Why was Caspar Wessel's geomemetrical representation of the

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					  Why was Caspar Wessel’s
 geometrical representation of
the complex numbers ignored
         at his time?
        Qu’est-ce que la géométrie?
           Luminy 17 April 2007

             Kirsti Andersen
            The Steno Institute
            Aarhus University

1. Wessel’s work

2. Other similar works

3.Gauss’s approach to complex numbers

4. Cauchy and Hamilton avoiding
geometrical interpretations

5. Concluding remarks

Jeremy Gray, “Exkurs: Komplexe Zahlen”
in Geschichte der Algebra, ed. Erhard
Scholz, 293–299, 1990.

Kirsti Andersen, “Wessel’s Work on
Complex Numbers and its Place in History”
in Caspar Wessel, On the Analytical
Representation of Direction, ed. Bodil
Branner and Jesper Lützen, 1999.
           Short biography of Caspar Wessel
Born in Vestby, Norway, as son of a minister
Started at the grammar school in Christiania,
now Oslo, in 1757

Examen philosophicum at Copenhagen
University 1764
Assistant to his brother who was a geographical
From 1768 onwards cartographer, geographical
surveyor, trigonometrical surveyor
Surveying superintendent, 1798
          Short biography of Caspar Wessel
1778 Exam in law – he never used it
1787 Calculations with expressions of the form
                         cos v   1sin v
1797 Presentation in the Royal Danish Academy
of Sciences and Letters of On the Analytical
Representation of Direction. An Attempt Applied
Chiefly to Solving Plane and Spherical Polygons

1799 Publication of Wessel’s paper in the
Transactions of the Academy

1818 Wessel died without having become a
member of the Academy
Short biography of Caspar Wessel
Short biography of Caspar Wessel

                   He calculated the
                   sides in a lot of
                   plane and
                   spherical triangles

                   He wondered
                   whether he could
                   find a shortcut
    On the Analytical Representation of Direction

Wessel’s aim: an algebraic technique for
dealing with directed line segments

In his paper he first looked at a plane, in
which he defined addition and

Addition: the parallelogram rule
     On the Analytical Representation of Direction

His definition of multiplication – could be
inspired by Euclid, defintion VII.15

A number is said to multiply a number
when that which is multiplied is added to
itself as many times as there are units in
the other, and thus some number is

Or in other words the product is formed by
the one factor as the other is formed by
the unit
      On the Analytical Representation of Direction
Wessel introduced a unit oe, and required
the product of two straight lines should in every
respect be formed from the one factor, in the
same way as the other factor is formed from
the ... unit
                           < eoc = <eoa +

                            oc | : | ob | = | oa | :
                            |                              |

                           oe |
                            |   oc | = | oa | ⋅ | ob   |
Advanced for its time? Continuation of
       On the Analytical Representation of Direction

Next, Wessel introduced another unite
and could then express any directed line
segment as       a  be = r(cosu  sinue)
The multiplication rule implies that

                     e  e = 1
        On the Analytical Representation of Direction

For             a  be = r(cosu  sinue )
                c  de = r'(cosv  sinve )
      (a  be )(c  de ) = rr'(cos(  v)  sin(u  v)e )

The addition formulae
cos(u+v) = cosucosv + sinusinv
sin(u+v) = cosusinv + sinucosv:

      (a  be )(c  de ) = (ac  bd)  (ad  bc)e
        On the Analytical Representation of Direction
Wessel: no need to learn new rules for calculating

He thought he was the first to calculate with
directed line segments

Proud and still modest

As he also worked with spherical triangles he
would like to work in three dimensions

He was not been able to do this algebraically, but
he did not give up
      On the Analytical Representation of Direction
A second imaginary unit         h = 1

When x  ye  zh    is rotated the angle v
around the - axis, the result is

  yh  xz' e = yh  (cosv  e sinv)(x  ze )
and rotating the angle u around the         -axis
gives a similar expression

In this way he avoided
     On the Analytical Representation of Direction

First a turn of
the sphere the
outer angle A
around the η-
On the Analytical Representation of Direction

     On so on six times,
     until back in starting
      On the Analytical Representation of Direction

Both in the case of plane polygons and
spherical polygons Wessel deduced a neat
universal formula

However, solving them were in general not
easier than applying the usual formulae
      On the Analytical Representation of Direction
Summary on Wessel’s work

He searched for an algebraic technique for
calculating with directed line segment

As a byproduct, he achieved a geometrical
interpretation of the complex numbers. He
did not mention this explicitly

However, a cryptic remark about that the
possible sometimes can be reached by
“impossible operations”
             Reaction to Wessel’s work

Nobody took notice of Wessel’s paper


Among the main stream mathematicians no
interest for the geometrical interpretation
of complex numbers in the late 18th and
the beginning of the 19th century!
                 Signs of no interest
○ If the geometrical interpretation of
complex numbers had been considered a
big issue, Wessel’s result would have been

○ After Wessel, several other
interpretations were published, they were
not noticed either

○ Gauss had the solution, but did not find it
worth while to publish it
                 Other geometrical interpretations

Abbé Buée 1806

Argand 1806, 1813

Jacques Frédéric Français 1813

Gergonne 1813

François Joseph Servois’s reaction in 1814
no need for a masque géométrique
directed line segments with length a and direction angle
α descibed by a function φ (a,α) with certain obvious
j (a,a ) = aea   1

         Other geometrical interpretations

Benjamin Gompertz 1818

John Warren 1828

C.V. Mourey 1828
Gauss claimed in 1831 that already in
1799 when he published his first proof of
the fundamental theorem of algebra he
had an understanding of the complex

In 1805 he made a drawing in a notebook
indicating he worked with the complex

A letter to Bessel from 1811 (on “Cauchy
integral theorem”) shows a clear
understanding of the complex plane

However, he only let the world know
about his thoughts about complex
numbers in a paper on complex integers
Cauchy Cours d’analyse (1821)

an imaginary equation is only a symbolic
representation of two equations between real

26 years later he was still of the same opinion.
He then wanted to avoid

the torture of finding out what is represented by
the symbol1        , for which the German
geometers substitute the letter i

Instead he chose – an interesting for the
time – introduction based on equivalence
classes of polynomials
               j ( x)  c ( x)     w[mod
                                     ( x)
when the the two first polynomials have the
            w (x)
same remainder after division by the

He then introduced i, and rewrote the above
equation as j (i ) = c (i )

                     w ( x) = x  1

he had found an explanation why

          (a  bi)(c  di) = ac  bd  (ad  bc)i

By 1847 Cauchy had made a large part of
his important contributions to complex
function theory – without acknowledging
the complex plane

Later the same year, however, he
accepted the geometrical interpretation
                       Similar to Cauchy’s
                       couples of real numbers
                       Hamilton introduced
                       complex numbers as a
                       pair of real numbers in
                       1837– unaware at the time
                       of Cauchy’s approach

He wished to give square roots of negatives a
without introducing considerations so expressly
geometrical, as those which involve the
conception of an angle
His approach went straightforwardly until he
had to determine the γs in
               (0,1)  (0,1) = (g 1 ,g 2 )
Introducing a requirement corresponding to
that his multiplication should not open for
zero divisors he found the necessary and
sufficient condition that 2
                       g1  g < 0
and then concluded that this could be
              g1 = 
obtained by setting 1        g 2 = 0 and

In other words Hamilton preferred an
inconclusive algebraic argument to a
geometrical treatment
                   Concluding remarks
When the mathematicians in the seventeenth
century struggled with coming to terms with
complex numbers a geometrical interpretation
would have been welcome

It might for instance have helped Leibniz in his
confusion about             1

By the end of the eighteenth century there was
the idea that analytical/algebraic problems should
be solved by
analytical/algebraic methods. Hence no interest
for Wessel’s and others’ interpretations of
                 Concluding remarks
A geometrical interpretation could at most
be considered an illustration, not a

Warren in 1829 about the reaction to his
book from 1828

... it is improper to introduce geometric
considerations into questions purely
algebraic; and that the geometric
representation, if any exists, can only be
analogical, and not a true algebraic

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