defined to be that unit of which the supplement is the progressive product |E.r|Eg. For instance, if n = 4, Er = e1e3, Es = e2e3e4, we have lEr|E, = ( - e2e4)( - ej = = |e3, consequently, by the rule of regressive multiplication, e e e e e =e l 3' 2 3 4 3 Applying the distributive law, we obtain, when r + » > re, ArBg = 2aEr2/3Eg = 2(a/8)ErEg, where the regressive products ErEs are to be reduced to units of species (r + s — re) by the foregoing rule. If A = 2aE, then, by definition, jA = 2a|E, and hence A|(B + C) = A|B + A]C. Now this is formally analogous to the distributive law of multiplication; and in fact we may look upon A|B as a particular way of multiplying A and B (not A and |B). The symbol A|B, from this point of view, is called the inner product of A and B, as distinguished from the outer product AB. An inner product may be either progressive or regressive. In the course of reducing such expressions as (AB)C, (AB){C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly one, partly the other. In the first two cases the product is said to be pure, in the third case mixed. A pure product is associative; a mixed product, .speaking generally, is not. The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Nab and Sab respectively. As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units. Only a few illustrations can be given here. Let a, b, c, d, e,f be quantities of the first species in- the fourth category; A, B, C . . . quantities of the third species in the same category. Then (de)(abc) = (abde)c + (cade)b + (bcde)a — (abce)d — (abcd)e, (a6)(AB) = (aA)(6B) - (aB)(6A) abc = (ac)b - {bc)a, (abcd) = (ac)(bd) - (ad)(bc). These may be compared and contrasted with such quaternion formulae as S(NabNcd) — SadSbc — SacSbd dSabc — aSbcd — bScda + cSadb where a, b, c, d denote arbitrary vectors. 8. An re-tuple linear algebra (also called a complex Linear number-system) deals with quantities of the type algebras. ^ = ^aiei derived from re special units ev e.2. . . en. The sum and product of two quantities are defined in the first instance by the formulae 2ae + 2/3e = 2(a + /3)e, x 2/^- = 2(ai/5?-)cl-^, so that the laws A, c, d of § 3 are satisfied. The binary products c; ej, however, are expressible as linear functions of the units e* by means of a “ multiplication table ” which defines the special characteristics of the algebra in question. Multiplication may or may not be commutative, and in the same way it may or may not be associative. The types of linear associative algebras, not assumed to be commutative, have been enumerated (with some omissions) up to sextuple algebras inclusive by B. Peirce. Quaternions afford an example of a quadruple algebra of this kind. If, in the extensive calculus of the reth category, all the units (including 1 and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2n - l)-tuple linear algebra, which, however, is not wholly associative. It
should be observed that while the use of special units, or extraordinaries, in a linear algebra is convenient, especially in applications, it is not indispensable. Any linear quantity may be denoted by a symbol (oq, a2,. . . an) in which only its scalar coefficients occur; in fact, the special units only serve, in the algebra proper, as umbrce or regulators of certain operations on scalars (see Number, § 33). This idea finds fuller expression in the algebra of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of scalars, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general, commutative. Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices. 9-_ The algebras discussed up to this point may be considered as independent in the sense that each of them deals with a class of symbols of quantity more or less homogeneous, and a set of operations S^hsid!ary applying to them all. But when an algebra is * £ebrasused with a particular interpretation, or even in the course of its formal development, it frequently happens that new symbols of operation are, so to speak, superposed upon the algebra, and are found to obey certain formal laws of combination of their own. For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold’s symbolical method in the calculus of invariants; and the like. In most cases these subsidiary algebras, as they may be called, are inseparable from the applications in which they are used; but in any attempt at a natural classification of algebra (at present a hopeless task), they would have to be taken into account. Even in ordinary algebra the notation for powers and roots disturbs the symmetry of the rational theory; and when a schoolboy illegitimately extends the distributive law by writing J(a + b)=l/a+ ^/b, he is unconsciously emphasizing this want of complete harmony. 10. The reader cannot fail to observe that this article is far from being an outline of universal algebra, in the sense ascribed to that term at the beginning; it is, rather, a brief presentation of some of the principal facts with which universal algebra has to deal. It may even be doubted whether any theory of universal algebra, except in a very restricted or provisional sense, is actually possible at present. It may, perhaps, be admitted that we have arrived at the conception that an “ algebraic quantity ” is a symbol defined merely by its formal relations; and that the symbols + and x are legitimately used when the first is commutative and associative, and the second distributive. But there is hardly any other general statement that may not be upset by future discoveries; and in fact even these are inconsistent with much current notation. The state of mathematical symbolism to-day may be fairly compared to that of botany, when the idea of a natural classification first began to suggest itself. Authorities.—A. be Morgan, “On the Foundation of Algebra,” Trans. Camb. P. S. vii. viii. 1839-1844 ; G. Peacock, Symbolical Algebra, Cambridge, 1845 ; G. Boole, Laws of Thought, London, 1854 ; E. Schroder, Lehrbuchder Arithmetiku. Algebra, Leipzig, 1873, Vorlesungen fiber die Algebra der Logik, ibfd. 1877 &c. ; A. F. Mobius, Der barycentrische Calcul, Leipzig, 1827 (reprinted in M.’s collected works, vol. i. Leipzig, 1885) ; W. R. Hamilton, Lectures on Qiiaternions, Dublin, 1853, Elements of Quaternions, ibid. 1866 ; H. Grassmann, Die lineale Ausdehnungslehre, Leipzig, 1844, Die Ausdehnungslehre, Berlin, 1862 (these are reprinted with valuable emendations and notes in II. G.’s Gesammelte math. u. phys. Werke, vol. i. Leipzig (2 parts), 1894, 1896), and papers in Grunert's Arch, vi., Crelle, xlix. Ixxxiv., Math. Ann. vii. xii. ; B. and C. S. Peirce, “ Linear Associative Algebra,” Amer. Journ. Math. iv. (privately circulated, 1871); A. Cayley, on Matrices, Phil. Trans, cxlviii., on Multiple Algebra, Quirt. M. Journ. xxii.; J. J. Sylvester, on Universal Algebra (i.e. Matrices), Amer. Journ. Math. vi. ; H. J. S. Smith,
