Physics 1687

Mathematical Principles of Natural Philosophy

Isaac Newton

Three laws of motion and one law of gravity govern the apple and the planet alike.

Choose your version

In depth · the introduction

Newton found that the same handful of rules explains a falling apple, the orbit of the Moon, and the swing of the tides.

The big idea

Newton's first move was to nail down how motion works at all, in three laws. First: things keep doing what they're doing — sitting still, or coasting in a straight line — unless a force acts on them. Second: a force changes their motion in proportion to its strength, and a heavier thing changes less for the same push. Third: every push comes with an equal push back. These three sentences are still the first thing every physics student learns.

His second, even bolder move was gravity. He proposed one force that every bit of matter exerts on every other bit — stronger for more massive things, and weaker with distance in an exact way (twice as far means a quarter as strong). One rule, reaching across all of space, ties the apple, the Moon, the planets, and the tides into a single system.

How it came about

In 1665 the plague closed Cambridge, and a young Newton retreated to his family farm at Woolsthorpe. In those "plague years" he later said he was at the height of his powers for invention — sketching the calculus, splitting light with a prism, and beginning to wonder whether the force that drops an apple might reach as far as the Moon. (An apple really did fall in that orchard; the bolt-from-the-blue version is a tidy legend, but the question was real.)

For two decades the idea sat largely unpublished. Then in 1684 the astronomer Edmond Halley visited and asked what curve a planet would follow under an inverse-square force. Newton replied at once: an ellipse — he had worked it out. Halley, astonished that the proof wasn't published, coaxed and then personally financed the writing. The result, the Principia, appeared in Latin in 1687, three dense volumes that remade the science of motion.

Why it mattered

Before Newton, the sky was the domain of mystery; after him, it was arithmetic. He showed that the universe runs on laws you can write down and use to predict the future — where a planet will be next year, when a comet will return, how high a tide will rise. That was the founding promise of modern science and engineering: that nature is lawful, and the laws are ours to find and to use.

A way to picture it

Here is the trick that makes an orbit make sense. Throw a ball and it arcs to the ground because gravity pulls it down while it moves forward. Throw it harder and it lands farther away. Now imagine throwing it so fast that, as it falls, the ground curves away beneath it just as quickly — the ball keeps falling but never gets any closer. That is an orbit: falling sideways so fast that you keep missing the Earth. The Moon is doing exactly that, forever.

What came next

Newton's universe ran like clockwork for two centuries. The next great unification came from James Clerk Maxwell, who in the 1860s folded electricity, magnetism, and light into a single set of field equations — extending Newton's dream of universal law to a wholly new domain.

Then, in 1905 and 1915, Albert Einstein revised Newton's very stage. He showed that absolute space and absolute time do not exist, and reimagined gravity not as a force reaching across a void but as the curving of spacetime by mass. Newton's laws survive as the everyday limit of Einstein's — astonishingly accurate for apples, bridges, and rockets, and still the mechanics we use to fly to the Moon.

The original document

Original source text

The Axioms, or Laws of Motion

Isaac Newton · Philosophiæ Naturalis Principia Mathematica · 1687 · Axioms, or Laws of Motion (Motte trans., 1729)

Law I

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air.

Law II

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively.

Law III

To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone.

Of Universal Gravitation (Book III)

Book III · Of the System of the World · Propositions on gravity

There is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.

That all the planets gravitate towards one another we have proved before; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the centre of the planet.

And this is the force by which the moon is retained in its orbit, and by which bodies fall toward the earth. The force which retains the celestial bodies in their orbits is the very same force we commonly call gravity; for the moon, were it deprived of all motion, would, by its gravity, fall toward the earth.

And therefore the force by which the moon is retained in its orbit is that very same force which we commonly call gravity.

The System of the World

From “A Treatise of the System of the World” · Newton's own thought experiment

The greater the velocity with which a body is projected, the farther it goes before it falls to the earth. We may therefore suppose the velocity to be so increased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it arrived at the earth, till at last, exceeding the limits of the earth, it should pass quite by without touching it.

A body projected from the top of a high mountain, parallel to the horizon, with a sufficient velocity, would not fall to the earth at all, but go forward into the celestial spaces, and proceed in its motion in infinitum. And the same reasoning that holds for the projectile holds also for the moon, which is perpetually drawn off from a rectilinear course toward the earth, and made to revolve in a curve.

And by the same principle the planets are kept in their orbits about the sun, and the satellites about their primary planets — the one force of gravity governing them all.

The General Scholium

The General Scholium · added to the 2nd edition, 1713

Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, and operates according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances.

I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses. … It is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.

“Hypotheses non fingo.” — General Scholium