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THE POPULAR SCIENCE MONTHLY.— SUPPLEMENT. 



Incidentally, it occurs to me to ask whether 

 Mill, in treating geometry, had not forgotten a 

 little sentence which sums up the conclusion of 

 the first section of his chapter on Names ? ' 

 Here he luminously discusses the question wheth- 

 er names are more properly said to be the names 

 of things, or of our ideas of things. After giv- 

 ing some reasons of apparent cogency, he con- 

 cludes emphatically in these words : " Names 1 

 therefore, shall always be spoken of in this work 

 as the names of things themselves, and not mere- 

 ly of our ideas of things." Here is really a diffi- 

 culty. Straight line is certainly a name, and yet 

 it can hardly be the name of a thing which is not 

 a straight line. It must then be the name either 

 of a real straight line, or of our idea of a straight 

 line. But Mill distinctly denied that there were 

 such things as straight lines, "in our planet at 

 least ; " hence the name (unless, indeed, it be the 

 name of lines in other planets) must be the name 

 of our ideas of straight lines. He promised ex- 

 pressly that names " in this work," that is, in the 

 " System of Logic," should always be spoken of 

 as the names of things themselves. It must have 

 been by oversight, then, that he forgot this em- 

 phatic promise in a later chapter of the same vol- 

 ume. We may excuse an accidental lapsus me- 

 morial, but a philosopher is unfortunate who 

 makes many such lapses in regard to the funda- 

 mental principles of his system. 



But let us overlook Mill's breach of promise, 

 and assume that we may properly employ ideal 

 experiments. We are told 2 that, though it is 

 impossible ocularly to follow lines " in their pro- 

 longation to infinity," yet this is not necessary. 

 " Without doing so we may know that if they 

 ever do meet, or if, after diverging from one an- 

 other, they begin again to approach, this must 



119), in the following words : " A process by which, I 

 will venture to affirm, not, a single truth ever was ar- 

 rived at, except truths of psychology, a science of 

 which Ideas or Conceptions are avowedly (along with 

 other mental phenomena) the subject-matter." These 

 words do not appear in the fifth and ninth editions. 

 Now, as Mill could not possibly pretend to include 

 geometry, a strictly physical science, under psychology, 

 we find him implying, or rather asserting, that not a 

 single truth ever was arrived at in geometry by the 

 very method of handling our ideas on which he de- 

 pends for the knowledge of the axioms of geometry. 

 The striking out of these words seems to indicate 

 that he had perceived the absolute conflict of his two 

 doctrines ; yet he maintains his opinion about the 

 cardinal error of handling ideas, and merely deletes a 

 too glaring inconsistency which results from it. 



1 Book I., chapter ii., section 1, near the end. 



a Book IT., chapter v., section 5, beginning of fourth 

 paragraph. 



take place not at an infinite, but at a finite dis- 

 tance. Supposing, therefore, such to be the case, 

 we can transport ourselves thither in imagina- 

 tion, and can frame a mental image of the ap- 

 pearance which one or both of the lines mu>t 

 present at that point, which we may rely on as 

 being precisely similar to the reality." Now, we 

 are also told • that " neither in Nature nor in the 

 human mind do there exist any objects exactly 

 corresponding to the definitions of geometry." 

 Not only are there no perfectly straight lines, but 

 there are not even lines without breadth. Mill 

 says, 2 " We cannot conceive a line without breadth ; 

 we can form no mental picture of such a line ; 

 all the lines which we have in our minds are lines 

 possessing breadth." Now I want to know what 

 Mill means by the prolongation of a line which 

 has thickness and is not straight. Let us examine 

 this question with some degree of care. 



In the first place, if the line, instead of being 

 length without breadth, according to Euclid's 

 definition, has thickness, it must be a wire ; if it 

 had had two dimensions without the third, it 

 would surely have been described as a surface, 

 not a line. But then I want to know how we are 

 to understand the prolongation of a wire. Is the 

 course of the wire to be defined by its surface or 

 by its central line, or by a line running deviously 

 within it? If we take the last, then, the line be- 

 ing devious and uncertain, its prolongation must 

 be undefined. If we take a certain central line, 

 then either this line has breadth or it has no 

 breadth ; if the former, all our difficulties recur ; 

 if the latter — Well, Mill denied that we could 

 form the idea of such a line. The same difficulty 

 applies to any line or lines upon the surface, or 

 to the surface itself regarded as a curved surface 

 without thickness. Unless, then, we can get rid 

 of thickness in some way or other, I feel unable 

 to understand what the prolongation of a line 

 means. 



But let us overlook this difficulty, and assume 

 that we have got Euclid's line — length without 

 breadth. In fact, Mill tells us 3 that "we can 

 reason about a line as if it had no breadth " be- 

 cause we have " the power, when a perception is 

 present to our senses, or a conception to our in- 

 tellects, of attending to a part only of that per- 

 ception or conception, instead of the whole." I 

 believe that this sentence supplies a good instance 



1 Book II., chapter v., section 1, beginning of third 

 paragraph. 



2 Same section, second paragraph, eleven lines 

 from end. 



3 Same paragraph, seventeen lines from end. 



