﻿154 Notices respecting New Boohs. 



the equation ^ 2 =— 1 if we read it " i repeated equals — 1," i. e., 

 equals complete reversal ; if we read it " i squared = — 1," our early 

 algebraic feelings are shocked — Mr. Heaviside is amazed at it 

 (p. 303) ! 



Hence with the anti-quaternionist Sa/3= + a6cos0. Again, 

 with Hamilton the vector of the product a/3, i. e. Ya/3, follows at 

 once naturally as a vector perpendicular to the plane of a and j3, 



of length a&sin0, because a/3=-w[, which is the operation of 



P 

 converting /3 -1 into a. Now this is done by rotation round the 

 perpendicular to their plane ; and as a vector stands for rotation — so 

 that e m signifies rotation through m right angles round the line 

 coinciding with the unit vector e — it is clear that Ya/3 is perpen- 

 dicular to both a and /3, and in a definite sense. "With the anti- 

 quaternionists Ya/3 does not follow naturally at all ; it is perfectly 

 arbitrary — provided, of course, that there is something at once 

 definite and consistent in its representation. It might, for 



example, be very well defined as being of length r tan 0, and as 



lying in the plane of the bisector of and the perpendicular to the 

 plane of a and /3, and inclined at (say) to this perpendicular. 

 However, the anti-quaternionists agree to make Yafi exactly what 

 Hamilton made it, and we need not stop to inquire whether any 

 adequate reasons for this can be found from their point of view. 



Hence, then, results Mr. Heaviside's system of Vector Analysis, 

 which we may call Heretical Vector Analysis, without necessarily 

 implying any censure, since what is orthodox is sometimes false 

 and bad, and what is heretical true and good. 



It is not very easy to discover whether Mr. Heaviside's opposi- 

 tion to Quaternions is absolute or not ; that is, whether he thinks 

 that the subject is one which may be studied as a branch of pure 

 mathematics by advanced speculators, but postponed to the study 

 of Vector Analysis. If, on the one hand, he thinks that the 

 Hamiltonian system (involving the versor property of vectors, &c.) 

 is not worthy of study, the quaternionists might, perhaps, show 

 what the system has achieved or is capable of achieving ; but even 

 this is not incumbent on them, for they may justify the study 

 simply for its value as a mental exercise — as other branches of pure 

 mathematics are accepted. The problem of disentangling the 

 quateruion q from the equation q 2 + aq + b=0, where a and b are 

 given quaternions, is a perfectly legitimate exercise of thought, 

 even though it has no application to telephony or to dynamo 

 machines. The utility of associating the versor property with 

 vectors is illustrated in a striking way in a fundamental case by 

 Hamilton (' Elements of Quaternions,' p. 370) thus : — If a, p, y are 

 the vectors from the centre of a sphere of unit radius to any three 

 points, A, B, C, on its surface, and if, in addition, these last 

 denote the magnitudes of the angles of the spherical triangle, we 

 have (with the versor meanings attached to a, /3, y) the equation 



