The history of science was much changed by an event that occurred the same year. Plague had decimated London; it spread to Cambridge, and college was dismissed. Newton returned to the farm. Farming did not interest him; instead he equipped a room with instruments he had brought from Cambridge and conducted experiments in optics and continued his mathematical studies. During this year he performed many of the experiments that led, forty years later, to the publication of his Opticks. But the year held even more revolutionary thoughts and events for Newton.
According to the traditional account, he was sitting one day in the shade of an apple tree when an apple fell on his head and he discovered the law of gravity. The account is not all wrong. He was sitting under an apple tree, and he did note an apple’s fall, though not on his head. He did not discover the law of gravity, because everyone had known for centuries that the force of gravity attracts all bodies toward the center of the Earth. What Newton discovered, then and there, was the law of universal gravitation, which is a very different thing.
What it occurred to him to ask, as he sat and watched the apple fall, was whether the gravitational force that drew the apple toward the Earth extended as far as the orbit of the Moon. If so, would it account for the Moon’s orbit around the Earth? He knew, from his reading of Galileo, the actual force of gravity at sea level, and that the force decreased with distance from the Earth’s center (gravity is weaker on a mountaintop than at sea level). He also knew how far the Moon was from the Earth, on the average. Quickly he scratched out some calculations. The theory worked. No other assumption needed to be made to explain the workings of the solar system than that the system was held together, and its motions controlled, by a single force that was everywhere the same and that affected all bodies according to a very simple formula: in which F is the gravitational force, m1 and m2 are the masses of two bodies interacting upon one another, and d is the distance between them. Gravity varies, in other words, directly as the product of the masses of the two bodies, and inversely as the square of the distance between them.
Newton, in short, had solved the greatest puzzle in the history of science up to his time—a puzzle that the most learned philosophers and scientists had been struggling with at least from the time of Ptolemy, fifteen hundred years before, a puzzle furthermore to which the best minds of the previous century had been devoted, from Copernicus to Gilbert to Kepler to Galileo. It was an exciting moment for this young man of twenty-two. He worked out half a dozen theorems and propositions that accounted not only for the orbit of the Moon about the Earth but also for the orbits of the planets—including the Earth—about the Sun. But in doing so he also thought he had discovered a new mathematical way of describing and measuring planetary orbits. He called it the method of fluxions; we call it the integral calculus. This new mathematical tool interested him even more than his solution of the problem of the ages, so he put that aside, placing the sheets of paper on which he had written out the solution in a portable desk, and began to work in earnest on the calculus. He told nobody what he had done.
It was obvious to all at Cambridge that he was the best mathematician in England. The professor of mathematics at the university resigned his post so Newton might have it, at the age of twenty-six. He built telescopes, experimented with optics, did mathematics. In the meantime others, notably Edmund Halley (the discoverer of Halley’s Comet), had come to some notion of the law of gravitation but were having no success in using it to explain the orbits of the planets. In 1684 Halley journeyed up to Cambridge to consult Newton. He explained the problem and asked whether Newton would consider helping him and his friends solve it.
“I have already solved that problem,” Newton said. “I did so twenty years ago.” It is one of the most incredible moments in the history of science, indeed of human thought. Halley was dumfounded but Newton scrambled among his papers and presented to Halley the sheets on which were written four theorems and seven problems. They are the nucleus of his major work.
It took him eighteen months to write, in Latin, Principia Mathematica Philosophiae Naturalis (Mathematical Principles of Natural Philosophy—Principia for short). The Royal Society could not afford to publish the work, so Halley personally footed the cost.
Probably there is no more daunting book to nonmathematicians. But if you have some courage and are willing to devote some effort this need not be so. Many books are harder to read than Newton’s Principia. Few books are so important to read, and so gratifying.
The work is divided into three books. Book I presents the general theory of gravitation; it deals with “The Motion of Bodies.” Book II treats “The Motion of Bodies in Resisting Mediums,” to show that the Cartesian theory of vortices was not tenable and incidentally creating the modern sciences of hydrostatics and hydrodynamics. Book III, with a grandeur typical of Newton, is titled “The System of the World.”
There is a great deal of mathematics in Books I and II but most of it can be skipped over and ignored, although it will be useful to read the statements, if not the proofs, of many of the theorems. Read the beginning of the Principia, with its remarkable set of Definitions; they establish the ground rules of classical mechanics and of Newtonian physics. Read also the following Scholium, or general note. Its half-dozen pages describe the point of view that will be adopted throughout the work. Following that, read the Axioms, or Laws of Motion. They are basic to all modern science. Read the half-dozen Corollaries that follow them, together with the Scholium following Corollary VI.
At this point Book I begins. Skip through it, noting the theorems and problems and reading the Scholiums and Lemmas, getting an idea of what is going on. Pay special attention to Proposition 66, Theorem 26; this is the famous three-body problem, which is unsolved (except for trivial instances) to this day.
Skip through Book II in the same way, trying as hard as you can to follow the general line of the argument even if you do not grasp all of its details. Read carefully the Scholium at the end of Section VIII, concerning the motions of light and sound. And read as carefully as you can the entire Section IX—only a few pages. It is the preparation for what follows in the third book.
It is also true that many pages of Book III need not be read carefully; they can be very quickly skimmed. But some of these famous pages should be dwelt on. The very first page of Book III, for instance; a short preface in which Newton explains what he will now do, which is to explain how the world as a whole works. Even more essential are the following two pages: “Rules of Reasoning in Philosophy.” These too are at the heart of modern science. No scientist of our day wants to break any of these four rules. They are the essence of a common sense view of the world and the epitome of the scientific empiricism that has dominated human thought ever since Newton.