The Net Advance of Physics RETRO:

The Rise and Fall of Nineteenth-Century Geometric Algebra
Part One

Argand's Plane

2013 January 14

Jean Robert Argand: a commoner, a book seller (like surprisingly many working-class intellectuals of the Eighteenth and Nineteenth Centuries), entirely self-educated and apparently unconnected to the scientific establishment. The details of his life, recent though it was (1768-1822), are largely unknown; he is as obscure a giant as the history of mathematics in modern times affords. Argand's 1806 proof of the fundamental theorem of algebra languished at the margins (and is not now available online). In the same year, he introduced the plane today called by his name in a self-published, anonymous booklet, later reprinted as Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques [Annales de Mathématiques Pures et Appliquées 4, 133 (1813)]. It is this work we will examine here (using Hardy's 1881 translation).

Negative numbers, according to Argand, exist only in the imagination: they are thus literally "imaginary". "If, for example, a represents a material weight, as a gram, the series ....... 4a, 3a, 2a, a, 0, cannot be extended beyond 0 ; for, while we may take 1 gram from 3, 2, or 1 gram, we cannot take it from 0." But what if we are talking about money? A debt is a negative credit: "Thus −100 francs, −200 francs, ....... which in the former case can express only imaginary quantities, here represent quantities as real as those denoted by positive expressions." A similar argument can relate negative values to motion rightward or leftward on a scale. A situation where negative values are possible, therefore, is a "complex" situation. Not only is the "simple" concept of number involved, but also a second concept: direction.

Of course, in Argand's time, the use of negative numbers was already well established; that of "even roots of negative quantities" remained controversial. Could the same idea which made negative numbers acceptable -- introducing direction -- work for this case as well? The square root of minus one is only "regarded imaginary, because we cannot assign it a place in the scale of positive and negative quantities," but perhaps this scale can be expanded to include some new, "third" direction. (For Argand, right and left were two directions.) The obvious choice: a direction perpendicular to both right and left.

Descartes' right-angled axes were familiar to all mathematicians in 1806; it is surprising that Argand makes no explicit reference to them. Instead, he inscribes circles like an ancient Greek, and constructs his perpendiculars within them. However, he makes an observation the Greeks did not: any radius drawn in the plane has an unique direction-angle as well as a magnitude, and thus must be distinguished from a mere directionless line-segment. Moreover, the geometrical relations obeyed by such a radius do not depend upon its extending from a particular origin, but are true of all similarly-directed segments with the same magnitude. Thus Argand introduces vectors, which Hamilton would independently rediscover (and name) a few years later.

The component of a vector parallel to the abcissa has a real magnitude; so does the perpendicular component. Argand associates this second real component with an imaginary number, although "these lines are quantities quite as real as the positive unit ; they are derived from it by the association of the idea of direction with that of magnitude, and are in this respect like the negative line, which has no imaginary signification. The terms real and imaginary do not therefore accord with the above exposition. It is needless to remark that the expressions impossible and absurd, sometimes met with, are still less appropriate. The use of these terms in the exact sciences in any other sense than that of not true is perhaps surprising."

So far, nothing too unexpected for the reader from a later era -- then Argand begins proposing new, less pejorative terminology: instead of "real", say "prime order", and instead of "imaginary", say "medial order". The complex objects containing both should be called "intermedials". More startlingly, he notes that the symbols + and − contain only straight lines, and introduces two new symbols corresponding to the modern i and − i, respectively ⊂ and ⊄. Actually, these are not the exact symbols he uses, which I cannot represent in ASCII, but they have the right typographical properties: the symbol corresponding to i contains a curved line, and that corresponding to − i one curved and one straight.

"Let every straight line, horizontal or vertical, in the signs to be multiplied, have a value 2, and every curved one a value 1 ; we shall have for the four signs the following values:

⊂=1, −=2, ⊄=3, +=4.
Then take the sum of the values of all the factors and subtract as many times 4 as is necessary to make the remainder one of the numbers 1, 2, 3, 4 ; this remainder will be the value of the sign of the product." A remarkable attempt to make the symbols of mathematics less arbitrary!

Although Argand does not use the term "operator", it is clear that such is the nature of each of his four "signs", and he uses them in the same way that we use "−" to represent both "subtraction" and "multiplication by negative one". Thus, for example, he writes:

(a ⊂ b)(a ⊄ b) = a²+b²

Argand next returns to vector analysis, giving a lengthly discussion of vector addition in the plane, and noting that any "directed line" may be decomposed into components along any two other distinct directed lines. Multiplication, which would so trouble vector algebraists for the next century, is the subject of his eleventh article: "To construct the product of two directed radii, lay off, from the origin of arcs, the sum of the arcs corresponding to each radius, and the extremity of the arc thus laid off will determine the position of the radius of the product ; this, as before, is logarithmic multiplication." The meaning of this opaque statement, in modern language, is as follows: Consider two unit vectors with angular coördinates A and B respectively. Their product will be a unit vector with angular coördinate A+B.

The use of trigonometric functions to express components is introduced next, and the roots of unity are discovered by surprisingly laborious means. The main application which interests Argand seems to be the summation of long trigonometric series ; thus, to cite one example out of many, he uses a vectorial argument to sum cos(a) + cos(a+b) + ... +cos(a+nb). "What precedes is sufficient to show that the method here presented may be employed in trigonometrical researches."

Many of Argand's ideas were "in the air" when he wrote, and some had already been independently discovered (as will be discussed in the second installment of this series). That does not detract from his courage and brilliance, as a mathematical outsider, in proposing them, and in weathering the subsequent storm of establishment criticism (see Part Three). Complex numbers would eventually fare better in the scientific community than their ill-starred children the quaternions, but both generations were rebels from birth.

A portrait of Argand? Yes -- but not of ours. This is Aimé Argand, no known relation, friend of the ballooning Brothers Montgolfier, inventor of a celebrated lamp:

The Rise and Fall of Nineteenth-Century Geometric Algebra