Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/120

104 where $$x$$ is any quantity or function. We may also have occasion to write But it is almost impossible to resist the tendency to express these relations in the form  in which the operators appear in a sense as quantities, i.e., as subjects of functional operation. Now since these operators are often of such nature that they cannot be perfectly specified by a single numerical quantity, when we treat them as quantities they must be regarded as multiple quantities. In this way certain formule which essentially belong to multiple algebra get a precarious footing where they are only allowed because they are regarded as abridged notations for equations in ordinary algebra. Yet the logical development of such notations would lead a good way in multiple algebra, and doubtless many investigators have entered the field from this side.

One might also notice, to show how the ordinary algebra is being saturated with the notions and notations which seem destined to turn it into a multiple algebra, the notation so common in the higher algebra This is evidently the same as Grassmann's internal product of the multiple quantities $$(a, b, c)$$ and $$(x, y, z),$$ or, in the language of quaternions, the scalar part, taken negatively, of the product of the vectors of which $$a, b, c$$ and $$x, y, z$$ are the components. A similar correspondence with Grassmann's methods might, I think, be shown in such notations as, for example, The free admission of such notations is doubtless due to the fact that they are regarded simply as abridged notations.

The author of the celebrated "Memoir on the Theory of Matrices" goes much farther than this in his use of the forms of multiple algebra. Thus he writes explicitly one equation to stand for several, without the use of any of the analytical artifices which have been mentioned. This work has indeed, as we have seen, been characterized as marking the commencement of multiple algebra,—a view to which we can only take exception as not doing justice to earlier writers. But the significance of this memoir with regard to the point which I am now considering is that it shows that the chasm so marked in the second quarter of this century is destined to be closed up. Notions and notations for which a Cayley is sponsor will not be