Felix Klein’s early contributions to anschauliche Geometrie

IF 0.7 2区 哲学 Q2 HISTORY & PHILOSOPHY OF SCIENCE Archive for History of Exact Sciences Pub Date : 2024-05-25 DOI:10.1007/s00407-024-00329-1
David E. Rowe
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Abstract

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein’s approach had roots in Poncelet’s principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann’s theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker’s work had a strongly qualitative character that imbued Klein’s early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet’s earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie).

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费利克斯-克莱因对anschauliche Geometrie的早期贡献
1873 年至 1876 年间,费利克斯-克莱因发表了一系列论文,这些论文后来被编入《费利克斯-克莱因文集》第二卷(1922 年)的 "anschauliche Geometrie "标题下。本研究不仅试图追踪这部著作的发展历程,还试图将其置于一个更大的历史背景中。在方法论上,克莱因的研究方法源于庞斯莱的连续性原则,但对他影响更直接的是他的老师普吕克和克莱布施。19 世纪 60 年代,克莱布施对黎曼的阿贝尔函数理论中的一些核心思想进行了再加工,从而获得了代数曲线系统的复杂结果,这些结果大多由黑塞和斯坦纳在早些时候发表。这些发现在枚举几何中发挥了重要作用,而普吕克的研究则带有强烈的定性特征,这也是克莱因早期研究的特色。在这些作品中,可以看到一个主题,即真实曲线与曲面之间的相互作用,而这正是它们的变换特性所反映的。19 世纪 70 年代初,克莱因和 Zeuthen 开始探索推导实三次方曲面和四次方曲线所有可能形式的可能性。他们使用的连续性方法让人想起庞斯莱早期的方法。两位作者还依赖于视觉论证,克莱因后来在直观几何(anschauliche Geometrie)的旗帜下推进了这一论证。
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来源期刊
Archive for History of Exact Sciences
Archive for History of Exact Sciences 管理科学-科学史与科学哲学
CiteScore
1.30
自引率
20.00%
发文量
16
审稿时长
>12 weeks
期刊介绍: The Archive for History of Exact Sciences casts light upon the conceptual groundwork of the sciences by analyzing the historical course of rigorous quantitative thought and the precise theory of nature in the fields of mathematics, physics, technical chemistry, computer science, astronomy, and the biological sciences, embracing as well their connections to experiment. This journal nourishes historical research meeting the standards of the mathematical sciences. Its aim is to give rapid and full publication to writings of exceptional depth, scope, and permanence.
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