472 ADDRESS DELIVERED AT A MEETING OF 



Thus we see that a quaternion is reducible to a binomial, the 

 component parts of which the scalar and the vector designate, 

 the one a number, the other a line. The whole tendency of the 

 later speculations of the author has been to realize this reduction, 

 and having determined the laws of operation upon scalars and vec- 

 tors, to dismiss altogether the consideration of the constituents of 

 the vector, and to treat it as a single integral quantity. It is easy 

 to see what amount of simplicity is thus, at one step, introduced 

 into the whole of Geometry and Mechanics. In place of the three 

 co-ordinates (rectilinear or polar) by which the magnitude and 

 direction of a line, or of a force, are ordinarily determined, the 

 theory of Sir William Hamilton deals with the line itself, or with 

 the force, directly ; and thus not only is the number of necessary 

 equations reduced at once, in the proportion of three to one, but 

 also the interpretation of the equations themselves is rendered 

 simpler and more direct. 



The scalar, or algebraically-real part of the quaternion, thus 

 appearing to have no direct geometrical significancy, geometers 

 seemed inclined to regard it as a sort of intruder in their domain ; 

 and I believe it was to the desire to exclude it, that we may, in 

 part, attribute the very elegant and ingenious theories of triplets 

 invented by Professor De Morgan and Professor Graves. The sca- 

 lar, however, is represented in mechanics by the time ; and even in 

 its application to pure geometry, Sir "William Hamilton has shown 

 that the introduction of this fourth quantity confers power and ge- 

 nerality upon the Calculus of Quaternions, inasmuch as no direction 

 in space is thus selected as eminent above another, but all are re- 

 garded as equally related to the extra-spatial, or scalar direction. 

 The calculus thus frequently admits of a simpler and more direct 

 application to geometrical problems than the Cartesian method of 

 co-ordinates, inasmuch as it demands no previous selection of arbi- 

 trary axes. 



I may observe, also, that in the triplet theories of Professor 

 De Morgan and Professor Graves, the law of the moduli is not pre- 

 served, if the term modulus be taken in its ordinary signification, 

 it being not generally true that the sum of three squares, multiplied 

 by the sum of three squares, is a sum of three squares. 



But whatever be thought of the principles of the Calculus of 

 Quaternions, its advantages as an instrument of mathematical 

 thought will undoubtedly be judged by the simplicity and ease 



