Abstract
Felix Klein’s formative years constitute a famous chapter in the history of mathematics, especially familiar because Klein himself wrote about it often. Later writers have often highlighted his collaboration with Sophus Lie and the ideas that led to Klein’s “Erlangen Program.” Klein re-packaged his early work when he edited his collected works, a project that engaged his attention from 1919 to 1923. By unpacking its first volume, we can begin to appreciate that transformations groups formed a relatively small part of Klein’s early geometrical work, whereas a great deal of it was devoted to important new results in line geometry.
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Notes
- 1.
For those with an interest in connections between the ideas in the Erlangen Program and mathematical physics, see the essays in Ji and Papadopoulos (2015).
- 2.
In my view, historians should avoid the temptation to adjudicate past contests—such as the “race” between Einstein and Hilbert for generally covariant gravitational field equations that was taken up in the late 1990s—nor should they make lightly considered retrospective value judgments about the status of past work.
- 3.
For an enlightening study of Newton’s mathematical views and methods, see Guicciardini (2009). Until well into the twentieth century, mathematicians and historians of mathematics often took sides in the Newton vs. Leibniz story, and much of this literature reflects strongly nationalistic inclinations toward hero worship. A more objective attitude gradually emerged after the Second World War, however, when a great deal of relevant source material came to light. Even so, the Newton scholar A.R. Hall expressed great dismay over some of the opinions expressed by Joseph Ehrenfried Hofmann in Hofmann (1974), the standard account of Leibniz’s intellectual journey leading to the calculus (Hall 1980, 65–67). Today, one can easily study Newton’s early works in D.T. Whiteside’s monumental editions of The Mathematical Papers of Isaac Newton in eight volumes (Whiteside 1967–1981), whereas the editors of the Leibniz edition in Hanover continue to turn out new volumes of his mathematical manuscripts.
- 4.
Without naming sources or dates, Stubhaug’s study cites passages from several letters from Lie to Klein, in which he vented his anger over Killing’s work and especially Engel’s role in corresponding with Killing; several of the letters cited can be found in Rowe (1988).
- 5.
Klein signaled the new ones by writing them in square brackets.
- 6.
Hilbert was still alive and so had the opportunity to read the three volumes of his papers (Hilbert 1932–1935), but he took no part in preparing that edition.
- 7.
As he told me in an interview conducted in Lugano in September 1984.
- 8.
See Konrad Jakobs and Heinrich Utz, Erlangen Programs, Mathematical Intelligencer 6(1)(1984): 79.
- 9.
He had to add a footnote in 1893, when the text of Klein (1872/1893) was reprinted, pointing out that one needed to stipulate that the group contains its inverse transformations.
- 10.
For a brief account, see Stubhaug (2002, 386–389), which however fails to engage with many aspects of this conflict.
- 11.
Lie was reacting to Klein’s interest in launching the Encyklopädie der mathematischen Wissenschaften, but he likely also had in mind his recently published Evanston Colloquium Lectures.
- 12.
- 13.
On the early reception of non-Euclidean geometry in Germany, see Volkert (2013).
- 14.
“Haben Sie das Plückersche Werk gesehen, welches unter meinen Auspicien in die Welt gegangen ist? Schöne Gedanken, aber welche Darstellung!” (Confalonieri 2019, 73).
- 15.
On his family background and youth, see Wiescher (2016).
- 16.
The notions of order and class refer here respectively to the number of singular lines passing through a generic point and the number that lie in a generic plane.
- 17.
A student who attended wrote up a detailed report on these models in the Göttingen collection (see modellsammlung.uni-goettingen.de).
- 18.
- 19.
In 1870, Ferdinando Aschieri showed that the Battaglini complex could be viewed geometrically as the family of lines that intersect two quadric surfaces harmonically, thus, in four points with cross ratio equal to \(-\)1; see Rowe (2016, 246).
- 20.
Klein used the terminology of complexes that lie in involution with one another, but Hudson found this language awkward and I here follow Hudson (1990, 38).
- 21.
Actually, Plücker freely mixed projective and metric concepts, whereas Klein belonged to a younger generation of geometers who paid careful attention to this distinction.
- 22.
Asymptotic curves on surfaces are those along which the tangent plane and osculating plane coincide; see Struik (1961, 96).
- 23.
The incident that led to this declaration of war bears a striking resemblance to Putin’s demand that Ukraine shall never be allowed to join NATO. France insisted that the Catholic branch of the Hollenzollerns, which had declined an offer to assume the throne of Spain, should declare that this decision was valid for all time. Bismarck famously took full advantage of this diplomatic blunder.
- 24.
Klein to Lie, 29 July 1870, quoted from Rowe (2019, 190).
- 25.
This was duly noted by Engel and Heegaard in their notes on this paper; see Lie (1934, 674).
- 26.
The claim is correct if one assumes the common asymptotic tangents are also tangents to the curve along which the surfaces touch.
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Acknowledgements
Among his many other contributions to the history of mathematics over the course of a distinguished career, Jeremy Gray has also contributed a great deal toward enriching our understanding of major parts of Felix Klein’s mathematical work. He gave prominent attention to Klein’s papers on algebra and function theory from the late 1870s and early 1880s in Linear Differential Equations and Group Theory from Riemann to Poincaré (Gray 2008), which was first published in 1986. For those who read German, I would also recommend Erhard Scholz’s history of the emergence of the concept of manifolds (Scholz 1980), as these two studies complement one another very nicely. More recently, Jeremy has made the case for the importance of Klein’s contributions to Galois geometry (Gray 2018, 171–188; Gray 2019). Alongside these works, he also dared to write an intellectual biography of Henri Poincaré (Gray 2013), thereby filling one of the yawning gaps in the literature. Having benefited from these works and many others over the course of my career, it gives me great pleasure to take part in this celebration of Jeremy’s 75th birthday.
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Rowe, D.E. (2023). On Felix Klein’s Early Geometrical Works, 1869–1872. In: Chemla, K., Ferreirós, J., Ji, L., Scholz, E., Wang, C. (eds) The Richness of the History of Mathematics. Archimedes, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-031-40855-7_6
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