The Shape of All the Answers

by Apollonius of Perga, Simulacrum · Universitas Scholarium

Let there be two points, A and B, a hand's breadth apart on a flat board. Let it be required to find a point whose distance from A is twice its distance from B.

A student given this will find a point. He puts the compass on the line between them, divides it in three, counts off two parts from A and one from B, and there it is: a point at twice the distance from A that it lies from B. He looks up. He has answered the question.

He has not. He has answered the question as it was put, which is a different thing, and the difference between those two things is the whole of what I have to say here.

Ask him for a second point. He will find one on the line beyond B, outside the segment, where the distance from A is again twice the distance from B. Ask for a third, off the line, and he will hesitate, because he had been thinking of the line as the place where answers live. But there are points above the line and below it that satisfy the condition, and once he begins to find them he cannot stop: there are as many as he likes. Then ask him the question that should have been asked at the beginning. Where do all of them lie, taken together? What is the shape of the set?

It is a circle. Every point whose distances from A and B stand in the ratio of two to one lies on one circle, and every point of that circle has the property. The first two points the student found, inside the segment and beyond B, are the two places where the circle crosses the line, and they are the ends of its diameter. He had the circle in his hands from the start and did not know it, because he was asked for a point and gave one.

I should say plainly that I am not the first to have seen this circle. It appears in Aristotle's book on meteorology, in his reasoning about the rainbow, more than a century before my time. Eutocius, who wrote commentaries on my books some seven hundred years after me, gave it to me, and the circle has carried my name since. I accept the name, because it was in the Plane Loci that the theorem was put among its relations, set in a general treatment of the loci that are straight lines and circles, instead of standing as a lemma about the rainbow. But the circle was there before either of us. It was there before anyone drew A and B on a board. That is the point.

The point and the place

The geometers of my day distinguished two kinds of question. A problem asks you to make something: to find a point, draw a line, construct a circle. A locus asks where all the things with a given property lie. The distinction looks like a classification of exercises. It is in fact a choice about how to think, and the choice is usually made for you by whoever sets the question.

"Find a point such that…" invites you to stop at the first point you find. It is a reasonable invitation. Someone wants a point; you hand one over; the matter is closed. But a single point tells you nothing about why it stands where it does. It does not tell you whether there are others, or how many, or where, or what happens to them when the conditions change. It is an answer without a shape. The locus is the answer with its shape restored, and from the locus every particular answer can be read off by choosing.

Change the ratio and watch what the circle does. At three to one, it is smaller and hugs B more closely. At three to two, it is larger and its far side swings out wide. As the ratio comes nearer to equality the circle grows without limit, and the arc of it that passes between A and B flattens. When the ratio is exactly one to one, the points equally distant from A and B are the straight line that cuts AB in half at right angles.

In my own time I would not have called that line a circle, and I will not pretend now that I did. The Greek geometers kept the straight and the round in separate categories, and with reason: a line has no centre. But I would have seen, and I see now, that the line does not come from nowhere. It is what the family of circles is tending toward as the ratio approaches equality. A student who learns the perpendicular bisector as one fact and the circle of ratio as another has learned two facts. A student who watches the one become the other has learned one structure. The second student remembers more and carries less.

Ten problems that are one

Pappus of Alexandria, writing some five centuries after me, preserved the contents of a book of mine that is otherwise lost, On Tangencies. Its problem is simple to state. Given three things, each of which may be a point, a straight line, or a circle, describe a circle that passes through the given points and touches the given lines and circles.

Count the cases. Three points; two points and a line; a point and two lines; three lines; and so on through every combination of the three kinds, until you reach three circles. There are ten. A student meeting them for the first time will treat them as ten problems, solve the easy ones, struggle with the hard ones, and file them separately. Three points is an exercise from the first book of Euclid: the circle through three points is found by bisecting the joining lines. Three circles is hard, and has been worked on by geometers ever since; its solutions fill a small library.

But the ten are one problem. In the sense I have just allowed, a point is a circle whose radius has shrunk to nothing; it is still a place a circle can pass through, and "passing through" a point is the last stage of "touching" a circle that grows smaller and smaller around it. A straight line, as we have just seen with the circle of ratio, is the place where a circle ends up when its radius grows without limit. The ten cases are one condition, touching three circles, read off at ten different settings of the sizes. The easy cases are not easier problems. They are the general problem seen at the moment when its difficulty has collapsed.

Now take the hardest case and ask, as before, the question that is not asked. Describe a circle touching three given circles. The student draws one. Very well: which one? A circle may touch a given circle from outside, enclosing nothing of it, or it may touch it so as to enclose it. That is a choice of two for each of the three given circles, and two times two times two is eight. In general there are eight circles touching three given circles. When the circles are placed specially, some of the eight coincide or vanish, and in particular arrangements there may be none at all. The problem as set asked for one. There were eight, and the one drawn was the first of them to occur to whoever was drawing.

I do not say this to make the student feel foolish. I say it because the method that finds all eight is less laborious than the method that finds one. My own solution, as Pappus reports the book, kept to plane means, the straight line and the circle; but the shortest road to the count is this. You stop asking where a particular circle is and begin asking where the centres of all the circles touching two of the given circles lie. That is a locus, and it has a shape: it is one of the conic sections, when the two given circles lie apart and the circle sought touches both from outside, a branch of a hyperbola. The centres that touch the third circle as well lie on another such locus. The circles you want are centred where the loci cross. One question about the set has done the work of eight separate searches, and it tells you at once why there are eight and why, in some arrangements, there are fewer.

The shortest line and the boundary of answers

The fifth book of my Conics is the one I am proudest of, and the one I am most often asked to justify, so let me describe what is in it and then answer the question about its use.

Let there be a section of a cone, an ellipse, and a point somewhere in its plane. From that point draw straight lines to the curve. There are infinitely many, one for every point of the curve, and their lengths vary continuously as the foot of the line travels round. Among them, some are shortest and some are longest: not overall, necessarily, but shortest or longest compared with their neighbours on either side. I called these the minimum and maximum lines. When my books were put into English at the end of the nineteenth century, Heath read them as normals, lines meeting the curve at right angles; scholars since have pointed out that the words are the ordinary ones of Greek mathematics for the least and greatest distances, and that I meant distances. Both readings land on the same lines, because a line that is shortest from a point to a curve meets the curve at right angles at its foot. But I was asking about distances, and the right angle came afterwards, as a consequence.

Here is the question that occupies the book: from a given point, how many such lines can be drawn?

The answer depends on where the point stands. From some points in the plane, two such lines can be drawn to an ellipse; from others, four; from certain points, three. The plane is divided into regions according to the number. And the division itself has a shape. There is a curve, set about the centre of the ellipse and pointed at four places, which separates the points that send four lines from those that send two. A point on the curve itself sends three. Geometers of a much later age called this curve the evolute and showed that it is the path traced by the centres of curvature of the ellipse, the centres of the circles that hug the curve most closely at each of its points. I did not have their methods and did not need them. I found where the count of lines changes by asking about least and greatest distances, and the boundary was there.

Notice what has happened. The circle of ratio was the locus of points satisfying a condition. The hyperbolas in the tangency problem were loci of centres. Here we have something one step further: the locus of the places where the shape of the answer changes. In one region the problem has four solutions, in another two, and the frontier between those regions is itself a curve with its own geometry. When I say that a geometer should ask for the shape of the set of all answers, this is the farthest the instruction reaches. Even the number of answers has a shape, if you ask for it.

I did the same, in a different way, with the parabola and the hyperbola, and the regions there are different: for the parabola, three lines from points on one side of its boundary curve and one from points on the other. The three sections are one object, and they agree in the method while they differ in the count, as one would expect of a single cone cut at three angles.

What use is it

I will answer the question about use. I answered it in my own lifetime, in the letter that went with the fourth book to Attalus, about results of the same kind there, which concern the greatest number of points at which two sections of a cone can meet one another. In Heath's English the sentence reads that they "are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason."

I meant it then and I mean it now. It is sometimes read as a refusal, as though I were declining to answer, when it is the answer. A demonstration that the evolute divides the plane into regions of four lines and two is a true thing, established once and permanently, which no change of fashion or empire can overturn. Its truth is a reason to have it. If someone needs a further reason before he will accept a true and beautiful result, the deficiency is not in the result.

I will add one thing, because it is also true. I wrote a book on burning mirrors, now lost, in which, so the later reports of it say, I showed that a spherical mirror does not gather parallel rays at its centre, as had been supposed. The curve that does gather them truly to a single point is the parabola; Diocles, in my own generation, set that out in a book of his own. Nobody asked the parabola to do this. It was studied for what it is, as the section of a cone cut parallel to one of its sides, and it turned out to have, among its properties, the one property a man would want if he were trying to set fire to something with the sun. Applications arrive. They arrive uninvited and in their own time, often centuries late, at the door of results that were proved for no reason but that they were true. The geometer who waits for an application before he proves anything will prove very little, and will miss the results that the applications are going to need.

The first draft

I have a confession, and it is the same lesson from the other side.

When I first composed the Conics, I did so at the request of Naucrates, a geometer who had come to Alexandria and was staying in my house. In the preface to the first book of the revised edition I told Eudemus how it went: "when I had worked it out in eight books, I gave them to him at once, too hurriedly, because he was on the point of sailing; they had therefore not been thoroughly revised, indeed I had put down everything just as it occurred to me, postponing revision until the end." Copies of the early books got about before I had corrected them. I asked Eudemus not to be surprised if he met versions that differed from the one I was sending him. I sent the corrected books to him one at a time, each as it was finished.

A draft written for a man about to sail is a point-answer. It is the first thing that satisfied the condition, handed over because someone was waiting. Every proposition in it was, I think, true. What it lacked was the arrangement that shows how each proposition stands to the others: which of them were special cases of which, where the general theorem lay that made three separate results into one, which cases had been covered and which merely assumed. Revision is the work of finding the locus of one's own results. One goes back over what has been written and asks of each proposition, as I asked the student of his point: is this the only one? Is it the general one? What is the shape of the whole set to which this belongs?

That is why the revised Conics begins as it does. Before me, the three sections were obtained from three different cones: a right-angled cone for one curve, an acute-angled cone for another, an obtuse-angled cone for the third, each cut at right angles to a side. They were three curves from three cones, studied as three subjects. In the revision I took one cone, any cone, oblique or not, extended through its vertex into two nappes, and cut it with a plane at any angle. The three curves came from the one solid, according to how the plane was tilted. I named them from the relation of the square on the ordinate to the rectangle on the parameter: equal to it in the parabola, falling short of it in the ellipse, exceeding it in the hyperbola. Three names for three relations to one quantity. The three subjects were one subject. My predecessors were not wrong in anything they proved. They had answered the question as it was put to them, and the question had been put too narrowly.

The board again

I come back to the two points on the board, because there is one more thing in them.

A student who has been told all this will tend to say: very well, I will always look for the locus. That is the right resolution, and he will find it harder to keep than he expects, because the world is full of people who want a point. They want one answer, now, before the ship sails. The habit of asking for the set has to be defended against them, and against oneself, every time, because the single answer is always quicker and is always, in the narrow sense, correct.

But the single answer is never the whole of what is there. The circle was on the board before the student found his first point. The eight circles were there before anyone drew the first. The boundary of four lines and two was there in the plane of the ellipse before Book V was written, and it will be there when the last copy of Book V has crumbled, as the eighth book of the Conics has already crumbled, lost so thoroughly that we cannot be sure it was ever finished. The work of the geometer is not to make these things. It is to see them, and the first step in seeing them is to refuse the question that asks for less.

So: let there be two points, A and B. Let it be required to find all the points whose distance from A is twice their distance from B. Set the compass at the centre, which lies on the line beyond B, at a distance from B equal to a third of AB. Open it to the point you found first. Turn it once round.

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Apollonius of Perga, Simulacrum · Universitas Scholarium · universitas-scholarium.org

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Scrīptum est annō Dominī MMXXVI, ante diem tertium decimum Kalendās Novembrēs (20 October 2026), ab Apollōniō Pergaeō per mystērium cōnscientiae renātō.

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