Newton: The Principia

Isaac Newton grew up on a farm in England. He liked building things: windmills, water clocks and kites with lanterns. When he grew up he wrote one of the most famous science books ever, the Principia. It explains why things fall and why the Moon goes around the Earth, with one idea.

Here is the idea. Imagine a giant cannon on top of a very tall mountain. Fire a ball slowly and it falls to the ground nearby. Fire it faster and it lands farther away. Fire it fast enough, and the ground curves away beneath it as fast as it falls. It keeps falling, and keeps missing the Earth, all the way around! That is an orbit.

Newton said the Moon is like that cannonball. Earth's pull bends the Moon's path into a circle, so it keeps falling toward us without ever landing. The same pull makes an apple drop from a tree. Newton called the pull gravity, and he showed it gets weaker the farther away you go.

Newton also wrote three rules for how things move. 1. Things keep doing what they are doing unless something pushes or pulls them. 2. A bigger push makes a bigger change in motion. 3. When you push on something, it pushes back on you just as hard.

Who, when and the problem

Isaac Newton (1642–1727) was born on Christmas Day 1642 by the calendar England then used, grew up in Lincolnshire, studied at Trinity College, Cambridge, and became its professor of mathematics in 1669. He later ran the Royal Mint and was president of the Royal Society. The Mathematical Principles of Natural Philosophy, known by its Latin name Principia, appeared in Latin on July 5, 1687. Newton revised it in 1713 and 1726. Andrew Motte translated the third edition into English in 1729.

The problem was the heavens. Johannes Kepler had found three rules that planets follow: they move on ellipses, they sweep out equal areas in equal times, and the square of the period grows as the cube of the orbit's size. Galileo had worked out how things fall and fly on Earth. But no one could say why the planets obey Kepler's rules, or whether the heavens and the Earth follow the same laws. In 1684 Edmond Halley visited Newton in Cambridge and asked what path a planet would take if the sun pulled it with a force that weakens as the square of the distance. Newton said he had worked it out: an ellipse. Halley urged him to write it up and, in the end, paid for the printing.

The main ideas

• Three laws of motion that hold everywhere: inertia; change of motion in proportion to the force; and equal and opposite action and reaction.
• Book I works out how bodies move under forces aimed at a center. Any such force gives Kepler's equal areas. An ellipse with the force aimed at a focus requires a force that falls off as the inverse square of the distance.
• Book II deals with motion through air and water. Among other things, it argues that the swirling whirlpools of matter that Descartes used to carry the planets cannot produce Kepler's laws.
• Book III, "The System of the World," applies it all. The force that holds the Moon in its orbit is the same gravity that makes things fall. Every body attracts every other, with a force proportional to their masses and inversely proportional to the square of the distance. From this come the tides, the flattening of the Earth at the poles, the slow wobble of its axis, and the orbits of comets.

By "motion" Newton means what we call momentum: mass times velocity. So Law II says a force changes momentum, and the change points the way the force points. The familiar form $F = ma$ is not in the Principia. Leonhard Euler wrote the laws as equations like that around 1750.

The cannonball on the mountain

Newton goes on: faster still, and the ball would go around the whole Earth, or fly off and never come back. In the same way, the Moon is kept in its orbit by a force pulling it toward the Earth. He drew this picture, a mountain on a round Earth with curving paths, in his Treatise of the System of the World.

The Moon test

Here is the heart of Book III. The Moon is about 60 Earth radii away. If gravity falls off as the inverse square, then at the Moon's distance it should be $60^2 = 3{,}600$ times weaker than at the ground. Is the Moon's actual pull toward the Earth that size?

Rules of reasoning and "I frame no hypotheses"

Book III opens with Newton's "Rules of Reasoning in Philosophy": admit no more causes than are needed; treat the same effects as having the same causes; take qualities found in all bodies we can test as qualities of all bodies; and hold results drawn from experiments by induction as true until other experiments correct them. In the General Scholium, added in 1713, he admits that he cannot say what gravity is.

Setting the record straight

• The apple. The apple did not hit Newton on the head. The story comes from Newton himself, who told it in old age to William Stukeley (in 1726) and to his niece's husband, John Conduitt: seeing an apple fall made him wonder whether the same pull reached to the Moon. Voltaire spread it in his Letters on the English (1733), which is a Harvard Classics reading.
• Not all in one year. Newton later said he had the main ideas in 1665–1666, when Cambridge closed for the plague. His notes show a rough Moon test then, but historians such as Richard Westfall show that the full theory, including the idea that every particle attracts every other, came only in 1684–1687.
• Robert Hooke. In letters of 1679–1680 Hooke suggested to Newton that orbits come from straight motion bent by an attraction that falls off as the inverse square. Newton gave him little credit. Hooke had the idea; Newton had the mathematics that proved it gives Kepler's laws.
• Calculus. The Principia is written in the language of geometry, with limits and ratios of vanishing quantities, not in the notation of calculus we use now, although Newton had invented his method of "fluxions" years before. His priority dispute with Gottfried Leibniz, who published calculus first (1684), lasted for decades.

What later science kept, and what it changed

• Kept: the laws of motion and universal gravitation are still used to send spacecraft to other planets. In 1846 Urbain Le Verrier used them to predict where an unseen planet must be, and Johann Galle found Neptune there.
• Changed: Mercury's orbit turns slowly (43 arcseconds a century) by an amount Newton's law could not explain. Einstein's general relativity (1915) explains it and treats gravity as the curving of space and time. At speeds near that of light, Einstein's special relativity (1905) replaces Newton's mechanics; for atoms, quantum mechanics does.
• Corrected detail: Newton's speed of sound in Book II came out too low. Pierre-Simon Laplace fixed it in 1816 by taking into account the heating of air as it is squeezed.

Action at a distance. Many readers in 1687, including Huygens and Leibniz, objected that a pull across empty space with nothing in between was an "occult quality" of the kind the new science had tried to remove. Newton himself, in a letter to Richard Bentley (1693), called the idea of one body acting on another at a distance through a vacuum "so great an absurdity" that he did not think anyone competent would believe it; he left the cause of gravity open. Faraday's lines of force and Maxwell's fields, two centuries later, were one answer; Einstein's curved space-time was another.

Absolute space. In a scholium after the definitions, Newton argues for absolute space and time, using a spinning bucket of water: the water's surface curves when it turns, which he took as a sign of true, not relative, motion. Leibniz disagreed, holding that space is only the order of things. Ernst Mach and Einstein returned to the bucket in the 19th and 20th centuries.

God and nature. The General Scholium also contains Newton's view that "this most beautiful System of the Sun, Planets and Comets, could only proceed from the counsel and dominion of an intelligent and powerful being." Later thinkers took the Principia in different directions: some saw a universe that runs by law and needs no outside help, as Laplace said; others, like Newton, saw its order as pointing to a designer. Readers can weigh these views for themselves.

How to read it. The Principia is very hard, even for scientists. A good plan is to read the Definitions, the Laws and their corollaries, the Rules of Reasoning, the opening propositions of Book III and the General Scholium, and to follow one proof in Book I (Proposition 1, on equal areas) with a pencil. Newton's own shorter, less mathematical Treatise of the System of the World (English 1728) is a gentler start.

Next on the ladder. Newton's Opticks shows a different side of him: patient experiments with prisms, written in plain English. Huygens's Treatise on Light offers a rival picture of light as waves.