Alexander Pope put it in a couplet: "Nature and Nature's Laws lay hid in night / God said 'Let Newton be,' and all was light." It's a wonderful line, but it slightly overstates the case — because the laws Newton illuminated had already been written down, empirically, by Johannes Kepler decades earlier. What Newton actually did, and what Spivak walks through with real rigor in Chapter 17 of Calculus, was show that Kepler's three laws are not independent facts about the heavens at all. They are all consequences of one single hypothesis: that the sun pulls on each planet with a force pointed straight at the sun, whose strength falls off as the inverse square of the distance. Everything else — the ellipses, the swept-out areas, the a³/T² relation — falls out of that one assumption by calculus alone.
Kepler's three laws, stated precisely
Before touching a single derivative, it's worth being exact about what Kepler actually claimed, since the popular versions tend to blur the details. Spivak states them the way Kepler published them: the first law, from 1609, says the planets move in ellipses, with the sun at one focus — not at the center, at a focus. The second law, published the same year, says that equal areas are swept out by the radius vector in equal times — the segment connecting sun to planet sweeps out area at a constant rate, so the planet moves faster near the sun and slower far away. The third law, which took Kepler another ten years to find, ties different planets together: if a is a planet's orbital major axis and T its period, then the ratio a³/T² is the same number for every planet in the solar system.
These are three separate empirical facts about three separate features of an orbit — its shape, its timing, and its scale relative to other orbits. There's no obvious reason they should be related. Newton's achievement was to show they all come from the same physical law.
The equal-area law is really about central forces
Spivak opens with Newton's own argument, which is delightfully unlike modern calculus: instead of a continuously varying force, Newton imagines the planet coasting in a straight line for a short time interval, then receiving an instantaneous kick directed straight at the sun, then coasting again. A short parallelogram argument on the resulting triangles shows that if the kick is aimed exactly at the sun, the two swept-out triangles have equal area. It's not a rigorous proof by modern standards, but as Spivak says, "in its own charming way it shows physically just why the result should be true."
The calculus version makes this exact. Describing the planet's path as c(t) = r(t)·e(θ(t)), where r is the distance from the sun and θ the angle, and writing e(t) = (cos t, sin t), a short computation with the determinant gives the area swept out up to time t as satisfying
A′(t) = ½ det(c, c′) = ½ r²θ′
Kepler's second law says exactly that A′ is constant, which is the same as saying A″ = 0, which turns out to be equivalent to det(c, c″) = 0 — meaning the acceleration vector c″ always points along c itself, straight at (or away from) the sun. This is Spivak's Theorem 1: Kepler's second law holds if and only if the force is central. The area law isn't really a separate fact about gravity at all — it's a fact about any force aimed at a fixed point, gravitational or not.
From central force to ellipse
The harder question is why a central, inverse-square force produces an ellipse specifically, rather than some other curve. Here Spivak's proof earns its keep. Writing the inverse square hypothesis as c″(t) = −(H/r²)e(θ(t)) for a constant H, and combining it with the central-force relation r²θ′ = M (constant), the trick is to stop thinking of c′ as a function of time and instead treat it as a function of the angle θ — Spivak calls this "the main trick." Doing so turns the equation into something that can be integrated directly, term by term, producing an explicit formula for c′ in terms of sin θ and cos θ and two constants of integration. Substituting that back into the area relation and simplifying (using a change-of-variable trick from an earlier chapter to eliminate a phase shift) yields the equation of the orbit itself:
r [1 + ε cos θ] = Λ
which is precisely the polar equation of a conic section with the origin — the sun — at a focus. Depending on the eccentricity ε, that conic is an ellipse (planets, periodic comets), a parabola, or one branch of a hyperbola (objects that pass through the solar system once and never return). This is Theorem 2, and Spivak also proves the converse (Theorem 4): if the orbit actually is a conic section with the sun at a focus, the force must obey an inverse-square law. Gravity and the ellipse determine each other.
Kepler's third law then falls out almost as a bonus. Since the area swept out is A(t) = ½Mt, running the planet through a full period T sweeps out the whole ellipse's area, giving M = 2πab/T. Feeding this into the constant from the orbit equation produces M²/Λ = 4π²a³/T² — and since the left side only depends on constants from Newton's force law, not on which planet you're looking at, the ratio a³/T² is forced to be the same for every planet, exactly as Kepler observed.
Why the rigor is the point
It would be easy to treat this chapter as physics dressed up in calculus notation, but that misses what's actually remarkable about it. Kepler found his three laws by grinding through Tycho Brahe's observational data — pure empirical pattern-matching, brilliant but unexplained. Newton showed, using nothing more than derivatives, determinants, and a differential equation, that all three patterns are forced by a single hypothesis about force. That's the kind of compression proof-based calculus is built to deliver: not "the planets happen to do this," but "anything obeying an inverse-square central force must do this, and only this." Chapter 17 is also, quite deliberately, a bridge — Spivak calls it a chapter that "diverges from the main stream of the book to demonstrate that we are already in a position to do some real physics," and it opens the door directly into the territory covered by his companion volume, Physics for Mathematicians, Mechanics I, which picks up exactly this kind of careful, calculus-first treatment of mechanics at greater length.
If you want to work through Newton's geometric parallelogram argument, the full determinant proof of Theorem 1, and the eccentricity calculation in Theorem 2 step by step, Chapter 17 is in Spivak's Calculus, 4th Edition — and it's one of the most satisfying payoffs in the whole book for the analytic machinery built up over the preceding sixteen chapters.
Further reading: How to self-study Spivak's Calculus · Table of contents · Index · Calculus, 4th Edition · Answer Book
Which piece of Newton's argument struck you as more surprising: that the equal-area law is really just a statement about central forces, or that the specific inverse-square law is exactly what's needed to force an ellipse? Tell us in the comments.
