The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue

By Matthew L. Jones | Go to book overview

CHAPTER FIVE
Forms of Expression

“The use of geometry rests in application,” Gottfried Wilhelm Leibniz explained to the famous jurist Hermann Conring—above all, the application of “exercising the intelligence.” Leibniz could attest that geometry “serves to accustom one to the solid and the certain,” for he had pursued mathematics “in France, more perhaps than is necessary for the stretching of the mind.” He had no regrets: “for, from that time in which I attended more diligently to geometry, I began to judge all things a bit more carefully”1 While Leibniz did not explicitly invoke Descartes, the Frenchman, like Plato, had clearly been right about the power of mathematics to exercise the mind.

From the manifold specimens of his mathematical work, Leibniz told Conring of a key preliminary to the invention of his calculus: his quadrature (finding the area) of the circle. In slightly anachronistic terms, Leibniz had demonstrated that

The magnitude of the circle, he wrote, “can most simply be expressed by this series, that is, the aggregate of fractions alternately added and subtracted.” While the series was useful for finding numerical approximates to the area, Leibniz ended his comments by emphasizing, “but this, as I said, is to be considered primarily for the exercising of the intelligence.”2

Leibniz developed the infinite series expressing the area of a circle early in his famously productive sojourn in Paris from 1672 to 1676. He suggested that such written expressions offer the only exact knowledge of the quadrature of the circle available to embodied human beings without

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The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue
Table of contents

Table of contents

  • Title Page iii
  • Contents vii
  • Illustrations ix
  • Acknowledgments xi
  • Abbreviations xv
  • A Note on Conventions xvii
  • Introduction 1
  • Part I - Descartes 13
  • Chapter One - Geometry as Spiritual Exercise 15
  • Chapter Two - A Rhetorical History of Truth 55
  • Part II - Pascal 87
  • Chapter Three - Mathematical Liaisons 89
  • Chapter Four - The Anthropology of Disproportion 131
  • Part III - Leibniz 167
  • Chapter Five - Forms of Expression 169
  • Chapter Six - Seeing All at Once 229
  • Epilogue 267
  • Notes 271
  • Bibliography 329
  • Index 363
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