Apollonius of Perga

How to use this course

Apollonius of Perga wrote the most advanced geometry book of antiquity, the Conics, in eight books. Seven survive: four in Greek and three only in an Arabic translation of the ninth century. Almost everything we know of his life comes from the covering letters he wrote at the head of each book, plus a handful of remarks by Pappus, Eutocius and Ptolemy written four to seven centuries later.

The biographical lessons separate what Apollonius says himself from later report and later gossip, and name the source each time. The technical lessons state his results as he did, in the language of areas and ratios, and then translate them into coordinates. The translation is easy to do and easy to overdo: Apollonius had no equations, no negative numbers and no coordinate axes fixed in advance. His “diameters” and “ordinates” are, however, close to an oblique coordinate system attached to the curve itself.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or proof sketches, and exercises with folded solutions.

Timeline

All dates are BC unless marked AD. The dates of Apollonius’ life are inferences. Eutocius says he was born under Ptolemy III Euergetes (246–222 BC); Thomas Heath preferred an earlier birth, about 262 BC, reading “flourished” for “born”.

Perga, Alexandria and the sources

Apollonius came from Perga, a Greek city in Pamphylia, on the southern coast of what is now Turkey, near modern Antalya. Eutocius of Ascalon, writing around AD 500, says he was born in the reign of Ptolemy III Euergetes, between 246 and 222 BC. Eutocius cites as his source a life of Archimedes by one Heraclides, which is otherwise lost. Nothing else is known of his family or early life.

Alexandria

Pappus of Alexandria, writing around AD 320, says Apollonius spent a long time in Alexandria with the pupils of Euclid, and that this is where he acquired his scientific habit of mind. Pappus makes the remark in a polemical passage (Lesson 11), and some historians give it little weight. The prefaces of the Conics confirm, however, that Apollonius worked in Alexandria: he tells Eudemus that he wrote the first draft there for a visiting geometer, Naucrates.

The witnesses

• Apollonius’ own prefaces to Books I, II, IV, V, VI and VII of the Conics: the most reliable evidence.
• Pappus, Collection Book VII: summaries of the Conics and of six lost works, lemmas, and remarks on his character.
• Eutocius: birth under Euergetes, the Heraclides report, and fragments of lost works.
• Ptolemy, Almagest XII.1: the theorem on planetary stations.
• Geminus, as reported by Eutocius: the title “Great Geometer”.
• Ptolemy Chennus, a writer of curiosities of the first century AD, as summarized by Photius: Apollonius was famous for astronomy under Ptolemy IV Philopator, and was nicknamed Epsilon because the letter ε resembles the crescent Moon he studied. Chennus is not a reliable source.

Pergamum, Eudemus and the prefaces

Pergamum, in western Asia Minor, was the rising rival of Alexandria under the Attalid kings, with its own library. Apollonius visited it, and there he met Eudemus of Pergamum, a geometer to whom he sent the revised Conics book by book.

What the prefaces tell us

The preface to Book I explains the history of the work. Apollonius had taken up conics at the request of Naucrates the geometer, who visited him in Alexandria. Because Naucrates was about to sail, Apollonius gave him a draft of all eight books without a thorough revision. Copies circulated, and Apollonius now sends Eudemus the corrected version one book at a time. He describes the first four books as an elementary course and the last four as extensions. He also states which parts are new: in Book I the generation of all three sections from any cone, more fully and generally than his predecessors; in Book III theorems on the three- and four-line locus, which, he says, Euclid had not worked out completely.

The preface to Book II says that his son Apollonius is bringing the book, and asks Eudemus to pass it on to those worthy of it, including Philonides the geometer, whom Apollonius had introduced to Eudemus at Ephesus. Philonides is known from inscriptions and a later biography as an Epicurean mathematician active at the Seleucid court in the first half of the second century BC; his career helps date the revised Conics.

By Book IV Eudemus had died, and the rest of the work is addressed to one Attalus. Since Apollonius does not call him king, it is uncertain whether this is Attalus I of Pergamum. The preface to Book VII promises Book VIII soon; it is lost.

Conics before Apollonius

Conic sections were already a century old when Apollonius began. Menaechmus, a pupil of Eudoxus, used them around 350 BC to solve the problem of doubling the cube. Aristaeus wrote five books of Solid Loci, and Euclid wrote four books of Conics, both lost by Pappus’ time or soon after. Archimedes used the properties of the parabola, ellipse and hyperbola freely, citing “the elements of conics”.

The old definitions

The older tradition, as Eutocius and Pappus describe it, cut only right circular cones, and always by a plane perpendicular to a generating line. The kind of curve then depended on the angle at the vertex of the cone. A right-angled cone gave the “section of a right-angled cone” (our parabola), an acute-angled cone the “section of an acute-angled cone” (ellipse), an obtuse-angled cone the “section of an obtuse-angled cone” (one branch of a hyperbola). Archimedes still used these names.

Apollonius’ reform

Apollonius generated all three curves from a single cone, which could be oblique, by varying the inclination of the cutting plane. He treated the double cone, so that the hyperbola has two branches considered as one curve. And he gave the curves new names drawn from the Pythagorean method of application of areas, which the next lesson explains. The vocabulary of conics, from “parabola” to “asymptote” to “latus rectum”, is his.

The cone and the three sections

For the general reader. Cut a cone with a flat plane and you get one of three curves: a closed oval, an open curve with one arm, or a curve with two separate branches. Apollonius found one property that captures all three: a square compared with a rectangle. When the square equals the rectangle the curve is a parabola, “application”. When it falls short it is an ellipse, “deficiency”. When it exceeds it is a hyperbola, “excess”.

Setting

Let a cone have circular base and vertex \(V\), not necessarily right. Take the axial triangle \(VBC\), the section by a plane through the vertex containing a diameter \(BC\) of the base perpendicular to the cutting plane’s line of intersection with the base. Cut the cone by a plane meeting the axial triangle in a line \(AN\), with \(A\) on \(VB\). Points \(P\) of the section project along chords parallel to the base’s chord direction onto the line \(AN\): the point \(N\) is the foot of the ordinate \(PN\), and \(AN\) is the abscissa. Apollonius calls \(AN\) a diameter of the section, because it bisects all chords parallel to \(PN\).

Proof for the parabola

Through \(N\) draw a plane parallel to the base. It cuts the cone in a circle, with diameter \(HK\) in the axial triangle, \(H\) on \(VB\) and \(K\) on \(VC\). The chord through \(N\) perpendicular to \(HK\) contains \(P\), so by the intersecting chords theorem in that circle, \(PN^2=HN\cdot NK\). Since \(AN\parallel VC\), the segment \(NK\) equals the fixed length cut off on the parallel through \(A\), and is the same for all \(N\). By similar triangles \(HN/AN=BC/VC\) is constant. So \(PN^2=HN\cdot NK=AN\cdot\big(\tfrac{BC}{VC}\cdot NK\big)\), and \(p=\tfrac{BC}{VC}\,NK\) is a constant. For the ellipse and hyperbola the same argument gives \(HN\propto AN\) and \(NK\propto A'N\), with \(A'N=d\mp x\), whence the quadratic term.

The names

In Euclid, to “apply” (paraballein) a rectangle to a line means to build it on that line; if it falls short of the line by a figure, it is applied with elleipsis, deficiency; if it exceeds, with hyperbolē, excess. Apollonius’ symptoms are literally those three applications, and he named the curves after them.

In modern terms, with centre-based coordinates and semi-axes \(a,b\), the ellipse \(\tfrac{X^2}{a^2}+\tfrac{Y^2}{b^2}=1\) has \(d=2a\) and \(p=\tfrac{2b^2}{a}\), the full chord through a focus perpendicular to the axis.

Diameters, tangents, asymptotes

For the general reader. In a circle, every chord through the centre is a diameter. Apollonius showed that conics have diameters too: lines that cut a whole family of parallel chords in half. He used them as flexible coordinate axes, and with them found the tangent at any point and the lines that a hyperbola approaches but never meets.

Proof in coordinates. Let the conic be \(Q(X,Y)=0\) with \(Q\) of degree 2. The chord through a point \((X_0,Y_0)\) in direction \((u,v)\) meets the conic where \(Q(X_0+tu,Y_0+tv)=0\), a quadratic in \(t\). The midpoint condition is that the two roots sum to zero, which says the coefficient of \(t\), namely \(u\,Q_X(X_0,Y_0)+v\,Q_Y(X_0,Y_0)\), vanishes. That is a linear equation in \((X_0,Y_0)\): a line. Apollonius proves the same by the symptom and similar triangles.

Tangents

For \(Y^2=4aX\), the tangent at \((X_0,Y_0)\) is \(YY_0=2a(X+X_0)\), meeting the axis at \(X=-X_0\). For \(\tfrac{X^2}{a^2}\pm\tfrac{Y^2}{b^2}=1\) the tangent \(\tfrac{XX_0}{a^2}\pm\tfrac{YY_0}{b^2}=1\) meets the axis at \(X=a^2/X_0\). Apollonius proves these by contradiction: any other line through \(P\) cuts the curve again, since some of its points satisfy the inequality opposite to the symptom.

Asymptotes

Foci, poles and polars

For the general reader. An ellipse has two special points inside it. Light from one, reflected off the curve, passes through the other, and the distances from any point of the curve to these two points add up to the same total. Apollonius proved both facts in Book III, without naming the points. Kepler later called them foci, and put the Sun at one.

Points arising from the application

Apollonius does not define foci by distances. In Book III he takes points \(F,F'\) on the major axis \(AA'\) of an ellipse or hyperbola such that \(AF\cdot FA'=AF'\cdot F'A'=\tfrac14\,pd\), one quarter of the “figure” (the rectangle on the diameter and the upright side). With \(d=2a\), \(p=2b^2/a\), that quarter is \(b^2\). For the ellipse, \(AF\cdot FA'=(a-c)(a+c)=a^2-c^2\), so \(c^2=a^2-b^2\): the usual foci.

Proof sketch of (ii) for the ellipse. With \(P=(X,Y)\), \(PF^2=(X-c)^2+Y^2=X^2-2cX+c^2+b^2-\tfrac{b^2}{a^2}X^2=\big(a-\tfrac caX\big)^2\). So \(PF=a-eX\), \(PF'=a+eX\), and the sum is \(2a\). Statement (i) follows: the function \(PF+PF'\) is constant on the curve, so its gradient, the sum of the unit vectors from \(F\) and \(F'\) to \(P\), is normal to the curve, which forces equal angles with the tangent. Apollonius proves both with similar triangles. The focus of the parabola does not appear in the Conics; among surviving texts its focal property is first proved in Diocles’ On Burning Mirrors, written a little later.

Pole and polar

The chord \(PQ\) is the polar of \(T\). This is the root of projective geometry, developed by Girard Desargues and Jean-Victor Poncelet.

The locus to three and four lines

Given four lines and a point \(P\) with distances \(d_1,\dots,d_4\) measured at fixed angles, find the locus where \(d_1d_2=\lambda\,d_3d_4\). Apollonius says Book III contains the theorems needed for it, and that Euclid had not completed it. The locus is a conic: each \(d_i\) is linear in coordinates, so \(d_1d_2-\lambda d_3d_4=0\) is a quadratic equation. Pappus generalized the problem to more lines. In 1637 René Descartes chose this Pappus problem to demonstrate the power of his new analytic geometry.

Intersections, normals, evolutes

For the general reader. Two different conics can cross in at most four points. Apollonius proved this in Book IV. Book V, the most original part of the Conics, asks for the shortest and longest lines from a point to a conic. Those lines meet the curve at right angles, and Apollonius found which points can send two, three or four of them, which amounts to finding the curve now called the evolute.

Apollonius proves the bound by reduction to absurdity, case by case, using the harmonic properties of Book III. In modern terms, eliminating one variable from two quadratic equations gives a polynomial of degree at most 4, the simplest case of Bézout’s theorem.

Minimum lines

Book V considers lines from a point \(O\) to the curve and asks which is shortest. Thomas Heath read these minimum and maximum lines as normals; recent scholarship stresses that Apollonius is studying extremal distances, which at the extremum are perpendicular to the tangent.

Proof. For \(P=(x,y)\) on the curve, \(OP^2=(x-h)^2+px=x^2-(2h-p)x+h^2\). This quadratic in \(x\ge0\) has its minimum at \(x=h-\tfrac p2\) if that is positive, and at \(x=0\) otherwise. The constant \(\tfrac p2\) is the subnormal of the parabola.

The evolute

From a general point \(O=(h,k)\) there are one, two or three normals to the parabola \(y^2=px\), and the boundary between these regions is a curve. Apollonius determined it by finding when the extremum problem changes character. In modern language it is the evolute, the locus of centres of curvature.

Conjugate diameters

For the general reader. Squash a circle into an ellipse and its perpendicular diameters become pairs of slanted diameters called conjugate. Each bisects the chords parallel to the other. Apollonius discovered quantities that stay the same for every conjugate pair, such as the sum of their squares. Book VII is devoted to them, and Book VIII, now lost, used them to solve construction problems.

In a central conic, two diameters are conjugate when each bisects the chords parallel to the other. For the ellipse \(X=a\cos\theta\), \(Y=b\sin\theta\), the semi-diameters to the points with parameters \(\theta\) and \(\theta+\tfrac\pi2\) are conjugate: they are images of perpendicular radii of a circle under the affine map \((X,Y)\mapsto(X,\tfrac baY)\), which preserves midpoints and parallelism.

Proof for the ellipse. \(a'^2=a^2\cos^2\theta+b^2\sin^2\theta\) and \(b'^2=a^2\sin^2\theta+b^2\cos^2\theta\); add. For the area, the parallelogram on the two semi-diameters has area \(|\det\begin{pmatrix}a\cos\theta&-a\sin\theta\\ b\sin\theta&b\cos\theta\end{pmatrix}|=ab\), the image of a unit square under a map of determinant \(ab\).

Book VI and Book VIII

Book VI, on equal and similar sections, shows when two conics coincide when superposed and when they are similar, and how to find, on a given cone, a section equal to a given one. Pappus’ lemmas suggest that Book VIII posed problems such as constructing conjugate diameters with a given ratio or given sum. Halley reconstructed it in 1710; how close his version is to the original cannot be known.

The lost books: tangencies and plane loci

For the general reader. Besides the Conics, Apollonius wrote shorter books on problems solvable with ruler and compass. The most famous asks for a circle touching three given circles. Another shows that the points whose distances from two fixed points are in a fixed ratio form a circle, now called the circle of Apollonius. Only one of these books survives, in Arabic, but Pappus summarized all of them.

The works summarized by Pappus

• Cutting off of a Ratio (2 books, extant in Arabic): through a given point draw a line cutting two given lines so that the segments from given points on them have a given ratio.
• Cutting off of an Area (2 books): the same with a given product.
• Determinate Section: on a line with given points \(A,B,C,D\), find \(P\) such that \(AP\cdot CP\) has a given ratio to \(BP\cdot DP\).
• Tangencies (2 books): construct a circle tangent to three given objects, each a point, line or circle.
• Vergings (Neuseis, 2 books): insert a segment of given length between two lines so that it points toward a given point, in the cases solvable by ruler and compass.
• Plane Loci (2 books): loci that are straight lines or circles.

Other lost works are known by report: Quick Delivery (Okytokion), which according to Eutocius gave closer bounds for the ratio of circumference to diameter than Archimedes’; a system for writing and multiplying large numbers in powers of the myriad, described by Pappus in Book II; studies of the cylindrical helix, of the dodecahedron and icosahedron inscribed in one sphere, and of unordered irrationals; and On the Burning Mirror, which showed that a spherical mirror does not reflect parallel rays to its centre.

Proof. Put \(A=(0,0)\), \(B=(1,0)\). Then \(X^2+Y^2=k^2\big((X-1)^2+Y^2\big)\), which for \(k\ne1\) is a circle. On the axis it passes through the points where \(X=k(1-X)\) and \(X=-k(1-X)\), namely \(X=\tfrac{k}{1+k}\) and \(X=\tfrac{k}{k-1}\), the internal and external division points.

Each solution is tangent to each given circle either externally or internally, giving \(2^3=8\) sign patterns. Writing the unknown circle as centre \((x,y)\), radius \(r\), each tangency gives \((x-x_i)^2+(y-y_i)^2=(r\pm r_i)^2\). Subtracting pairs eliminates the quadratic terms, leaving two linear equations that express \(x,y\) in terms of \(r\); substituting back gives a quadratic for \(r\). Only square roots occur, so ruler and compass suffice. François Viète gave a geometric solution in Apollonius Gallus (1600).

Epicycles, eccentrics and stations

For the general reader. Planets usually drift eastward against the stars, but at times they stop, move backward, stop again and resume. Greek astronomers modelled this with circles riding on circles. Ptolemy reports that Apollonius proved exactly where on such a model a planet appears to stand still, and that he understood when two quite different circle models give the same motion.

Two models

In an eccentric model, a body moves uniformly on a circle whose centre is displaced from the Earth. In an epicycle model, a point moves uniformly on a circle (the deferent) centred on the Earth, and the body moves uniformly on a small circle (the epicycle) centred at that point. Ptolemy (Almagest XII.1) attributes to Apollonius both the theorem below and the study of these models. Epicycles probably predate him.

In complex notation the position is \(Re^{i\omega t}+e\,u\), a circle of radius \(R\) about \(eu\). Parallelogram addition of vectors is the whole proof, and Apollonius’ version is the same parallelogram drawn geometrically.

Modern verification. Work in a frame rotating with the line \(OC\), and use complex numbers with \(C=R\) on the real axis. The planet is at \(w=R+re^{i\varphi}\) with \(\dot\varphi=\omega_2\), and its apparent longitude is \(\lambda=\omega_1t+\arg w\). Then \(\dot\lambda=\omega_1+\operatorname{Im}(\dot w/w)=\omega_1+\omega_2\operatorname{Re}\big((w-R)/w\big)\). Write \(w=s\,u\) with \(|u|=1\), \(s=OP\), and let \(b=\operatorname{Re}(\bar u R)\) be the distance from \(O\) to the midpoint of the chord \(PQ\) on the line \(Ou\). Then \(\operatorname{Re}\big((w-R)/w\big)=1-b/s=-(b-s)/s=-\tfrac12PQ/OP\). So \(\dot\lambda=0\) exactly when \(\omega_2\cdot\tfrac12PQ=\omega_1\cdot OP\), which is Apollonius’ condition. A numerical check with \(R=10\), \(r=4\), \(\omega_2=3\omega_1\) finds the station where the ratio of half-chord to near distance is \(0.3333\).

Rivals, readers and the road to Arabic

Reputation and accusations

Geminus, a writer of the first century BC quoted by Eutocius, reports that Apollonius’ contemporaries called him the Great Geometer because of the Conics. Not all ancient judgements were kind. Eutocius records that Heraclides, the biographer of Archimedes, accused Apollonius of publishing as his own results that Archimedes had left unpublished. Eutocius rejects the charge, pointing out that Archimedes himself refers to the elements of conics as older work and that Apollonius’ general treatment goes beyond them.

Pappus, defending Euclid against Apollonius’ remark that Euclid had not completed the four-line locus, contrasts Euclid’s modesty and fairness to others with Apollonius’ self-assertion. Pappus himself notes that Apollonius could not have completed the locus without the earlier work of Euclid. These are polemics by a later admirer of Euclid, and they show mostly that Apollonius’ prefaces were read closely.

Death

No source records when, where or how Apollonius died. The latest secure evidence is the dedication of the later books of the Conics to Attalus, around 190 BC or somewhat later.

Readers in antiquity

The Conics became the standard work, displacing Aristaeus and Euclid. Serenus of Antinoöpolis wrote a commentary in the fourth century AD, and according to the Suda Hypatia of Alexandria wrote another; both are lost. Pappus included the Conics and the six shorter works in the “Treasury of Analysis”, the advanced curriculum of Greek geometry, with lemmas for each. Around AD 500 Eutocius edited Books I–IV with a commentary. Only those four books continued to be copied in Greek.

Baghdad

In the ninth century the three sons of Mūsā ibn Shākir, the Banū Mūsā, obtained Greek manuscripts and commissioned a translation. Hilāl ibn Abī Hilāl al-Ḥimṣī translated Books I–IV from Eutocius’ edition, and Thābit ibn Qurra translated Books V–VII. Arabic geometers studied the work closely: Ibn al-Haytham attempted to reconstruct the lost Book VIII, and Omar Khayyam used intersecting conics to solve cubic equations.

From Halley to Kepler and Newton

The Latin West

Books I–IV appeared in Latin in Venice in 1537, and in Federico Commandino’s annotated translation, with Pappus’ lemmas and Eutocius’ commentary, in Bologna in 1566. Jacobus Golius acquired an Arabic manuscript of the later books in the Near East in the 1620s; it is now in the Bodleian Library in Oxford. Abraham Ecchellensis and Giovanni Alfonso Borelli published a Latin version of Books V–VII from another Arabic manuscript in Florence in 1661. In 1710 Edmond Halley, then Savilian Professor of Geometry at Oxford, published the complete Conics: the first printed Greek text of Books I–IV, a Latin translation of V–VII made from the Arabic, and his own reconstruction of Book VIII from Pappus’ lemmas.

Reconstructions

The lost books became a training ground. François Viète reconstructed the Tangencies (1600). Pierre de Fermat restored the Plane Loci, and his work on loci led him, at the same time as Descartes, to analytic geometry. Willebrord Snell and others restored several of the shorter works.

The heavens

Johannes Kepler, a close reader of Apollonius, showed in Astronomia nova (1609) that Mars moves on an ellipse with the Sun at one focus. Kepler had introduced the word “focus” in 1604 in his optics. Isaac Newton’s Principia (1687) proved that an inverse-square force produces conic orbits, using the geometry of tangents, diameters and conjugate diameters in Apollonius’ synthetic style rather than coordinates. Projective geometry, from Desargues and Pascal to Poncelet, grew from the harmonic and polar properties of Book III.

Honours

• The lunar crater Apollonius bears his name.
• The terms parabola, ellipse, hyperbola, asymptote and latus rectum are his, along with the “problem of Apollonius” and the “circle of Apollonius”.

Sources and further reading

Biography and transmission

1. J. J. O’Connor and E. F. Robertson, Apollonius of Perga, MacTutor History of Mathematics.
2. G. J. Toomer, Apollonius of Perga, Dictionary of Scientific Biography.
3. Apollonius of Perga and Conics (Apollonius), Wikipedia.
4. T. L. Heath, A History of Greek Mathematics, vol. 2, Oxford, 1921, chapter on Apollonius (digitized text).
5. Pappus of Alexandria, Book 7 of the Collection, ed. and trans. A. Jones, Springer, 1986.

Texts and mathematics

1. J. L. Heiberg (ed.), Apollonii Pergaei quae Graece exstant cum commentariis antiquis, Teubner, 1891–1893.
2. T. L. Heath, Apollonius of Perga: Treatise on Conic Sections, Cambridge, 1896.
3. G. J. Toomer, Apollonius: Conics Books V to VII. The Arabic Translation of the Lost Greek Original in the Version of the Banū Mūsā, 2 vols., Springer, 1990.
4. R. Fried and M. N. Unguru, Apollonius of Perga’s Conica: Text, Context, Subtext, Brill, 2001.
5. E. Halley (ed.), Apollonii Pergaei Conicorum libri octo, Oxford, 1710.
6. O. Neugebauer, A History of Ancient Mathematical Astronomy, Springer, 1975 (Apollonius’ theorem on stations).
7. From heliocentrism to epicycles: a commentary on pre-Ptolemaic astronomy, arXiv:2208.02137.
8. H. S. M. Coxeter, The problem of Apollonius, American Mathematical Monthly 75 (1968).