﻿Notices respecting New Boohs. 155 



20 2B 2A 



y" ^^ a v =-l, 

 or what is perhaps more obvious, 



2(tt-C) 2(ti— B) 2(ti— A) 



y 13 a =+1; 



from which we have at once 



2(ir-C) 2(tt-B) -2(w— A) 



7T ^ 7T 7T 



y p =a » 



which, as Hamilton observes, contains at a glance the whole 

 doctrine of spherical triangles. But if, on the other hand, we are 

 to understand that the study of Quaternions is to be merely post- 

 poned, then Heretical Vector Analysis will not serve as an 

 introduction. No system which makes i 2 = + 1 and Sa/3= -f ah cos 6 

 can possibly serve as a preliminary to Quaternions ; and as there 

 is nothing intrinsically base in the sign minus, or in the notation 

 for a scalar, the reason for replacing Orthodox by Heretical Vector 

 Analysis disappears. 



Among the changes of quaternionic notation made by Mr. Heavi- 

 side we may further notice that he proposes (p. 157.) to denote 

 the tensor of the vector of a/3, i. e. TVa/3, by the symbol V a/3, 

 which surely is quite inappropriate and incapable of adoption. 



Again, he says (p. 135) " no amount of familiarity will make 

 Quaternions an easy subject." True — and necessarily true, more- 

 over, when we consider the wideness of the field of thought in 

 which Quaternions work. But neither is it all child's play in 

 Heretical Vector Analysis. The expression a/3 is no more the 

 same as /3a in this department than in Quaternions, and the same 

 careful picking of steps is necessary in both. 



Most students of Professor Tait's treatise on Quaternions will, 

 I think, be satisfied that Mr. Heaviside's criticisms on this work 

 are not well founded. " After muddling my way somehow through 

 the lamentable quaternionic Chapter II., the third chapter was 

 tolerably easy" (p. 174). Farther on (p. 289) the objection 

 becomes more definite : " There is the fundamental Chapter II. 

 wherein the rules for the multiplication of vectors are made to 

 depend upon the difficult mathematics of spherical conies, com- 

 bined with versors, quaternions, and metaphysics. " This language 

 is surely a little loose and unjust. One article in Tait's Chapter 

 II., marked with a star to indicate its postponement, and there- 

 fore unessential character for the reader, proves the simple 

 property of spherical conies involved. Moreover, to be accurate, 

 it is not the multiplication of vectors that is dealt with here, but 

 the multiplication, &c, of quaternions in general. There is a good 

 deal of difficult mathematics associated with spherical conies, as 

 any reader of Salmon's ' Geometry of Three Dimensions ' will see ; 

 but not one particle of it is involved in this short Article of Tait's 



