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sense of the term, it is in some sense imaginary, when contemplated 

 on the geometrical side. This part of the quaternion is denomi- 

 nated by Sir William Hamilton the scalar. 



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 ne- 

 cessary equations reduced at once, in the proportion of three to 

 one, but also the interpretation of those equations 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 gene- 

 rality upon the calculus of quaternions, inasmuch as no direction in 

 space is thus selected as eminent above another, but all are regarded 

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

 calculus thus frequently admits of a simpler and more direct appli- 

 cation to geometrical problems than the Cartesian method of co- 

 ordinates, inasmuch as it demands no previous selection of arbitrary 

 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, — 



