History of Mathematics: Non-Euclidean Geometry in the 19th Century

Transcription

1 History of Mathematics: Non-Euclidean Geometry in the 19th Century Abstract Title: History of Mathematics Non-Euclidean Geometry in the 19th Century Core Audience: beginning graduate/advanced undergraduate: pure Course Format: extended (10 hours at 2 hours per week) Course description Key words: Riemannian metrics on a surface, intrinsic curvature, isometric group actions, geodesic, primary and secondary sources. 1 The course gives an introduction to non-euclidean geometry through its history, and to the methods for conducting research in the history of modern mathematics. No previous knowledge of either subject is assumed. The discovery of a possible geometry of space that differed from Euclid s was one of the great breakthroughs of the 19th century, and it ushered in a transformation not only of geometry but our ideas about the truth of mathematics. We study the origins of non-euclidean (or hyperbolic) geometry in the work of Bolyai, Lobachevskii, and Gauss in the 1830s, then look at the successful and much more rigorous accounts given by Riemann and Beltrami in the 1850s and 1860s, and conclude by learning how to use the simple but powerful techniques introduced by Poincaré in the 1880s. We meet the fundamental notions of Riemannian geometry, including intrinsic (Gaussian) curvature, and the idea of groups acting isometrically on a manifold. At the same time we look in detail at several original sources, discuss the nature of sources (primary and secondary; good, bad, and even contradictory), and consider how to write a historical essay. There are opportunities to discuss problems, both mathematical and historical. The final assessment consists of writing a 2,000 word essay on a historical subject that brings together the topics and skills presented in the course.

2 2 Format Six lectures and four discussion classes focussed on problems (mathematical and historical). Original sources in translation will be provided in class and electronically most are accessible through digital libraries on the web. Recommended reading: Jeremy Gray, Worlds out of nothing, 2nd ed. 2011, Springer Stillwell, J. 1996, Sources of hyperbolic geometry, American and London Mathematical Societies, HMath 10. Lecturers: June Barrow-Green and Jeremy Gray. Lecturers home institution: Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA Notes Week One Lecture 1: Introduction to the history of non-euclidean geometry. A very quick look at the pre-history from Euclid to Lambert gives a hint of the long history of the subject. Lambert s work in the 1760s is an indication that belief in the truth Euclidean geometry was beginning to weaken. An overview from Bolyai and Lobachevskii via Riemann to Poincaré indicates the three phases of the discovery of non-euclidean geometry in the 19th century. First, we look at the unsuccessful attempts of Bolyai in Hungary and Lobachevskii in Russia in the 1830s, and offer a quick comparison with Gauss s unpublished ideas. Second, we describe the rigorous accounts by Riemann and Beltrami independently, grounded in the techniques of Gaussian and Riemannian differential geometry. Third, we look at the conformal disc model proposed by Poincaré, and the clear introduction of isometric group actions. Lecture 2: Group work on the texts below reading and class discussion.

3 3 1. Schweikart s memo to Gauss 2. Gauss on Janòs Bolyai s Appendix 3. Lobachevskii s theory of parallels Opening Remarks 4. Correspondence between Wolfgang and Janòs Bolyai 5. Riemann s Lecture On the hypotheses which lie at the foundations of geometry 6. Beltrami s Essay on the interpretation of geometry 7. Poincaré on the disc model of non-euclidean geometry Texts will be available in class and electronically. After a period for reading the texts, which will be divided among groups of students, students will discuss what they have read, what each text seems to say, and problems that the texts raise. Find a text, either online or in library, to help with course, e.g. Gray Worlds out of Nothing. If not the latter, let us know what you have found. Week Two Lecture 3: Context and content: The work of Bolyai, Lobachevskii, Gauss. We will concentrate on Lobachevskii s account of 1840 (available in English translation on the web) and which is (famously) similar to Bolyai s independent account. Lobachevskii gave an account of a new geometry (different from Euclidean geometry) on spaces of two and three dimensions, and we shall consider how it works, what is convincing about it and what is not, and why it failed in Lobachevskii s lifetime. We shall also examine Gauss s claims to have discovered non-euclidean geometry before Bolyai and Lobachevskii, and his role in promoting their discoveries. Lecture 4: Discussion of historical sources: Historians distinguish between primary and secondary sources. Primary sources, which may be published or unpublished, are original records of what was said or discovered. They include letters, notes, and published papers

4 4 and books, and may include accounts written by biographers and historians as well as participants. Secondary sources, which may be good, bad, and contradictory, are accounts written by others. They include reports, obituaries, and historians accounts and analyses. The use of sources is a delicate matter in the history of mathematics. Primary sources are not an unvarnished record of the past. They can be partisan, selective, and misleading in many ways, but they offer the best evidence of what took place. They may, of course, be lost. Secondary sources should be based in primary sources, and contain elements of interpretation. They may be our only source of some information, and they may be inaccurate, either wilfully or inadvertently. They are likely to contain an element of reading between the lines of primary sources, and they may make more sense. A choice of sources will be offered for discussion, and will be available in class and electronically. Read texts by Gauss (his introduction of the concept of intrinsic curvature), Riemann s On the hypotheses which lie at the bases of geometry, and Beltrami s Essay on the interpretation of non-euclidean geometry (see the list of sources below). Week Three Lecture 5: The discoveries of Riemann and Beltrami changed the story of non- Euclidean geometry. In 1854, as part of his post-doctoral examination, Riemann proposed a remarkable extension of Gaussian differential geometry to spaces of any dimension. Independently, knowing only the outlines of Riemann s work, which was not published until 1867, Beltrami proposed a similar account (with a different metric) in These publications, and the posthumous publication of Gauss s notes on the subject, persuaded mathematicians but not philosophers to accept the new geometry. We examine the historical context: Riemann s relationship to Gauss at Göttingen, ideas about surfaces of constant curvature, and what little was known by then of the work of Bolyai and Lobachevskii. We also consider the fundamentals of Gaussian differential geometry, the concept of a metric on a surface, and the grounds for accepting the accounts by Riemann and Beltrami as rigorous Lecture 6:

5 5 Time will be divided between two class discussions, one on the mathematics in the texts by Riemann and Beltrami, and the other on the mathematical problems they raise other than learning the mathematics per se. There are issues about the interpretation of mathematical arguments in context, and how to write about it. The mathematics of the Poincaré disc model: Read a text in which it is introduced (there are several, including Gray Worlds out of nothing, Chapter 25) and work through enough mathematical questions to be able to bring problems and answers next week. Week Four Lecture 7: We introduce Poincaré in his historical context, give a brief description of his career, and indicate the main conclusions of his long involvement with non-euclidean geometry. These include his conjecture of the uniformisation theorem, which says that almost all Riemann surfaces are quotients of the non-euclidean plane, and his ideas about human cognition and our knowledge of geometry (his philosophy of geometrical conventionalism). Lecture 8: The Poincaré disc model allows us to do non-euclidean geometry in a particularly clear way, to state precisely how geodesics in the space are represented, to prove that the angle sum of non-euclidean triangles is less than π, to define isometries in the space, to derive the formulas of hyperbolic trigonometry discovered by Bolyai and Lobachevskii, to define the equidistant curve to a non-euclidean geodesic, to define the horocycle, and much more. In three dimensions, the Poincaré ball allows us to define horocycles, prove that the Euclidean plane can be isometrically embedded in non- Euclidean space, and to give a rigorous reconstruction of Lobachevskii s arguments. We will raise questions about groups and geometry: examples of groups and isometries, the congruence theorems of classical Euclidean geometry. Week Five Lecture 9:

6 6 Groups of isometries are fundamental in geometry. In this lecture we briefly indicate some properties of the isometry groups of Euclidean and non-euclidean geometry in two and three dimensions. Lecture 10: Reprise of Lecture 1: Why non-euclidean geometry was so exciting and important, and why it still is. Assessment Write an essay of about 2000 words on one of the following: 1. Lobachevskii, N.I Geometrical researches on the theory of parallels; 2. Beltrami, E Essay on the interpretation of non-euclidean geometry ; 3. Poincaré, H. On the Foundations of Geometry, Monist 9, LTTC Sources Beardon, A.F The geometry of discrete groups, Springer. Beltrami, E Saggio de interpetrazione della Geometria non-euclidea, Giornale de Matematiche 6, , in Opere matematiche, vol. I, nr. 24, The Internet Archive and Gallica offer this in Italian. English translation Essay on the interpretation of non-euclidean geometry, in (Stillwell 1996, 7 34), extract in F&G Bolyai, Wolfgang and Janòs, Correspondence, in F&G Bonola, R Non-Euclidean geometry, repr. Dover The book also contains Halsted s English translations of Bolyai s Appendix, Lobachevskii s Geometrical researches on the theory of parallels, Gauss on Bolyai s Appendix, and Schweikart s memo to Gauss. Ewald, W.B. (ed.) From Kant to Hilbert: a source book in the foundations of mathematics, Clarendon Press, Oxford. Fauvel, J. and Gray J.J (F&G) The History of Mathematics: a Reader, Macmillan and Open University. Gauss, C.F on Janos Bolyai s Appendix in (Bonola 1912, 100) and F&G Gauss, C.F. General investigations of curved surfaces of 1827 and 1825 ; transl. with notes and a bibliography by J.C. Morehead and A.M. Hiltebeitel, The Princeton University Library, 1902, available through the Digital Mathematics Library from Michigan. Repr. P. Dombrowski, 150 years after

7 7 Gauss disquisitiones generals circa superficies curvas, astérisque 62, 1979 with a new introduction. Repr. P. Pesic (ed.) General investigations of curved surfaces, Dover Lobachevskii, N.I Geometrical researches on the theory of parallels, available through the Digital Mathematics Library. Note that the DML says 1914, the date of the English translation. Nikulin, V.V. and Shafarevich, I.R Geometries and groups, Springer. Poincaré, H Théorie des groupes fuchsiens, Acta Mathematica, 1, 1 62, in Oeuvres 2, English translation in Poincaré, H Papers on Fuchsian Functions, transl. J. Stillwell, Springer, , extract in (Stillwell 1996, ). Poincaré, H On the Foundations of Geometry, Monist 9, 1 43, available on the Internet Archive (search for Monist and find volume 9). Repr. in (Ewald 1996, vol. 2, ). Riemann, G.B.F. 1854/1867 Über die Hypothesen, welche der Geometrie zu Grunde liegen, K. Ges. Wiss. Göttingen 13, 1 20, in Bernhard Riemanns gesammelte Mathematische Werke und wissenschaftliche Nachlass, R. Dedekind and H. Weber (eds.), with Nachträge, M. Noether and W. Wirtinger (eds.), 3rd ed. R. Narasimhan (ed.), Springer, New York, 1990, st ed. R. Dedekind, H. Weber (eds.) Leipzig Transl. Clifford, W.K. 1873, On the hypotheses which lie at the bases of geometry, Nature, repr in Mathematical papers, 55 71, Chelsea reprint, New York, Available through the Digital Mathematics Library from Michigan. Also in Riemann Collected Papers, transl. R. Baker, C. Christenson, H. Orde, 2004, Schweikart, K.F Memo to Gauss, in (Bonola 1912, 76) and F&G 522. Stillwell, J. 1996, Sources of hyperbolic geometry, American and London Mathematical Societies, HMath 10. Stillwell, J Geometry of surfaces, Springer.