Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/108

But the idea of double algebra, once received, although as it were unwillingly, must have suggested to many minds more or less distinctly the possibility of other multiple algebras, of higher orders, possessing interesting or useful properties.

The application of double algebra to the geometry of the plane suggested not unnaturally to Hamilton the idea of a triple algebra which should be capable of a similar application to the geometry of three dimensions. He was unable to find a satisfactory triple algebra, but discovered at length a quadruple algebra, quaternions, which answered his purpose, thus satisfying, as he says in one of his letters, an intellectual want which had haunted him at least fifteen years. So confident was he of the value of this algebra, that the same hour he obtained permission to lay his discovery before the Royal Irish Academy, which he did on November 13, 1843.^[1] This system of multiple algebra is far better known than any other, except the ordinary double algebra of imaginary quantities,—far too well known to require any especial notice at my hands. All that here requires our attention is the close historical connection between the imaginaries of ordinary algebra and Hamilton's system, a fact emphasized by Hamilton himself and most writers on quaternions. It was quite otherwise with Möbius and Grassmann.

The point of departure of the Barycentrischer Calcul of Möbius, published in 1827,—a work of which Clebsch has said that it can never be admired enough,^[2]—is the use of equations in which the terms consist of letters representing points with numerical coefficients, to express barycentric relations between the points. Thus, that the point $S$ is the centre of gravity of weights, $a,b,c,d,$ placed at the points $A,B,C,D,$ respectively, is expressed by the equation

$(a+b+c+d)S=aA+bB+cC+dD.$

An equation of the more general form

$aA+bB+cC+{\text{etc.}}=pP+qQ+rR+{\text{etc.}}$

signifies that the weights $a,b,c,$ etc., at the points $A,B,C,$ etc, have the same sum and the same centre of gravity as the weights $p,q,r,$ etc., at the points $P,Q,R,$ etc., or, in other words, that the former are barycentrically equivalent to the latter. Such equations, of which each represents four ordinary equations, may evidently be multiplied or divided by scalars,^[3] may be added or subtracted, and may have

1. Phil. Mag. (3), vol. xxv, p. 490; North British Review, vol. xlv (1866), p. 57.
2. See his eulogy on Plücker, p. 14, Gött. Abhandl., vol. xvi.
3. I use this term in Hamilton's sense, to denote the ordinary positive and negative quantities of algebra. It may, however, be observed that in most oases in whioh I shall have occasion to use it, the proposition would hold without exclusion of imaginary quantities,—that this exclusion is generally for simplicity and not from necessity.