274
ALGEBRA
on the other hand, the “ equations ” of chemistry, although symbolical in form, are not deductions from a limited number of fundamental equivalences, and for this reason are not regarded as algebraical. It is conceivable that, in course of time, a new algebra may be invented, suggested by chemistry and admitting of a chemical interpretation, and in fact a certain analogy has been observed between the graphic symbols of organic chemistry and the umbral notation introduced by Aronhold into the theory of algebraic forms. 2. What will here be called “ ordinary ” algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of arithmetical problems and the statement of arithmetical facts. Although the distinction is one which cannot be ultimately maintained, it is convenient to classify the signs of algebra into symbols of quantity (usually figures or letters), symbols of operation, such as +, J, and symbols of distinction, such as brackets. Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary arithmetical meanings. It could not escape notice that one and the same symbol, such as ,J(a — 6), or even (a - b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved. This led to a prolonged controversy on the nature of negative and imaginary quantities, which was ultimately settled in a very curious way. The progress of analytical geometry led to a geometrical interpretation both of negative and also of imaginary quantities; and when a “ meaning,” or, more properly, an interpretation, had thus been found for the symbols in question, a reconsideration of the old algebraic problem became inevitable, and the true solution, now so obvious, was eventually obtained. It was at last realized that the laws of algebra do not depend for their validity upon any particular interpretation, whether arithmetical, geometrical, or other; the only question is whether these laws do or do not involve any logical contradiction. When this fundamental truth had been fully grasped, mathematicians began to inquire whether algebras might not be discovered which obeyed laws different from those obtained by the generalization of arithmetic. The answer to this question has been so manifold as to be almost embarrassing. All that can be done here is to give a sketch of the more important and independent “ special algebras ” at present known to exist. 3. Although the results of ordinary algebra will be taken for granted, it is convenient to give the principal rules upon which it is based. They are (a-h b)+ c = a + (b + c) (a) (a x b) x c = a x (b x c) (a') a+b=b+a (c) axb = b x a (o') a(b +-c) = ab + ac (i>) (a -b) + b = a (i) (a + b) xb = a (i') These formulae express the associative and commutative laws of the operations + and x , the distributive law of x , and the definitions of the inverse symbols — and -j-, which are assumed to be unambiguous. The special symbols 0 and 1 are used to denote a —a and a-^a. They behave exactly like the corresponding symbols in arithmetic; and it follows from this that whatever “meaning” is attached to the symbols of quantity, ordinary algebra includes arithmetic, or at least an image of it. Every ordinary algebraic quantity may be regarded as of the form a + fi J - , where a, [3 are “ real ” ; that is to say, every algebraic equivalence remains valid when its symbols of quantity are interpreted as complex numbers of the type a + (3 J - (cf. Number). But the symbols of ordinary algebra do not necessarily
denote numbers; they may, for instance, be interpreted as coplanar points or vectors. Evolution and involution are usually regarded as operations of ordinary algebra; this leads to a notation for powers and roots, and a theory of irrational algebraic quantities analogous to that of irrational numbers. 4. The only known type of algebra which does not contain arithmetical elements is substantially due to Boole. Although originally suggested by formal logic, it is most simply interpreted as an algebra of Nott' regions in space. Let i denote a definite region of space; and let a, b, etc., stand for definite parts of i. Let a + b denote the region made up of a and b together (the common part, if any, being reckoned only once), and let a x 6 or ab mean the region common to a and b. Then a + a = aa = a‘, hence numerical coefficients and indices are not required. The inverse symbols -, -f- are ambiguous, and in fact are rarely used. Each symbol a is associated with its supplement d which satisfies the equivalences a + d = i, ad = 0, the latter of which means that a and d have no region in common. Finally, there is a law of absorption expressed hy a + ab = a. From every proposition in this algebra a reciprocal one may be deduced by interchanging + and x, and also the symbols 0 and i. For instance, cc + y = x + xy and xy = x(x + y) are reciprocal. The operations + and x obey all the ordinary laws a, c, d (§ 3). 5. A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol «A. The sum of two weighted points aA, /3B is, by Mobius’s definition, the point (a + /3)G, where G divides calculus™ AB so that AG : GB = /? : a. It can be proved by geometry that (aA + /3B) + yC = aA + (/3B + yC) = (a + (3 + y)P, where P is in fact the centroid of masses a, (3, y placed at A, B, C respectively. So, in general, if we put aA + /3B + yC + . .. + AL — (a + /3-t-y-l- ... + A)N. X is, in general, a determinate point, the barycentre of aA, /3B, etc. (or of A, B, etc. for the weights a, (3, etc.). If (a + /3 + ... + A) happens to be zero, X lies at infinity in a determinate direction; unless - aA is the barycentre of /?B, yC, . . . AL, in which case aA + /IB + . . . + AL vanishes identically, and X is indeterminate. If ABCD is a tetrahedron of reference, any point P in space is determined by an equation of the form (a /I + y + S)P = aA + /IB + yC + 8D : a, [3, y, 8 are, in fact, equivalent to a set of homogeneous co-ordinates of P. For constructions in a fixed plane three points of reference are sufficient. It is remarkable that Mdbius employs the symbols AB, ABC, ABCD in their ordinary geometrical sense as lengths, areas, and volumes, except that he distinguishes their sign; thus AB = - BA, ABC= -ACB, and so on. If he had happened to think of them as “products,” he might have anticipated Grassmann’s discovery of the extensive calculus. From a merely formal point of view, we have in the barycentric calculus a set of “special symbols of quantity” or “extraordinaries” A, B, C, etc., which combine with each other by means of operations + and - which obey the ordinary rules, and with ordinary'algebraic quantities by operations x and -r, also according to the ordinary rules, except that division by an extraordinary is not used. 6. A quaternion is best defined as a symbol of the type q = ~ases = a(je() + a1e1 + o.^e2 + a3e3, Hamilton’s where e0 . . . e3 are independent extraordinaries quaterand a0 . . . a3 ordinary algebraic quantities, which n,oasmay be called the co-ordinates of q. The sum and pro-
