Kepler’s Harmonies: Feynman, the Ellipse, and the Poetry of the Planets

Exploring Chapter 7 of The Feynman Lectures on Physics, and the profound simplicity behind Kepler’s Laws of Planetary Motion

When Richard Feynman opens a discussion on Kepler’s laws in The Feynman Lectures on Physics, he takes us not only into the realm of orbital mechanics, but also into the history of science itself—a time when careful observation, mathematical intuition, and imagination transformed our understanding of the heavens. The passage quoted from Lecture 7-1 (Kepler’s Laws) provides a compact but deeply rich account of Kepler’s three planetary laws, each rooted in the genius of observational astronomy and each resonating through centuries of physics and mathematics.

Let us take Feynman’s overview as a springboard, and explore more deeply the scientific and historical context of each law, tracing their discovery, interpretation, and impact.

Kepler’s First Law: The Shape of Orbits – From Circles to Ellipses

“Kepler found that each planet goes around the sun in a curve called an ellipse, with the sun at a focus of the ellipse.”

For millennia, the heavens were thought to be governed by divine geometry—most notably, the circle. The ancient Greeks, from Plato to Ptolemy, envisioned planetary orbits as perfect circles, each nested in crystalline spheres that turned with eternal uniformity. This was not superstition, but rather an aesthetic principle: the circle was seen as the most “perfect” shape, reflecting harmony and divine order.

Enter Johannes Kepler (1571–1630), a brilliant and deeply spiritual astronomer working in the shadow of his mentor Tycho Brahe. Tycho, an obsessively precise observer, had amassed the most accurate naked-eye data on planetary positions ever recorded. Kepler inherited this treasure trove after Tycho’s death in 1601—and with it, the formidable challenge of explaining the orbit of Mars.

For years, Kepler wrestled with the “Martian problem.” Mars simply didn’t behave as it should if it moved in a circle. It was through this painstaking data-fitting exercise that Kepler made a revolutionary discovery: Mars, and indeed all the planets, move in elliptical orbits, not circles. In 1609, he published this result as the first of his now-famous laws.

An ellipse is a conic section—a shape you get by slicing a cone at an angle. Feynman gives a beautifully accessible method for drawing one: fix two tacks into a piece of paper (these are the foci), loop a string around them, and pull the loop taut with a pencil. As you trace around, keeping the string tight, you draw an ellipse. The sum of the distances from the pencil to each tack remains constant—a simple geometric constraint that gives rise to the ellipse’s elegant symmetry.

Crucially, the Sun sits at one of the two foci—not at the centre. This means that the planet’s distance from the Sun is not constant, but varies throughout its orbit. The result demolished two millennia of circular dogma and opened a new, more accurate way of describing planetary motion.

Feynman notes that the ellipse is “not just an oval”—he’s right to emphasise this. The ellipse is not a vague or arbitrary shape but a precise mathematical curve. Its discovery marked a shift from philosophical speculation to data-driven modelling, a turning point in the scientific revolution.

Kepler’s Second Law: Equal Areas in Equal Times – A Law of Motion, Not Just Geometry

“The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time.”

This second law, also published in 1609, is far more than a geometric curiosity—it’s a law of motion. It tells us that a planet moves faster when it is closer to the Sun (perihelion) and slower when it is farther away (aphelion), but in such a way that the area “swept out” by the radius vector (a line from the Sun to the planet) is always the same over equal time intervals.

To grasp this visually, imagine plotting a planet’s position week by week on an elliptical orbit. If you connect the week-one and week-two positions with the Sun at one focus, the triangle formed has a certain area. Now do the same with two positions a week apart elsewhere in the orbit, say near aphelion—the triangle is longer and thinner, but Feynman tells us the area is identical. The planet compensates for its increased distance by moving more slowly.

This observation has enormous physical significance. It implies the conservation of angular momentum, a concept later formalised by Newton and refined by Lagrange and Euler. Angular momentum, the “rotational equivalent” of linear momentum, remains constant when there are no external torques. For planetary motion, this conservation law arises naturally from Kepler’s second law.

Historically, Kepler arrived at this without Newtonian mechanics. He observed the behaviour from data, and guessed a principle. But Newton, almost 80 years later, showed that this law is a necessary consequence of a central force (gravity) acting between two bodies. In his Principia Mathematica, Newton proved that any object moving under an inverse-square central force would obey Kepler’s laws—an astounding unification of celestial mechanics and terrestrial physics.

Kepler’s Third Law: Harmony Among Planets – The Mathematical Music of the Spheres

“The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits.”

Kepler’s third law came in 1619, a full decade after the first two. He had grown increasingly obsessed with numerical relationships between the planets—what he called the harmonies of the celestial spheres. In Harmonices Mundi (The Harmony of the World), he published this final law:

where T is the orbital period and a is the semi-major axis (half the longest diameter of the ellipse).

This is not a law about a single orbit, but a comparative law—a harmony among the planets. It shows that more distant planets take longer to orbit the Sun, and do so at slower average speeds.

This was the first hint of a universal scaling law—a proportion that holds not just for Earth or Mars, but for all the planets. In fact, this same law (with different constants) works for moons orbiting Jupiter, artificial satellites orbiting Earth, and even stars orbiting the centre of the Milky Way.

Kepler’s third law was the most mathematical and speculative of the three—and it was Newton who gave it its deepest physical meaning. By applying his Law of Universal Gravitation, Newton derived Kepler’s third law as a logical consequence. In doing so, he showed that gravity was the hidden hand behind all planetary motion—a single force governing both apple and planet.

The Significance: From Kepler to Newton to Einstein

Feynman’s crisp summary of Kepler’s three laws belies the vast intellectual journey they represent. Together, they marked a decisive shift from a qualitative, often mystical cosmology to a quantitative, mathematical physics. They laid the groundwork for Newton’s entire gravitational theory. And from Newton, the lineage continues to Einstein’s general relativity, where orbits become geodesics in curved spacetime.

Let’s not forget the technological and cultural implications either. Our ability to launch satellites, explore the solar system, and predict eclipses—all rest on these foundational principles. The ellipses traced by spacecraft like Voyager, or the precise orbital corrections of the James Webb Space Telescope, are modern echoes of Kepler’s geometry.

A Closing Thought: The Poetry of Precision

Feynman admired not just the accuracy of physics, but its aesthetic. Kepler’s laws exemplify this dual beauty: empirical laws that are not only useful but elegant. That the chaotic dance of the planets could be captured in three concise rules speaks to the underlying simplicity of nature—what Einstein called the “subtle and not malicious” logic of the universe.

And perhaps that is what draws us to Feynman’s treatment of Kepler: his delight in clarity, in the fact that centuries-old data, scribbled by candlelight, could yield truths so enduring that we teach them to undergraduates today, and use them to navigate probes to Saturn.

So the next time you trace an orbit, remember the tacks and string—and the mind of Kepler, who dared to believe that the heavens, if studied with care and imagination, would yield their secrets.