162 



QUATERNIONS AND THE AUSDEHNUNOSLEHRE. 



Geometrical addition in three dimensions is common to the two 

 systems, and seems to have been discovered independently both by 

 Hamilton and Grassmann, as well as by several other persons about 

 the same time. It is not probable that any especial claim for priority 

 with respect to this principle will be urged for either of the two with 

 which we are now concerned. 



The functions of two vectors which are represented in quaternions 

 by Sa/3 and Va/3 are common to both systems as published in 1844, 

 but the quaternion is peculiar to Hamilton's. The linear vector 

 function is common to both systems as ultimately developed, although 

 mentioned only by Grassmann as early as 1844. 



To those already acquainted with quaternions, the first question will 

 naturally be : To what extent are the geometrical methods which are 

 usually called quaternionic peculiar to Hamilton, and to what extent 

 are they common to Grassmann ? This is a question which anyone 

 can easily decide for himself. It is only necessary to run one's eye 

 over the equations used by quaternionic writers in the discussion of 

 geometrical or physical subjects, and see how far they necessarily 

 involve the idea of the quaternion, and how far they would be in- 

 telligible to one understanding the functions Sa/3 and Va/3, but having 

 no conception of the quaternion a/3, or at least could be made so by 

 trifling changes of notation, as by writing S or V in places where 

 they would not affect the value of the expressions. For such a test 

 the examples and illustrations in treatises on quaternions would be 

 manifestly inappropriate, so far as they are chosen to illustrate 

 quaternionic principles, since the object may influence the form of 

 presentation. But we may use any discussion of geometrical or 

 physical subjects, where the writer is free to choose the form most 

 suitable to the subject. I myself have used the chapters and sections 

 in Prof. Tait's Quaternions on the following subjects : Geometry of 

 the straight line and plane, the sphere and cyclic cone, surfaces of 

 the second degree, geometry of curves and surfaces, kinematics, statics 

 and kinetics of a rigid system, special kinetic problems, geometrical 

 and physical optics, electrodynamics, general expressions for the action 

 between linear elements, application of V to certain physical analogies, 

 pp. 160-371, except the examples (not worked out) at the close of the 

 chapters. 



Such an examination will show that for the most part the methods 

 of representing spatial relations used by quaternionic writers are 

 common to the systems of Hamilton and Grassmann. To an extent 

 comparatively limited, cases will be found in which the quaternionic 

 idea forms an essential element in the signification of the equations. 



The question will then arise with respect to the comparatively 

 limited field which is the peculiar property of Hamilton, How im- 





