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Descartes's Mathematical Thought

Chikara Sasaki
Kluwer Academic Publishers
Publication Date: 
Number of Pages: 
Boston Studies in the Philosophy of Science 237
[Reviewed by
Annette Imhausen
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René Descartes is considered to be a central figure not only in one but several disciplines, including philosophy, physics and mathematics. In regard to the latter, he is also known as the “father of modern mathematics”, and the Cartesian coordinates are named after him. Yet his writings in mathematics are not especially numerous — in the strictest sense, there is only one, and it is not an opus of many pages.

La Géométrie was first published in 1637 (this 1637 edition is the focus of chapter 5) as one of three appendices to his major work Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences. In this first edition, La Géométrie only had 118 pages (pp. 295–413). Sasaki sketches its immediate reception, especially the debate with Jean Beaugrand, who had accused Descartes of plagiarizing several other mathematicians, e.g. Viète and Harriot (p. 235 ff.), an accusation which may have led to a delay in the work’s success. La Géométrie proved influential enough, however, to merit a translation into Latin (by Frans van Schooten) and the addition of various commentaries (1st edition 1649, 2nd edition in two parts in 1659 and 1661). The first volume of the second Latin edition comprised more than 500 pages and was the one that Newton read as an undergraduate at Cambridge in 1664 (p. 275).

Sasaki begins his study of Descartes earlier, however, with his education at the Jesuit college La Flèche (chapter 1), where he studied from 1607 to 1615 (p.13), followed by an account of the mathematics of Christoph Clavius (chapter 2), which influenced the mathematical education in Jesuit colleges (p. 45), and finally of Descartes’s meeting with John Pell (p. 46–47). Chapters 3 and 4 describe Descartes’ predecessors’ attempt to restructure the system of mathematics — a goal Descartes would ultimately achieve with La Géométrie.

The second part of Sasaki’s book focuses on the concept of “mathesis universalis” – looking at the concept in Aristotle (chapter 6), and then in the 16th (chapter 7) and 17th (chapter 8) centuries — in an effort to place Descartes’ contribution within its cultural context.

Numerous books and articles about the mathematical and scientific contributions of Descartes have been published in history and philosophy of mathematics, most notably the recent publication by Henk Bos, Redefining Geometrical Exactness : Descartes' Transformation of the Early Modern Concept of Construction (Springer, 2001). What is it, then, that distinguishes the book under review from other publications?

Descartes’s Mathematical Thought is the revised and enlarged version of the author’s dissertation, which earned him a Ph.D. from Princeton University in 1989. He began the work for this dissertation in 1978 (p. vii). Thus this book is the outcome of about thirty years worth of research. The diligence that went into its creation is obvious at first glance: numerous footnotes provide references to a wide variety of primary sources as well as secondary literature, and the bibliography comprises more than 164 primary sources.

My impression is that the focus of Sasaki’s work is not so much an overview or a critical analysis of Descartes’s mathematics, as an effort to place him within his historical and cultural context. This has been attempted by collecting what Sasaki considers possible relevant primary sources and it succeeds in establishing a very detailed account of Descartes and his contemporary culture. It is this level of detail that makes the book valuable. While the dense detail makes the book a demanding read, the time and energy spent with it are likely to be rewarding.

Annette Imhausen is an historian of Egyptian mathematics. She is currently a research fellow at the University of Cambridge.



List of Abbreviations and a Note on the Quotation and Translation.

Introduction. René Descartes and Modern European Mathematics.

I: The Formation of Descartes's Mathematical Thought.

1. Descartes and Jesuit Mathematical Education. 1. Descartes and the Jesuit College of La Flèche. 2. The Curriculum at La Flèche. 3. Mathematical Studies in the Ratio Studiorum 4. Motives for the Teaching of Mathematics in the Jesuit Colleges.

2: The Mathematical Thought of Christoph Clavius. 1. Descartes and Clavius. 2. The Philosophy of Mathematics of Clavius. 3. Pappus in the Works of Clavius. 4. Diophantus in the Works of Clavius. 5. Descartes's Mathematical Background before the Encounter with Beeckman.

3: The First Attempt at Reforming Mathematics. 1. "An Entirely New Science": The Idea for the Unification of Arithmetic and Geometry. 2. The Mathematics in the Cogitationes privatae. 3. The De Solidorum Elementis. 4. Descartes's Mathematical and Philosophical Dream of 1619.

4: The Mathematical Background of the Regulae Ad Directionem Ingenii. 1. The Old Algebra: The First Fruit of "An Entirely New Science". 2. The Mathematics in the Regulae ad Directionem Ingenii. 3. Mathesis Universalis .

5. The Géométrieof 1637 . 1. The Pappus Problem. 2. The Composition of the Géométrie. 3. Descartes's Place in the Formative Period of the Modern Analytic Tradition. 4. Beyond Cartesian Mathematics. Interim Consideration. Descartes and the Beginnings of Mathematicism in Modern Thought.

II: The Concept of `Mathesis Universalis' in Historical Perspective.

6. `Universal Mathematics' In Aristotle. 1. Aristotle's Metaphysics and Posterior Analytics. 2. Greek Commentators: Alexander of Aphrodisias and Asclepius of Tralles. 3. Medieval Commentators: Ibn Rushd (Averroës), Albertus Magnus, Thomas Aquinas, and the Scotist Antonius Andreae. 4. Renaissance Commentators: Agostino Nifo and Pedro da Fonseca. 5. The Status of Mathematics in the Aristotelian Scheme of Learning.

7: `Mathesis Universalis' in the Sixteenth Century. 1. Proclus Diadochus and Francesco Barozzi. 2. Adriaan van Roomen.

8: `Mathesis Universalis' in the Seventeenth Century. 1. Reviewing Descartes's Concept of `Mathesis Universalis' from His Philosophy of Mathematics. 2. The Leibnizian Synthesis.

Conclusion: Descartes and the Modern Scheme of Learning.