JOEX STUART MILL'S PHILOSOPHY TESTED. 



2S7 



of a non sequitur, being in conflict with the sen- 

 tence which immediately follows. Mill holds that 

 we learn the properties of lines by experimenta- 

 tion on ideas in the mind ; these ideas must surely 

 be conceived, and they cannot be conceived with- 

 out thickness. Unless, then, the reasoning about 

 a line is quite a different process from experiment- 

 ing, I fail to make the sentences hold together at 

 all. If, on the other hand, we can reason about 

 lines without breadth, but can only experiment 

 on thick lines, would it not be much better to 

 stick to the reasoning process, whatever it may be, 

 and drop the mental experimentation altogether ? 

 But let that pass. Suppose that, in one way 

 or other, we manage to attend only to the direc- 

 tion of the line, not its thickness. Now, the line 

 cannot be a straight line, because Mill tells us 

 that neither in Nature nor in the human mind is 

 there anything answering to the definitions of 

 geometry, and the second definition of Euclid de- 

 fines a straight line. If not straight, what is it? 

 Crooked, I presume. What, then, are we to un- 

 derstand by the prolongation of a crooked line ? 

 If the crooked line is made up of various portions 

 of line tending in different directions, if, in short, 

 it be a zigzag line, of course we cannot prolong ii 

 in all those directions at once, nor even in any 

 two directions, however slightly divergent. Let 

 us adopt, then, the last bit of line as our guide. 

 If this bit be perfectly straight, there is no diffi- 

 culty in saying what the prolongation will be. 

 But then Mill denied that there could be such a 

 bit of straight line; for the length of the bit 

 could scarcely have any relevance in a question 

 of this sort. If not a straight line, it may yet be 

 a piece of an ellipse, parabola, cycloid, or some 

 other mathematical curve. But if a piece of an 

 ellipse, do we mean a piece of a perfect ellipse ? 

 In that case one of the definitions of geometry 

 has something answering to it in the mind at 

 least ; and if we conceive the more complicated 

 mathematical curves, surely we can conceive the 

 straight line, the most simple of curves. But if 

 these pieces of line are not perfect curves, that 

 is, do not fulfill definite mathematical laws, what 

 are they ? If they also are crooked, and made 

 up of fragments of other lines and curves, all the 

 difficulty comes over again. Apparently, then, 

 we are driven to the conception of a line, no por- 

 tion of which, however small, follows any definite 

 mathematical law whatever. For if any portion 

 has a definite law, the last portion may as well 

 be supposed to be that portion; then we can 

 prolong it in accordance with that law, and the 

 result is a perfect mathematical line or curve, of 



which Mill denied the existence either in Nature 

 or in the human mind. We are driven, then, to 

 the final result that no portion of any line follows 

 any mathematical law whatever. Each line must 

 follow its own sweet will. What, then, are we to 

 understand by the prolongation of such a line ? 

 Surely the whole thing is reduced to the absurd. 



But in this inquiry we must be patient. Let 

 us forget the non-existence of straight lines, the 

 cardinal error of mental experimentation, and 

 whatever little oversights we have yet fallen 

 upon. Let us suppose there really are geomet- 

 rical figures which we can treat in the manner 

 of " a strictly physical science," such as geometry 

 seems to be. What lessons can we draw from 

 Mill's Logic as to the mode of treating the fig- 

 ures ? A plain answer is contained in the follow- 

 ing extract from the second volume : 



"Every theorem in geometry," he says, 1 "is 

 a law of external Nature, and might have been as- 

 certained by generalizing from observation and 

 experiment, which in this case resolve themselves 

 into comparison and measurement." 



Here we are plainly told that the solution of 

 every theorem in geometry may be accomplished 

 by a process of which measurement is, to say the 

 least, a necessary element. No doubt a good 

 deal turns upon the word " generalizing," by 

 which I believe Mill to mean that what is true of 

 the figure measured will be true of all like figures 

 in general Give him, however, the benefit of 

 the doubt, and suppose that, after measuring, we 

 are to apply some process of reasoning before 

 deciding on the properties of our figure. Still it 

 is plain that, if our measurements are not accu- 

 rate, we cannot attain to perfect or unlimited ac- 

 curacy in our results, supposing that they depend 

 upon the data given by measurement. Now, I wish 

 to know how Mill would ascertain by generalizing 

 from comparison and measurement that the ratio 

 of the diameter and circumference of a circle is 

 that of one to 3.141592653589'79323846. . . . 



Some years ago I made an actual trial with a 

 pair of compasses and a sheet of paper to ap- 

 proximate to this ratio, and, with the utmost care, 

 I could not come nearer than one part in 540. 

 Yet Mr. W. Shanks has given the value of this 

 ratio to the extent of T07 places of decimals, 2 

 and it is a question of mere labor of computation 

 to carry it to any greater length. It is obvious 



1 Book III., chapter xxiv., section 7, beginning of 

 second paragraph. 



a Proceedings of the Royal Society (1872-'73), vol 

 xxi., p. 319. 



