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The Key to Newton's Dynamics

University of California Press

Isaac Newton's Philosophiae Naturalis Principia Mathematica (The mathematical principles of natural philosophy), hereafter referred to as the Principia , justifiably occupies a position as one of the most influential works in Western culture, but it is a work more revered than read. Three truths concerning the Principia are held to be self-evident: it is the most instrumental, the most difficult, and the least read work in Western science. A young student who passed Newton on the streets of Cambridge is reported to have said, "There goes the man who writ the book that nobody can read." It fits Mark Twain's definition of a classic as a work that everyone wants to have read but that nobody wants to read. The essential core of the Principia , however, does not lie beyond the reach of any interested and open-minded individual who is willing to make a reasonable effort.

In 1693, Richard Bentley, a young cleric who was later to become Master of Newton's college, wrote to ask Newton for advice on how to master the work. Newton suggested a short list of background materials, and then, concerning the Principia itself, advised Bentley to read only the first three sections in Book One (i.e., the first sixty pages of the four hundred pages that make up the first edition). These sections provide the theoretical background for the astronomical applications that Newton presented in Book Three and regarded as of popular scientific interest. In the introduction to Book Three, Newton repeated the advice that he had given to Bentley:

I had composed the third book in a popular method so that it might be read by many. But since those who had not sufficiently entered into the principles could not easily discern the strength of the consequences nor put aside long-held prejudices, I chose to rework the substance of that book into the form of propositions in the mathematical way, so that they might be read only by

those who had first mastered the principles. Nevertheless, I do not want to suggest that anyone should read all of these propositionswhich appear there in great numbersince they could present too great an obstacle even for readers skilled in mathematics. It would be sufficient for someone to read carefully the definitions, laws of motion, and the first three sections of the first book; then let [the reader] skip to this [third] book.1

Newton's sage advice to the general reader to concentrate on the first three sections of Book One of the Principia appeared in the first edition of 1687 and remained unchanged in the two revised editions published in 1713 and 1726, all during Newton's lifetime. It is the third and final edition that has been reproduced in many subsequent editions and translated into many other languages. Because this third edition is readily available and because it is seen to represent Newton's most fully developed views, it is almost exclusively taken as a basis for the study of Newton's dynamics. The general reader, however, should not begin with this final edition and its many additions and revisions, but rather with the first edition and its relatively straightforward presentation.

In 1684, Newton sent to London a tract entitled On the Motion of Bodies in Orbit (On Motion) that was to serve as the foundation for the first edition of the Principia of 1687. This comparatively short tract presents in a clean and uncluttered fashion the basic core of Newton's dynamics and its application to the central problem of elliptical motion. The brief set of definitions that appeared in On Motion was expanded in the Principia into a much larger set of definitions, laws, and corollaries. Further, the first four theorems and four problems in On Motion were expanded into fourteen lemmas and seventeen propositions in the Principia . (Theorem 1 of On Motion is Proposition 1 of the Principia but Problem 4 of On Motion is Proposition 17 of the Principia ). The expanded framework of numbered propositions by itself, however, does not tell the entire story. Even more troublesome for the general reader is Newton's practice of adding new material to the old framework. Having established the expanded set of propositions and lemmas in the early draft of the first edition, Newton elected to hold to that framework as he inserted additional material in his published revised editions. Even in the preface to the first edition, Newton apologized to his readers for such insertions.

Some things found out after the rest, I chose to insert in places less suitable, rather than to change the number of the propositions as well as the citations. I heartily beg that what I have done here may be read with patience.2

After the publication of the first edition, Newton began work on a grand radical revision of the Principia in which many of the propositions would have been renumbered and retitled. In contrast to the single method of the first edition, Newton clearly presented three alternate methods of dy-

namic analysis in this projected revised scheme, each method set forth in a new proposition. Unfortunately, Newton never implemented this new scheme of renumbering the propositions and lemmas in the published revisions. If the challenge of renumbering the propositions and correcting the cross-references was too much in the limited first edition, then it was apparently overwhelming in the expanded revised editions. The new material added to the published revised editions simply was inserted into the old structure of the first edition. The third method of dynamic analysis, so clearly differentiated in the projected revision, was distributed throughout the theorems and problem solutions of the second and third sections of the published revisions. The reader of On Motion and, to a lesser extent, of the first edition is not faced with this difficulty. In those works, Newton clearly explicates his analysis with a single method applied uniformly to several problems; until the reader understands his original method and his unpublished restructuring, however, Newton's additions to the much studied revised third edition appear as distractions rather than enrichments.

A Simplified Solution

The story of Isaac Newton and the apple is a familiar one. We have all seen the portrayal of an English gentleman who is sitting under a tree and is struck on the head by a falling apple. In a flash, he leaps to his feet and runs off shouting about the theory of universal gravitation. The story has its foundation in Newton's own telling and is attested by a number of memoranda written by those close to him in his later years. The setting is the garden of his country home, the time is 1666, and Newton, a young man of twenty-four, is home after a few years at university. The apple tree that provides his inspiration stands in his front garden, and the fruit it bears is a yellow-green cooking apple called the Flower of Kent. One version of the story, told by Newton in his later years and recorded by an associate, John Conduitt, includes the following statement:

Whilst he was musing in a garden it came into his thought that the power of gravity (which brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much farther than was usually thought. Why not as high as the moon said he to himself and if so that must influence her motion and...