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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.

Inhaltsverzeichnis

Frontmatter

The Manuscript

Frontmatter

1. Introduction

Abstract
This essay presents the text and translation, with comments, of a Latin work of Descartes which exists only in a copy, made by Leibniz, but not known until 1860. The manuscript treats two subjects and is notable in several respects, aside from being a work of Descartes.
P. J. Federico

2. History of the Manuscript

Abstract
A few dates and events in the life of Descartes will be noted, before giving an account of the history of the manuscript, a description, and a discussion of its date.
P. J. Federico

3. Description of the Leibniz Copy

Abstract
The Leibniz copy of the Descartes manuscript is in the Niedersächsische Landesbibliothek at Hanover.12 It is written on a double folio sheet folded in two, making four folio pages measuring a little over 20 X 30 cm (8 X 12 in.). Page 1, which starts with the title, is completely filled with writing, from very close to the top to 1 cm from the bottom. Page 2 begins an entirely different topic without any warning mark or word, with the page not entirely full; it has two small figures inserted in the text and a number of closely packed small tables. Page 3 continues the subject of page 2 and the writing occupies less than 60% of the sheet; it ends with a large closely packed table (which is printed on a folding plate in the Oeuvres). Page 4 is blank. The handwriting is small and the lines are crowded; in many places it is quite difficult to make out letters or words. A photographic copy of the manuscript is reproduced pp. 11–21.
P. J. Federico

4. Date of the Original Descartes Manuscript

Abstract
While the date of 1676 for the Leibniz copy is known, the date of the manuscript of Descartes is not known.
P. J. Federico

Solid Geometry: The Elements of Solids

Frontmatter

5. Some Geometric Background

Abstract
This section supplies the necessary basis for statements made and things referred to in the following sections. It introduces and uses some of the terms used by Descartes so that their significance may become clear. The use of terminology and ideas not in the manuscript and not current at the time of the manuscript is avoided as far as possible. Except where otherwise indicated or obvious from the context, the discussion is limited to things which, it is believed, would have been known generally to mathematicians at the time of the manuscript, and to explanations of these things. (In the explanations and derivations we do not limit ourselves to seventeenth century terminology).
P. J. Federico

6. Translation and Commentary, Part I36

Abstract
1 A solid right angle is one which embraces the eighth part of the sphere, even though it is not formed by three plane right angles. But all the angles of the planes by which it is bounded, taken together, equal three right angles.
P. J. Federico

7. General Comments

Abstract
The preceding section discussed the individual paragraphs of the manuscript, primarily from the standpoint of explanation and derivation. The present section offers some general comments and observations.
P. J. Federico

8. Note on the Euler Papers of 1750 and 1751

Abstract
We next give a brief review of Euler’s work on the general theory of polyhedra. Two papers are relevant here: the first, giving some general results and stating his theorem, was read on November 25, 1750, and the second, giving proofs, was read on September 9, 1751. Both were included in the proceedings of the St. Petersburg Academy for the year 1752–1753, which was published in 1758.65 The first paper was preceded by a few weeks by a letter to Goldbach summarizing some of the results it contained.66
P. J. Federico

9. Descartes and Euler

Abstract
Our analysis of and comments on the manuscript in Sections 6 and 7 did not mention Euler’s work, and the summary of Euler’s work in Section 8 did not mention Descartes; this was done in order that all comparative remarks might be considered at the same time.
P. J. Federico

Number Theory: Polyhedral Numbers

Frontmatter

10. The Figurate Numbers of the Greeks

Abstract
The figurate numbers of the Greeks go back to the time of Pythagoras and are frequently referred to by Greek authors. The natural way to represent numbers was by a set of units, shown by dots in sand or by pebbles, and these were arranged in patterns of geometrical figures. In writing, a dot or the letter α would be used for each unit. The figurate numbers and various relations and problems concerning them form a substantial part of Greek number theory (arithmetic). Surviving works which treat figurate numbers are by Nicomachus of Gerasa (c. 100 A.D.), Theon of Smyrna (c. 130 A.D.),99 Diophantus of Alexandria (c. 250 A.D.),100 and Iamblichus (c. 283–330 A.D.).101 Summaries are given by Heath and Dickson, and some relevant extracts by Cohen and Drabkin.102 This section will review some of this material, with some added matter, before presenting the text of the second part of the manuscript, which deals with figurate numbers.
P. J. Federico

11. Translation and Commentary, Part II

Without Abstract
P. J. Federico

12. General Comments

Abstract
The first question to be considered is the indebtedness, if any, of Descartes to Faulhaber.
P. J. Federico

Backmatter

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