Green
Lion PressHooke claimed to be able to demonstrate all the laws of celestial motion by assuming a power inversely as the square of the distance between the celestial bodies.
This "inverse square" relation had been observed in connection with the spreading of light and the action of magnets, and many people, including Wren and Halley, were speculating that it might apply to celestial actions. Wren had discussed the possibility of an inverse square force law with Isaac Newton as early as 1677.
Wren responded to Hooke's claim in that conversation by offering a prize for anyone who could produce a demonstration that an inverse square force law would lead to the motions of the planets described by Kepler. No such demonstration was forthcoming from Hooke, at least nothing that satisfied Wren, and the question stood.
In August of that year, Halley visited Newton at his home in Cambridge and mentioned the challenge, asking Newton whether he knew what sort of orbit an inverse square force law would produce. Newton answered that it was an ellipse, and that he had demonstrated it. Halley, excited, urged him to send the demonstration to him as soon as he could.
The first version Newton sent was a nine-page document entitled De motu corporum in gyrum (On the Motion of Bodies in Orbit), submitted to the Royal Society in November 1684. It not only demonstrated the planetary ellipses but also showed how all of Kepler's laws may be seen as consequences of physical forces.
It was obvious to Halley that this was a momentous contribution to placing the mathematics of planetary motion on a sound physical foundation. It went beyond the work of Kepler in two primary ways. First, it was universal, not depending on different plans for actions between planets and the sun and actions between planetary matter and the planet itself. Newton was able to show that terrestrial heaviness and the forces that move the planets were a single phenomenon. Second, Newton's system required fewer contrivances and ad hoc assumptions than Kepler's. It didn't require reference to imaginary or hypothetical entities. For example, Kepler supposed two powers in the sun neither of which could be observed other than by their effect on planetary motion. A kind of magnetism had to be supposed for the sun that contradicted the sorts of magnetism found on earth. The sorts of actions Newton relied on could be easily found and tested on earth.
Halley was eager to have the document published. But Newton wanted to develop it further, and urged Halley to hold off on publication until he could re-work and expand it. All his prodigious intellectual energies were consumed with this expansion. The manuscript of Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), was initially delivered to the Royal Society for publication in April 1686.
However, Newton still continued to work on the book, expanding it into three books. The final version of Book III was delivered to Halley for publication a year later in April 1687. Book III applied mathematical demonstrations of the two earlier books of Principia to our world and derived from these foundations the principles of universal gravitation and the elliptical planets of orbits (along with many other things, including the motions of the tides and the paths of comets).
Principia was written in a stupendous burst of creative energy. Newton lived during this period like one possessed. He often forgot to eat. "When he has sometimes taken a Turn or two [in the garden], has made a sudden stand, turn'd himself about, run up ye Stairs,... fall to write on his Desk standing, without giving himself the Leasure to draw a Chair to sit down in."
In such circumstances, the compass of the Principia could not be restrained. Among the many dazzling insights for which he sketched out proofs were: the conceptual equivalent of conservation of kinetic and potential energy; general expressions for orbits under any arbitrary force law; expressions for attractions between finite spheres under any force law; attempts to treat the motion of multiple mutually attracting bodies using approximations and qualitative arguments; the motion of pendulums; the motion of waves in water; the motion of fluids (including a refutation of the Cartesian theory of vortices; explanations of tides and of the nutation of the earth's axis and the motion of the moon's nodes; and laws governing the orbits of comets.
Buried within this heap of brilliant propositions is a central jewel, the establishment of universal gravitation and its use to demonstrate the elliptical orbits of the planets, which constitutes the main argument of Principia. It is this central jewel of an argument which this guidebook takes us through.
This is not only a supremely important step from the point of view of the history of science, but in addition it is Newton's practical demonstration of his theory of how science could be done in a way that could yield certainty, being (as he saw it) purely deductive.
This attempt to give science a logically sound deductive basis constituted a radical departure from Francis Bacon's inductive method, which was very influential at the time. Bacon advocated collecting many and varied instances of the phenomena under study and trying to see patterns.
By contrast Newton used minimal experimental data. His main experimental foundations, the "Phenomena" of Book III, were (as we shall see) very far from being pure observations, but they were based on observations and theory generally accepted. Everything was deduced, using mathematical demonstrations, from these few observation-based conclusions about how our world works.
In his "Preface to the Reader" Newton describes this revolutionary method
thus:
And on that account we present these [writings] of ours as the mathematical principles of philosophy. For the whole difficulty of philosophy appears to turn upon this: that from the phenomena of motion we may investigate the forces of nature, and then from these forces we may demonstrate the rest of the phenomena. ... In the third book,..., we present an example of this procedure, in the unfolding of the system of the world. For there, from the celestial phenomena, using the propositions demonstrated mathematically in the preceding books, we derive the forces of gravity by which bodies tend to the sun and the individual planets. Then from the forces, using propositions that are also mathematical, we deduce the motions of the planets, of comets, of the moon, and of the sea. In just the same way it would be possible to derive the rest of the phenomena of nature from mechanical principles by the same manner of argument.
This is the Newtonian version of confirmation theory, which is ingenious
and is significantly different from both contemporary and modern theories
of scientific explanation.
Kepler had previously shown that the planetary orbits are elliptical. Without using Kepler's laws, building only on his own foundations, Newton successfully used his method to derive the elliptical orbits of planets. This is a test of the method, showing that it arrives independently at true results. Newton's central argument provides this test of the method.
Because this guidebook concentrates upon the central argument, we will be looking here not at the whole of Principia but at what Newton explicitly delineates as the core sequence.
This core sequence consists of the following parts of Principia. The mathematical foundations for the development of universal gravitation and celestial mechanics are found in the Definitions, Laws of Motion, and some basic hypothetical mathematical propositions, the first seventeen propositions of Book I. That is where we start.
Then we go to Book III, where he introduces what he calls Phenomena, a small number of conclusions about our world. We look at these carefully and satisfy ourselves about them.
Then we follow the primary propositions of the application to our universe in the first thirteen propositions of Book III. (In the process we dip back into Books I and II for a few extra propositions and augment our observation-based data with one result from experiments on pendulums.) Here lies the thrilling derivation of universal gravitation: the discovery that the moon is falling just like a rock (or a pendulum bob), that inertial and gravitational mass are quantitatively the same, that every particle attracts every other particle inversely as the square of their distance. Having established these things, we can use them to prove that the planets will move in ellipses. This will be our exhilarating finale. This very procedure, and this very selection, is recommended by Newton in his preface to Book III.
At the end of the study undertaken by this guidebook, we will have established
the following from the phenomena of our world using the mathematical tools
of the first two books:
Newton incorporated many interesting diversions into Principia in his great burst of creative energy, diversions which lie outside the central argument, and which are not included in this guidebook. In parts of Book I Newton explores some very interesting attributes of conic sections. In other parts of Book I, he investigates the instability that results from the fact that our universe contains more than two bodies (he does not,as is sometimes carelessly said, "solve the three-body problem"). Book II lays down influential propositions of fluid mechanics. Even in the early propositions in Book I, a series we look at in detail, some propositions, which in themselves are stimulating and entertaining, are omitted as not essential to the line of argument.
It is important to remember that Principia is not just a cookie jar of entertainment and exercise for the mathematical faculties. It has a coherent argument which leads some place in particular, an argument line to which Newton explicitly draws our attention. Knowing that, and if we respect Newton and care to understand his enterprise, we must want to follow that argument to Newton's conclusion. Conveniently, this central argument is just right for one semester of study.
It is greatly tempting, and would be only too easy, to use up all our time for Newton in the early, hypothetical, mathematical demonstrations, and give only a rushed perusal to Book III. Yet Book III is the place where Newton gives his main argument, the place where he actually constructs the edifice for which the early propositions laid out the building blocks. The propositions of the first books are pro-theorems, preparation, for his conclusions about the System of the World.
In the first two books of Principia, Newton lays a mathematical foundation for the "philosophical" arguments. (When Newton says "philosophy" he means natural philosophy, which we would understand as physics and astronomy.) The propositions he lays out here are hypothetical. They do not show how things must be; rather, they show how things will be if certain conditions hold. For example, if it be supposed, as the conditions of a proposition, that a center of force exists in a certain place, and a body moves around it in some specified path, the proof of the proposition shows what forces would be required to produce that path. Such a path might never actually appear in nature, but that is a matter of complete indifference from a mathematical perspective. These hypothetical propositions are then available to be used, should their conditions be shown to hold, to demonstrate things about our actual universe. This is done in Book III, which he calls "System of the World," for certain of the hypothetical propositions.
Book III, the System of the World, is not an afterthought, not mere application of what has already been proved. Nothing about our world has been proved in the early books. It is in Book III that the breakthrough discoveries are made about gravitation, and it is in Book III that we find that the planets do move in ellipses.
If the impressive insights of Book III look obvious, it is only because we have lived with those conclusions since Newton's time, and it is evidence that we need to give them a more careful reading so that the significance and originality may make themselves felt.
There are two reasons, then, for keeping the careful focus of the guidebook selection. The first is to bring out the coherent line of the central argument. The second is to make sure that there is time for adequately thoughtful consideration of that central line of argument.
Those who have more than a semester to spend on Newton can profitably work through this central line of development and then follow out some of the intriguing side paths (either in the earlier books or later in Book III) on a firm foundation of understanding of the primary endeavor.
Green
Lion Press