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Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time?. Qu’estce que la géométrie? Luminy 17 April 2007 Kirsti Andersen The Steno Institute Aarhus University. Programme 1. Wessel’s work 2. Other similar works 3.Gauss’s approach to complex numbers
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Qu’estce 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 1745
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 surveyor
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
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
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 multiplication
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 produced
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
Advanced for its time? Continuation of Descartes?
< eoc = <eoa + <eob
oc : ob = oa : oe 
or
oc  = oa  ⋅ ob 
On the Analytical Representation of Direction
e
Next, Wessel introduced another unit and could then express any directed line segment as
The multiplication rule implies that
a
+
b
e
r
(cos
u
+
sin
u
e
)
=
e
×
e
=

1
On the Analytical Representation of Direction
For
The addition formulae
cos(u+v) = cosucosv + sinusinv
sin(u+v) = cosusinv + sinucosv:
a
+
b
e
=
r
(cos
u
+
sin
u
e
)
c
+
d
e
=
r
'
(cos
v
+
sin
v
e
)
(
a
+
b
e
)(
c
+
d
e
)
=
rr
'
(cos(
u
+
v
)
+
sin(
u
+
v
)
e
)
(
a
+
b
e
)(
c
+
d
e
)
=
(
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
h
A second imaginary unit ,
When is rotated the angle v around the  axis, the result is
and rotating the angle u around the axis gives a similar expression
In this way he avoided
h
y
h
+
x
¢+
z
'
e
=
y
h
+
(cos
v
+
e
sin
v
)(
x
+
z
e
)
e
h
On the Analytical Representation of Direction
First a turn of the sphere the outer angle A around the ηaxis
On the Analytical Representation of Direction
On so on six times, until back in starting position
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”
Nobody took notice of Wessel’s paper
Why?
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!
○ If the geometrical interpretation of complex numbers had been considered a big issue, Wessel’s result would have been noticed
○ 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
○ Cauchy and Hamilton explicitly were against a geometrical interpretation
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 properties

has these, but there could be more functions
j
j
a

1
(
a
,
a
)
=
ae
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 plane
In 1805 he made a drawing in a notebook indicating he worked with the complex plane
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 published in 1831
Cauchy Cours d’analyse (1821)
an imaginary equation is only a symbolic representation of two equations between real quantities
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 symbol , for which the German geometers substitute the letter i
Instead he chose – an interesting for the time – introduction based on equivalence classes of polynomials
[mod ]
when the the two first polynomials have the same remainder after division by the polynomial
He then introduced i, and rewrote the above equation as
j
(
x
)
º
c
(
x
)
w
(
x
)
j
(
i
)
=
c
(
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 meaning
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
Introducing a requirement corresponding to that his multiplication should not open for zero divisors he found the necessary and sufficient condition that
and then concluded that this could be obtained by setting and
(
0
,
1
)
×
(
0
,
1
)
=
(
g
,
g
)
1
2
1
2
g
+
g
<
0
1
4
g
=
0
g
=

1
2
1
In other words Hamilton preferred an inconclusive algebraic argument to a geometrical treatment
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
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 complex numbers
A geometrical interpretation could at most be considered an illustration, not a foundation
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 representation of the roots