Until the 17th century, rigor and exactness in mathematics meant geometry and Euclid. Other means of confirming results, such as computation, were considered inferior to the traditional constructions using ruler and compass. In 1637 Descartes introduced what is now called analytical geometry, which made algebraic methods equal to geometry in the methods of mathematics. In this detailed study, Bos explores the origins of what is meant by "rigor" in mathematics, and how that definition evolved to include the use of new geometric and algebraic methods.

1 General introduction.- 2 The legitimation 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.- 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 demarcationof 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.- List of problems.- Name Index.