Falling Moons and Mathematical Insight: Feynman on Newton’s Law of Gravitation

In Chapter 7-4 of The Feynman Lectures on Physics, Richard Feynman takes us on a journey not just through physics but through the intellectual revolution triggered by Sir Isaac Newton in the 17th century. Titled Newton’s Law of Gravitation, the lecture is a lucid and imaginative exposition that connects planetary motion with a simple fact: objects fall.

What begins as an inquiry into Kepler’s laws blossoms into Newton’s powerful realisation that the force keeping the moon in orbit is the same force that pulls apples to the ground. It’s a chapter rich in insight, narrative flair, and deeply revealing examples.

From Kepler’s Laws to Newton’s Generalisation

Feynman starts with Newton’s appreciation of Kepler’s second law, the principle that planets sweep out equal areas in equal times. This seemingly geometric truth holds a deeper message: it tells us that the force acting on a planet must be centrally directed, pointing straight at the Sun. This deduction, elegant and precise, allowed Newton to reason about the nature of the force acting on celestial bodies, long before the notion of gravitational fields or Newton’s own formulation of force as F = ma had matured.

But the real brilliance emerges as Newton turns to Kepler’s third law, which relates the period of a planet’s orbit to its distance from the Sun. By examining the proportional relationship between distance and orbital time, Newton demonstrated that the force between a planet and the Sun must decrease with the square of the distance. This is the beginning of the famous inverse-square law—one of the most influential relationships in the history of science.

Feynman remarks on Newton’s genius for generalisation: from planets orbiting the Sun, to moons orbiting planets, and ultimately to everything pulling on everything else. Here, Newton’s law becomes universal—a breathtaking leap that connects the heavens and the Earth.

The Moon Falls, Too

One of Feynman’s most striking examples is the idea that the moon is falling, just like an apple.

This might seem absurd. After all, the moon doesn’t get any closer to Earth. But Newton’s key insight—beautifully retold by Feynman—is that the moon is in perpetual free fall, pulled towards the Earth but continually missing it due to its tangential velocity. This is the core of orbital motion.

Feynman walks us through Newton’s own method: comparing how far the moon “falls” in one second with how far an object on Earth falls in the same time. The object near the Earth falls 16 feet in one second. The moon, located approximately 60 Earth radii away, falls just 1/20 of an inch in a second—an amount consistent with the inverse-square law:

This prediction originally gave Newton pause: his calculation didn’t match the observed motion. The discrepancy was so significant that he set the idea aside for years. Only after more accurate measurements of the Earth’s radius (from the work of the French astronomer Jean Picard) did Newton return to the problem. With the corrected figures, the prediction worked—beautifully. In this moment, the universal law of gravitation was confirmed.

Falling Sideways: The Bullet Thought Experiment

To illustrate how something can “fall” without ever hitting the ground, Feynman uses a classic thought experiment: fire a bullet horizontally, and it will fall 16 feet in the first second, just like a dropped object. Fire it faster, and it travels further before hitting the ground—but it still falls. Fire it fast enough, and it falls “around” the Earth, staying in perpetual free fall. This is orbital motion.

Feynman gives us a back-of-the-envelope calculation. The Earth’s radius is about 4000 miles, and if a bullet falls 16 feet in a second, how fast must it travel so that the Earth curves away beneath it at the same rate?

The answer: 5 miles per second.

This is orbital velocity. It’s the principle that underlies satellites and space travel. Feynman invokes Yuri Gagarin, the first human to orbit the Earth, to bring this point to life. Gagarin stayed in space by moving fast enough that he kept “missing” the Earth as he fell.

The Tides and Misunderstandings

From celestial mechanics, Feynman transitions to tides—one of the most misunderstood consequences of gravity.

Before Newton, many tried to explain tides as the water being pulled up by the moon. But that would produce a single tide per day, not two. Others proposed the Earth being pulled away from the water, which produced a similar mismatch.

Newton explained that both Earth and Moon orbit a common centre of mass, and the tides arise because of differential gravitational force. The side of the Earth nearest the moon is pulled more strongly than the centre, and the far side less strongly. The flexible oceans respond by bulging both towards and away from the moon. This produces the familiar pattern of two high tides per day, not one.

Feynman explains this beautifully, showing that the true cause is the imbalance of force—not an absolute pulling or pushing. It’s a subtle but profound difference, and a testament to Newton’s clarity of thought.

From Insight to Prediction

As Feynman notes near the end, a great law must be fruitful—we should get out more than we put in. Newton used Kepler’s laws to derive the law of gravitation, but from that he could make new predictions: the moon’s motion, the elliptical shape of orbits, and the behaviour of the tides.

Eventually, Newton’s law of gravitation would underpin the entire structure of classical physics—a foundation that would remain unshaken until Einstein’s general theory of relativity offered a deeper, though more complex, view of gravity.

Final Thoughts: Feynman as Interpreter

Feynman’s lecture is not just a retelling of Newton’s work—it’s a celebration of scientific imagination. His thought experiments, particularly the bullet orbit and the falling moon, convey difficult ideas with striking clarity.

He shows us that physics is not just about equations, but about the interplay of logic, observation, and creativity. Newton’s genius lay not only in his mathematics, but in his ability to see the same lawoperating from the apple tree to the stars.

And through Feynman’s eyes, that genius comes alive again.