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Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction

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Sources and Studies in the History of Mathematics and Physical Sciences
[Reviewed by
Fernando Q. Gouvêa
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And here is a history of mathematics book that takes both mathematics and history with deep seriousness. Bos is interested in how the concept of what a "construction" is and what makes a mathematical construction "exact" affected the evolution of mathematics in the early modern period. As at several other points in the evolution of mathematics, this period was marked, among other things, by a "foundational" debate. The question was basically this: when is a curve (or other geometrical object) "known"? The answer, at the time, was the Greek answer: when we can construct it. But that, of course, leads to the question of what counts as a construction. The "Euclidean" answer, that only constructions with lines and circles were acceptable, was clearly not a good answer. But finding an answer to the question was problematic. For example, Omar Khayyam had shown that one could solve a cubic geometrically by intersecting two conic sections. Does this count as a solution of the cubic?

In early modern times, the problem was made more acute by the fact that new curves, related to the study of motion, were being discovered. (From a modern point of view, these curves were solutions of differential equations.) In what sense could these curves be said to be "known"? When is a construction to be considered "exact" (as opposed to approximate)?

Bos argues that these issues had a real impact on early modern mathematics. He also argues that the work of Descartes is a decisive turning point in this story, and therefore he focuses his study on the period from 1590 to 1650, bounded by the publication of Pappus' Collection at one end and by the death of Descartes at the other. A forthcoming book will deal with the rest of the story, up to about 1750.

One interesting point about the question Bos is studying is that it was never solved. Instead, it was discarded, rendered irrelevant by a change in how people understood the existence of mathematical objects. That in itself makes this a fascinating book. Bos tells us in the introduction that he started working on this book in 1977, and it shows: the book is detailed, careful, and most of all interesting. Anyone who wants to do serious historical work on this period will need to read this book, and the rest of us may want to read it too.

Fernando Q. Gouvêa ( is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

Preface. 1. General introduction. 2. The legitimization of geometrical procedures before 1590. 3. 1588: Pappus' "Collection." 4. The early modern tradition of geometrical problem solving; survey and examples. 5. Early modern methods of analysis. 6. Arithmetic, geometry, algebra and analysis. 7 Using numbers in geometry - Regiomontanus and Stevin. 8. Using algebra - Viète's analysis. 9. Clavius. 10. Viète. 11. Kepler. 12. Molther. 13. Fermat. 14. Geometrical problem solving - the state of the art c. 1635. 15. Introduction to Part II. 16. Construction and the interpretation of exactness in Descartes'studies of c. 1619. 17. Descartes'general construction of solid problems c.1625. 18. Problem solving and construction in the "Rules for the direction of the mind" (c. 1628). 19. Descartes' first studies of Pappus' problem (early 1632). 20. The Geometry, introduction and survey. 21. Algebraic operations in geometry. 22. The use of algebra in solving plane and indeterminate problems. 23. Descartes'solution of Pappus' problem. 24. Curves and the demarcation of geometry in the "Geometry." 25. Simplicity and the classification of curves. 26. The canon of geometrical construction. 27. The theory of Equations in the "Geometry." 28. Conclusion of Part II. 29. Epilogue.