412 



NA TURE 



[March 2, 1S99 



point of view, a complex number of the form (A), in which 

 j,y, k, w obey the relations (i.) and (ii.) with w'^ = o in 

 the place of u-' = — i. From the geometrical point of 

 view it may be regarded as the ratio of tuo velocity 

 systems in ordinary space. 



The shorthand system expounded by the author for 

 dealing with this algebra follows closely the lines of that 

 used in quaternions, but is, as might be expected, con- 

 siderably more complicated. Since a is permutable with 

 i,j, and k, the octonian (A) may be written in the form 

 a', + .r,/ + _)',/ + jj ,{• + a{7c'.^ + x.i + y.J + z.j!), 

 or, ^1 + o)jf^„ 



where q-^ and q.^ are two quaternions. The phraseology 

 and notation adopted turn largely on this division of 

 an octonion into two parts, .^n octonion for which 

 both Ti'i and w.^ are zero is called a motor. Like a 

 vector in quaternions it may be regarded either as an 

 operation or as an object to be operated on. From the 

 latter point of view it is, when interpreted kinematically, 

 a velocity-system. Corresponding to the linear and 

 vector function of a vector in quaternions, there is 

 here the linear and motor function of a motor ; i.e. 

 from the algebraical standpoint a matrix of six rows 

 and columns. A self-conjugate function is defined 

 somewhat as in quaternions, but it is not the case 

 that a symmetric matrix corresponds to a self-con- 

 jugate function. The least convincing part of the 

 book is, perhaps, that in which the author extends to 

 octonions some of the conceptions of the Ausdehmings- 

 lehre. The inner product of five and of six motors are 

 defined so that the first is a motor and the other a 

 number; and it is apparently implied that the inner product 

 of any given number of motors is a definite quantity of 

 some kind. No expressions, however, for such a product 

 are given except in the two special cases mentioned; and 

 the idea involved does not seem to be utilised in the 

 chapter of applications with which the book ends. This 

 chapter consists of more or less well-known results in- 

 vestigated or expressed in the octonion notation ; and it 

 inevitably suggests the question : Is the method one 

 which will lend itself to the purposes of research ? It is 

 of course too soon to say. In the meantime it is clear 

 that octonions cannot be used, any more than quaternions, 

 for the purposes of numerical calculations ; and that the 

 newer method cannot, from the very nature of the case, 

 claim the beauty and symmetry that are such dis- 

 tinguishing marks of quaternion analysis. 



W. BURNSIDK. 



THE ALPINE GUIDE. 

 Ball's Alpine Guide: lite IVeslern Alps. New edition, 

 reconstructed and revised on behalf of the Alpine 

 Club by W. A. B. Coolidge. Pp. xlix -I- 612. (London: 

 Longmans, (jreen, and Co., 1898.) 



THE issue of the first volume of the new edition of 

 "Hall's Alpine (luide" will be welcomed by all 

 those who seek their relaxation in the Alps. Though it 

 is nearly ten years since this enterprise was taken in 

 hand by the Alpine Club as a fitting memorial to John 

 Ball, its first President, the character of the revision is 

 so thorough, and the incorporated matter has been so 

 NO. I 53 I, VOL. 59] 



judiciously assimilated with the original, that any delay 

 will be readily forgiven. The task of producing this 

 edition has fallen to Mr. W. A. I!. Coolidge ; and that 

 this has been to him a labour of love, carried out on the 

 lines laid down by Ball, there can be no question. What 

 distinguished " Ball's Guide" from all others, and rendered 

 it pre-eminent, was that in it one had, as it were, a com- 

 panion leading one through the finest scenery in Europe, 

 a man of culture with a true feeling for the mountains, 

 and an accomplished naturalist who found an especial 

 fascination in topography and plant distribution. Nor 

 has Mr. Coolidge fallen from that high plane. With a 

 skill so admirable has he interwoven in the original 

 fabric the results of Alpine exploration of the last quarter 

 of a century, that we have a book absolutely up to date 

 and yet without any indication of patchwork or com- 

 posite origin. .\nd for this he merits the sincere 

 gratitude of mountain lovers. 



The present volume deals with the Western Alps, i.e. 

 the Maritime, Cottian (including Uauphine) and Graian 

 -■Mps, the range of Mont Blanc, the Central and Eastern 

 Pennines. It is divided into six chapters, dealing with 

 the six main areas or ranges, and these fall into a 

 varying number of sections (there are twenty-one in all) 

 treating of the districts. As examples of the districts 

 may be mentioned the Pelvoux, Grand Paradis, Grand 

 Combin, and Monte Rosa districts. Each district has 

 its introduction, in which especially matters relating to 

 the topography of the district in question are lucidly set 

 forth. "Ball's tluide" being written by an Alpine 

 wanderer for wanderers, the subject-matter is unfolded 

 in the form of routes, i.e intersecting lines selected with 

 a view to serve as a scaffolding for the descriptive 

 matter. In this way some ten or twelve routes serve to 

 exhaust a district. All matter off the line of journey in 

 any route — as, for instance, an account of a peak or some 

 adjacent valley — is enclosed in heavy square brackets. 

 This system works out very well in practice, as indeed it 

 should, seeing that about one-third of the book consists 

 of matter thus enclosed. For the convenience of such 

 as are not wanderers, but who remain for days or weeks 

 at some centre, at least one "route" in each district is 

 largely occupied with notices of the various expeditions 

 that may be conveniently made from its chief centre. 

 And in this matter Mr. Coolidge has kept pace with 

 present-day requirements, for in addition to Zermatt 

 and Chamonix we find the newer centres of .AroUa, \'al 

 d'ls^re, Cogne, La Berarde, &c., thus treated. In this 

 way the country in each district is very minutely worked 

 out, but concisely and without tedious elaboration. Mr. 

 Coolidge exercises a wise selection in the peaks which he 

 describes : he gives detail when detail is desirable ; whilst 

 subordinate peaks, unless of topographical or special 

 mountaineering importance, are merely named. Though 

 there is no pretence to the full detail of a "Climbers' 

 Guide," practically all interesting routes are sketched in 

 or suggested. There is no scamping of out-of-the-way 

 spots. Take, for instance, Grand Paradis District, 

 Route E., Cogne to Pont Canavese by the \'al Soana. 

 Here, within the limits of two pages, we find mentioned 

 all the essential features of a tract of country that it 

 could not take less than three weeks to explore, and these 

 stated briefly, clearly and correctly. Vet it is doubtful 



