# Newton’s Principia:

The Central Argument

## Translation, Notes, and Expanded Proofs

### Notes and expanded proofs by Dana Densmore

Translations and diagrams by William H. Donahue

###### 7" x 10", 589 pages, more than 300 diagrams, index and bibliography

Publication date: October 2003.

For pricing and ordering information, see the ordering section below.

Newton's Principia: The Central Argument makes the great adventure of Principia available not only to modern scholars of history of science, but also to nonspecialist undergraduate students of humanities. It moves carefully from Newton's definitions and axioms through the essential propositions, as Newton himself identified them, to the establishment of universal gravitation and elliptical orbits.

The guidebook unfolds what is implicit in Newton's words as he himself would have filled in the steps and completes the argument in ways that are authentic and not anachronistic, exactly following Newton's thinking rather than substituting tools of modern calculus or the formulations of modern physics. It is Newton in his own terms, allowing students to reconstruct Newton's propositions authentically. It is not a commentary or a presentation of Newton's propositions as they might appear in a modern textbook. Rather, this guidebook unfolds what is implicit in Newton's words as he would have filled in the steps, while completing the argument in ways that are not anachronistic.

### Third Edition Features

The third edition of this guide to Newton’s magnum opus has been completely redesigned for ease of study. Newton’s text appears in color with a distinctive antique font, while the notes and expanded proofs are set in the highly-legible Stone Sans typeface.

In response to requests by numerous readers, this new edition has been significantly expanded to include and discuss additional portions of the Principia. Among these additions are:

- Notes to Newton’s Definitions and Laws of Motion
- Proofs for the Corollaries to the Laws of Motion
- Complete Scholia to the Definitions and Laws
- The Three-Body Problem

In this edition, the treatment of several propositions and lemmas has been made more straightforward. In general, wherever readers have reported difficulties, notes have been reworked to respond to their needs. The result is a major revision with many improvements.

Newton's Principia: The Central Argument presents Newton's original text (the selections newly translated for this edition), offers notes and questions for pondering, and then expands Newton's sketched proofs step by step. Following his original proofs exactly eliminates the common confusions and misinterpretations of what Newton assumed and what he proved in the course of the development of his great work.

Densmore's painstaking reconstruction of Newton's original thought processes makes this work a significant contribution to Newtonian scholarship. Most works of Newtonian scholarship from his time through the present have bypassed the difficulty of true reconstruction by translating Newton's proofs into algebra and modern calculus. This misses the essence of Newton's masterpiece (he deliberately chose not to use algebra or calculus) and sometimes leads to outright mistakes. Readers and scholars who want to know what Newton really said, as opposed to how one might prove the same things in a different way, will find the full proofs nowhere else.

Dana Densmore discussing the distinction between inherent force and applied force.

Photo by David Trozzo.

### Reviews

Densmore's commentary has a directness, an intelligence and infectious energy that takes readers through all the difficulties to a very satisfying accomplishment... I cannot emphasize too strongly what an achievement it is.

— Curtis Wilson, Editor

General History of Astronomy

This is a wonderful book. Taking Newton in his own terms, it insists on the full rigor of the demonstrations and does not hesitate to point out where full rigor appears to be lacking. The flavor of the book can be sampled in its treatment of the phenomena cited at the beginning of Book 3, where Densmore pauses to explain in what sense generalizations not directly observable can be called phenomena (for example, Kepler's third law applied to the satellites of Jupiter) and how the data for them were collected in the late seventeenth century. ... As she says in the Preliminaries, 'we understand Newton only in understanding why he proved things as he did' (p. xxiv). Students are not the only ones who can profit from the exercise.

— Richard S. Westfall, Indiana University

Author of Never at Rest: A Biography of Isaac Newton

Review in Isis

(Isis is the journal of the History of Science Society, published by the University of Chicago Press).

The stress is on encouraging students to reconstruct Newton's proofs in their original geometric form, rather than translating them into the more familiar symbolic calculus. This is particularly interesting because Newton's geometric style informs our geometric and physical intuition in a way which is complementary to the understanding achieved via analytical tools. Historians of science have a great deal to learn from it. A first class work.

— Niccolo Guicciardini

Mathematical Reviews, Issue 99

READ MORE by viewing or downloading a PDF file with more reviews.

### Sample pages

Below are links to PDF versions of the Table of Contents, Foreword and Book I Proposition 1.

You may need to open these PDF documents in Adobe Reader or an equivalent program.