NATURE 



[November 22, 1894 



QUA TERN IONS. 

 Anwciidiing dcr Ouitternionen auf die Geometric. \on 



Dr. P. Molenbroek. (Leyden : E. J. Brill, 1S93.) 

 The Outlines of Quaternions. By Lieut. -Colonel 



H. W. L. Hime. (London: Longmans, Green, and 



Co., 1S94.) 



IN these books we have evidence of the growing 'de- 

 mand for quaternion literature. Dr. Molenbroek's 

 work is the promised sequel to his first volume on the 

 Theory of < Quaternions, and contains many admirable 

 e.>camples of the application of the method to geometry. 

 .•Ul who are familiar with Hamilton's and Tait's classics 

 on the subject will recognise many of these examples as 

 old friends, taken almost verbatim from their original 

 sources. In not a few of the applications, however, Ur. 

 Molenbroek ventures into fresh fields, and shows that he 

 can use quaternions with ease and power. It is interest- 

 ing to notice the occasional eti'ective use of the conjugate 

 quaternion, an invention of the great master which is 

 apt to be lost sight of after the foundations of the 

 calculus have been laid. The treatment throughout is 

 on the familiar Hamiltonian lines, the author's aim being 

 development and not fancied improvements. The book 

 consists of six chapters, in which are taken up — to name 

 a few of the most important applications — spherical 

 trigonometry, the plane and sphere, quadric surfaces, 

 surfaces in general, curves in space, and the theory of 

 rectilinear rays. The elementary properties of the 

 remarkable operator V> and 'he integration of partial 

 differential equations of the first and second orders, are 

 discussed as part of the general theory of surfaces. In 

 the same chapter. Dr. Molenbroek, by means of two new 

 differentiating operators, obtains a simple symbolic re- 

 presentation for the first, second, and higher polars of a 

 point with regard to a given surface. These remarks 

 will indicate sufficiently the scope of a work which, 

 though not altogether above criticism in minor details, 

 is a distinct addition to quaternion literature, and 

 deserves a wide circulation. 



Colonel Hime's work is a much more modest produc- 

 tion; being intended for the mere beginner. In general 

 scope it might be compared to the first nine chapters of 

 Kelland and Tait's " Introduction to Quaternions. ' The 

 book contains many good examples in the simpler applica- 

 tions of quaternions to the geometry of triangle, plane, 

 sphere, conic section, cone, Ovc, but it is less satisfactory in 

 the exposition of the fundamental principles of the cal- 

 culus. For example, the identification of unit vector and 

 right versor is stated, but the reason for this identification 

 is nowhere distinctly given. Again, the truth that the 

 familiar Hamiltonian symbols /' j k may be regarded as 

 in a sense im.ti^iitaries, because i- = J- = k- — — I, is 

 supposed to lead to the equation 

 i=j = k = \/ - I = - i = -j= - k (Eq. 8, p. 40). 

 This seems to be playing sad havoc with one's very 

 definitions. Then on page 76 we find what is virtually 

 the equation Aji 'd = y transformed into (i- — y^, the 

 non-commutative principle being wholly ignored, and in 

 consequence a quaternion and a scalar equated ! These 

 errors, especially the latter, are very surprising in a book 

 whose author is a true disciple of Hamilton. Of minor 

 blemishes we might refer to the appeal to Cartesian 

 NO. 1308, VOL. 51] 



expansions in order to demonstrate (?) the associative 

 principle in multiplication. Xor do we quite understand 

 Colonel Hime's system of referring to authorities. For 

 example, why should Prof. Hardy be quoted as the 

 authority for the statement that every versor may be 

 represented by a power of a unit vector : for is it not all 

 in Hamilton (see " Ele.nents,'' § 309) .' .Again, Dr. ()dstrcil 

 is credited with a proof that the three angles of a plane 

 triangle are together equal to two right angles, the proof 

 being an obvious particular case of Hamilton's remark- 

 able expression for the product of the versor arcs of a 

 spherical triangle. But surely the theorem regarding the 

 angles of a plane triangle underlies the fundamental 

 properties of quaternions and versors ; so that the sup- 

 posed proof is really reasoning in a circle. Dr. Odstrcil 

 is worthy of higher praise than this. These blemishes 

 apart, however, and leaving out of account the two errors 

 already noted, we find in Colonel Himes book a ser- 

 viceable exposition of the elementary applications of 

 quaternions. A careful study of its pages will go far to fit 

 the reader for the arduous task of grappling with the 

 higher and more characteristic developments to be found 

 in the writings of the masters of the quaternion calculus. 



OUR BOOK SHELF. 



Sir Victor Brooke, Sportsman and Naturalist. By O. 



L. Stephen. (London: John Murray, 1S94.) 

 The late Sir Victor Brooke was an excellent example 

 of a combination of sportsman and naturalist. In this 

 book his life as a sportsman predominates; but a chapter 

 on his researches in natural history, by Sir William 

 Flower, shows that he possessed the keenness of 

 observation required in a man of science. His most 

 important contribution to science was an exhaustive 

 paper, published in the Proceedings of the Zoological 

 .Society, the subject being the classification of the 

 Cen'ida:. At one time he was an enthusiastic student 

 of natural history, but the state of Lady Brooke's 

 health having compelled him to live out of England 

 for the greater part of the year, he could not con- 

 veniently carry on his researches. From about iSSo 

 his life was chiefly devoted to foreign travel and sport. 

 The extracts from his letters and journals are full 

 of stirring adventures, and contain some interesting 

 observations on animal life and habits. Mr. Stephen 

 prefaces these extracts with a memoir of his dead friend. 

 The book is beautifully printed, and is illustrated by 

 ten fine plates. It appeals particularly to those who 

 were acquainted with Sir Victor Brooke, and who ad- 

 mired his character ; nevertheless, such of the public 

 as read it will find the contents interesting. 



A Text-book 0/ Dynamics. A Text-liook of Statics. By 

 William Briggs and G. H. Bryan. (The University 

 Tutorial Series.) (London: W. B. Clive, 1894.) 

 These books belong to the elementary class, and a 

 perusal of them shows that they will prove excellent 

 additions to this series of useful text-books. 



In both the authors have assumed little or no know- 

 ledge of trigonometry, and they have been written 

 so that either may be read first. The treatment is con- 

 spicuous for its clearness and conciseness, and is all that 

 a student about to enter a course could desire. The 

 figures are neatly drawn, and many new ones are 

 noticeable in the latter book. 



Notwithstanding the fact that these text-books are 

 published to meet the requirements of candidates for 

 certain examinations, they may still be used by others, 

 who are making themselves acquainted with these sub- 



