The recognition and the constitution of the theorems of closure
type: Item
instance of: scholarly article
title: The recognition and the constitution of the theorems of closure
zbMATH Open document ID: 1408.01007
author: François Lê
published in: Historia Mathematica
publication date: 2018-09-07T00:00:00Z
review text: 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 aut…
reviewed by: Karl Heinz Schlote
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 01A85
Mathematics Subject Classification ID: 14-03
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6934128
zbMATH Keywords: history of geometry
zbMATH Keywords: closure theorems
zbMATH Keywords: geometric theorems
zbMATH Keywords: porisms
zbMATH Keywords: categorization
MaRDI profile type: Publication
full work available at URL: j.hm.2018.06.001
OpenAlex ID: W2813136201
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DOI: 10.1016/J.HM.2018.06.001
description: scientific article; zbMATH DE number 6934128