

Notices respecting New Books. 263 



"A right line considered as having not only length but also direct >on, 

 is said to be a vector. " But this, they say, is obviously inadequate, 

 44 since a linear velocity at a point is a vector, and of course a velocity 

 is not a line, further, not all straight lines possess the attributes 

 requisite to vectors." These reasons are quite irrelevant. For 

 Hamilton never said that a velocity was a line in their assumed 

 sense of the term ; nor did he even say that a velocity was a 

 vector. Also Hamilton never said that all straight lines, in any 

 sense in which other writers might be pleased to regard them, 

 were vectors. Nor does his definition imply it. He said that 

 right lines considered as having not only length but also direction 

 were vectors — a totally different statemeut. Obviously, however, 

 our critics are psychologically incapable of understanding the 

 plain meaning of Hamilton's words ; for two pages further on 

 they make the following remarkable utterance : " Even so high an 

 authority as Hamilton states that a right line is a vector; a view 

 which is quite untenable." Hamilton certaiuly never made such a 

 statement, except when he carefully guarded himself by saying 

 that the right line was considered as having not only length but also 

 direction. Following this second emasculated statement wrongly 

 ascribed to Hamilton, the authors go on to declare that a right 

 line remains the same so long as its length remains constant, even 

 though it rotate like the spoke of a wheel into a new position. 

 Now — so runs the argument — the vector of this rotating " right 

 line " does not remain the same. Therefore, they conclude, this 

 " right line " cannot be a vector. But whoever said it was ? 

 Their statement that Hamilton said so is absolutely ludicrous. 

 In the course of two pages these critics of Hamilton, with 

 wondrous wisdom, drop out of the definition certain all-important 

 words, and then, with superb simplicity, develop their attack by 

 first giving to the term right line a meaning which, as their own 

 previous words prove, was not Hamilton's meaning. They commit 

 the double sin of misquotation and bad logic. 



We are nowhere told distinctly what they themselves consider 

 a vector to be. They seem to follow the ordinary custom of 

 making it apply as a class name to certain very different physical 

 quantities, such as displacement, velocity, acceleration, iforce» 

 angular velocity, moment of momentum, aud the like — a use which 

 compels them to talk profoundly of vectors of different sorts. 

 For example, on page 76 they say: "In Quaternions, vectors of 

 described straight lines and of points constitute one sort of 

 applicate quantities ; linear velocities at points, another ; linear 

 accelerations at points, a third ; and so on." In Hamilton's 

 Quaternions this is not so. Must we repeat it ? A vector is a 

 right line considered as having not only length but also direction. 

 This is the opening statement of the Elements of Quaternion^ 

 and is the foundation of the whole calculus. Vectors, so defined^ 

 are found to satisfy an importaut law, the law of vector addition. 

 Hamilton does not speak of a velocity or an acceleration as being 

 a vector. When he applies his calculus to dynamics and physics 

 he finds certain quantities which can be represented or symbolized 



