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THE POPULAR SCIENCE MONTHLY.— SUPPLEMENT. 



that the result does not and cannot depend on 

 measurement at all, or else it would be affected 

 by the inaccuracy of that measurement. It is 

 obviously impossible, from inexact physical data, 

 to arrive at an exact result, and the computa- 

 tions of Mr. Shanks and other calculators are 

 founded on a priori considerations ; in fact, upon 

 considerations which have no necessary connec- 

 tions with geometry at all. The ratio in ques- 

 tion occurs as a natural constant in various 

 branches of mathematics, as, for instance, in the 

 theory of error, which has no necessary connec- 

 tion with the geometry of the circle. 



It is amusing to find, too, that Mill himself 

 happens to speak of this same ratio, in his " Ex- 

 amination of Hamilton," 1 and he there says, 

 " This attribute was discovered, and is now 

 known, as a result of reasoning." He says noth- 

 ing about measurement and comparison. What 

 has become, in this critical case, of the empirical 

 character of geometry which it was his great ob- 

 ject to establish ? A few lines further on (p. 

 372) he says that mathematicians could not have 

 found the ratio in question " until the long train 

 of difficult reasoning which culminated in the dis- 

 covery was complete." Now, we are certainly 

 dealing with a theorem of geometry, and if this 

 could have been solved by comparison and meas- 

 urement, why did mathematicians resort to this 

 long train of difficult reasoning ? 



I need hardly weary the reader by pointing 

 out that the same is true, not merely of many 

 other geometrical theorems, but of all. That the 

 square on the hypothenuse of a right-angled tri- 

 angle is exactly equal to the sum of the squares 

 on the other sides ; that the area of a cycloid is 

 exactly equal to three times the area of the de- 

 scribing circle ; that the surface of a sphere is ex- 

 actly four times that of any of its great circles ; 

 even that the three angles of a plane triangle 

 are exactly equal to two right angles — these and 

 thousands of other certain mathematical theorems 



1 Second edition, p. 371. 



cannot possibly be proved by measurement and 

 comparison. The absolute certainty and accuracy 

 of these truths can only be proved deductively. 

 Reasoning can carry a result to infinity — that is 

 to say, we can see that there is no possible limit 

 theoretically to the endless repetition of a pro- 

 cess. Thus it is found, in the 117th proposition 

 of Euclid's tenth book, that the side and diagonal 

 of a square are incommensurable. No quantity 

 however small, can be a sub-multiple of both ; or, 

 in other words, their greatest common measure 

 is an infinitely small quantity. It has also been 

 shown that the circumference and diameter of a 

 circle are incommensurable. Such results cannot 

 possibly be due to measurement. 



It may be well to remark that the expression 

 " a false empirical philosophy," which has been 

 used in this article, is not intended to imply that 

 all empirical philosophy is false. My meaning is 

 that the phase of empirical philosophy upheld by 

 Mill and the well-known members of his school, 

 is false. Experience, no doubt, supplies the ma- 

 terials of our knowledge, but in a far different 

 manner from that expounded by Mill. 



Here this inquiry must for the present be in- 

 terrupted. It has been shown that Mill under- 

 takes to explain the origin of our geometrical 

 knowledge on the ground of his so-called " Em- 

 pirical Philosophy," but that at every step he in- 

 volves himself in inextricable difficulties and self- 

 contradictions. It may be urged, indeed, that 

 the groundwork of geometry is a very slippery 

 subject, and forms a severe test for any kind of 

 philosophy. This may be quite true, but it is no 

 excuse for the way in which Mill has treated the 

 subject ; it is one thing to fail in explaining a 

 difficult matter : it is another thing to rush into 

 subjects and offer reckless opinions and argu- 

 ments, which, on minute analysis, are found to 

 have no coherence. This is what Mill has done, 

 and he has done it, not in the case of geometry 

 alone, but in almost every other point of logical 

 and metaphysical philosophy treated in his works. 

 — Contemporary Review. 



