Nov. 2b, 18S5J 



NA TURE 



87 



spending to the various values of X. Tliis series we call 

 a " straight hne." But we here make no implication of 

 any geometrical character attached to the line. It is 

 merely a collocation of points, where each point is a 

 group of four symbols and nothing more. 



Each point on the line is thus correlated with a specific 

 value of the numerical magnitude X, and thus if we have 

 two points we may refer to them as the points X and /ix 

 respectively. 



As we are free from any geometrical meaning of our 

 symbols, we can only introduce the expression " distance 

 between two points" by defining precisely what is meant. 

 The distance will be a function of X and /i, whose form is 

 to be decided by the properties which we desire to attri- 

 bute to it. We may therefore select certain laws which 

 we desire this function shall obey, and then discover what 

 function will satisfy the conditions necessary. 



We may take -a hint from our familiar geometry as 

 to the conditions to be imposed upon the distance function. 

 If A, B, C be three points upon a straight line, then there 

 is no more fundamental notion of distance than that 

 implied in the equation 



AB + BC=AC. 

 We shall accordingly insist that our distance function 

 shall be obedient to this law (which we may call Law I.). 

 If, therefore, X, /i, c be three points upon our straight line, 

 and if /(X, /j) denote the distance from X to ji, then we 

 find 



/(X,^)+/(;x, .)=/(\, .), 

 and as jx must disappear from this equation we have 



The first step in the construction of an appropriate dis- 

 tance function has thus been taken, but we have still a 

 wide range of indeterminateness, for </)(X) may of course 

 be any conceivable function of X. It will therefore be 

 competent for us to select some additional law and to 

 insist upon obedience to it also. 



Again we revert to our familiar geometry for a sugges- 

 tion. In that geometry it is assuredly obvious that the 

 distance between two points cannot be zero unless the 

 two points are coincident. Trite as this condition may 

 appear, it is yet sufficient to clear every trace of indeter- 

 minateness from the form of <f> : we shall term this 

 Law II. 



Combining Laws I. and II. it will be easy to show 

 that if P be a point on the line then there can only be 

 one point, Q, on the line at a given distance from P, for, 

 suppose that there was a second point, Q', then we have, 

 by Law I., 



PQ + QQ' = PQ ; 



but if P(2 be equal to PQ, then 



QQ = o, 

 from which, by Law II., we see that Q and (2 must be 

 identical. 



If the point P be defined by X, and the point Q at a 

 given distance therefrom be defined by /i, theia the relation 

 between X and ^ must be of the one-to-one type. The 

 distance given, we must therefore have some equation of 

 the form 



AXfx. -{■ B\-\- Cii-\- D = o. 

 Any constant values for A, B, C, D will be consistent 

 with the conditions, but we can without loss of generality 

 simplify this equation. If we make X = /i we obtain the 

 quadratic 



^X--+(5+C)X+Z? = o. 



We thus see that there are in general two critical 

 points on the line corresponding to the roots of this 

 equation. If we choose these two points for x and y, 

 which is of course possible without sacrifice of generality, 

 the roots of this equation should be o and 00, or in other 

 words the constants A and D must be each zero. We 



thus see that by an appropriate choice of the fundamental 

 points the relation between X and ji assumes the simple 

 type 



B\ + C> = o, 

 or, finally, 



X = kij., 



in which /i is a function of the particular distance between 

 X and jx. 



We have, however, seen that the distance is also to be 

 expressed in the form 



0(X) - 4>{^). 



This must therefore be a function of k, that is, of X -^ ^, 

 and thus we have 



c^(X) - ,^(m) = /^(-). 



From this equation the particular value of the distance 

 has disappeared. It must therefore be true for all values 

 of X and all values of /x. It must remain true if differen- 

 tiated either with respect to X or /i. We therefore have 



<^'(X) 



^jX- 





\^/ 



whence 



but as X and i>. are independent this requires 

 H 



x' 



<^'(X) 



4>{\) = H log X, 

 whence finally we see that the distance between the two 

 points X and /j. is 



//log^, 

 A' 

 where i/ is a constant. 



There seems nothing arbitrary in this process. We 

 have set out with the two laws I. and II., and we have 

 without any other assumption been conducted to the 

 logarithmic conception of distance which lies at the 

 foundation of the elliptic geometry. We might almost be 

 tempted to ask how any other conception of distance can 

 be reasonable. The two laws assumed are obviously true 

 on any intelligible conception of distance, and yet they 

 conduct to the logarithmic expression and apparently to 

 nothing else. 



It remains to shovv' where the assumption made in 

 ordinary geometry comes in. Hitherto we have not 

 restricted the generality of the constants A, B, C, D 

 which enter into the equation between X and jx. Euclid, 

 however, demands that the expression 

 {B+Cy - 4AC 

 shall be equal to zero. This has the effect of rendering 

 the quadratic equation a perfect square. The logarithmic 

 theory is accordingly evanescent, and we have to resort 

 to the specialised conception of ordinary distance. 



Robert S. Ball 



NOTES 

 The Council of the Royal Society at their last meeting 

 awarded the Copley Medal to Auguste Kekule, of Bonn (For. 

 Mem.R.S.), for his researches in organic chemistry, and the 

 Davy Medal to Jean Servais Stas, of Brussels (For.Mem.R.S.), 

 for his researches on the atomic weights. At the same meeting, 

 Prof. D. E. Hughes, F.R.S., and Prof. E. Ray Lankester, 

 F. R.S., were nominated for the Royal Medals — the former 

 eminent for his electric researches, and the latter for his services 

 to embryology and animal morphology. Her Majesty has since 

 signified her approval of these nomination?. 



