Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/172

representing in length the product of their lengths and the sine of the angle which they include. This is denoted by in quaternions. How these notions are represented in my pamphlet is a question of very subordinate consequence, which need not be considered at present. The importance of these notions, and the importance of a suitable notation for them, is not, I suppose, a matter on which there is any difference of opinion. Another function of $\alpha$ and $\beta ,$ called their product and written $\alpha \beta ,$ is used in quaternions. In the general case, this is neither a vector, like ${\text{V}}\alpha \beta ,$ nor a scalar (or ordinary algebraic quantity), like ${\text{S}}\alpha \beta$ but a quaternion—that is, it is part vector and part scalar. It may be defined by the equation—

$\alpha \beta ={\text{V}}\alpha \beta +{\text{S}}\alpha \beta .$

The question arises, whether the quatemionic product can claim a prominent and fundamental place in a system of vector analysis. 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 $\alpha +\beta$ represents the third side of a triangle as determined by the sides $\alpha$ and $\beta$ ${\text{V}}\alpha \beta$ represents in magnitude the area of the parallelogram determined by the sides $\alpha$ and $\beta ,$ and in direction the normal to the plane of the parallelogram. ${\text{S}}\gamma {\text{V}}\alpha \beta$ represents the volume of the parallelopiped determined by the edges $\alpha ,\beta ,$ and $\gamma .$ 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 ${\text{V}}\alpha \beta$ and ${\text{S}}\alpha \beta$ represent the sine and the cosine of the angle included between $\alpha$ and $\beta ,$ 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 favor of the quaternionic product of two vectors as a fundamental notion in vector analysis, which does not appear trivial or artificial in comparison with the above considerations. The same is true of the quatemionic quotient, and of the quaternion in general.

How much more deeply rooted in the nature of things are the functions ${\text{V}}\alpha \beta$ and ${\text{S}}\alpha \beta$ than any which depend on the definition of a quaternion, will appear in a strong light if we try to extend