Descartes on Polyhedra: A Study of the "De Solidorum Elementis"The present essay stems from a history of polyhedra from 1750 to 1866 written several years ago (as part of a more general work, not published). So many contradictory statements regarding a Descartes manuscript and Euler, by various mathematicians and historians of mathematics, were encountered that it was decided to write a separate study of the relevant part of the Descartes manuscript on polyhedra. The contemplated short paper grew in size, as only a detailed treatment could be of any value. After it was completed it became evident that the entire manuscript should be treated and the work grew some more. The result presented here is, I hope, a complete, accurate, and fair treatment of the entire manuscript. While some views and conclusions are expressed, this is only done with the facts before the reader, who may draw his or her own conclusions. I would like to express my appreciation to Professors H. S. M. Coxeter, Branko Griinbaum, Morris Kline, and Dr. Heinz-Jiirgen Hess for reading the manuscript and for their encouragement and suggestions. I am especially indebted to Dr. Hess, of the Leibniz-Archiv, for his assistance in connection with the manuscript. I have been greatly helped in preparing the translation ofthe manuscript by the collaboration of a Latin scholar, Mr. Alfredo DeBarbieri. The aid of librarians is indispensable, and I am indebted to a number of them, in this country and abroad, for locating material and supplying copies. |
Contents
Introduction | 3 |
History of the Manuscript | 5 |
Description of the Leibniz Copy | 8 |
Facsimile of the Manuscript | 11 |
Transcription of the Manuscript | 22 |
Date of the Original Descartes Manuscript | 30 |
Some Geometric Background | 35 |
Translation and Commentary Part I | 43 |
Descartes and Euler | 72 |
Addendum to Section 9 | 75 |
The Figurate Numbers of the Greeks | 83 |
Translation and Commentary Part II | 92 |
General Comments | 116 |
Notes | 121 |
| 137 | |
| 143 | |
Other editions - View all
Descartes on Polyhedra: A Study of the De Solidorum Elementis P. J. Federico No preview available - 2011 |
Common terms and phrases
algebra analogy angle sum formula angulorum planorum angulum Archimedean solids arithmetical series base Branko Grünbaum convex polygons convex polyhedra corners cossic symbols cube derived Descartes manuscript dodecahedron double sides equal equation Euler's formula Euler's theorem ex omnibus angulis exterior angles exterior solid angles faciebus facies Faulhaber figurate numbers Foucher de Careil Géométrie geometry given gives gnomon hence hexagonal Jonquières later latera Leibniz mathematical minus notes number of dots number of edges number of faces number of plane number of sides number of solid numero angulorum numerus facierum octahedron Oeuvres omnes omnibus angulis planis Paragraph pentagonal number plane angles plane figures polygonal numbers polyhedral numbers polyhedron proof Proposition Prouhet pyramidal numbers quadratis quam radix refers regular polygons regular solids Rhombicuboctahedron Section sentence solid body solidi sphaera sphere spherical polygon square tetrahedron topology translation triangular faces triangular number triangulis truncated truncated octahedron truncated tetrahedron vertices
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