A Dozen Episodes from the Mathematics of the Nineteenth Century ...

A Dozen Episodes from the Mathematics of theNineteenth CenturyWeek 8: Complex numbers throughout thenineteenth centuryHenrik Kragh SørensenHistory of Science Department, University of Aarhus22nd February 2001http://home.imf.au.dk/hkragh/hom/episoder/LecturesThe lectures in week 8 will treat the conceptions of complex and hypercomplex numbersthroughout the 19th century. Starting with the geometrical interpretations ofcomplex numbers advocated by Wessel and Argand, both on the mathematicalperiphery (Andersen 1999, Wessel 1799), we will turn to the proofs of the FundamentalTheorem of Algebra (FTA) by Gauss, and the symbolic interpretation ofcomplex numbers with Cauchy and Hamilton. Finally, the further developmentof generalized complex numbers, leading to hypercomplex numbers and algebras,will be traced, (van der Waerden 1985, Hoffmann 2000).The background literature will consist of (Kline 1972, chapters 27.3, 32).ExercisesFor the exercises, we will read part of William Rowan Hamilton’s Theory ofconjugate functions or algebraic couples (text 4) as well as a short passage fromThomas Hankins’ biography of Hamilton (text 5).John and Erik have prepared the following guiding questions for the exercises:1. What does Hamilton actually introduce in the text?2. What is the general outline of the selection? What are Hamilton’s intentions?3. What are the differences (from a modern perspective) between step and momentcouple?4. What do pure primary and pure imaginary translate into in modern terminlogy?1

5. Where are the differences between Hamilton and modern mathematics themost/the least apparent? The focus is here on the presentation of mathematicalmaterial.6. Based on the introduction, would Hamilton be classified as a pure or as anapplied mathematician?7. Compare Hamilton’s way of presenting mathematics to the modern tradition.Are they different? How?8. At a central point, Hamilton introduced two constants, γ 1 and γ 2 . What aretheir roles? How could they be interpreted from a modern perspective (thinkgeometrically).9. Importantly, Hamilton fixed γ 1 = −1, γ 2 = 0. Why? What did he achieve?What were his arguments?10. Based on the texts, discuss the Kantian foundation of Hamilton’s philosophyof numbers.ReferencesAndersen, K. (1999). Wessel’s work on complex numbers and its place in history.In B. Branner and J. Lützen (Eds.), Caspar Wessel. On the Analytic Representationof Direction. An Attempt Applied Chiefly to Solving Plane andSpherical Polygons. Copenhagen: C. A. Reitzel.Gericke, H. (1996). Talbegrebets historie. Århus: Matematiklærerforeningen og Institutfor de Eksakte Videnskabers Historie, Aarhus Universitet. Danish editiontranslated and edited by Kirsti Andersen and Kate Larsen.Guld, T. L. (1997). Hamilton og kvaternionerne. Speciale, Institut for de EksakteVidenskabers Historie, Aarhus Universitet, Aarhus.Hamilton, W. R. (1837). Theory of conjugate functions, or algebraic couples; witha preliminary and elementary essay on algebra as the science of pure time. InThe mathematical papers of Sir William Rowan Hamilton, Volume 3, pp. 3–96.Cambridge: at the University Press. First published Trans. Roy. Irish Acad.vol. XVIII (1837), pp. 293–422. Read on 4 November 1833 and 1 June 1835.Hoffmann, M. (2000, June). Fra tal til algebraer. Den britiske begyndelse i1830’erne og aspekter af den tyske fortsættelse op til 1890. Speciale, Institutfor Videnskabs–historie, Aarhus Universitet, Aarhus.Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford:Oxford University Press. Reprinted in three volumes 1990.van der Waerden, B. L. (1985). A History of Algebra. From al-Khwārizmī to EmmyNoether. Berlin, Heidelberg, New York, Tokyo: Springer-Verlag.Wessel, C. (1799). Om Directionens analytiske Betegning, et Forsøg, anvendtfornemmelig til plane og sphæriske Polygoners Opløsning. Nye Samling af detKongelige Danske Videnskabernes Selskabs Skrifter 5, 469–518.2