Chapter 15
Acceptance of the Negative, the General Irrational, and the Imaginary as Numbers

118. Descartes’ Géométrie and the Negative. The Géométrie of Descartes appeared in 1637. This famous little treatise enriched geometry with a general and at the same time simple and natural method of investigation: the method of representing a geometric curve by an equation, which, as Descartes puts it, expresses generally the relation of its points to those of some chosen line of reference.⁶¹ To form such equations Descartes represents line segments by letters,—the known by $a,b,c,$ etc., the unknown by $x$ and $y$. He supposes a perpendicular, $y$, to be dropped from any point of the curve to the line of reference, and then the equation to be found from the known properties of the curve which connects $y$ with $x$, the distance of $y$ from a fixed point of the line of reference. This is the equation of the curve in that it is satisfied by the $x$ and $y$ of each and every curve-point.⁶² To meet the difficulty that the mere length of the perpendicular ($y$) from a curve-point will not indicate to which side of the line of reference the point lies, Descartes makes the convention that perpendiculars on opposite sides of this line (and similarly intercepts ($x$) on opposite sides of the point of reference) shall have opposite algebraic signs.

This convention gave the negative a new position in mathematics. Not only was a “real” interpretation here found for it, the lack of which had made its position so difficult hitherto, but it was made indispensable, placed on a footing of equality with the positive. The acceptance of the negative in algebra kept pace with the spread of Descartes’ analytical method in geometry.

119. Descartes’ Geometric Algebra. But the Géométrie has another and perhaps more important claim on the attention of the historian of algebra. The entire method of the book rests on the assumption—made only tacitly, to be sure, and without knowledge of its significance—that two algebras are formally identical whose fundamental operations are formally the same; i. e. subject to the same laws of combination.

For the algebra of the Géométrie is not, as is commonly said, mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; but symbols which there represent numbers here represent line segments. Not only is this the case with the letters $a,b,x,y,$ etc., which are mere names (noms) of line segments, not their numerical measures, but with the algebraic combinations of these letters. $a+b$ and $a-b$ are respectively the sum and difference of the line segments $a$ and $b$; $ab$, the fourth proportional to an assumed unit line, $a$, and $b$; $\frac{a}{b}$, the fourth proportional to $b,a,$ and the unit line; and $\sqrt{a},\sqrt[3]{a}$, etc., the first, second, etc., mean proportionals to the unit line and $a$.⁶³

Descartes’ justification of this use of the symbols of numerical algebra is that the geometric constructions of which he makes $a+b,a-b,$ etc., represent the results are “the same” as numerical addition, subtraction, multiplication, division, and evolution, respectively. Moreover, since all geometric constructions which determine line segments may be resolved into combinations of these constructions as the operations of numerical algebra into the fundamental operations, the correspondence which holds between these fundamental constructions and operations holds equally between the more complex constructions and operations. The entire system of the geometric constructions under consideration may therefore be regarded as formally identical with the system of algebraic operations, and be represented by the same symbolism.

In what sense his fundamental constructions are “the same” as the fundamental operations of arithmetic, Descartes does not explain. The true reason of their formal identity is that both are controlled by the commutative, associative, and distributive laws. Thus in the case of the former as of the latter, $ab=ba$, and ; for the fourth proportional to the unit line, $a$, and $b$ is the same as the fourth proportional to the unit line, $b$, and $a$; and the fourth proportional to the unit line, $a$, and $bc$ is the same as the fourth proportional to the unit line, $ab$, and $c$. But this reason was not within the reach of Descartes, in whose day the fundamental laws of numerical algebra had not yet been discovered.

120. The Continuous Variable. Newton. Euler. It is customary to credit the Géométrie with having introduced the continuous variable into mathematics, but without sufficient reason. Descartes prepared the way for this concept, but he makes no use of it in the Géométrie. The $x$ and $y$ which enter in the equation of a curve he regards not as variables but as indeterminate quantities, a pair of whose values correspond to each curve-point.⁶⁴ The real author of this concept is Newton (1642–1727), of whose great invention, the method of fluxions, continuous variation, “flow,” is the fundamental idea.

But Newton’s calculus, like Descartes’ algebra, is geometric rather than purely numerical, and his followers in England, as also, to a less extent, the followers of his great rival, Leibnitz, on the continent, in employing the calculus, for the most part conceive of variables as lines, not numbers. The geometric form again threatened to become paramount in mathematics, and geometry to enchain the new “analysis” as it had formerly enchained the Greek arithmetic. It is the great service of Euler (1707–1783) to have broken these fetters once for all, to have accepted the continuously variable number in its purity, and therewith to have created the pure analysis. For the relations of continuously variable numbers constitute the field of the pure analysis; its central concept, the function, being but a device for representing their interdependence.

121. The General Irrational. While its concern with variables puts analysis in a certain opposition to elementary algebra, concerned as this is with constants, its establishment of the continuously variable number in mathematics brought about a rich addition to the number-system of algebra—the general irrational. Hitherto the only irrational numbers had been “surds,” impossible roots of rational numbers; henceforth their domain is as wide as that of all possible lines incommensurable with any assumed unit line.

122. The Imaginary, a Recognized Analytical Instrument. Out of the excellent results of the use of the negative grew a spirit of toleration for the imaginary. Increased attention was paid to its properties. Leibnitz noticed the real sum of conjugate imaginaries (1676–7); Demoivre discovered (1730) the famous theorem

Euler also, practising the method of expressing complex numbers in terms of modulus and angle, formed their products, quotients, powers, roots, and logarithms, and by many brilliant discoveries multiplied proofs of the power of the imaginary as an analytical instrument.

123. Argand’s Geometric Representation of the Imaginary. But the imaginary was never regarded as anything better than an algebraic fiction—to be avoided, where possible, by the mathematician who prized purity of method—until a method was discovered for representing it geometrically. A Norwegian, Wessel,⁶⁵ published such a method in 1797, and a Frenchman, Argand, the same method independently in 1806.

As +1 and -1 may be represented by unit lines drawn in opposite directions from any point, $O$, and as $i$ (i. e. $\sqrt{-1}$) is a mean proportional to +1 and -1, it occurred to Argand to represent this symbol by the line whose direction with respect to the line +1 is the same as the direction of the line -1 with respect to it; viz., the unit perpendicular through $O$ to the 1-line. Let only the direction of the 1-line be fixed, the position of the point $O$ in the plane is altogether indifferent.

Between the segments of a given line, whether taken in the same or opposite directions, the equation holds:

Given, therefore, a complex number, $a+ib$; choose any point $A$ in the plane; from it draw a line $AB$, of length $a$, in the direction of the 1-line, and from $B$ a line $BC$, of length $b$, in the direction of the $i$-line. The line $AC$, thus fixed in length and direction, but situated anywhere in the plane, is Argand’s picture of $a+ib$.

Argand’s skill in the use of his new device was equal to the discovery of the demonstration given in §54, that every algebraic equation has a root.

124. Gauss. The Complex Number. The method of representing complex numbers in common use to-day, that described in §42, is due to Gauss. He was already in possession of it in 1811, though he published no account of it until 1831.

To Gauss belongs the conception of $i$ as an independent unit co-ordinate with 1, and of $a+ib$ as a complex number, a sum of multiples of the units 1 and $i$; his also is the name “complex number” and the concept of complex numbers in general, whereby $a+ib$ secures a footing in the theory of numbers as well as in algebra.

He too, and not Argand, must be credited with really breaking down the opposition of mathematicians to the imaginary. Argand’s Essai was little noticed when it appeared, and soon forgotten; but there was no withstanding the great authority of Gauss, and his precise and masterly presentation of this doctrine.⁶⁶