Fundamental Notions in Vector Analysis

Abstract

This method of approach to the quaternion vector analysis is practically that adopted by Prof. Joly in his “Manual of Quaternions.” The method is unsatisfactory, because it makes too great a demand at the outset upon the learner's faith. Why should She be put equal to —abcosθ? The answer is, of course, because that is the simplest way of getting a vector algebra applicable to Euclidean space, and at the same time associative in its vector products. But the existence of so many varieties of non-associative vector algebra shows how absolutely unimportant this latter consideration is to many who find vector analysis useful. In these varieties not only is there no explicit recognition of a quantity αβ, where α and β are vectors, but there is a perfect hatred of the mere suggestion of it as a quantity worthy of general discussion, except (be it noted) in the particular case in which α is perpendicular to β. Mr. Ray shows, by a simple Cartesian process, how easily we may arrive at the recognition of this product if we start with the geometrical definitions of Hamilton's Vαβ and Sαβ. But the method is unconvincing to the man who prejudges the whole matter by barring out the quantity or symbolic form αβ as being fundamentally foreign to any well-regulated system of vector analysis! If they would not listen to Hamilton, Tail, or Joly, will they listen to any other quaternionist, charm he never so wisely?