Why was Caspar Wessel

1
Why was Caspar Wessels geometrical
representation of the complex numbers ignored at
his time?

• Quest-ce que la géométrie?
• Luminy 17 April 2007
• Kirsti Andersen
• The Steno Institute
• Aarhus University

2
Programme 1. Wessels work 2. Other similar
works 3.Gausss approach to complex numbers 4.
Cauchy and Hamilton avoiding geometrical
interpretations 5. Concluding remarks
3
References Jeremy Gray, Exkurs Komplexe
Zahlen in Geschichte der Algebra, ed. Erhard
Scholz, 293299, 1990. Kirsti Andersen,
Wessels 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.
4
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
5
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 Wessels
paper in the Transactions of the
Academy 1818 Wessel died without having become a
member of the Academy
6
Short biography of Caspar Wessel
7
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
8
On the Analytical Representation of Direction
Wessels 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
9
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
10
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?
lt eoc lteoa lteob oc ob oa
oe or oc oa ob
11
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
r
u
u


e
e
(cos
sin
)

e
e


-
1
12
On the Analytical Representation of Direction
For The addition formulae cos(uv)
cosucosv sinusinv sin(uv) cosusinv
sinucosv
a
b
r
u
u



e
e
(cos
sin
)
c
d
r
v
v



e
e
'
(cos
sin
)
(
)(
)
'
(cos(
)
sin(
)
)
a
b
c
d
rr
u
v
u
v






e
e
e
(
)(
)
(
)
(
)
a
b
c
d
ac
bd
ad
bc



-


e
e
e
13
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
14
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
x
z
y
v
v
x
z
h
e
h
e
e






'
(cos
sin
)(
)
e
h
15
On the Analytical Representation of Direction
First a turn of the sphere the outer angle A
around the ?-axis
16
On the Analytical Representation of Direction
On so on six times, until back in starting
position
17
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
18
On the Analytical Representation of Direction
Summary on Wessels 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
19
Reaction to Wessels work
Nobody took notice of Wessels 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!
20
Signs of no interest
? If the geometrical interpretation of complex
numbers had been considered a big issue, Wessels
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
21
Other geometrical interpretations
Abbé Buée 1806 Argand 1806, 1813 Jacques
Frédéric Français 1813 Gergonne 1813 François
Joseph Servoiss reaction in 1814 no need for a
masque géométrique directed line segments with
length a and direction angle a descibed by a
function f (a,a) with certain obvious properties
has these, but there could be more functions
j
j
a
a
(
,
)
a
ae

-
1
22
Other geometrical interpretations
Benjamin Gompertz 1818 John Warren 1828 C.V.
Mourey 1828
23
Gauss
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
24
Cauchy
Cauchy Cours danalyse (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
25
Cauchy
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
c
(
)
(
)
x
x
º
w
(
)
x
j
c
(
)
(
)
i
i

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Cauchy
Setting he had found an explanation why
w
(
)
x
x


2
1
(
)(
)
(
)
a
bi
c
di
ac
bd
ad
bc
i



-


27
Cauchy
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
28
Hamilton
Similar to Cauchys couples of real
numbers Hamilton introduced complex numbers as a
pair of real numbers in 1837 unaware at the time
of Cauchys 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
29
Hamilton
His approach went straightforwardly until he had
to determine the ?s 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
g
g
2
0

lt
1
4
g
0

g
1

-
2
1
30
Hamilton
In other words Hamilton preferred an inconclusive
algebraic argument to a geometrical treatment
31
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 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 Wessels and others interpretations
of complex numbers
32
Concluding remarks
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