Vorlesungen über Die Entwicklung Der Mathematik Im 19. Jahrhundert

(German for 'Lectures on the Development of Mathematics in the 19th Century') is a book by Felix Klein that was published posthumously in two volumes (volumes 24 and 25 of ) in 1926 and 1927.

Felix Klein had lectured on the development of mathematics in the 19th century and then on relativity during World War I. The books were created from the notes of these lectures and edited by Richard Courant and Otto Neugebauer for the first volume and Courant and Stefan Cohn-Vossen for the second. Some content that Klein had originally envisioned as part of the text is missing.

The book has been enthusiastically received and widely praised. The first volume has been translated into Russian in 1937 and into English in 1979; in 1989, a second Russian translation appeared, followed in 2003 by a translation of the second volume. Both volumes have also been translated into Chinese.

Background and publication

Felix Klein (1849–1925) was a German mathematician best known for his Erlangen program, which emphasised the use of groups in geometry. From 1886 to 1913 he was professor at the University of Göttingen, which became one of the leading centres of mathematical research under his leadership.

Klein was interested in the history of mathematics and bought relevant books for the Göttingen library. One of his students, , studied for a PhD in history of mathematics under Klein and was later awarded the first habilitation degree in Göttingen in this topic area. Klein was responsible for the mathematical content in an encyclopaedic project called ('The Culture of the Present') published by in Leipzig, a publishing house he had longstanding ties with. The history of mathematics was supposed to be covered in three volumes; H. G. Zeuthen was responsible for the period up to the middle ages and Paul Stäckel for the time from 1500 to 1800. For covering 19th century applied mathematics, Klein tried to convince Heinrich Weber and Carl Runge, but he eventually accepted he had to do it himself.

Klein planned to lecture on the development of mathematics in the 19th century in the winter semester of 1910/11 and again in the winter semester of 1912/13, but both times was unable to do so. The classes were then taught privately during World War I. The winter 1914/15 courses were not announced in the (list of lectures), while the summer 1915 and winter 1915/16 courses were announced as " and free of charge". The first semester of lectures was attended by 24 people, including 13 male students, 9 faculty and 2 women. The women were Iris Runge and Klein's daughter Elisabeth Staiger. The first two semesters of lecture notes were edited and typed by the recently widowed Staiger, while the third course was worked on by Käthe Heinemann and Helene Stähelin. Stähelin's part, completed in Basel in 1918, included figures drawn by Erwin Voellmy. For a few years, the lecture notes were only available as these typescripts. After Klein's death, Richard Courant and Otto Neugebauer edited the notes and published the first volume in 1926 in the Springer Verlag's "yellow series", . In their preface, they explained that they had changed as little as possible in the original text and admitted it was closer to a draft than to a thorough and balanced presentation of the history. While they thanked several other mathematicians for their help, they did not mention the three women who had prepared the typescripts from the original lectures anywhere in the book.

Possibly because the ''Kultur der Gegenwart'' project was running into financial difficulties caused by the war, or because of his personal interests, Klein's lectures from 1916 onwards were concerned with relativity, and he postponed working on content regarding the works of Poincaré and Sophus Lie. Starting in 1916, Klein taught his lectures in his own house to avoid having to walk to the university. The notes were taken by Klein's assistant Walter Baade. Klein's lectures on "selected aspects of newer mathematics" were concerned with Einstein, the special theory of relativity on an invariant basis, and the foundations of general relativity. In 1916, his audience of 14 included the professors Runge and Carathéodory and the Swiss student Paul Finsler. Hilbert's assistant Emmy Noether and Käthe Heinemann were among the five women in attendance, and there were two blind students, Willi Windau and Friedrich Mittelsten Scheid. In winter 1916/17, there were seven attendees: Baade, Richard Bär, Josef Engel, Vsevolod Frederiks, Heinemann, and Windau. Klein sent a version of his summer 1917 lectures on general relativity to Albert Einstein, who dismissed the approach as overemphasising the formal over the heuristic point of view. In 1927, the lectures were published as the second volume of , with the additional subtitle ('The Fundamental Concepts of Invariant Theory and Their Infiltration into Mathematical Physics'). The editors were Courant and especially Stefan Cohn-Vossen, who admitted the fragmentary character of the book in their introduction.

Content

The first volume is organised in eight chapters; the first chapter, simply titled "Gauß", is concerned with the life and work of Carl Friedrich Gauss and split into a section on applied and one on pure mathematics. The second chapter, "France and the École Polytechnique in the First Decades of the Nineteenth Century", contains sections on mechanics and mathematical physics (including the work of Poisson, Fourier and Cauchy), geometry (Monge and his followers), analysis and algebra (focusing on Cauchy and Galois). The third chapter "The Founding of Crelle's Journal and the Rise of Pure Mathematics in Germany" is concerned with mathematicians connected to ''Crelle's Journal'' (the analysts Dirichlet, Abel and Jacobi and the geometers Moebius, Plücker and Steiner), is followed by a chapter on "The Development of Algebraic Geometry after Moebius, Plücker and Steiner", with sections on projective geometry, invariant theory and ''n''-dimensional spaces and general complex numbers, including content on Graßmann and Hamilton. In the fifth chapter, "Mechanics and Mathematical Physics in Germany and England until about 1880", Klein discusses Hamilton's and Jacobi's works on mechanics and works of English mathematicians including Green, Stokes and Maxwell. Chapter 6, "The General Theory of Functions of Complex Variables according to Riemann and Weierstraß" contrasts Riemann's approach to complex analysis with that of Weierstraß, and is followed by Chapter 7, "Deeper Insight into the Nature of Algebraic Varieties and Structures", which discusses algebraic geometry (including contributions by Clebsch and Noether) and algebraic number theory (Kummer, Dedekind and Hilbert, among others). In the final chapter, "Group Theory and Function Theory; Automorphic Functions", Klein discusses first group theory in connection with the works of Lagrange

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