92 MULTIPLE ALGEBRA. 



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.* 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 Mobius and Grassmann. 



The point of departure of the Barycentrischer Galcul of Mobius, 

 published in 1827, a work of which Clebsch has said that it can 

 never be admired enough,! 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 8 is the centre of gravity of weights, a, 6, c, d, placed at the 

 points A, B, C, D, respectively, is expressed by the equation 



An equation of the more general form 



a A + bB + cC+ etc. =pP +qQ+rR + 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,J may be added or subtracted, and may have 



* Phil. Mag. (3), vol. xxv, p. 490 ; North British fieview, vol. xlv (1866), p. 57. 



fSee his eulogy on Pliicker, p. 14, Gott. Abhandl., vol. xvi. 



$ 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 cases in which 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. 



