macfarlane: algebra of physics 



333 



Principles of the algebra of physics. Proc. Amer. Assoc. Adv. 

 of Science, 40: 65-117. I thus enunciated the problem as it 

 appeared to me then (p. 65) : 



I am convinced that the notation of quaternions can be improved; 

 that the principles require to be corrected and extended; that there is 

 a more complete algebra which unifies quaternions, Grassmann's method, 

 and determinants, and applies to physical quantities in space. The 

 guiding idea in this paper is generalization. What is sought for is an 

 algebra which will apply directly to physical quantities, will include and 

 unify the several branches of analysis, and when specialised will become 

 ordinary algebra. 



The fundamental rules of quaternions were investigated by 

 considering (1) the product of two quadrantal versors, (2) the 

 product of a quadrantal versor and a unit vector, (3) the product 

 of two unit-vectors. 



Let h, j, k (fig. 1) be a righthanded 

 system of orthogonal axes, and let h n/2 

 denote a quadrant round h. Then if 

 we take the quadrants in the cyclical 

 order 



h w/2 f /2 = k v+v ! 2 



Why do we take the angle greater than 

 7r, not the smaller angle? Because the 

 equivalent rotation round k which will 

 bring j into coincidence with h must be righthanded, and there- 

 fore amounts to Sir/2. But when we take the anti-cyclic order, 

 we get 



r /2 v' 2 = r /2 



r/y. J. 



