Magnificent Principia: Exploring Isaac Newton's Masterpiece

Colin Pask

Publisher:

Prometheus

Publication Date:

2013

Number of Pages:

526

Format:

Paperback

Price:

18.00

ISBN:

978-1-63388-568-4

Category:

Sourcebook

BLL Rating:

BLL

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

MAA Review
Table of Contents

[Reviewed by

Richard J. Wilders

, on

10/13/2019

]

This is a paperback edition of a previously issued hardcover. It is, in my opinion, among the very best of a large collection of works about Newton’s most famous work: Mathematical Principles of Natural Philosophy, often known just as Newton’s Principia. Among those, S. Chandrasekhar’s Newton’s Principia for the Common Reader is most similar to the present work. What distinguishes Pask’s book is its style and tone. Using the most welcoming writing style I have ever encountered in such a work, Pask invites the reader into an encounter with perhaps the most important work of science of all time. Albert Einstein had this to say about Newton:

The whole evolution of our ideas about the processes of nature … might be regarded as an organic development of Newton’s work. (p. 11)

So, why read Pask rather than going directly to the Principia? Because the Principia is (by design, many think) very difficult to read. I spent a fair portion of a sabbatical reading the Cohen and Whitman translation (1999, University of California Press) aided by Cohen’s 370 page “Guide to Newton’s Principia.” While I’m glad I did so, Magnificent Principia offers a gentler, though no less rigorous, guide to the intricacies of Newton’s seminal work.

Pask puts Principia into context, allowing his readers to see how and why Principia was (and is) such an important book. While there are several other books with similar goals, Pask has created a real page-turner: Magnificent Principia is hard to put down. Because of his clever organization (see more below) you don’t need to read the book straight through to profit from it. Clever chapter and section titles encourage one to explore. Here are two section titles: “Big Bodies and Superb Theorems” and “Time and a mathematical gem.” Can you guess what they are about? Don’t you want to know?

Each chapter has an introduction that sets the stage for the mathematics to come as well as a set of concluding remarks which provide a summary of the material in the chapter. Those with some familiarity with Principia could use these to decide where to delve into Magnificent Principia for further information. Reading just the introductions and the summaries provides an intellectually honest introduction to Principia which can be understood with little more than a high school math background. That being said, I think the book deserves to be read in full by anyone with an interest in Newton, even those who have read other similar works.

At the end of each chapter is a collection of suggested further readings. For example, on page 146 we find a list of many of the other guides to reading Newton. Each set of suggested further readings provides a very tempting list of works to explore. In particular, he points the reader toward biographies of the persons discussed, such as Richard Westfall’s classic Never at Rest. All of the recommended works with which I am familiar are very well done — one could do worse than to spend a year or so reading Pask and a sample of his suggested further reading!

Part 1, titled “Introductory Material,” provides a wonderful introduction to Newton and his Principia. Following a very nice short biography of Newton, Pask provides an introduction to the history of science just prior to Newton’s time including the important questions asked and the state of the answers which had thus far been found. Newton himself told us that he stood on the shoulders of giants and Pask introduces us to the most important of those shoulders. He then provides a brief summary of Newton’s plan of attack in Principia. The material in Part 1 would make a wonderful supplement for a history of mathematics course.

Following this, Pask leads us on a well-guided tour through the Principia. Each idea is presented first in conceptual form before we encounter Newton’s formulation and subsequent solution. The general plan of attack for each topic looks like this:

Statement of the problem in informal language including background information as to why the problem was of interest and what was known about its solution prior to Newton.
Newton’s statement of the problem.
Statement of the problem using modern mathematics.
Newton’s typically geometric solution.
Solution using modern mathematics.
Commentary.

It’s possible to obtain a good understanding of what Newton was about without delving into the intricacies of 17th century mathematics. Just skip items 2 and 4 above.

Here is an example: Newton’s Proposition XXXIX Problem XXVII discusses the classic first-year calculus problem of an object moving along a straight line under the influence of a known force, usually given in terms of the acceleration of the object.

Supposing a centripetal force of any kind, and granting quadratures of curvilinear figures; it is required to find the velocity of a body, ascending or descending in a right line, in the several places through which it passes, as also the time in which it will arrive at any place; and vice versa. (page 251)

As is usual for Principia, it’s not immediately obvious what Newton is talking about. Pask begins by formulating the problem in modern terms. Given \(s’’(t)\) and information about the position and velocity at some point in time find a formula for \(s(t)\). Pask then outlines Newton’s approach, including Newton’s figures (drawn throughout from Motte’s 1884 translation as published again by Prometheus Books in 1955). For those unfamiliar with the language of 17th century mathematics Pask offers modern equivalents. As an example, he explains that “granting quadratures of any curvilinear figures” means that Newton is assuming the ability to compute any required integrals. Newton expresses the results of the required integrals as the areas under curves rather than our modern analytic form. That is, Newton treats distance as the area under the velocity curve. Pask follows Newton’s geometric analysis and solution with a modern treatment problem as the solution of a second-order differential equation.

As usual, Newton plays his cards close to the vest, always being fearful of criticism or plagiarism. He does not explain how he obtained his results, saying merely that

All these things follow … by the quadrature of a certain curve, the invention of which, as being easy enough, for brevity’s sake I omit. (page 260)

In modern terms, Newton is telling us he omitted the work required to compute the needed integrals because it is “easy enough.” This led some of his rivals to conjecture that perhaps Newton did not have the techniques he describes as “easy enough.” However, there is evidence that Newton did have the modern techniques by this point. In particular, Pask provides a manuscript page from 1694 (page 261) showing Newton’s solution of possible paths for an object being acted upon by a central force governed by the inverse square law.

Book III of Principia presents Newton’s “system of the world”: his model of the solar system. It includes some really wonderful work which is admirably explicated by Pask. Here are two examples.

Masses of the Planets: While the universal gravity law provides, in theory, all we need to know to deduce the detailed structure of the solar system, the gravity constant, \(G\) was not known in Newton’s time. This meant that the masses of the planets could not be directly computed. Newton got around this issue in much the same way as Archimedes computed areas without using \(\pi\): he expresses the masses of planets in terms of the mass of the sun. To do so requires that we have the orbital data for both the planet and one of its moons. Supposing that \(m_p, T_p, a_p\) are the mass, orbital period and semi major axis of a planet and that \(T_m, a_m\) are the same parameters for a moon, we can deduce (as a consequence of Kepler’s laws, which Newton deduces from his inverse square law) that \[ \frac{m_p}{m_{\text{sun}}} = \frac{T_p^2 a_m^3}{T_m^2 a_p^3}.\] Compare this to Archimedes result that the volume of a cylinder is 3/2 the volume of the inscribed sphere. In each case an unknown quantity (\(G\) for Newton, \(\pi\) for Archimedes) is cleverly avoided.

Center of the Universe: Among the major questions of ancient astronomy was the location of the center of the universe, a question we now consider to be meaningless. Prior to Copernicus, it was assumed by many to be at the center of the Earth. With Copernicus the center moves to the Sun. Newton Changes the question to that of finding the center of the Solar System which he defines based on the following result:

Proposition XI: That the common centre of gravity of the Earth, the Sun, and all the planets is immovable. (page 415)

He concludes: “Hence the common centre of gravity of the Earth, the Sun, and all the planets, is to be esteemed the centre of the world.” (page 415) As we now know the center of gravity is not stationary but, absent any external forces, the solar system moves at constant velocity.

Magnificent Principia ends with a discussion of the reception of and reaction to Principia. This includes a nice summary of several early works which reformulated Newton’s geometric results into the language of calculus. There are more complete treatments of this material, but there is certainly enough here to understand how important Newton’s work was in the development of modern science.

Pask concludes as follows:

The theories that Newton gave us in Principia are often incomplete, formulated in obscure ways, and tied up in models and analogies that only such a genius could invent. History has shown us that the basic ides are correct… The Principia remains a great and simulating document; it is with that book that the science of dynamics truly began. (Page 506)

Magnificent Principia is, and should remain for some time, “a great and stimulating” book itself. It introduces the Principia using an organizational scheme that allows the casual reader to understand what Newton accomplished and why his work still matters. For those willing to work a little harder, Pask offers finely nuanced descriptions of the details in both Newton’s and the modern formulations This book deserves a spot on the shelf of anyone who teaches physics or calculus as well as anyone who wants to understand one of the most important books ever written. Pask has demonstrated in remarkably clear terms the truth of perhaps the most famous of all quotes about Newton:

Nature and Nature’s laws, lay hid in night: God said, Let Newton be! and all was light. (Alexander Pope, as quoted on page 480)

For many the light shed by Principia seems dim and obscure at first reading — Pask shines a spotlight on a work of remarkable depth allowing his readers to explore Principia in whole or in part.

Richard Wilders is Marie and Bernice Gantzert Professor of the Liberal Arts and Sciences and Professor of Mathematics at North Central College in Naperville, IL. He teaches mathematics and its history and has taught courses in the history and philosophy of science. He and his wife enjoy the rich Chicago theater and arts scene as well as watching their beloved Ohio State Buckeyes. rjwilders@noctrl.edu.

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Tags:

History of Mathematics