€6 G. FLEURI. 



I think that this is one of the principal causes why the calculus 

 of quaternions has not yet been more extensively studied. More- 

 over, a wrong idea, according to me, seems to prevail even amongst 

 those who are best acquainted with the methods of quaternions, 

 as to what direction should be given to that study. In Mr. P. G. 

 Tait's and in Mr. A. McAulay's papers for instance, quaternion 

 methods are proposed to replace Cartesian methods, and comparison 

 is continually made between quaternions and Cartesian geometry. 

 This is equivalent to proposing complex quantities to replace 

 Cartesian plane geometry. That quaternions and complex 

 quantities give ready solutions for certain classes of elementary 

 questions of geometry the works of Bella vitis (method of equi- 

 pollences) and Tait (quaternions) sufficiently prove. They furnish 

 also very often for that purpose what is really a method of 

 abridged notation. But I do not see very well how we could 

 easily manage with them when we have to introduce elliptic and 

 in general Abelian functions (see for instance, Clebsh Analytical 

 Geometry, 3rd part).* 



If quaternionic methods were only useful for the purpose of 

 replacing Cartesian geometry, I should consider Mr. Heaviside 

 perfectly right in adopting some different hypothesis for the 

 establishment of a quaternion calculus. For, why are Hamilton's 

 hypotheses more natural than those of Mr. Heaviside'? And here 

 we find one of the main reasons why I have undertaken the 

 present work. Quaternions are already fifty years old, and to-day 

 a physicist proposes to replace their calculus by a different one ! 

 And why 1 



When we come to examine the thing, we easily see that Mr. 

 Heaviside has been completely mistaken as to the real nature of 

 complex quantities of ordinary algebra ; */—i is for him a symbol 

 without existence arising from consideration of equations of second 

 degree and at any price he wants to get rid of it.f Unfortunately 



* Of course it may be simply a want of custom on my part, but even so, 

 I do not see the use of working with quaternions in that direction. 



f I do not speak about his separation of the scalar and vector products 

 of two vectors whose reunion forms the quaternion, for it does not in 

 reality change anything in the theory of Hamilton. 



