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LIX. Notices respecting New Books. 



The Outlines of Quaternions. By Lieut.-Colonel II. \V. L. IIime. 

 (Longmans, Green, and Co. 1894.) 



TT is becoming more and more clearly recognized that a vector 

 -*- algebra is the proper mathematical method for treating physical 

 problems ; and it may confidently be said that in Hamilton's 

 Quaternions we find such a vector analysis of a most powerful 

 and flexible character. O'Brien, a contemporary of Hamilton, 

 constructed a vector analysis similar in many respects to Hamil- 

 ton's, but lacking the solidarity and flexibility which apparently 

 the quaternion alone can give. Recent attempts to follow where 

 O'Brien led have served but to bring out in stronger light the 

 transcendent superiority of the Hamiltonian system. This is said 

 advisedly ; for the recent vector analyses which have been pre- 

 sented to the world are all unconscious plagiarisms of O'Brien's. 



But there is another class of mathematicians who object to the 

 quaternion because it does not seem to fit in with the general drift 

 of modern analysis. It is undeniable that, on the purely mathe- 

 matical side, there has been little systematic advance in quater- 

 nions since Hamilton's own time. But is not this due rather to 

 the timidity of man than to any inherent weakness in the system ? 

 In the later chapters of Tait's treatise, which abound in original 

 and striking applications, there are hints and suggestions that 

 might easily be developed into memoirs and even treatises by the 

 purely analytical mind. The truth is that many a seemingly simple 

 quaternion expression or equation, when translated into ordinary 

 analytical symbols, assumes a form that mocks at the bravest 

 analyst; and certain quaternion equations, sufficiently interpret- 

 able, have never yet been expressed in Cartesian coordinates. 



If we except Hamilton's, Tait's, and McAulay's papers and 

 treatises, most of the literature of the subject has been a restate- 

 ment of known results. Lieut.-Colonel Hime's book confessedly 

 belongs to this class. It is intended for the student. A know- 

 ledge of geometry and algebra such as may easily be obtained in a 

 secondary school is sufficient to enable the learner to read intelli- 

 gently nearly the whole of the book. The processes of differen- 

 tiation are discussed in a short chapter; but they are not used to 

 any marked extent in the geometrical illustrations given further 

 on. Within the limits assigned, Colonel Hiine has given us a 

 book which cannot fail to be of service in popularising the study 

 of quaternions. Its value would have been enhanced had it 

 contained at the end of every chapter a selection of exercises for 

 the student's private work. As the author himself points out 

 in the closing paragraph, it is not in simple geometrical appli- 

 cations that the peculiar power of quaternions is displayed. It 

 is certain, however, that no candid mind can read the sections 

 devoted to spherical geometry and trigonometry without being 



