The book that ends this story does something no book had done before: it predicts, from one rule, a set of quantities that had never been connected, and it declines to say why the rule holds.
The previous lesson got as far as a comparison of two numbers, the fall of a stone and the fall of the moon, agreeing to about one per cent under an inverse square law. That is a good result about the earth and its satellite. This lesson is about what happens when the same rule is applied to every pair of bodies in the universe, what it bought, and the bill that came with it.
How it got written
In August 1684 Edmond Halley travelled to Cambridge and asked Isaac Newton what path a planet would follow under an inverse square attraction. Newton said an ellipse, and that he had calculated it, and then could not find the paper. He reworked it and sent Halley a nine-page tract, De motu corporum in gyrum, that November. Halley saw what it was and pushed for more.
Two and a half years later the Philosophiae Naturalis Principia Mathematica was printed, in July 1687. The Royal Society had agreed to publish it and then discovered it had no money, having spent its budget on Francis Willughby's History of Fishes, so Halley paid for the printing himself out of a modest income. Something like three or four hundred copies were made. Halley also wrote the review, and the Latin ode at the front.
The structure is worth knowing. Definitions and axioms come first, including the three laws of motion, of which the third, that action and reaction are equal and opposite, does the work that turns attraction into a mutual relation. Book I develops the dynamics of bodies moving without resistance, and contains the proof that an inverse square force gives conic sections and the shell theorem of the previous lesson. Book II treats motion through resisting fluids, and ends by demonstrating that the vortices of the mechanical philosophy cannot reproduce Kepler's third law. Book III, the System of the World, applies all of it to the solar system.
Weighing a planet
The central claim of Book III is that every body attracts every other with a force proportional to the product of their masses and inversely as the square of the distance between them. Newton never wrote it as an equation with a constant in front, and the constant itself was not measured until Henry Cavendish weighed the earth with a torsion balance in 1798, obtaining a value equivalent to about in modern units. Everything in the Principia is done in ratios, which is why it can determine the relative masses of the sun and planets without knowing any of them absolutely.
The method is the third law applied twice. For a small body circling a large one, the same substitution as in the previous lesson gives , so the combination measures the mass of the central body. Compare the value for a moon of Jupiter with the value for a planet of the sun, and the masses of Jupiter and the sun come out in ratio.
Example. Io orbits Jupiter at km with a period of days. What is Jupiter's mass as a fraction of the sun's?
Convert into astronomical units and years, so that the earth's own orbit gives for the sun. Io's distance is AU and its period is years. Then
which is one part in . The modern value is one part in . Newton's own figure in the third edition was one in , limited by the accuracy of Io's measured distance from Jupiter. Nobody had ever weighed anything that was not on a balance.
Now you. Titan orbits Saturn at km in days. Find Saturn's mass in the same way.
Answer
Titan's distance is AU and its period is years, so , or one part in about . The modern value is one part in . Note what is being used: only a distance and a period, both obtained from a telescope, plus the assumption that the same law governs Saturn's system as governs the sun's. The result is a property of the planet itself. This is the moment the solar system stops being a set of positions and becomes a collection of objects with quantities attached, and it is the reason later astronomers could tell that something was pulling Uranus off course and calculate where the unseen planet had to be.
Three predictions nobody else could make
The case for the Principia is not that it explains gravity, since it does not, but that one law delivers a list of unrelated things that had never been connected to anything.
The tides get their first correct account. The moon pulls harder on the near side of the earth than on the centre, and harder on the centre than on the far side, so the water is drawn up in two bulges on opposite sides. Two high tides per day follow, and the timing follows the moon, which is exactly the fact Galileo threw away, as the lesson on the trial described. The sun does the same thing more weakly, in a ratio of about to in the moon's favour, so tides are large when the two act together at new and full moon and small when they act at right angles. Every part of that is a consequence of the inverse square law and none of it was available before.
The precession of the equinoxes gets its first explanation. Hipparchus had discovered around 130 BC that the equinox drifts slowly round the zodiac, taking about years, a rate of some arcseconds a year. Nobody had ever proposed a cause. Newton showed that the earth's equatorial bulge gives the sun and moon something to take hold of, and that the resulting torque makes the axis wheel round like a slow top.
And the earth turns out not to be round.
Example. A rotating fluid body bulges at its equator. Newton calculated a flattening of about 1 part in 230, and the true value is 1 in 298. What does the difference amount to on the ground, and how was the question settled?
A flattening of on a radius of km means the equatorial radius exceeds the polar one by about km; Newton's would make it km. Either way the earth is an oblate spheroid, wider than it is tall. The observable consequence is that the length of one degree of latitude, measured along the surface, is longer near the poles, where the surface curves less sharply: km at the pole against at the equator, a difference of km per degree. This mattered because the Cassini family, from measurements within France, had concluded the opposite, that the earth is a lemon rather than an orange. The Académie settled it by sending two expeditions: Maupertuis to Lapland in 1736, and Godin, Bouguer and La Condamine to Peru from 1735, an eight-year ordeal at high altitude. The Lapland degree came back longer than the French one. Newton was right, Cassini was wrong, and Voltaire congratulated Maupertuis on having flattened both the earth and the Cassinis.
Now you. Why is Newton's too large?
Answer
Because he treated the earth as a fluid of uniform density. A real earth has a dense iron core and a lighter mantle and crust, so more of its mass sits near the centre than a uniform model assumes. Mass concentrated at the centre is less affected by the rotation and pulls the outer layers in more strongly, so the body resists bulging and the flattening comes out smaller. The discrepancy is therefore not an error in the theory but a measurement of something else: the degree to which the earth's density increases with depth. That is the same pattern as the residual half per cent in the moon's period from the previous lesson. A law that predicts a quantity to within thirty per cent, and whose shortfall turns out to be a measurement of the interior of a planet, is doing more work than a law that fits exactly for reasons nobody can name.
The comet
The most public vindication came from Halley. In his Synopsis of the Astronomy of Comets of 1705 he computed orbits for 24 comets using Newton's methods, and noticed that those of 1531, 1607 and 1682 had nearly identical elements and were spaced about 76 and 75 years apart. He proposed that they were one object on a long ellipse, and predicted its return around 1758.
That prediction is unlike anything before it. It is specific, it concerns a class of object regarded for millennia as portents outside the order of nature, it could not be hedged, and it would be tested long after its author was dead. Halley died in 1742.
Example. As 1758 approached, Alexis Clairaut, working with Joseph Lalande and Nicole-Reine Lepaute, spent six months computing how much Jupiter and Saturn would delay the comet, and announced perihelion around mid-April 1759, with a stated uncertainty of about a month. The comet passed perihelion on 13 March 1759. How good is that?
The prediction was about 32 days early relative to the event, inside the uncertainty Clairaut had quoted. Measured against the orbital period of roughly 76 years, or days, the error is per cent. What deserves emphasis is not the accuracy but the structure of the exercise: a law inferred from planetary motions was used to calculate a perturbation of one body by two others, applied to an object last seen in the previous century, and checked against an event nobody could influence. There is no way to fake that and no way to explain it away if it fails.
Now you. Why does the perturbation calculation matter more than the bare prediction of a return?
Answer
Because a bare return is weakly diagnostic. Three sightings 76 and 75 years apart already suggest a periodic object, and almost any theory that allows long closed orbits would let you extrapolate the next one to within a year or two. The perturbation calculation uses the specific content of the law: it takes the masses of Jupiter and Saturn, obtained by the method earlier in this lesson from their own moons, works out how much each deflects the comet as it passes, and converts that into a shift of days in the return. That number is meaningless under any rival account, because no rival can even state the question. Clairaut also announced in advance the uncertainty he attached to it, which is what makes the outcome a test rather than a claim: a result 32 days from a prediction quoted to a month is a pass, and the same 32 days against a prediction quoted to a week would have been a failure. Stating in advance how wrong you expect to be is one of the strongest habits the new method acquired.
The price
Set against all this is one thing, and every competent mechanical philosopher in Europe saw it at once. Gravitation acts between bodies separated by empty space, instantaneously, in proportion to a quantity of matter, with no intervening medium and no contact whatever. The lesson on the mechanical philosophy set out the prohibition that everyone in the field had signed: no action at a distance, no influence across a gap, no property of a body that is not the size, shape or motion of its parts. The Principia breaks it on the first page of Book III.
Huygens, who admired the mathematics without reservation, wrote that the principle of attraction seemed to him absurd. Leibniz called it a return to the occult qualities of the schoolmen and, later, a perpetual miracle. The charge was not obscurantism: it was that Newton had stopped doing physics and started doing mathematics, describing the phenomena in equations without giving any account of how the world produces them.
Newton agreed with them about the problem. In a letter to Richard Bentley in 1693 he wrote that for gravity to be innate and essential to matter, so that one body could act upon another at a distance through a vacuum without anything to carry the action, was so great an absurdity that no one competent in philosophical matters could fall into it. He spent decades looking for a medium that would carry the force, and published none of it.
What he did instead is in the General Scholium added to the second edition in 1713. He has not been able to discover the cause of gravity from the phenomena, and he frames no hypotheses, because whatever is not deduced from the phenomena is to be called a hypothesis and hypotheses have no place in experimental philosophy. It is enough, he says, that gravity really exists and acts according to the laws set out, and suffices to account for all the motions of the heavenly bodies and of the sea.
That is the price, and it should be stated plainly, because it is the largest single change in this whole subject. Before the Principia, an explanation meant an account of how something is brought about. After it, a mathematically exact law with a wide range of confirmed predictions counts as knowledge whether or not anyone can say what produces it. Understanding is demoted; prediction is promoted. Everything that has come since has been done on those terms, and the people who objected were not fools defending superstition. They were defending the older and more ambitious idea of what it means to explain something, and they lost.
How long it took
Britain accepted the Principia quickly, helped by the fact that Newton was its author and by a run of textbooks and public lectures. On the Continent it took roughly fifty years. French universities taught Cartesian vortices into the 1740s, and the shift came from a small number of advocates and two decisive results: Voltaire's Lettres philosophiques of 1734 and his Elements of Newton's Philosophy of 1738, Émilie du Châtelet's French translation with her own commentary, finished shortly before her death in 1749 and published in full ten years later, the Lapland expedition's confirmation of the earth's shape in 1737, and the comet in 1759.
Notice what did the persuading. Not the elegance of the argument, and not the metaphysics, on which Newton was and remained vulnerable. What settled it was that the theory kept producing quantities that could be checked, and the checks kept coming out right, in cases the theory had not been built to handle. That is a claim about method as much as about gravity, and drawing out what the method turned out to be, and when it was actually adopted, is the business of the last lesson.