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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**

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:*

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:

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

- 2013 Jan. 14: Part 1: Argand's Plane
- To Be Continued ....