DEEP READ · 93
Philosophiæ Naturalis Principia Mathematica · Isaac Newton · 1687
Before this book, the heavens and the earth were two different places, running on two different sets of rules. Newton took three laws of motion and one equation for gravity and showed that whatever pulls an apple down is the very same thing that keeps the Moon from flying off — and that you can calculate both with one formula. Along the way he demonstrated something even larger: a new kind of knowledge that declines to say why, states only what, states it in mathematics exact enough to predict — and that habit is what we have called science ever since.
Newton (1642–1727) was Lucasian Professor of Mathematics at Trinity College, Cambridge. The Principia appeared in Latin in 1687, printed at the personal expense of the astronomer Edmond Halley. It is the terminus of the Scientific Revolution: it gathers the scattered findings of Copernicus, Kepler and Galileo into a single mathematical frame, and for the next two centuries physics largely wrote footnotes to it.
Start with what Newton demolished. In Aristotle's physics, motion is a state that must be continuously fed: push a cart and it moves; let go and it stops. Hence no force, no motion. This is not a stupid view — it describes with great accuracy everything that happens in a world full of friction, which is why it ruled for twenty centuries.
Newton's First Law, in his words (paraphrased): every body perseveres in its state of rest, or of uniform motion in a straight line, unless compelled by impressed forces to change it. The operative word is unless. Rest and steady motion are put in the same box: both are the default, neither needs explaining. Only change does. The cart stops not because motion naturally runs down, but because friction is a force decelerating it.
The practical power of that inversion is that it instantly disposes of the strongest argument against Copernicus. If the Earth really spins and hurtles around the Sun, people objected, you should land dozens of metres away when you jump, and a stone dropped from a tower should land off the mark — and it doesn't. In Aristotle's frame that objection is unanswerable. Inertia answers it in one line: you and the stone are already moving with the Earth, nothing acts to change that shared velocity, so from where you stand nothing looks different. It is why you can pour coffee steadily on an aircraft doing 800 km/h. Modern astronomy could only stand up once terrestrial mechanics had been rewritten underneath it.
| Aristotle | Newton | |
| What needs explaining | Motion itself | The change of motion |
| Default state | Rest (things seek their natural place) | Rest and uniform straight-line motion, equivalent |
| Heaven and earth | Two physics: perfect circles above, decay below | One physics, from apple to comet |
| Role of force | Sustains motion | Changes motion (produces acceleration) |
Where the Principia's first cut lands: it replaces not the answer but the question.
The book opens not with laws but with eight definitions, and the first two matter most. Definition I gives mass: the quantity of matter, density times volume. Definition II gives quantity of motion, what we call momentum: mass times velocity. The Second Law then says (paraphrased): the change of motion is proportional to the impressed force, and takes place along the line in which that force acts.
Here is a point every schoolchild now misses: Newton never wrote F = ma in the Principia. He wrote that force produces a change of momentum — closer to what we call the impulse form. The differential version came later, tidied up by Euler and others. This is not pedantry: the impulse form is the more general one, and for rockets venting gas, bullets hitting targets, or any collision where mass changes or the time is vanishingly short, "change of momentum" is far cleaner than "acceleration".
The depth charge buried in the Second Law is that it prises mass apart from weight, which until then had been one muddled thing — heavy was just heavy. In Newton's frame, mass is how stubbornly a body resists being accelerated (kick a shot put and a football with equal force and watch), while weight is the force gravity happens to exert on it. Only once they are separated does "on the Moon you weigh a sixth as much but your mass is unchanged" become a sentence one can even utter, and only then can you explain why an astronaut floating weightless is still hard to shove. Taking a concept that everyday life has glued together and cutting it into two quantities with separate jobs — that is the central craft of theoretical physics.
The Third Law (paraphrased): to every action there is always an equal and opposite reaction. The standard misreading is "then they cancel and nothing could ever move." The answer hides in a detail people skip: the two forces act on two different bodies, so they never add and never cancel. The Earth pulls the apple; the apple pulls the Earth. One force accelerates the apple downward, the other accelerates the Earth toward the apple — the Earth is simply so massive that its acceleration is unmeasurable.
Think of walking. You move forward not because your leg is magic but because your foot pushes backward on the ground and the ground pushes you forward. On smooth ice you can't extract that reaction, so you slip in place. A rocket in vacuum is the same law: it doesn't need air to push against, it only needs to throw exhaust backward, and the exhaust throws it forward. The deeper content of the law is conservation of momentum: in a system with no external force, total momentum never changes — a principle that has outlasted Newton's own mechanics and is among the most regime-change-proof ideas in all of physics.
First, clear away a confusion: Newton did not "discover gravity." That things fall has been known as long as there have been people. Newton did three other things: he claimed the force is universal (not just the Earth pulling, but any two masses pulling each other), he gave its exact mathematical form (proportional to the product of the masses, inversely proportional to the square of the distance), and he proved that this one law alone suffices to compute the entire solar system.
The calculation that convinced him is remembered as the Moon test. The logic: if the Earth's pull does not stop at the ground but reaches all the way to the Moon, diluting as the square of the distance, then — the Moon sits about 60 Earth-radii from the Earth's centre, so gravity there should be 1/60², i.e. 1/3600 of its strength at the surface. The Moon takes 27.3 days to go round, and from that circular motion you can work out how far it "falls" toward the Earth each second. The number lands exactly on one three-thousand-six-hundredth of the acceleration of a falling stone. Two completely unrelated roads — one from dropping things on the ground, one from the phases of the Moon — arrive at the same figure. At that moment heaven and earth were welded into a single system.
Schematic: double the distance and the same amount has to cover four times the area, leaving a quarter per patch — the inverse square isn't a fudge factor Newton chose; it is what spreading out from a point in three dimensions geometrically must do.
And there is a hurdle nearly every popular account skips, though it held Newton up for years: what entitles you to treat the Earth as a point? The ground under you is a ball twelve thousand kilometres across, and every chunk of it pulls you from a different distance and a different direction; summing them is a vicious integral. In Book I Newton proved what we now call the shell theorem: a uniform sphere attracts an outside body exactly as though its whole mass were concentrated at its centre. Only that theorem makes "60 Earth-radii from centre to centre" a legitimate move at all. He delayed publishing for years over it — not from modesty, but because he knew that without this step the whole edifice hangs in mid-air.
The most beautiful thing in the Principia is not the inverse-square law itself, but that it proves the road runs in both directions between that law and the observations.
Kepler had already extracted three laws of planetary motion from Tycho Brahe's data: orbits are ellipses with the Sun at one focus; the line from Sun to planet sweeps equal areas in equal times; the square of the period goes as the cube of the semi-major axis (T² ∝ a³). But these were empirical — shapes fitted to numbers, with no account of why. Newton did two things:
What lifts this from a restatement to a theory is the surplus: it yields more than was fed in. In Book III the same law explains and predicts, in one sweep, the tides (Moon and Sun pulling the oceans together, hence spring and neap tides); the precession of the equinoxes (the Earth is not a true sphere, and the pull on its equatorial bulge makes the spin axis wander like a top, once round in about 26,000 years); the oblate Earth (rotation bulges the equator and flattens the poles — confirmed decades later by expeditions sent to Lapland and Peru to measure it); the mutual perturbations of planets; and comets moving on very elongated closed orbits — which is exactly what let Halley declare that the comet of 1682 would return in 1758. Halley did not live to see it. The comet came back. A framework that merely redescribes known facts never predicts something it has not seen; the ones that do are called theories.
After publication the fiercest attacks were not that Newton had miscalculated but that he was talking nonsense. The Continent favoured Descartes' vortex theory: space is packed with invisible subtle matter swirling in great whirlpools, and planets are carried along by them. That is a picture you can imagine, in which everything happens by contact and shoving. Newton, by contrast, said the Sun reaches across a hundred and fifty million kilometres of nothing and pulls. Pulls how? There is nothing in between. Leibniz and Huygens denounced this as a relapse into the medieval occult quality — the equivalent of saying opium makes you sleep because it possesses a dormitive virtue: a name dressed up as an explanation.
In the General Scholium added to the second edition (1713), Newton gave his famous reply: Hypotheses non fingo — I frame no hypotheses. Meaning: I genuinely do not know what causes gravity; but I can deduce its law precisely from the phenomena, and that law suffices to account for the motions of the heavens and the tides of the sea. Whatever is not deduced from the phenomena I call a hypothesis, and hypotheses have no place in experimental philosophy.
That sentence is far more than a shield in a quarrel; it redefined the threshold of science. Before it, to "explain" meant to supply a mechanical picture — to show how the gears mesh. After it, to "explain" means to supply a mathematical relation exact enough to predict, with the question of what underlies it left open. It was an enormous and painful concession: science traded the ambition of saying why for the certainty of calculating how much. The trade turned out to be the right one. Two centuries later, Maxwell's fields, Einstein's curved spacetime and quantum mechanics' probability amplitudes offer no intuitive mechanical picture whatsoever, and physics could keep moving only because Newton had already torn down the requirement that you must have a picture before you may use a formula. Worth noting: privately Newton was not at peace with it either. In a letter to the theologian Richard Bentley he wrote (paraphrased) that for gravity to be innate to matter and to act across a vacuum with no intermediary was, to him, so great an absurdity that no competent thinker could fall into it. He simply refused to write his own guesses into the book as knowledge.
For the three laws to mean anything, Newton first had to answer an awkward question: uniform straight-line motion relative to what? In a smoothly moving railway carriage, are you still or moving? Still, relative to the carriage; moving, relative to the ground. If all motion is merely relative, the First Law says nothing at all.
In the Scholium following the definitions he posits absolute space, a container independent of all bodies, forever still and uniform, and absolute time, a river flowing evenly and unrelated to any event. The laws hold with respect to these.
His argument for them is the celebrated bucket. Hang a bucket of water by a rope, twist the rope, release it. At first the bucket spins while the water does not, and the surface is flat. Then the water is dragged into rotation and its surface curves into a hollow — at which point the water is at rest relative to the bucket. So the hollow cannot be caused by the water's motion relative to the bucket. Relative to what, then, is it turning? Newton answers: absolute space. The force of the argument is that it turns a metaphysical claim into something you can point at — you are not debating a word, you are staring at a dished surface that demands explanation.
It took two centuries of criticism for that. Leibniz objected that a space which is absolutely still, everywhere identical and in principle undetectable makes no difference to any observation, and therefore says nothing. Mach offered a sharper alternative: the surface curves because the water is rotating relative to the total mass of the distant stars — inertia is conferred not by space but by all the matter in the universe (later called Mach's principle), an idea that fed directly into Einstein. General relativity eventually abolished both containers, replacing them with a spacetime that mass can bend and that takes part in the action. But Newton did not posit them out of confusion: he saw clearly that without some non-rotating reference the laws will not stand. The question survives; only the answer changed.
Newton co-invented calculus, which he called the method of fluxions, and many results in the Principia were first obtained that way. Yet the book itself is almost entirely classical geometry: figures, construction lines, ratios. To a modern reader this is punishing — what a student today does in two lines, Newton does in half a page of diagram.
Two reasons. The mathematical community of the day accepted only Euclidean demonstration as rigorous, and a reader confronted with an unfamiliar new algorithm was entitled to withhold assent. And Newton himself was not entirely comfortable with the foundations of fluxions — what exactly is an infinitesimal? So he built a substitute floor at the start of Book I, his method of prime and ultimate ratios: never speak of infinitely small quantities, only of the limiting value approached by the ratio of two quantities as both vanish. This is a precocious first appearance of the limit concept, more than a century before Cauchy and Weierstrass made it rigorous. The price is a book so hard that the best mathematicians of the age needed months to chew through it.
Three books, a chain running from the abstract to the concrete and cashing itself out at each stage:
Compressed to a line: define exactly what you are talking about, derive a whole body of mathematical consequences from the fewest possible axioms, then throw those consequences at the real world and see whether they predict something you have never seen. That is not only the structure of the Principia; ever since, it has been what a science looks like.
Read only the Principia and you get a lopsided Newton: austere, all mathematics and no speculation. The real one was far stranger.
Opticks (1704) is his other mountain, and almost its opposite in style: written in English rather than Latin, built on experiment rather than geometric proof, and genuinely readable. At its centre is what he called the experimentum crucis. Everyone assumed a prism dyes white light with colour. Newton sent white light through one prism to spread a spectrum, then used a screen to let just one colour through to a second prism — and that beam bent again but never changed colour. The conclusion inverts the assumption: colour is not added by the prism; white light is already a mixture, and the prism merely sorts it by how much each part bends. The shape of that reasoning is worth keeping: not observing more, but designing a situation in which two explanations are forced to give different results.
Tellingly, in the Queries at the end of Opticks Newton drops the "I frame no hypotheses" discipline entirely and fires off dozens of bold speculations — whether light is corpuscle or wave, whether bodies and light act on one another, what the ether is, even how matter relates to mind. The contrast between his two books reveals his actual position: speculation is permitted and even necessary, but it must be labelled as speculation and must not walk around dressed as a theorem.
Then there are the millions of words of theological and chronological manuscript. He sincerely believed scripture held another code God had left for humanity, and that decrypting the universe and decrypting the text were one enterprise. In his own eyes the Principia was not a book about the laws of nature but a book about how God designed; the passage on the supreme Lord in the General Scholium is not a courtesy, it is his reason for writing. (He also left Cambridge in 1696 to run the Royal Mint, personally hunting counterfeiters and overseeing the recoinage — and was very good at it.)
1. The detonation in this book is a single word: universal — the apple and the Moon obey one law, and heaven and earth become one world.
2. What requires explaining is not motion but the change of motion. For two thousand years people asked "what keeps it going?"; Newton says the question is wrong.
3. Inertia instantly kills the best argument against a moving Earth: you and the stone are already moving with it, nothing changes that, so you feel nothing.
4. Forces always come in pairs, but they act on two different bodies, so they never cancel — you walk forward because the ground pushes you forward.
5. The Moon test is the book's heartbeat: 60 times farther, so 3600 times weaker; the number from dropping stones and the number from the Moon's period land on the same spot.
6. The inverse square isn't a fudge factor; it is what radiating from a point must do — spread the same amount over four times the area and each patch keeps a quarter.
7. What makes it a theory rather than a restatement is the surplus: it coughs up tides, precession, the Earth's bulge and the return of a comet — predicting what you haven't seen is the test.
8. "I frame no hypotheses": here science trades the ambition of saying why for the certainty of calculating how much — and only because that threshold fell could fields, curved spacetime and probability amplitudes later walk in.
9. The dished water in the bucket forced "absolute space" into being; Mach questioned the answer and Einstein replaced it, but the question — uniform motion relative to what? — is still with us.
10. Newtonian mechanics was not overthrown but subsumed: we still build bridges and launch satellites with it, and its boundary showed only in Mercury's extra 43 arcseconds a century — behind which stood general relativity.