420 Notices respecting New Books. 



quaternions," was the somewhat curious remark once made by a 

 vector analyst. He might as well have claimed full knowledge of 

 Modern Greek and absolute ignorance of Ancient Greek. 



Looking back over half a century of scientific progress, we 

 cannot but be struck with the remarkable way in which Hamilton 

 prepared, a mathematical method peculiarly suitable to the physical 

 theories that find their source in Maxwell's great work. It is 

 from this point of view that the utility of quaternions is at present 

 to be judged. Whatever may be the merits of other similar 

 systems, one thing is incontrovertible. Hamilton, in his quaternions, 

 gave to the world the first complete symmetric and workable 

 system of analysis directly applicable to vector quantities. 

 O'Brien, one of Hamilton's brilliant contemporaries, constructed 

 a vector analysis which has been reproduced with modified 

 notations by Professor Willard Gibbs and Dr. Heaviside. It has 

 a certain superficial similarity to Hamilton's method, and, in so 

 far, is effective enough, but it lacks the homogeneity and solidarity 

 so characteristic of the latter. In this connection it is well to 

 bear in mind the verdict both of Tait and of Professor Joly, that 

 any endeavour to improve or modify the calculus of quaternions 

 should be made with extreme caution. 



Surely we are justified in hoping that, now that Hamilton's 

 Elements has been made accessible to all students, there will be a 

 real endeavour on the part both of mathematicians and physicists 

 to " get the quaternion mind." Eor certain special t7pes of 

 academic problems the quaternion method has no special fitness — 

 it " degenerates " simply into ordinary scalar coordinates ; but for 

 takiug a direct hold of the essential nature of a general problem, 

 whether of tridimensional geometry or of dynamics in the widest 

 sense of the term, the quaternion method is facile princeps. 

 Take as an example Hamilton's own treatment of the curvature 

 of surfaces, to which in one of his notes Professor Joly has given 

 a most interesting extension. 



We have already referred to the linear vector function, which 

 comes to the front at the very outset when the simplest quaternion 

 equation is considered. Hamilton developed its theory with great 

 power, and Tait discovered other curious and important properties 

 of it. One of Tait's latest utterances was that there was a vast 

 deal more in (j> than had yet been made out — its depths were not 

 yet sounded. Professor Joly devotes five of his notes to various 

 aspects of linear vector functions, and materially extends our 

 knowledge, especially in regard to their invariants. 



By far the longest of the notes in the Appendix is that on V- 

 Professor Joly's discussion is very instructive, covering a good 

 deal of the ground already familiar to readers of Tait's and 

 McAulay's books and papers, but developing along different lines. 

 A comparison of the modes in which these three quaternionists 

 attack the same problem shows well the extraordinary variety in 

 detail of the quaternion method. It is a variety which, like that 

 of organisms, springs naturally from the fundamental vital 

 principle of the whole. 



