264 



Notices respecting JVew Boohs. 



by right lines regarded as having not only length but also direction, 

 and which, when so represented, satisfy the law of vector addition. 

 These representative lines are Hamilton's vectors. Analytically 

 there is only one sort. He speaks of the vector of a point, the 

 vector of a velocity, the vector of a force ; but he nowhere speaks 

 of the point, or the velocity, or the force, as being the vector. For 

 example, what is commonly called the vector product of two vectors 

 has various meanings according to the kind of quantities symbolized 

 by the vectors. Thus, if b be the vector of the force acting at 

 the point whose vector is a, then in Hamilton's view, ab is a 

 quaternion, whose vector part, Vab, represents the vector of the 

 moment of the force about the origin, and whose scalar part, Sab, 

 measures what Clausius subsequently called the virial. Again, if 

 a is the vector of the angular velocity, the expression Vab is the 

 vector of the velocity of the point whose vector is b. In their 

 misapprehension and misrepresentation of Hamilton's meaning 

 and method are not our authors confusing symbol and quantity ? 



On page 132 we read : " Quaternions is defective in that it is 

 not possible to multiply a vector by a quaternion. Thus, if the 

 relation of one vector to another is the quaternion q, it does not 

 necessarily follow that we can multiply a third vector by q and 

 obtain a vector." The second clause of the first sentence is simply 

 not true. As regards the second sentence, which is probably 

 meant to explain the first, why should we expect, or why should 

 we desire, that any quaternion multiplying any vector should give 

 a vector ? The truth is that a quaternion multiplying a vector 

 gives, in general, another quaternion. "Why should a calculus be 

 called defective because by the fundamental laws of its being it 

 cannot always satisfy a law which it satisfies occasionally and 

 under specially restricted conditions ? If the quaternion is 

 admitted at all, why desire to get rid of it by setting it in product 

 combination with any vector? In uniplanar quaternions this 

 law is satisfied ; but then uniplanar quaternions is so defective a 

 form of the calculus as to cease to be quaternions at all. 



"Whatever important additions this volume may make to mathe- 

 matical thought, the authors' claim to be the children of the light 

 is seriously menaced by their misquotation and misrepresentation 

 of the meaning of the words of the author they criticize, and by 

 their elaboration of an argument which is either irrelevant or a 

 begging of the question. Nevertheless they have a real appre- 

 ciation of the value of the quaternion and speak warmly of the 

 great services rendered by Hamilton. That they are also capable 

 of doing good things is shown in their discussion of the meaning 

 of Function, which occupies a comparatively small part of the 

 volume, and of which more will be given in the succeeding parts. 

 Here, we are glad to say, they give a definition, which in regard 

 to length and style recalls Euclid's famous definition of proportion. 

 In this discussion also, as in the discussion on variables, almost 

 every mathematician who is named is found sadly lacking in 

 accuracy ; but the ordinary mathematical reader will be cheered 

 by the thought that should he likewise fall into error he will still 

 be in the company of the immortals. 



