Newton's Principia and the Laws of Motion

In 1687 a reclusive Cambridge professor showed that the apple and the moon obey the same equation. Physics has not been the same since.

There is a kind of book that does not simply add to its field but draws a line through history with itself: before, and after. Isaac Newton's Philosophiæ Naturalis Principia Mathematica, published in 1687, is the clearest example we have. Before the Principia, motion in the heavens and motion on Earth were two different problems studied by two different traditions. Aristotle had taught that things on Earth seek their natural place; the stars move because the heavens are made of a different, perfect substance with its own rules. Galileo had cracked terrestrial motion. Kepler had charted the orbits of planets with three empirical laws. Nobody had glued the two halves together.

Newton glued them with a single insight: the force that pulls an apple to the ground is the same force that bends the Moon in its orbit.

Three laws on a page

The bulk of physics — the parts that matter when you push a chair across a floor, throw a ball, design a bridge, or send a probe to Saturn — is contained in three statements that fit on a single page.

First law. A body in motion stays in motion, in a straight line, at constant speed, unless something pushes it. A body at rest stays at rest. This is not obvious. Aristotle thought objects naturally come to rest. They do not. They appear to because there is always friction, drag, contact. Strip those away and motion is conserved. Galileo had glimpsed this; Newton stated it as a law.

Second law. A force on an object changes its motion in proportion to the force: F = ma. Push harder, accelerate more. Push the same on a heavier object, accelerate less. The deep move here is the introduction of mass — a quantity that resists acceleration, distinct from weight. The second law is the load-bearing equation of classical mechanics. Almost everything that follows is F = ma applied to some specific force.

Third law. When you push on something, it pushes back with equal force in the opposite direction. The Earth pulls you down; you pull the Earth up by exactly the same amount. (The Earth barely budges because of the second law: same force, much more mass.) The third law is what makes rockets work — fire mass downward, get pushed upward — and what keeps your chair holding you up instead of dropping you through the floor.

One force to rule the planets

The genius of the Principia is not the three laws on their own. Galileo and Descartes were already circling them. The genius is what Newton did next.

Kepler had found, by squinting at Tycho Brahe's data for decades, that planetary orbits are ellipses with the Sun at one focus, that planets sweep out equal areas in equal times, and that the cube of the orbital radius is proportional to the square of the period. Three laws, no explanation. Why ellipses and not circles? Why these particular numbers?

Newton supposed there is a single force — gravity — that any two masses exert on each other, proportional to the product of their masses and inversely proportional to the square of the distance between them. Then he did something physics had never quite seen: starting from this single force and the three laws, applying the calculus he had partly invented, he derived all three of Kepler's laws as theorems. The ellipses, the equal areas, the period law — all consequences. Not measured, not curve-fit. Deduced.

And the same force, the same constant, governs the falling apple. Newton checked the numbers: the Moon's acceleration toward the Earth, computed from the size of its orbit, agreed with the acceleration of an apple at the Earth's surface, scaled by the inverse square of their distances. The same physics in heaven and on Earth.

"I do not feign hypotheses"

Newton was deeply uneasy about one thing. He had given the world an equation for gravity that worked — planets, tides, comets, all of it — but he had no idea what gravity was. How does a force reach across empty space? What mediates it? He refused to invent a story.

“I have not been able to discover the cause of those properties of gravity from phenomena, and I feign no hypotheses... It is enough that gravity does really exist, and acts according to the laws which I have explained.”

That refusal is a kind of intellectual style we now take for granted. Predictive equations first; mechanism, maybe, eventually. It is the style of modern physics. Quantum mechanics works the same way: nobody is sure what the wave function is; the equation predicts everything anyway.

What the Principia really did

Three things change permanently after 1687.

First, the cosmos becomes one place. The same laws operate everywhere. The Moon and the cannonball obey the same equation. The split between celestial and terrestrial physics — two thousand years old — disappears. There is just physics.

Second, mathematics becomes the language of nature. Earlier natural philosophy was largely qualitative. Newton's book is wall-to-wall geometry and proof. The implicit claim — that the universe is the kind of thing you can deduce, not just describe — becomes the working assumption of every science that follows.

Third, determinism takes hold. If F = ma and you know all the forces, then in principle the entire future of any mechanical system is calculable from its present state. Pierre-Simon Laplace, a century later, pushed this to its logical conclusion: imagine an intellect that knew the position and velocity of every particle. To it, “nothing would be uncertain and the future just like the past would be present before its eyes.” Laplace's demon is a child of the Principia.

It is fashionable to point out that Newton was eventually superseded. He was. Mercury's perihelion drifts in a way F = ma cannot explain; you need general relativity. Electrons do not have definite trajectories; you need quantum mechanics. Newtonian mechanics is the low-mass, low-speed limit of more complete theories.

But notice how it was superseded. Einstein's general relativity, applied to a falling apple, gives Newton's answer to a precision no apple-scale experiment can distinguish. Quantum mechanics, applied to billiard balls, gives back Newton. The newer theories did not throw the Principia away. They embedded it as a special case. That is the deepest compliment one theory can pay another — and it is a sign that Newton was, in the regimes where he applied his equations, simply right.

Three laws. One force. A single mathematical universe. Three hundred and forty years later, every engineer, every astronomer, every physicist still starts there.