Chapter 16
Recognition of the Purely Symbolic Character of Algebra. Quaternions. Ausdehnungslehre.

125. The Principle of Permanence. Thus, one after another, the fraction, irrational, negative, and imaginary, gained entrance to the number-system of algebra. Not one of them was accepted until its correspondence to some actually existing thing had been shown, the fraction and irrational, which originated in relations among actually existing things, naturally making good their position earlier than the negative and imaginary, which grew immediately out of the equation, and for which a “real” interpretation had to be sought.

Inasmuch as this correspondence of the artificial numbers to things extra-arithmetical, though most interesting and the reason of the practical usefulness of these numbers, has not the least bearing on the nature of their position in pure arithmetic or algebra; after all of them had been accepted as numbers, the necessity remained of justifying this acceptance by purely algebraic considerations. This was first accomplished, though incompletely, by the English mathematician, Peacock.67

Peacock begins with a valuable distinction between arithmetical and symbolical algebra. Letters are employed in the former, but only to represent positive integers and fractions, subtraction being limited, as in ordinary arithmetic, to the case where subtrahend is less than minuend. In the latter, on the other hand, the symbols are left altogether general, untrammelled at the outset with any particular meanings whatsoever.

It is then assumed that the rules of operation applying to the symbols of arithmetical algebra apply without alteration in symbolical algebra; the meanings of the operations themselves and their results being derived from these rules of operation.

This assumption Peacock names the Principle of Permanence of Equivalent Forms, and illustrates its use as follows:68

In arithmetical algebra, when a > b,c > d, it may readily be demonstrated that

(a b)(c d) = ac ad bc + bd.

By the principle of permanence, it follows that

(0 b)(0 d) = 0 × 0 0 × d b × 0 + bd, or(b)(d) = bd.

Or again. In arithmetical algebra aman = am+n, when m and n are positive integers. Applying the principle of permanence,

(ap q )q = ap q ap q  toq factors = ap q+p q+ toq terms = ap,  whenceap q = apq.

Here the meanings of the product (b)(d) and of the symbol ap q are both derived from certain rules of operation in arithmetical algebra.

Peacock notices that the symbol = also has a wider meaning in symbolical than in arithmetical algebra; for in the former = means that “the expression which exists on one side of it is the result of an operation which is indicated on the other side of it and not performed.”69

He also points out that the terms “real” and “imaginary” or “impossible” are relative, depending solely on the meanings attaching to the symbols in any particular application of algebra. For a quantity is real when it can be shown to correspond to any real or possible existence; otherwise it is imaginary.70 The solution of the problem: to divide a group of 5 men into 3 equal groups, is imaginary though a positive fraction, while in Argand’s geometry the so-called imaginary is real.

The principle of permanence is a fine statement of the assumption on which the reckoning with artificial numbers depends, and the statement of the nature of this dependence is excellent. Regarded as an attempt at a complete presentation of the doctrine of artificial numbers, however, Peacock’s Algebra is at fault in classing the positive fraction with the positive integer and not with the negative and imaginary, where it belongs, in ignoring the most difficult of all artificial numbers, the irrational, in not defining artificial numbers as symbolic results of operations, but principally in not subjecting the operations themselves to a final analysis.

126. The Fundamental laws of Algebra. “Symbolical Algebras.” Of the fundamental laws to which this analysis leads, two, the commutative and distributive, had been noticed years before Peacock by the inventors of symbolic methods in the differential and integral calculus as being common to number and the operation of differentiation. In fact, one of these mathematicians, Servois,71 introduced the names commutative and distributive.

Moreover, Peacock’s contemporary, Gregory, in a paper “On the Real Nature of Symbolical Algebra,” which appeared in the interim between the two editions of Peacock’s Algebra,72 had restated these two laws, and had made their significance very clear.

To Gregory the formal identity of complex operations with the differential operator and the operations of numerical algebra suggested the comprehensive notion of algebra embodied in his fine definition: “symbolical algebra is the science which treats of the combination of operations defined not by their nature, that is, by what they are or what they do, but by the laws of combination to which they are subject.”

This definition recognizes the possibility of an entire class of algebras, each characterized primarily not by its subject-matter, but by its operations and the formal laws to which they are subject; and in which the algebra of the complex number a + ib and the system of operations with the differential operator are included, the two (so far as their laws are identical) as one and the same particular case.

So long, however, as no “algebras” existed whose laws differed from those of the algebra of number, this definition had only a speculative value, and the general acceptance of the dictum that the laws regulating its operations constituted the essential character of algebra might have been long delayed had not Gregory’s paper been quickly followed by the discovery of two “algebras,” the quaternions of Hamilton and the Ausdehnungslehre of Grassmann, in which one of the laws of the algebra of number, the commutative law for multiplication, had lost its validity.

127. Quaternions. According to his own account of the discovery,73 Hamilton came upon quaternions in a search for a second imaginary unit to correspond to the perpendicular which may be drawn in space to the lines 1 and i.

In pursuance of this idea he formed the expressions, a + ib + jc,x + iy + jz, in which a, b, c, x, y, z were supposed to be real numbers, and j the new imaginary unit sought, and set their product

(a + ib + jc)(x + iy + jz) = axby cz + i(ay + bx) + j(az + cx) + ij(bz + cy).

The question then was, what interpretation to give ij. It would not do to set it equal to a + ib + jc, for then the theorem that the modulus of a product is equal to the product of the moduli of its factors, which it seemed indispensable to maintain, would lose its validity; unless, indeed, a = b = c = 0, and therefore ij = 0, a very unnatural supposition, inasmuch as 1i is different from 0.

No course was left for destroying the ij term, therefore, but to make its coefficient, bz + cy, vanish, which was tantamount to supposing, since b,c,y,z are perfectly general, that ji = ij.

Accepting this hypothesis, denial of the commutative law as it was, Hamilton was driven to the conclusion that the system upon which he had fallen contained at least three imaginary units, the third being the product ij. He called this k, took as general complex numbers of the system, a + ib + jc + kd,x + iy + jz + kw, quaternions, built their products, and assuming

i2 = j2 = k2 = 1 ij = ji = k jk = kj = i ki = ik = j, found that the modulus law was fulfilled.

A geometrical interpretation was found for the “imaginary tripletib + jc + kd, by making its coefficients, b,c,d, the rectangular co-ordinates of a point in space; the line drawn to this point from the origin picturing the triplet by its length and direction. Such directed lines Hamilton named vectors.

To interpret geometrically the multiplication of i into j, it was then only necessary to conceive of the j axis as rigidly connected with the i axis, and turned by it through a right angle in the jk plane, into coincidence with the k axis. The geometrical meanings of other operations followed readily.

In a second paper, published in the same volume of the Philosophical Magazine, Hamilton compares in detail the laws of operation in quaternions and the algebra of number, for the first time explicitly stating and naming the associative law.

128. Grassmann’s Ausdehnungslehre. In the Ausdehnungslehre, as Grassmann first presented it, the elementary magnitudes are vectors.

The fact that the equation AB + BC = AC always holds among the segments of a line, when account is taken of their directions as well as their lengths, suggested the probable usefulness of directed lengths in general, and led Grassmann, like Argand, to make trial of this definition of addition for the general case of three points, A, B, C, not in the same right line.

But the outcome was not great until he added to this his definition of the product of two vectors. He took as the product ab, of two vectors, a and b, the parallelogram generated by a when its initial point is carried along b from initial to final extremity.

This definition makes a product vanish not only when one of the vector factors vanishes, but also when the two are parallel. It clearly conforms to the distributive law. On the other hand, since

(a + b)(a + b) = aa + ab + ba + bb,  and(a + b)(a + b) = aa = bb = 0, ab + ba = 0, orba = ab, the commutative law for multiplication has lost its validity, and, as in quaternions, an interchange of factors brings about a change in the sign of the product.

The opening chapter of Grassmann’s first treatise on the Ausdehnungslehre (1844) presents with admirable clearness and from the general standpoint of what he calls “Formenlehre” (the doctrine of forms), the fundamental laws to which operations are subject as well in the Ausdehnungslehre as in common algebra.

129. The Doctrine of the Artificial Numbers fully Developed. The discovery of quaternions and the Ausdehnungslehre made the algebra of number in reality what Gregory’s definition had made it in theory, no longer the sole algebra, but merely one of a class of algebras. A higher standpoint was created, from which the laws of this algebra could be seen in proper perspective. Which of these laws were distinctive, and what was the significance of each, came out clearly enough when numerical algebra could be compared with other algebras whose characteristic laws were not the same as its characteristic laws.

The doctrine of the artificial numbers regarded from this point of view—as symbolic results of the operations which the fundamental laws of algebra define—was fully presented for the negative, fraction, and imaginary, by Hankel, in his Complexe Zahlensystemen (1867). Hankel re-announced Peacock’s principle of permanence. The doctrine of the irrational now accepted by mathematicians is due to Weierstrass and G. Cantor and Dedekind.74

A number of interesting contributions to the literature of the subject have been made recently; among them a paper75 by Kronecker in which methods are proposed for avoiding the artificial numbers by the use of congruences and “indeterminates,” and papers76 by Weierstrass, Dedekind, Hölder, Study, Scheffer, and Schur, all relating to the theory of general complex numbers built from n fundamental units (see page 40).

SUPPLEMENTARY NOTE, 1902. An elaborate and profound analysis of the number-concept from the ordinal point of view is made by Dedekind in his Was sind und was sollen die Zahlen? (1887). This essay, together with that on irrational numbers cited above, has been translated by W. W. Beman, and published by the Open Court Company, Chicago, 1901.

The same point of view is taken by Kronecker in the memoir above mentioned, and by Helmholtz in his Zählen und Messen (Zeller-Jubeläum, 1887).

G. Cantor discusses the general notion of cardinal number, and extends it to infinite groups and assemblages in his now famous Memoirs on the theory of infinite assemblages. See particularly Mathematische Annalen, XLVI, p. 489.

Very recently much attention has been given to the question: What is the simplest system of consistent and independent laws—or “axioms,” as they are called—by which the fundamental operations of the ordinary algebra may be defined? A very complete résumé of the literature may be found in a paper by O. Hölder in Leipziger Berichte, 1901. See also E. V. Huntington in Transactions of the American Mathematical Society, Vol. III, p. 264.