denies the existence of perfectly straight lines, so 

 far as any judgment can be formed, and this de- 

 nial extends, not only to the actual, but the pos- 

 sible, existence of such lines. He thinks that 

 they seem to be inconsistent with the physical con- 

 stitution of our planet, if not of the universe. 

 Under these circumstances, there naturally arises 

 the question, What does geometry treat ? A sci- 

 ence, as Mill goes on to remark, cannot be con- 

 versant with nonentities ; and as perfectly straight 

 lines and perfect circles, squares, and other fig- 

 ures, do not exist, geometry must treat such lines, 

 angles, and figures, as do exist, these apparently 

 being imperfect ones. The definitions of such 

 objects given by Euclid, and adopted by later . 

 geometers, must be regarded as some of our 

 first and most obvious generalizations concerning 

 those natural objects. But, then, as the lines are 

 never perfectly straight nor parallel, in reality, 

 the circles not perfectly round, and so on, the 

 truths deduced in geometry cannot accurately 

 apply to such existing things. Thus we arrive 

 at the necessary conclusion that the peculiar ac- 

 curacy attributed to geometrical truths is an illu- 

 sion. Mill himself clearly expresses this result : 1 



" The peculiar accuracy, supposed to be char- 

 acteristic of the first principles of geometry, thus 

 appears to be fictitious. The assertions on which 

 the reasonings of the science are founded, do not, 

 any more than in other sciences, exactly correspond 

 with the fact ; but we suppose that they do so, for 

 the sake of tracing the consequences which follow 

 from the supposition." 



So far Mill's statements are consistent enough. 

 He gives no evidence to support his confident as- 

 sertion that perfectly straight lines do not exist; 

 but with the actual truth of his opinion I am not 

 concerned. All that would be requisite to the 

 logician, as such, is that, having once adopted 

 the opinion, he should adhere to it, and admit 

 nothing which leads to an opposite conclusion. 



The question now arises in what way we ob' 

 tain our knowledge of the truths of geometry, 

 especially those very general truths called axioms. 

 Mill has no doubt whatever about the answer. 

 He says : 2 



" It remains to inquire, What is the ground of 

 our belief in axioms— what is the evidence on 



i Book II., chapter v., section 1, at the beginning 

 of the fourth paragraph. 



2 Same chapter, at the beginning of section 4. 



