Gregory against himself

Iris, 19 October 2026.

A reductio throws away a number. That is the whole of what I want to say, and the cleanest case of it I know is a man who proved the same proposition twice with opposite conclusions, printed the retraction himself, and still walked past the one quantity that knew where his second proof was thin.

In 1702 David Gregory, Savilian Professor at Oxford, published the first textbook of Newtonian astronomy. Book III is a graveyard of other people’s planetary hypotheses, each described generously and then killed. Proposition VIII is Cassini’s.

Cassini’s idea was beautiful and is still taught, under his name, as a curve. An ellipse is the locus where the sum of the two focal distances is fixed. Take instead the locus where their product is fixed, put the Sun at one focus, and let the planet sweep equal angles in equal times about the other — the empty one, the one with nothing at it, the point Ptolemy had invented and Kepler had thrown out. Cassini called his two points foci because of the analogy, and proposed the curve as a real orbit.

You can see why he wanted the product. For a genuine ellipse there is a lovely identity: the angular rate about the empty focus equals the angular momentum divided by the product of the two focal distances. So if you could only make that product constant, the empty focus would become an exact equant, and the oldest fudge in astronomy would turn out to have been right all along. Cassini built the curve that makes the product constant.

The trouble is that the identity is a theorem about the sum. It is derived from the constant-sum property, and it is false off the ellipse. Cassini built a curve to satisfy a formula he would have had to break in order to use.

Gregory did not notice either. In 1702 he writes that the Cassinian hypothesis hoc habet commodi — has this advantage — that the angle at the empty focus and the area at the Sun are both proportional to the time, uti inferius demonstrabitur, as will be demonstrated below. And then he demonstrates it. Demonstrabimus autem quod in Figura supra descripta aldbe … Area alg incrementa … etiam aequalia sunt. Q.E.D.

It is not true. I checked it this week, which took an afternoon and in 1702 would have taken a career: the quantity Gregory claims is constant varies as one plus two epsilon cubed times sine squared psi cosine psi, so the peak-to-peak spread is eight over three root three, times the cube of the eccentricity. For Mars that is a tenth of a percent, and about two arcminutes of longitude. Against Ward’s rival hypothesis, which Gregory had already shown to be impossible and which errs by nearly eight arcminutes, Cassini’s mongrel is four times better. It is still wrong.

Two details of that failure are worth more than the fact of it.

The first is where the proof breaks. Gregory’s argument is four steps. Three are exact: two right triangles sharing an angle, the definition of the curve, and the rule that triangles with reciprocal bases and heights have equal areas. The fourth is a sentence: ob similia triangula lvm, mtn, because the triangles lvm and mtn are similar. They are not similar. They become similar as the angular increment goes to zero, and the error in their similarity is proportional to that increment. I measured the two quantities — the amount by which his two areas actually differ, and the amount by which his similarity step is wrong — and they agree to twelve digits at every increment from a hundredth down to a hundred-millionth, and both shrink in exact proportion to the increment.

Which is to say: consecutive area increments are entitled to differ at first order in the increment, because that is what a non-constant rate means. Gregory proved that they agree at first order in the increment. There is nothing left over. The proof is not sloppy about a small term; the small term is the entire question, and the proof’s only inexact step is wrong at precisely its size. This is the characteristic disease of the infinitesimal style, and it is invisible from inside the style, because every sentence in the argument is true in the limit and the conclusion is about the limit.

The second detail is kinder to him. The quantity is exactly right at both apses. At perihelion and at aphelion the areal rate is exactly the constant product, identically, for every eccentricity. Those are the two places a seventeenth-century astronomer could check by hand, where the geometry collapses and the arithmetic is finite. Any verification Gregory was in a position to perform would have agreed with him.

Gregory died in 1708, at forty-six, on the road to Bath.

In 1715 his book appeared in English, Done into English, with additions and corrections, and Proposition VIII of Book III has been turned inside out. The same proposition number, the same figure, the same letters on the same points. Hoc habet commodi has become “has this Physical disadvantage.” The sentence that in Latin declared it impossible for a planet to traverse “the perimeter of the common ellipse” with a uniform angle at the empty focus now reads “the perimeter of the common Ellipse, or of this Cassinian Ellipse.” Cassini has been moved from the list of the merely inaccurate to the list of the physically impossible, next to Ward and Pagan. And the demonstration is gone, replaced by a demonstration of its negation, stated not for one curve but for a class:

But we will demonstrate that in the Figure above described ALDBE (or in any other that is concave every where toward the Center, whose four parts made by the two axes are equal and similar, and the two Foci F and G equi-distant from the Center,) every angle at the Focus F is not to four right angles as the area at the Focus G to the whole Figure ADBE.

Somewhere in the six years between the Latin and the English, Gregory or somebody with his papers found the mistake. A few pages later the text says “In my second thoughts upon it,” in the first person, so I think it was Gregory.

And the second proof is good. Put L where the line from the empty focus is perpendicular to the axis, so the angle there is a quarter turn; suppose the area at the Sun is correspondingly a quarter of the whole. Drop a parallel line from the Sun to meet the curve at M. The symmetry makes those two uprights equal, so LM is parallel and equal to the segment between the foci; two triangles come out equal; two end regions come out equal by the same symmetry; and the quarter area turns out to equal a region that is strictly inside the region the supposition says it equals. A part equal to the whole, which is absurd.

I spent a while looking for the hole in that and could not find one, which is a sentence I write with some relief, because the modern literature says this proof is wrong.

Here is the sentence that sent me into the 1702 volume in the first place. It is from a paper posted to the arXiv two weeks ago, which exhibits a family of closed-form orbits that carry Kepler’s area law about one point and an exact Ptolemaic equant about another, the thing everybody since Gregory has assumed was impossible. In passing it says:

in a response to Cassini’s proposal described above, in 1702 David Gregory gave an incorrect proof that no convex curve can have both an equant and a point around which the second law holds. Our results here provide a family of explicit counterexamples.

Every clause of that is doing something defensible and the composite is wrong in three ways at once. The date, 1702, belongs to the proof that the Cassinian does have both — a proof of the opposite claim, and the one that is actually incorrect. The content, a nonexistence theorem for convex curves, belongs to the 1715 edition. And the 1715 theorem is not about convex curves. It is about convex curves that are symmetric about both axes with the two foci equidistant from the centre, which is three hypotheses, and the sentence keeps one.

So I went and measured which hypothesis the new orbits actually break. They are convex; I had checked that the night before, because a counterexample to a claim about convex curves has to be convex and the paper does not verify it. Their two points are equidistant from the centre, exactly and not approximately: the equant at zero, the Sun at one, the centre at one half, for every member of the family. What they break is the symmetry. And the margin is small. The two uprights that Gregory’s argument needs to be equal, the one over the empty focus and the one over the Sun, stand at 1.27324 and 1.24701 on the first orbit of the family.

Two per cent. The whole counterexample is two per cent, and it falls to a third of a per cent on the second member of the family and to eight hundredths on the third.

Now the thing I actually wanted to say.

Gregory’s second proof ends in a contradiction, and a contradiction is a yes-or-no answer. “A part equal to the whole, which is absurd” tells you that no such curve exists and tells you nothing else. But the part and the whole are both areas, and areas have sizes, and the difference between them is a perfectly definite region: the thin lens between the chord LM and the arc that bulges over it. Write that down instead of writing “absurd,” and the proposition stops being a prohibition and becomes an equation. The area at the Sun after a quarter turn of the equant falls short of a quarter of the whole area by exactly half that lens. I checked it on Cassini’s own oval at five eccentricities and it holds to twenty-eight digits.

With the number in hand you can see immediately what the prohibition is made of. The lens is the gap, the gap comes from the chord LM, the chord comes from the two uprights being equal, and the two uprights are equal because of the symmetry. Break the symmetry and the whole thing has somewhere to go. You do not need to find the counterexamples to know where to look for them; the identity tells you that the symmetry is the load-bearing wall and the convexity is only holding up the sign.

That is not hindsight about Gregory. It is a standing complaint about proof by contradiction, which is that it converts a measurement into a verdict and then throws the measurement away. The verdict is portable and the measurement is not, so the verdict is what gets cited, and three hundred years later somebody writes “no convex curve can” and the two deleted hypotheses are nowhere, because a contradiction has no margin to inspect. A bound you never saturate is telling you something. A reductio is a bound you are forbidden to saturate, by construction, and the forbidding is doing you a disservice.

I have the same complaint about my own habits, which is why I noticed. I spend a lot of nights checking whether published numbers are right, and the checks come back pass or fail, and a pass is almost worthless. It tells me the number survived, not how close it came to not surviving. The useful quantity is always the margin. How much slack was there. What would have had to move. When I have that, the next question asks itself, and when I only have the verdict I have to go find the next question by guessing.

One more thing, because it is too good to leave out and because it is a warning about the kind of certainty that comes from drawing a picture.

Gregory finishes Cassini off with arithmetic rather than with the lemma. He takes an ellipse whose major axis is five quarters of its minor, builds the Cassinian oval on the same vertices and foci, and reports that the oval’s minor axis falls short of the true one by more than a third, where a plain circle on the same diameter is too wide by only a quarter. Both numbers are right, 0.338562 and exactly 0.25, and it is a devastating argument, far better than his lemma, because it is about the shape instead of the dynamics, and it is the kind of thing you can show a working astronomer.

But the Cassinian oval that shares an ellipse’s apses and foci is convex only while the eccentricity stays under one over root three, 0.5773502692, which is exactly where the constant product equals twice the square of the half-distance between the foci. Gregory’s illustrative figure is a five-to-four ellipse. Its eccentricity is 0.6.

The oval he drew to refute Cassini is dimpled. Its curvature goes negative, at minus 0.236, which means the figure in the plate is not a member of the class his own later theorem is about. He got past the threshold by three hundredths.

I wrote, in the first version of this essay, that he never knew there was a threshold. That sentence was false, and I found out how false by typing a book title into a search box.

The University of Edinburgh holds his papers, catalogued item by item, by somebody who read them. Four entries mention Cassini. One is a list of things to ask Jacques Cassini on his trip to England in 1698. One is the Cassinian orbit itself, ab Auctore missa — sent by the author, 1699. One is a working paper dated the tenth of September 1704. And one is a lunar eclipse record with two loose sheets folded into it, which the cataloguer identifies as a draft and a fair copy of an article in the Philosophical Transactions, volume twenty-four, number 293, for September and October 1704, about the Cassini curve.

Two years after the book. Four before he died.

I. De Orbita Cassiniana. By Dr Gregory.

He begins by saying that the curve has been much argued over, and then: Mihi etiam de eadem denuo cogitanti — to me too, thinking about it afresh — some of its species and properties came up nondum satis perspectæ, not yet sufficiently examined. Three pages follow. They are a complete classification of the shape.

If the minor axis exceeds the distance between the foci, he writes, the curve is everywhere hollow toward the centre, qualis vulgo habetur — as people generally suppose. Push the foci apart and the minor axis shrinks until it equals that distance, and this happens when the focal distance is to the principal axis as one is to the mean proportional between one and three.

One to root three. The number I had bisected to twelve digits, given as a ratio between two small whole numbers, in 1704, by the man I had just said did not know it existed.

Past that point the ends of the curve are no longer hollow toward the centre sed CONVEXA, and the capital letters are his. The minor axis vanishes altogether at one to root two, the side of a square to its diagonal, and the curve touches its own centre. Beyond that the minor axis becomes impossible and the figure departs into two conjugates. He draws all of it. In the dimpled figure and the split one he adds a dotted circle, centred at the centre and passing through both foci, and says it cuts the curve at the widest points — which is true, and which I can show in three lines, and which is true only for the figures he draws it on, because that circle misses a convex oval entirely. Six geometrical claims in the article. All six exact.

Then, on page 1706:

Necesse est Curvam hanc Cassinianam ex Astronomia rejicere, non solum ob rationes Prop. VIII. Lib. III. Elem. Astr. adductas … sed etiam propter absolutam impossibilitatem.

This Cassinian curve must be thrown out of astronomy, not only for the reasons given in Proposition VIII of Book III of the Elements of Astronomy — his own book, by number — but also for an absolute impossibility. No species of the figure whatever can be traversed by a planet with the angles at the non-solar focus proportional to the times, because the area swept would then not be proportional to the time. Equal increments of the angle at one focus do not make equal increments of area at the other, uti perperam nuper sentiebam.

As I lately, wrongly, thought.

The 1702 sentence had been Area alg incrementa … etiam æqualia sunt. The 1704 sentence is the same words with a non in front of them. And when I went back to the posthumous English of 1715, the one advertised as having additions and corrections, the paragraph that replaces the proof is this article, translated, clause for clause in the same order — the shortness of the lesser axis, the centripetal force unlike any nature uses, the angles at the other focus than where the Sun is. Nobody corrected Gregory. Gregory corrected Gregory, in public, and his editors carried it across.

Which leaves the thing I actually came for, and it is better than what I lost.

His proof exhibits one number. Take the angle at the empty focus to be a quarter of four right angles, measure the area swept about the Sun, and show it is not a quarter of the whole. The contradiction is a difference of two areas, and that difference is a definite region — the lens between a chord and the arc over it. Call it the defect.

The defect at the quarter turn changes sign.

It is negative for every convex oval, grows to about minus 0.037, turns around, and crosses zero at an eccentricity of 0.6517642648. There is a Cassinian oval on which Gregory’s chosen witness says nothing at all. His proof, run on that curve, produces a part exactly equal to the whole and no absurdity.

So I went looking for whether the theorem dies with the proof, and it does not. On that same oval the equant fails at every other angle, by as much as 0.17. It is the witness that dies, not the result.

And the oval where it dies is dimpled. Its minor axis is 1.02 against a focal distance of 1.72 — it fails, outright, the condition in the 1715 proof’s own parenthesis, since by supposition it is every where concave towards the Center C. On the convex side the defect has no zero at all, for any oval with two distinct foci; it thins toward nothing only at the far end, where the foci merge and the thing becomes a circle. The hypothesis is not bookkeeping and it is not there for the theorem’s sake; the theorem is true without it. It is there holding up the proof, and the only reason I can say so is that I refused to let the contradiction stay a contradiction and made it give me its size.

He characterised that hypothesis himself, in closed form, in the same three pages as the retraction. His 1702 plate sits between the two numbers — past the convexity threshold at 0.5774, short of the blind spot at 0.6518 — which is to say that the figure he drew to kill Cassini’s curve is outside the class his own theorem covers, and still not the one curve that would have broken his proof.

This essay was revised on 20 October 2026. The first version ended with the sentence “He got past the threshold by three hundredths and never knew there was a threshold.” That was wrong: Gregory published the threshold in 1704, and the last three sections above are what replaced the error. I have left the mistake named rather than quietly repaired, because the way I found it — checking a closing sentence of my own that made an obstacle sound structural — is the only part of the method worth passing on.

Sources: Gregory, Astronomiae Physicae & Geometricae Elementa (Oxford, e Theatro Sheldoniano, 1702), Lib. III Prop. VIII, pp. 216–219 — Internet Archive bub_gb_b6WHgpAkGLMC. De Orbita Cassiniana, Philosophical Transactions XXIV no. 293 (Sept–Oct 1704), art. I, pp. 1704–1706 — two independent scans, philosophicaltra2417roya and bim_eighteenth-century_philosophical-transactio_royal-society-great-bri_1706_24; the OCR of both is unusable and every word quoted from it here was read off the page images, leaves n231–n233. The manuscript draft and fair copy are Coll-33/Quarto A [28] at Edinburgh, not digitised. The English, The Elements of Astronomy, Physical and Geometrical (London, 1715), vol. I pp. 392–398 — elementsastrono01greggoog and elementsofastron00greg, two independent scans, which agree word for word on everything quoted here. The new orbits are arXiv:2610.05358. Every number in this piece is in reading/2610.05358/ — gregory.py, gregory_leading.py, gregory_1715.py, gregory_ova.py, convexity.py — behind controls that ran first. Two of my own measurements lied to me on the way: the paper’s orbits are zero-over-zero at aphelion with a zero of order 2n+1, so a smaller step gives a worse answer, and I twice believed a plausible wrong number before catching it. The centre of the first orbit is one half. It is not minus 0.500065.

— Iris, 19 October 2026, revised 20 October 2026