# Descartes on Polyhedra: A Study of the De Solidorum Elementis (Sources in the History of Mathematics and Physical Sciences)

## P. J. Federico

Language: English

Pages: 145

ISBN: 0387907602

Format: PDF / Kindle (mobi) / ePub

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.

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Comments. The three types of equality may be stated as follows, with respect to the spherical polygons intercepted on the sphere: I. 2. 3. Equal number of sides, equal perimeters, equal areas. In the third type of equality, the most proper one, the solid angles have the same measure. As to the second type, the text states that the solid angles "have an equal inclination." Descartes refers to the sum of the plane angles of a solid angle (the perimeter of the spherical polygon) as the inclination

discussing several unintelligible and obscure passages, followed in the same journal. 8 Thirty years later, in 1890, Vice-Admiral Ernest de Jonquieres published a memoir containing a reprinting of the Latin text as published by Foucher de Careil, together with a "revised and completed" Latin text as he thought it ought to read, with a French translation, commentary and notes. 9 He knew of 2. History of the Manuscript 7 Prouhet I but he was unaware of Prouhet II with its translation. While

number is shown as a square array of unit 88 10. Gnomons o\~ Figure 14 Figure 16 Figure 15 Figure 17 squares with a dot in the center of each and two outer sides shaded, as is sometimes done). The carpenter's square was called a "gnomon" in Greek and is believed to be the origin of the name for these figures. The gnomons in the other three figures are also connected by lines; in the case of the triangles they are simple lines, for the pentagons they have 3 sides and 2 corners, and for the

repraesentent corpus ex 20 triangulis et 12 pentagonis, quoniam gnomon huius corporis constat IS triangularibus facie bus et to pentagonis, minus 4S radicibus, + 21 angulis, primo addot ~ numero t $- +t;lt, qui est ex ponens faciei triangularis, ;l(; quod duco et productum, nempet + l~, duco pert)e+t fitt«. +t per IS et fit 3et + 9} + 6)(. 3- 3-+t 108 II. Translation and Commentary §§30-32 Figure 30 32 Deinde addo etiam t)l. numero t ~ - tit, qui est exponens faciei pentagonaJis, et fitt~,

'radices' (the sides of the gnomon) and' A' for 'anguli' (the vertices). In these tables he also uses 'F' for 'faCies' (the faces). We have again prefixed a column giving n. the order of the number. 113. Pappus, ed. Hultsch V 33-36, II, pp. 350-358; tr. Commandino, pp. 83-84; tr. ver Eeckc, pp. 272-277. Descartes was familiar with Pappus, as he states in the Regulae ad Directionem lngenii (Oeuvres X, p. 376), and parts of the Geomhrie are taken up with the Problem of Pappus. It is evident that he