The claim at the centre of the Principia is that the force pulling an apple to the ground and the force holding the Moon in its orbit are the same force, following the same rule.
That is a very large claim, and it can be checked with one division. The check is known as the Moon test, and it is the moment when universal gravitation stops being a speculation and becomes a number that either agrees or does not.
The Test
Suppose gravity weakens as the inverse square of the distance from the centre of the Earth.
The Moon is about sixty Earth radii away. Sixty squared is 3,600, so the acceleration of the Moon towards the Earth should be one three thousand six hundredth of the acceleration of a falling body at the surface.
Surface gravity gives about 9.8 metres per second squared. Divide by 3,600 and you predict 0.00272 metres per second squared for the Moon.
Now measure the Moon’s actual acceleration, which requires only its distance and how long it takes to go round. For a circular orbit the centripetal acceleration is four pi squared times the radius, divided by the square of the period. The distance is 384,000 kilometres and the period is 27.32 days.
The answer is 0.00272 metres per second squared.
The two numbers, arrived at from completely unrelated measurements, agree. One comes from dropping something in a room, the other from watching the sky for a month.
Why That Agreement Is Worth More Than It Looks
The deep point is not the arithmetic, it is what the arithmetic joins together.
Before Newton, the physics of the heavens and the physics of the Earth were separate subjects. In the Aristotelian scheme, terrestrial matter was made of the four elements and moved in straight lines towards its natural place, while the heavens were made of a different substance and moved eternally in circles. Falling objects and orbiting objects had nothing in common; asking whether the same law covered both was close to a category error.
Newton’s Moon test says they are one phenomenon. The Moon is falling. It is falling all the time, exactly as an apple does, and the only reason it does not arrive is that it also moves sideways fast enough that the Earth’s surface curves away beneath it.
He had the picture early. His notebooks and his later accounts describe thinking about it in 1665 and 1666, at Woolsthorpe during the plague years, in what he called the prime of his age for invention. The apple story he told in old age to William Stukeley and to John Conduitt is probably a real prompt for a real train of thought, and the detail about being struck on the head is an embellishment of later centuries.
The First Attempt Was Off
When Newton did the calculation in the 1660s, the answer did not come out cleanly. His own phrase, written decades later, is that he found the forces answer pretty nearly.
Pretty nearly was doing a good deal of work. The discrepancy was something like fifteen or twenty per cent, and the reason was not his physics but his geography: the size of the Earth.
The Moon’s distance was known reasonably well in Earth radii, from parallax, so the entire test depends on how big an Earth radius is in ordinary units. Newton used the common English estimate of sixty miles to a degree of latitude. The true figure is nearer sixty nine and a half. That error propagates straight into surface gravity’s comparison, and the two numbers failed to match.
In 1669 and 1670 the French astronomer Jean Picard carried out a proper geodetic measurement of an arc of meridian and produced a much better value for the Earth’s size. With Picard’s figure the test comes out right.
The Real Reason for the Twenty Year Delay
It is often said that the bad Earth radius made Newton abandon gravitation for twenty years, and that he returned to it only when better data arrived. The manuscripts do not support that.
Two genuinely hard obstacles stood in the way, and neither was a measurement.
The first was the orbit problem. Knowing that a force falls off as the inverse square does not tell you what path a body follows under it. Demonstrating that an inverse square force produces an ellipse, with the centre of force at a focus, exactly as Kepler had found the planets to move, required mathematics Newton had to build. Others, Hooke among them, had the inverse square intuition. Nobody else could do this.
The second obstacle is more subtle and is the one that makes the Moon test legitimate at all.
When you write down the distance from the Earth to the Moon as sixty Earth radii, you are quietly assuming that the Earth attracts as though its entire mass were concentrated at its centre. Why should it? The Earth is an extended body, and the rock under your feet is a few thousand kilometres nearer to you than the rock on the far side. Every part of it pulls on the apple, in a different direction and at a different distance, and the sum of all those pulls is not obviously anything tidy.
Newton proved that it is. For a sphere with density depending only on distance from the centre, the total attraction on an external body is exactly the same as if all the mass sat at the centre. This is the shell theorem, Propositions 71 and following in Book I of the Principia, and he only obtained it in 1685.
He is reported to have been delighted, and he had reason: without that theorem the Moon test compares a clean number with a meaningless one, and the whole structure of the book is built on being allowed to treat planets as points.
How It Appears in the Book
The test is Proposition 4 of Book III, where Newton states that the Moon gravitates towards the Earth and is retained in its orbit by the force of gravity.
His version of the argument is more elegant than the modern one. Rather than comparing accelerations in metres per second squared, units he did not have, he asks how far the Moon would fall in one minute if you stopped its orbital motion. He computes it, and then compares that distance with the distance a body near the Earth’s surface falls in the same time, scaled by the inverse square, and shows they agree. His figure, in the units of the time, is about fifteen and one twelfth Paris feet.
For surface gravity he uses the best available measurement, which came from pendulums. Huygens had worked out the relationship between a pendulum’s length and its period, and a pendulum that beats seconds gives the local acceleration of gravity with real precision. So the terrestrial half of the most famous calculation in physics rests on a clock.
Newton also does something that is easy to miss and shows how carefully he worked. The Moon is not attracted only by the Earth: the Sun pulls on it too, and that perturbation slightly reduces the Earth’s apparent hold. He estimates the correction and applies it. The agreement he reports is not a rough one, it is the residual after he has accounted for the largest disturbance he knows about.
What the Test Made Possible
Once the same law is known to govern the Moon and the apple, it can be pointed at anything.
Within the Principia Newton uses it on the satellites of Jupiter, the tides, the precession of the equinoxes and the orbits of comets. Seventy years later, three people computing by hand used it to predict the return of a comet to within a month, which is the moment the theory demonstrated that it could say what would happen rather than explain what had.
The same sum, in the same form, is still what puts a satellite where it needs to be. An object in low Earth orbit is one Earth radius or so from the centre and takes about ninety minutes. Geostationary satellites sit at 6.6 radii, which by the inverse square puts their acceleration at about a fortieth of surface gravity, which gives a period of exactly one day. That is not a coincidence, it is the Moon test run backwards to find the distance where the period comes out right.
Seeing It Where It Was Written
Book III of the Principia is the part where the mathematics of the first two books is turned on the actual solar system, and Proposition 4 is where the turn happens. Reading it in place, after the definitions, the laws and the lemmas, makes the architecture of the argument visible: everything before it is preparation, and everything after it is consequence.
Our edition of the Principia keeps that structure intact, including the propositions on the attraction of spheres in Book I that the Moon test silently depends on. There is more on the law itself and on how the three books fit together elsewhere on the blog.
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