Classical Mechanics

Isaac Newton's Principia (1687) supplied it. Three laws of motion (inertia, $F = ma$, action and reaction) and one law of gravity, an attraction between every pair of masses falling off as the square of the distance, reproduce Kepler's ellipses exactly. The Moon is falling toward the Earth just as an apple does. It simply moves sideways fast enough to keep missing.

The achievement was contested from the start. Robert Hooke insisted he had suggested the inverse-square law to Newton in their correspondence, and Newton answered by striking nearly every mention of Hooke from his book. Others objected to gravity itself: a force acting instantly across empty space, with no mechanism. Newton famously declined to explain it: hypotheses non fingo, "I frame no hypotheses". That unease about how gravity acts at a distance would be answered only by general relativity.

Over the next century and a half, Euler, Lagrange, Laplace and Hamilton rewrote Newton's mechanics in more powerful mathematical forms. Laplace imagined an intellect that, knowing every position and velocity at one instant, could compute the whole future.

The theory's greatest triumph came in 1846. Uranus was drifting from its predicted path, and Urbain Le Verrier worked backwards to the position of an unseen planet pulling on it. Johann Galle pointed his telescope there and found Neptune on his first night. In England, John Couch Adams's unpublished calculations were later put forward as a rival prediction, and how seriously to take them is still argued.

Newton's key test of universal gravitation, which he said he first tried in the plague years of 1665–66, needs only a few numbers. If the same force that pulls an apple also holds the Moon, and it weakens with the square of distance, then the Moon's acceleration towards the Earth should be the apple's divided by the square of how much farther away it is.

The Moon orbits at about 384,400 km from the Earth's centre, about 60 times the Earth's radius of 6,371 km. So gravity there should be $60^2 = 3{,}600$ times weaker than at the surface:

$$\frac{9.81 \text{ m/s}^2}{3600} \approx 0.00272 \text{ m/s}^2 .$$

Now measure the Moon's actual acceleration, from its orbit alone. A body moving in a circle of radius $r$ with period $T$ accelerates towards the centre at $4\pi^2 r / T^2$. The Moon's period is 27.32 days, or $2.36 \times 10^6$ seconds:

$$\frac{4\pi^2 \times 3.844 \times 10^8 \text{ m}}{(2.36 \times 10^6 \text{ s})^2} \approx 0.00272 \text{ m/s}^2 .$$

The two agree. The fall of an apple in an orchard and the orbit of the Moon are the same phenomenon, given one rule about how gravity weakens with distance.

Put differently, in one second the Moon falls about $\tfrac12 \times 0.00272 \approx 1.4$ millimetres towards the Earth, while moving about a kilometre sideways. The curve of its path is exactly that fall. Newton's first attempt did not match as well, partly because the Earth's radius was poorly known. The close agreement, published in the Principia, showed that a law found on Earth governs the heavens.

The same method failed with Mercury. In 1859 Le Verrier found that its orbit swings around the Sun slightly faster than the other planets' pulls allow: 43 arcseconds per century by modern measurement. He predicted another unseen planet, Vulcan, inside Mercury's orbit. Astronomers looked for decades and never found it. The problem was not a missing planet but Newton's gravity itself.

A second crack was quieter. Galileo's relativity says that velocities simply add, so light should travel at different speeds for observers moving at different speeds. But electromagnetism was about to predict a single, fixed speed of light. The two theories could not both be right, and their collision produced special relativity.

Classical mechanics was not overthrown so much as bounded. Within its domain it remains exact enough to fly spacecraft and forecast weather, and even there it holds open problems. Whether the equations of fluid flow always have smooth solutions is still unknown.