414 



NATURE 



[September 28, 191 1 



gress in electric wave signalling is indicated even in 

 the title, which is now so worded as tu cover the 

 subject of Hertzian telephony, at the time of the first 

 edition so much in its infancy as not to be worthy of 

 inclusion. To this subject Prof. Fleming now devotes 

 a short final chapter, in which he reviews briefly the 

 special difficulties in transmission and summarises the 

 present position. (In the last paragraph, by an obvious 

 oversight, telegraphy is written in place of telephony.) 

 Much of the volume has had to be revised on 

 account of the progress which has been made in all 

 directions. The author has acted wisely in curtailing 

 the historical portions and devoting himself mainly to 

 the explanation of the scientific principles on which 

 the art of wireless telegraphy is based, and on which 

 the numerous instruments now used are founded. 

 The purely historical side of wireless telegraphy is 

 now more or less a matter of the past : it has entered 

 into a period of development which if less sensational 

 is of more benefit to mankind. From the more or less 

 crude empirical art of ten years ago wireless tele- 

 graphy is now firmly based on a solid scientific founda- 

 tion, exact methods of measurement have been de- 

 veloped, and steady progress, not less rapid because 

 of its steadiness, is possible. Prof. Fleming's book 

 still deserves to rank as the best existing treatise on 

 the subject, at any rate in the English language, and 

 il the same industry is shown in the future in keeping 

 it up to date it should continue for long to hold this 

 premier position. M. S. 



Die Anwendung der stereogr aphis chen Projektion bet 

 kristallographischen Untersuchungen. By Prof. 

 H. E. Boeke. Pp. viii+ 58+ plate. (Berlin : 

 Gebfuder Borntraeger, 1911.) Price 2.60 marks. 

 The stereographic is the form of plane projection of 

 the sphere ordinarily in use in crystallographic work, 

 and during recent years it has come much into 

 vogue, not merely for showing the zonal relations 

 subsisting between the poles corresponding to the faces 

 of a crystal, but also as a means of checking tin 

 accuracy of the calculations involved in the goniometric 

 measurement of a crystal. Accordingly, various 

 methods by means of nets or protractors have been 

 devised to facilitate the use of the projection, many of 

 which have scarcely yet found their way into the 

 text-books. Penfield provided for English readers in a 

 of brilliant papers that appeared in The 

 American Journal of Science a clear and concise 

 account of the best and most practicable methods, 

 and, moreover, designed various diagrams to aid the 

 student in plotting the positions of the poles. 

 _ In the present volume Prof. Boeke aims at providing 

 similar privileges for German readers. He givi 

 clear account of the properties of the projection, and 

 discusses at some length its use as an aid in computa- 

 tion in the case of the several kinds of systems of 

 tlline symmetry, both geometrical and "graphical 

 proofs being given of the fundamental propositions. 

 The application of the projection to crystal drav 

 ■ "id tli.> determination of the optical charai 

 ' cplained. \ pocket in the cover contains one of riot. 

 Wulff's stereographic nets, which are graduated in 

 distance and azimuth referred to a pole in the equa- 

 torial /one for every other degree, the size of the 

 sphere being the same a- that selected b P 

 viz. 14 cm. 



The book is one that may he commended For the 



use of students of crystallography, hut it might advan- 



msly have included an adequate description oi the 



rties and use of the gnomonic projection which 



enl is merely alluded to in a brief paragraph, 



even though some slight alteration of the title would 



have been involved. 



2 1S7. VOL. 87] 



Quaternions as the Result of Algebraic Operations. 

 By Dr. A. L. Baker. Pp. ix + 92. (London: Con- 

 stable and Co., Ltd., 19 11.) Price 6s. net. 



In this book the author establishes the principles of 

 quaternions by the use of the six operations — addition, 

 subtraction, multiplication, division, reversion, and 

 mean reversion. By the introduction of reversion, we 

 pass from arithmetic to algebra. Complex functions 

 depend on the recognition of mean reversion, the 

 operands still being scalars ; and when the operands 

 are scalars and vectors, the method becomes general- 

 ised into quaternions. 



The key of the argument is the conception 

 reversion, that operation which twice repeated re- 

 verses the quantity operated on. There is nothing 

 new in this, but Dr. Baker applies the conception in 

 an unusual way to the representation of a scalar as a 

 sphere in space, which, as possessing perfect sym- 

 metry and therefore devoid of direction in space, is 

 the only available ideographic symbol for a scalar. He 

 finds that a mean reversed scalar is represented in all 

 its properties by a directed magnitude in space, that is, 

 by a vector. The algebraic representation of mean 

 reversal is, of course, V — 1, leading to the usual 

 Argand diagram ; and the same idea enters into the 

 constitution of any vector. The argument that »= — 1 

 may be accepted as sufficiently sound; but it may be 

 doubted if the rule for the product of perpendicular 

 vectors, viz. ij = k, &c, can be rigorously deduced on 

 the assumption that the operation of a vector a upon 

 a perpendicular vector /3 must be the same in kind, 

 but as far removed in detail from that which would 

 have been used li.<<\ $ 1 een parallel to a- We certainly 

 I a* fer Hamilton's own somewhat metaphysical argu- 

 ment. 



It is clear that Dr. Baker has no regard for the 

 views of those self-styled purists who deny that a 

 vector can have versor properties. Having established 

 the well-known i, j, k rules, he develops in a satis- 

 factory manner the important properties of quaternions, 

 find ends his discussion by a useful account of the 

 lini ar vector function. Students new to the method 

 will probably find the argument in the earlier chapters 

 difficult to follow; thereafter all is plain sailing. 



C. G. K. 



Lessons on Soil. By Dr. E. J. Russell. Pp. xv+ 132. 



(Cambridge University Press, 191 1.) Price is. 

 A course of lessons on soil provides an essential 

 sequence to the formation of school gardens if it is 

 desired to make the best use of the latter. Teachers 

 contemplating such a course are strongly recommended 

 to consult this excellent primer, in which Dr. Russell 

 presents a series of lessons evolved from practical 

 classes conducted for children in the higher standards 

 at an elementary school and in an intermediate form 

 at a secondary school. The earlier chapters contain 

 simple experiments for observing the properties of 

 clay, sand, and other soil constituents; pot cultures are 

 introduced to compare the food value of soil and 

 subsoil, as also the action of water in soils; methods 

 are described for detecting the presence of soil organ- 

 isms and for demonstrating the advantages of hoeing. 

 Finally, the practical bearing oi the lessons, which 

 it should be noted are confined to soil physics, is 

 indicated, not only with reference to agriculture, but 

 is they si rve to explain the aspects ol the country- 

 id other natural features such as the connection 

 between stream, ford, and village. The primer is not 

 Olllj practical and informative, but is designed to 

 arouse the inquiring instinct. It is the first volume of 

 1 ntemplated bj the Syndics of the 

 < I imbridgi I nivei sity Press. 



