# The Cambridge Companion to Newton

By I. Bernard Cohen; George E. Smith | Go to book overview

9
Analysis and synthesis in
Newton's mathematical work
NICCOLÒ GUICCIARDINI

The opposition between analytical and synthetic proof methods has an intriguing and complex role in the history of Western mathematics. In Antiquity analytical method (in brief, analysis) was conceived of as a method of discovery, or problem solving: it starts from what is sought as if it had already been achieved, and, working step by step backwards, it eventually arrives at what is known. This and similar rather vague definitions were aimed at describing in a general way a whole apparatus of geometric problem solving procedures developed by the Greeks. Synthesis goes the other way round: it starts from what is known and, working through the consequences, it arrives at what is sought. The axiomatic and deductive structure of Euclid's Elements was the model of the synthetic method of proof. Analysis (or resolutio) was often thought of as a method of discovery preliminary to the synthesis (or compositio), which, reversing the steps of the analytical procedure, achieves the true scientific demonstration. Analysis was thus the working tool of the geometer, but it was with synthesis that one could demonstrate things in an indisputable way. In the Middle Ages this pattern of definitions became bound up with the philosophical and logical tradition. A question which was often raised concerned the relationship between the mathematical proof methods and other accepted forms of deductive proof, typically those codified in Aristotle's Organon.1

Publication in the sixteenth century of new editions of the Greek classics sparked new interest in the analytical method. Most notably, in 1588 Federico Commandino published his Latin translation of the Mathematicae Collectiones, a synopsis of Greek geometry compiled by the fourth-century mathematician Pappus. The attitude of Renaissance culture towards the classics, whether in sculpture,

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The Cambridge Companion to Newton

• Title Page *
• Contents v
• Figures vii
• Contributors ix
• Preface xiii
• Introduction 1
• Notes *
• 1 - Newton's Philosophical Analysis of Space and Time 33
• Notes *
• 2 - Newton's Concepts of Force and Mass, with Notes on the Laws of Motion 57
• Notes *
• 3 - Curvature in Newton's Dynamics 85
• Notes *
• 4 - The Methodology of the Principia 138
• Notes *
• 5 - Newton's Argument for Universal Gravitation 174
• Notes *
• 6 - Newton and Celestial Mechanics 202
• Notes *
• 7 - Newton's Optics and Atomism 227
• Notes *
• 8 - Newton's Metaphysics 256
• Notes *
• 9 - Analysis and Synthesis in Newton's Mathematical Work 308
• Notes *
• 10 - Newton, Active Powers, and the Mechanical Philosophy 329
• Notes *
• 11 - The Background to Newton's Chymistry 358
• Notes *
• 12 - Newton's Alchemy 370
• Notes *
• 13 - Newton on Prophecy and the Apocalypse 387
• Notes *
• 14 - Newton and Eighteenth-Century Christianity 409
• Notes *
• 15 - Newton Versus Leibniz: from Geometry to Metaphysics 431
• Notes *
• 16 - Newton and the Leibniz-Clarke Correspondence 455
• Notes *
• Bibliography 465
• Index 481
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