The recognition and the constitution of the theorems of closure
The article is a detailed investigation of a class of theorems centralized and unified under the name of ``theorems of closure'' (Schließungssätze). Starting with Poncelet's theorem on polygons inscribed in a conic and circumscribed about another conic as well as their prehistory the author follows the development of this class of problems during the 19th century up to the first decade of the 20th century. He wants to show how these diverse geometric theorems, that were primarily considered to be disconnected from each other, have been progressively gathered and unified as a larger class of problems and theorems labelled as ``Schließungssätze'' or ``Schließungsprobleme''. The author speaks of the category of the theorems of closure and his analysis is concentrated on the process of emerging of this category. He divides the process into different stages: (i) several theorems proved by J.-V. Poncelet, J. Steiner, C. G. J. Jacobi in the first half of the 19th century and A. Cayley, A. Clebsch, as well as some others in the 1850ies and 1860ies, (ii) the first mentions of a certain class of problems and theorems, called ``Schließungsprobleme'', by A. Clebsch, J. Rosanes, M. Pasch, and E. Weyr in the short period between 1864 and 1870, (iii) the search for unifying technical frameworks of the theorems of closure represented by articles of F. August, M. Simon and A. Hurwitz in the second half of the 1870ies, and (iv) the forming of the category of closure theorems by the publication of many works revisiting the already-known theorems of closure or creating and investigating extensions of these theorems. This last stage ends with the contributions of M. Zacharias, F. Dingeldey, O. Staude, and G. Kohn to the ``Encyklopädie der mathematischen Wissenschaften'' at the beginning of the 20th century. The articles of these mathematicians contained one or more sections regarding problems of closure or theorems of closure. In order to contour the domain of his investigations the author used the references given by the above mentioned mathematicians in their articles in the ``Encyklopädie'' as well as by \textit {M. Simon} in the section ``Schließungsproblem'' of his book on the history of elementary geometry [in: Über die Entwicklung der Elementar-Geometrie im XIX. Jahrhundert. Leipzig: B. G. Teubner. 108--109 (1906)]. In this way, a solid basis of 130 texts written by 87 authors has been created. The author of the paper under review gives a thorough analysis of the four stages. He characterizes lucidly the different approaches to a unification of the various problems of closure and finishes with some theoretical reflections upon the formation process of the category of closure theorems. The latter is referred to Wittgenstein's concept of ``family resemblance'' and strictly distinguished from the classical view on categories as defined by a set of essential features that were shared by all members of the category.
scientific article; zbMATH DE number 6934128
scientific article