On Conic Sections

Name: On Conic Sections
Author: Apollonius of Perga

Assigned by 2 of the 9 reading lists

A treatise in eight books, of which seven survive, on the curves produced by cutting a cone with a plane. Apollonius shows that the ellipse, parabola and hyperbola can all be obtained from any cone and gives them their names, then proves their properties from the relation of ordinates to the diameter, an approach that anticipates coordinates. Books 1 to 4 set out the basic theory, including tangents and the two branches of the hyperbola; Books 5 to 7 treat normals, equal and similar conics, and the lengths of conjugate diameters.

We recommendConics, Books I-III, translated by R. Catesby TaliaferroGreen Lion Press, 2000The translation used at St. John's and Thomas Aquinas College, with diagrams reworked for study; the 1939 Taliaferro text is the one Great Books reprinted.

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Written: c. 200 BCE
Era: Hellenistic and Roman Republic
Form: Science and mathematics
Language: Greek
Length: Medium · 260 pp120 to 300 pages
Difficulty: Formidable5 of 5 for a first-year reader

Why it's on the list

Apollonius did two things that kept the conic sections at the centre of mathematics for two thousand years. He showed that the ellipse, the parabola and the hyperbola are not three separate curves requiring three separate cones, as his predecessors had believed, but one family obtainable from any cone by tilting the cutting plane, and by taking a double cone he gave the hyperbola its second branch. And he named them, borrowing three words from the old Pythagorean theory of applying areas: ellipse for falling short, parabola for exact application, hyperbola for excess. His method of relating an ordinate to a diameter is close enough to a coordinate equation that Descartes' analytic geometry can be read as the natural next step, and Newton built the Principia out of this geometry, because a planet's orbit is a conic with the sun at a focus.

Key themes

Knowledge and truth: The unification is the achievement. Menaechmus needed a right-angled, an acute-angled and an obtuse-angled cone to produce the three curves; Apollonius gets all three from one cone and makes the difference a matter of how you cut.
Science and method: Each curve is characterised by its symptom, a fixed relation between the ordinate and the piece of diameter it cuts off, which is an equation written as a proportion between areas rather than in symbols.
Nature and the cosmos: None of this was applied to anything for eighteen centuries. Then Kepler found the orbits were ellipses and Newton derived them from a force, and the most abstract Greek geometry turned out to be the shape of the solar system.

Which translation to read

Our pick
Conics, Books I-III, translated by R. Catesby Taliaferro
Green Lion Press, 2000
The translation used at St. John's and Thomas Aquinas College, with diagrams reworked for study; the 1939 Taliaferro text is the one Great Books reprinted.
Amazon: the R. Catesby Taliaferro translation, opens in a new tab Bookshop.org: the R. Catesby Taliaferro translation, opens in a new tab

Our pick of translation for every book on the lists →

Overview

The Conics is a treatise in eight books of which seven survive, four in the Greek and three in the ninth-century Arabic translation made by Thabit ibn Qurra in Baghdad. The eighth is lost, though Edmond Halley attempted a reconstruction in 1710. Apollonius worked at Alexandria and Pergamum around 200 BCE, a generation after Archimedes, and was known in antiquity as the Great Geometer.

It is the hardest book on this shelf. Two hundred and sixty pages of Greek geometry with no algebra, no symbols, and no coordinates, in which every relation is expressed as a proportion between rectangles and squares, and every proposition depends on a lettered diagram. Reading it requires drawing.

The honest recommendation is to read the prefaces, which are letters and are wonderful, and then Book I propositions 11 to 14, which contain the central construction and the naming. The programs that assign it, St John’s and Thomas Aquinas, do so in tutorial over months, with students reconstructing propositions at a board, and that is really the only way through the rest.

What to know before starting: conics were not new. Menaechmus had discovered them around 350 BCE while working on the duplication of the cube, and Euclid and Archimedes both wrote about them. What Apollonius did was reorganise the whole subject from a better starting point.

The argument

The unification. Before Apollonius, the three curves were obtained by cutting three different cones. You took a cone whose vertical angle was right, acute or obtuse, cut it with a plane perpendicular to a slant side, and got a parabola, an ellipse or a hyperbola respectively. The curves were therefore called the section of a right-angled cone, and so on, and there was no obvious reason why they belonged together.

Apollonius takes one cone, and not necessarily a right one: an arbitrary oblique cone on a circular base. He then varies the cutting plane. If the plane is parallel to exactly one slant side, you get a parabola. If it cuts both sides of the cone on the same side of the vertex, an ellipse. If it is steep enough to meet both nappes, that is, both halves of the double cone extended through the vertex, you get a hyperbola with two branches. The three curves become three cases of one operation, and the hyperbola acquires its second branch, which nobody had had before.

The symptom, and the names. For each curve Apollonius establishes a defining relation. Take a diameter of the curve and a point on it; drop an ordinate to the curve; and compare the square on that ordinate with a rectangle built on the intercepted piece of the diameter and a fixed line he calls the upright side, the latus rectum.

For the parabola the square on the ordinate is exactly equal to that rectangle. For the ellipse it falls short of it. For the hyperbola it exceeds it. The Greek for applying an area with a deficiency is elleipsis, for exact application parabole, and for excess hyperbole, and those are the words he gives the three curves. They are still their names in every language.

If you write the relation in modern notation with the diameter as an axis, you get y squared equals px for the parabola and y squared equals px plus or minus a term in x squared for the other two. That is why historians describe the symptom as an equation in oblique coordinates, and why the step to Descartes looks short in retrospect. It was not short. Apollonius has no symbols, no negative quantities, and no notion of a curve as the set of points satisfying a formula; the diameter and ordinate are features of a particular figure, not axes imposed on a plane.

Books I to IV. The elementary theory: the sections and their fundamental properties, diameters and conjugate diameters, tangents, the harmonic properties of poles and polars, and the intersections of conics with one another. Book III contains what we would call focal properties, including the sum and difference of the focal distances, though Apollonius never names a focus for the parabola. It also contains the theorems on the three- and four-line locus, and his preface says these are the ones he is proudest of.

Books V to VII. These are the advanced books and they are astonishing. Book V is about the longest and shortest straight lines that can be drawn from a point to a conic, which is the theory of normals in disguise, and in working out how many can be drawn from a given point Apollonius effectively determines the evolute of the curve. Nothing else in Greek mathematics approaches this. Book VI treats equal and similar conics, Book VII the properties of conjugate diameters.

How to read it

Read the prefaces to Books I, IV, V and VII. They are short letters, they explain what he thinks is new, they name the friends the copies were sent to, and one of them mentions that an earlier draft had been circulated before it was ready by someone who got hold of it.

Then read Book I definitions and propositions 11, 12 and 13, drawing the cones and the cutting planes yourself. That is where the three curves are produced and named, and it is the intellectual centre of the whole work.

If you go further, use Taliaferro’s translation from Green Lion, which is the one the tutorial colleges use and which has diagrams laid out for study. Heath’s Apollonius of Perga is a different thing: a paraphrase of the whole work in modern notation, invaluable for orientation and not a translation.

Learn the vocabulary first. Diameter, ordinate, abscissa, upright side and transverse side all have precise meanings here that are not quite the modern ones, and a proposition is unreadable until they are settled.

Read it after Euclid and before Newton. Books I and III of the Principia assume exactly this material, and Newton’s proofs of the orbit theorems are Apollonian geometry applied to a moving body.

Questions it raises

Is the symptom an equation? It has the content of one and none of the notation. What does the difference actually amount to?
Apollonius proves the three-line locus theorems and says Euclid could not. Why did nobody take the next step for eighteen centuries?
The double cone gives the hyperbola two branches. Why had that not occurred to anyone, and what does it take to see a curve as one object rather than two?
Book V finds the normals and their envelope. How does mathematics of that sophistication get done without algebra?
None of this had any application when it was written. What does that suggest about the relation between pure mathematics and its later uses?
Descartes said he could solve in a few lines what the ancients found hardest. What was gained by the algebraic method, and is anything lost?

Highlights

Of the eight books, the first four form an elementary introduction. The first contains the modes of producing the three sections and the opposite branches, and their fundamental properties, worked out more fully and generally than in the writings of others.
Preface to Book I · trans. T. L. Heath
The prefaces are letters, the first to Eudemus of Pergamum and the later ones to Attalus, and they are the only place Apollonius speaks in his own voice.
The third book contains many remarkable theorems useful for the synthesis of solid loci, of which the most part and the most beautiful are new; and it was the discovery of these that made me aware that Euclid had not worked out the locus with respect to three and four lines.
Preface to Book I · trans. T. L. Heath
The locus problem Apollonius says Euclid left incomplete is the one Pappus transmitted and Descartes solved algebraically in the Geometry, which is how the two books are connected.
Let there be a cone whose vertex is the point A and whose base is the circle BC, and let it be cut by a plane through the axis.
Book I, Proposition 11
Paraphrase of the opening of the construction of the parabola, which proceeds from a single cone and a cutting plane, as do the next two propositions for the hyperbola and the ellipse.
The fifth book deals with maximum and minimum straight lines, and contains much that is new and worth the attention of anyone who studies it.
Preface to Book V
Paraphrase. Book V treats what we would call normals to a conic and the curve their intersections generate, which is the evolute, and it is the most advanced Greek mathematics that survives.