4i8 



NATURE 



[September 2, 1897 



" Whatever can be divided, and has parts, possesses 

 some thinghood, and must, therefore, contain two 

 ultimate units, the whole namely, and the smallest 

 element possessing thinghood." 



The mathematical continuum contains no "smallest 

 element," and there is, accordingly, no necessity for a 

 thing which can be divided, and which has parts, to 

 contain such an element. This remark may perhaps 

 offer the key for the solution of the problem set by Mr. 

 Russell, the problem namely of determining the proper- 

 ties of a. form of externality. It is conceivable that, in 

 arriving at the axioms of projective geometry as con- 

 stituting a statement of these properties, he has assumed 

 the solution of a problem in the theory of majtifolds just 

 as Helmholtz, in arriving at the axiom of constant 

 space-curvature as necessary to congruence, assumed 

 the solution of a problem in the theory of groups. In 

 the latter case the weapon needed to attack the problem 

 was forged at a much later date by Lie. In the case of 

 Mr. Russell's problem the appropriate engine of dis- 

 covery is still undeveloped, the mathematics of the 

 manifold being at present limited to numerical aggre- 

 gates. No one has yet done for the science of space 

 what Dedekind did for the science of number. 



Mr. Russell is happier in his treatment of the axioms 

 of metrical geometry, and he has done real service to 

 mathematics in pointing out the essential weakness of 

 the Riemann-Helmholtz method. This method started 

 from the consideration of space as a numerical aggregate, 

 whose points are determined by coordinates, and then 

 sought for the condition of the possibility of measure- 

 ment. This condition was found in the uniformity of 

 the measure of space-curvature, and it was shown, on the 

 one hand, to imply the possibility of the straight line, 

 and, on the other, to be equivalent to the statement 

 that figures which can be brought to congruence are 

 equal. The argument, as Mr. Russell shows, really 

 involved a vicious circle. For space can be regarded as 

 a numerical aggregate only if we have the means of 

 assigning to points coordinates which have some spatial 

 import, and coordinates which have such import pre- 

 suppose measurement. The conclusion arrived at by 

 Mr. Russell is that the essential postulate of metrical 

 geometry is the axiom of free mobility, or the assertion 

 of the possibility of equal figures in different places, and 

 he has shown that the denial of this axiom would lead to 

 logical and philosophical absurdities. In this con- 

 nection it is only fair to Riemann to remember that 

 his essay " Ueber die Hypothese, welche der Geometric 

 zu Grunde liegen " remained unpublished until after his 

 death, a fact which points to the belief that he was not 

 satisfied with it. 



Leaving to philosophers by profession the task of 

 appreciating and criticising Mr. Russell's philosophy of 

 space, we may attempt to estimate the value of his book 

 for mathematics. It has already been pointed out that 

 in his criticism of Riemann and Helmholtz he has 

 brought forward considerations which are mathematically 

 important, and this is not the only place where he has 

 had occasion to point to examples of the special philo- 

 sophical vice of the mathematician, the tendency namely 

 to mistake the sign for the thing signified {cf. Couturat 

 " De rinfini mathematique," p. 331). To mathematicians 

 NO. 1453, VOL. 56] 



also his book should be interesting on account of its 

 acute and novel treatment of familiar topics : thus — pro- 

 jective coordinates are numbers arbitrarily but system- 

 atically assigned to points of space "like the numbers of 

 houses in a street" (p. 119). The ambiguity in the 

 definition of distance, which is unavoidable on projective 

 principles, does not show that distance is ambiguous, 

 but that projective methods cannot adequately deal with 

 distance (p. 35). The distinction between real and 

 imaginary points is the distinction between quantities 

 to which points correspond and quantities to which no 

 points correspond (p. 44). The book is throughout well 

 written, and is for the most part free from obscurity, 

 and it may be recommended to all who wish to have 

 clear ideas on matters of fundamental importance in 

 mathematics. A. E. H. L- 



OUR BOOK SHELF. 

 A Bibliography of Gilbert White, the Natural Historian 



and Antiquarian of Selborne. Ey Edward A. Martin, 



F.G.S. Pp. xiii -1- 274. (Westminster: The Roxburghe 



Press, 1897.) 

 There are many places in England prettier than the 

 little Hampshire village of Selborne, but none of them are 

 so full of interest to the outdoor naturalist as the home of 

 Gilbert White. Though more than a century has passed 

 away since the simple student of nature's ways in the 

 sleepy hollow of Selborne first gave the world the benefit 

 of his observations and impressions, the book in which 

 these notes are published is as fresh now as ever it was. 

 The reason for this is, it seems to the writer, that Gilbert 

 White was usually content to record facts as he found 

 them, and he did not regard nature from the point of view 

 of a pre-conceived theory. .A.ccurate observations of 

 natural objects and phenomena live for ever, but the ex- 

 planation of such facts must alter from time to time as 

 wider knowledge of the laws of nature is obtained. 



The success of White's " Selborne " has had two un- 

 fortunate effects : it has made every country clergyman 

 who can distinguish a martin from a swallow think that 

 he is a Gilbert White, and it has caused the literary world 

 to be deluged with so-called popular natural history 

 works, which are often more remarkable for thoughts 

 about nothing than for observations of something. We 

 can, however, forgive the authors of such rhapsodies for 

 inflicting their musings upon a busy world, because of the 

 real naturalists which White's " Selborne " has created. 



How large and widespread is the public to which the 

 book appeals may be seen by the volume before us. Mr. 

 Martin has found no less than seventy-three separate 

 editions of our natural history classic ; so the aggregate 

 number of volumes published must be very great. The 

 features of each of these editions are described in detail ; 

 hence Selbornites are now provided with interesting 

 particulars of the various volumes which have refreshed 

 the mind and administered to the intellectual enjoyment 

 of thousands of nature-lovers the world over. Mr. Martin 

 has not, however, confined his work to a mere list of 

 editions of the "Natural History of Selborne"; he 

 describes the naturalist himself and the main facts of 

 his life, points out some of the chief observations and dis- 

 coveries, gives a chapter on the village of Selborne, and 

 devotes another to White's old house, " The Wakes." 

 The work is thus more than a bibliography ; it is a guide 

 to the study of Gilbert White and his natural history, 

 and as such will be prized by many of his disciples. 



Reference is made on p. 71 to a suggestion of White's 

 that entomology required some " neat plates " for its ad- 

 vancement, and it is stated that the idea has been carried 

 out by the Science and Art Department. Surely there is 

 a mistake here. 



