Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time?

1. 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

2. Programme 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

3. References 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.

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 Wessel’s 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 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

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? < eoc = <eoa + <eob |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 e r (cos u + sin u e ) = e × e = - 1

12. 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

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 h + x ¢+ z ' e = y h + (cos v + e sin v )( x + z e ) e h

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

19. Reaction to Wessel’s work 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!

20. Signs of no interest ○ 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

21. 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

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 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

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 ( x ) º c ( x ) w ( x ) j ( i ) = c ( i )

26. Cauchy Setting he had found an explanation why 2 w ( x ) = x + 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 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

29. Hamilton 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

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 Wessel’s 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