214 Proceedings of Royal Society of Edinburgh. [sess. 
It certainly does not hold any such place among the fundamental 
geometrical conceptions as the geometrical sum, the scalar product, 
or the vector product. The geometrical sum a + /5 represents the 
third side of a triangle as determined by the sides a and /3. Va/3 
represents in magnitude the area of the parallelogram determined by 
the sides a and /3, and in direction the normal to the plane of the 
parallelogram. SyVa/5 represents the volume of the parallelopiped 
{sic) determined by the edges a, /I, and y. These conceptions are 
the very foundations of geometry. 
“We may arrive at the same conclusion from a somewhat narrower 
but very practical point of view. It will hardly be denied that 
sines and cosines play the leading parts in trigonometry. Now the 
notations Va/5 and represent the sine and cosine of the 
angle included between a and combined in each case with certain 
other simple notions. But the sine and cosine combined with these 
auxiliary notions are incomparably more amenable to analytical 
transformation than the simple sine and cosine of trigonometry, 
exactly as numerical quantities combined (as in algebra) with the 
notion of positive or negative quality are incomparably more 
amenable to analytical transformation than the simple numerical 
quantities of arithmetic. 
“ I do not know of anything which can be urged in favour of the 
quaternionic product of two vectors as a f undMinental notion in vector 
analysis, which does not appear trivial or artificial in comparison 
with the above considerations. The same is true of the quaternionic 
quotient, and of the quaternion in general.” 
Now, what does the argument of the second paragraph quoted 
amount to ? Certainly no quaternionist ever denied the importance of 
the sine and cosine in trigonometry ; and Hamilton was unquestion- 
ably the first to show forth the analytical power of the functions Sa/1 
and Va/1. But, because these functions are so incomparably more 
amenable to analytical transformation than are their trigonometrical 
ghosts, are we to infer that they are necessarily more fundamental than 
anything else ? And why should the angle itself be so unceremoniously 
left out? On the principle of answering a wise man according to his 
wisdom, might we not continue : It will hardly be denied that 
angles and their functions play the leading part in trigonometry. 
Now, the notation represents the angle included between a and 
