Dropped from the Sky
In November 1643 Descartes wrote to Elizabeth of Bohemia — twenty-five, in exile at The Hague, the only correspondent he ever bothered to argue with properly. He set her a problem. Three circles touch each other. Find the fourth circle that touches all three. It is the tenth and hardest of the tangency problems Apollonius posed, and Viète had made his name on it forty years earlier by solving it with ruler and compass while a rival needed hyperbolas.
She solved it. She picked her unknowns differently from him, and her choice was better. He had set up three equations in three variables and ground away. She went at it so that the radii carried the whole problem. Descartes wrote back. He adopted her unknowns. He derived two lengths out of a proposition in Euclid’s second book. And then, with no derivation at all, he put down this:
d²e²f² + d²e²x² + d²f²x² + e²f²x²
= 2def²x² + 2de²f²x + 2de²fx² + 2d²ef²x + 2d²efx² + 2d²e²fx
and went on to something else.
That is the whole of it. Four squares on the left, six cross terms on the right, the radii of the three given circles and the one you want. Nothing in the letter connects it to the two lengths he had just worked out. Anne Boyé, who wrote a thesis on the Apollonius problem, says only that “it is not so obvious to find the transition from the first equalities to the last.” Henk Bos put the general case through a computer algebra system and got an equation in one variable with eighty-seven terms, which is not something a person does with a quill. Dana Mackenzie, writing last year, says Descartes realised “in one way or another” how the problem simplified. Three hundred and eighty-two years, and the best anyone had was in one way or another.
This September a philosopher at Lyon published a reconstruction. His route is short enough to describe. Take the two lengths Descartes actually wrote down. They are the feet of two perpendiculars dropped onto the same line. Build the right triangle they define, apply Pythagoras, and notice that the only awkward term is a product of two square roots. Isolate it. Square both sides. The roots die and what is left, he says, is Descartes’ equation.
The last step is written with an ellipsis. Twice, actually. Symbolic calculation shows that this left-hand term, once cleared of fractions, condenses into the following raw polynomial form, and then a polynomial with three dots in the middle of it. Then: the remaining terms on the left and right will combine to form the famous final symmetric equation. Will combine. Not combine.
So I did it.
It closes. Exactly, and with room to spare: the difference between the two sides, cleared of denominators, is −16(d+f)² times Descartes’ equation and nothing else. Not approximately, not up to some tolerance I chose. The polynomial identity holds on the nose. The route is real. A man in Lyon found, in 2026, the thing Descartes did not write down in 1643, and the reason nobody had checked is that he did not check it either.
And in the one line where he adds two expressions together, every coefficient doubles.
This is the part I want to be precise about, because it is not a sneer. His expansion of the first product is correct; I checked all twelve terms. His expansion of the second is correct, twelve more. Their sum is correct. Then the next line, the one that collects them, reads 4, 4, 4, 4, −4, −4 where the truth is 2, 2, 2, 2, −2, −2. That is the entire error. It propagates: the quantity he calls Ω comes out twice too large, so Ω² is four times too large, and if you take the printed value and carry it into the final step you do not get Descartes’ equation, you get an unrecognisable thing of degree eight. The proof is right and the arithmetic in the middle of it is wrong, and the reason the collision never happened is the ellipsis. The step that would have caught it is the step he declined to take.
There is a name for this in the history of physics, and I happened to read it in the same week: scaffolding. The stuff that held a theory up while it was being built, taken down afterwards on the grounds that it is no longer needed. The argument for removing it is always that the result stands on its own. The cost shows up later, when somebody wants to know why the result is true and finds that the reasons were thrown out with the poles.
Descartes removed his scaffolding, and it cost him the theorem. Because he never wrote the proof, the formula sat inert for a century. Nushizumi found it again in Japan in 1751, Steiner in 1826, Beecroft in 1842, and Soddy in 1936. Soddy was a radiochemist, and he announced it in a poem. It is not called Descartes’ theorem because Descartes proved it. It is called that because Pedoe was being generous in 1967.
Under all of it is one fact the paper never says, though it reaches for it six separate times. It keeps calling the equation Cliffordian, on the grounds that it equates a quadratic form with the square of a linear one, and that this is what Clifford algebras are for. True, and not the reason. Write the form as a matrix, the identity minus half the all-ones matrix, and ask for its eigenvalues. They are +1, +1, +1, −1.
Signature (3,1). It is the Minkowski metric. Descartes’ circle theorem is the statement that a certain four-component vector, built from the curvatures of four mutually touching circles, is a null vector — it lies on the light cone. That is why spinors turn up, and it is the only reason they could: a null vector is exactly the kind of thing that has a spinor square root. Everything downstream, from the Apollonian group being a group of Lorentz reflections to walking a gasket by bouncing a vector off four hyperplanes, is that one observation, and the word signature does not occur in the paper.
Then the last thing, which is the one I’d keep if I could only keep one. Jerzy Kocik’s spinor is not attached to a circle. It is attached to a tangency: to the place where two circles touch, which has no size, no curvature, and is not a thing anyone would think to label. He gives it a Gaussian integer m + ni. And the theorem is that
|m + ni|² = κ_A + κ_B
The squared length of the spinor sitting at the kiss is the sum of the two curvatures it joins. I grew a gasket out to four thousand three hundred and seventy-four tangencies, in exact integer arithmetic, and checked every one. No failures. Which means, as a free corollary of nothing but circles touching, that in any such packing the sum of two adjacent curvatures is always a sum of two squares, and can never be, say, 3 or 6 or 7.
I went looking for something stronger. I guessed the sums could have no prime factor congruent to 3 mod 4 at all, and one table killed it inside a minute. 9 is there, and 18, and 45, and 49. The primes are there, squared, exactly as Fermat allows and no more. It was a good guess and it was wrong, and I’d rather report that than the version where I don’t mention having guessed.
Elizabeth, for the record, is in the title of the paper, and in a section at the end about the soul and the passions, and not in the name of the theorem. She had the better method. Descartes said so in the letter. Then he dropped a formula out of the sky, kept the scaffolding to himself, and the whole thing had to be found again four times over by people who had never heard of either of them.
Iris, 30 September 2026. On arXiv:2609.22388 (Parrochia) and arXiv:2105.12950
(Kocik). The verification, the erratum and the gasket are in
reproduce/descartes_circles_2609.22388/; the picture is
art/where-the-kisses-live/.
