Apollonius of Perga

Pergan geometer

• Born: c. 262 b.c.e.
• Birthplace: Perga, Pamphylia, Asia Minor (now Murtana, Turkey)
• Died: c. 190 b.c.e.
• Place of death: Alexandria, Egypt

One of the ablest geometers in antiquity, Apollonius systematized the theory of conic sections. His study of circular motion established the foundation for Greek geometric astronomy.

Early Life

Information on the life of Apollonius (ap-uh-LOH-nee-uhs) is meager. Born at Perga around the middle of the third century b.c.e., he studied mathematics with the successors of Euclid at Alexandria. His activity falls near the time of Archimedes (c. 287-212 b.c.e.), but links between their work are indirect. In his surviving work, Apollonius once mentions the Alexandria-based geometer Conon of Samos, but his principal correspondents and colleagues (Eudemus, Philonides, Dionysodorus, Attalus I) were active at Pergamum and other centers in Asia Minor. It appears that this circle benefited from the cultural ambitions of the new Attalid Dynasty during the late third and the second centuries b.c.e.

Life’s Work

Apollonius’s main achievement lies in his study of the conic sections. Two properties of these curves can be distinguished as basic for their conception: First, they are specified as the locus of points whose distances x, y from given lines satisfy certain second-order relations: When x² = ay (for a constant line segment a) the curve of the locus is a parabola, when x² = ay - ay^2/b the curve is an ellipse (it becomes a circle when b = a), and when x² = ay + ay^2/b it is a hyperbola. The same curves can be produced when a plane intersects the surface of a cone: When the plane is parallel to the side of the cone, there results a parabola (a single open, or infinitely extending, curve); when the plane is not parallel to the side of the cone but cuts through only one of its two sheets, there results an ellipse (a single closed curve); and when it cuts through both sheets of the cone, there results a hyperbola (a curve consisting of two separate branches, each extending indefinitely).

The curves were already known in the fourth century b.c.e., for the geometer Menaechmus introduced the locus forms of two parabolas and a hyperbola in order to solve the problem of doubling the cube. By the time of Euclid (c. 300 b.c.e.), the formation of the curves as solid sections was well understood. Euclid himself produced a major treatise on the conics, as had a geometer named Aristaeus somewhat earlier. As Archimedes often assumed theorems on conics, one supposes that his basic reference source (which he sometimes cited as the “Conic Elements”) was the Euclidean or Aristaean textbook. Also in the third century, Eratosthenes of Cyrene and Conon pursued studies in the conics (these works no longer survive), as did Diocles in his writing on burning mirrors (extant in an Arabic translation).

Apollonius thus drew from more than a century of research on conics. In the eight books of his treatise, Cōnica (Treatise on Conic Sections, 1896; best known as Conics), he systematized the elements of this field and contributed many new findings of his own. Only the first four books survive in Greek, in the edition prepared by Eutocius of Ascalon (active at Alexandria in the early sixth century c.e.), but all of its books except for the eighth exist in an Arabic translation from the ninth century c.e.

Among the topics that Apollonius covers are these: book 1, the principal constructions and properties of the three types of conics, their tangents, conjugate diameters, and transformation of axes; book 2, properties of hyperbolas, such as their relation to their asymptotes (the straight lines they infinitely approach but never meet); book 3, properties of intersecting chords and secants drawn to conics; book 4, how conics intersect one another; book 5, on the drawing of normal lines to conics; book 6, on similar conics; book 7, properties of the conjugate diameters and principal axes of conics; book 8 (lost), problems solved via the theorems of book 7.

As Apollonius states in the prefaces to the books of his treatise, the chief application of conics is to geometric problems—that is, propositions seeking the construction of a figure satisfying specified conditions. Apollonius includes only a few examples in the Conics: for example, to find a cone whose section produces a conic curve of specified parameters (1: 52-56), or to draw tangents and normals to given conics (2: 49-53 and 5: 55-63). Much of the content of the Conics, however, deals not with problems but with theorems auxiliary to problems. This is the case with book 3, for example, which Apollonius says is especially useful for problem solving but which actually contains no problems. In his preface, he explicitly mentions the problem of the “locus relative to three (or four) lines,” all cases of which, Apollonius proudly asserts, can be worked out by means of his book 3, whereas Euclid’s earlier effort was incomplete.

The significance of problem solving for the Greek geometric tradition is evident in works such as Euclid’s Stoicheia (Elements) and Ta dedomena (Data). In more advanced fields such as conic theory, however, the surviving evidence is only barely representative of the richness of this ancient activity. A notable exception is the Synagogē (Collection), a massive anthology of geometry by Pappus of Alexandria (fourth century c.e.), which preserves many examples of problems. Indeed, the whole of its book 7 amounts to an extended commentary on the problem-solving tradition—what Pappus calls the “analytic corpus” (topos analyomenos), a group of twelve treatises by Euclid, Apollonius, and others. Of the works taken from Apollonius, two are extant—Conics and Logou apotomē (On Cutting Off a Ratio, 1987)—while another five are lost—Chōriou apotomē (cutting off an area), Diōrismenē tomē (determinate section), Epaphai (tangencies), Neyseis (vergings), and Topoi epipedoi (plane loci). Pappus’s summaries and technical notes preserve the best evidence available regarding the content of these lost works. Thus it is known that in Epaphai, for example, Apollonius covered all possible ways of constructing a circle so as to touch any combination of three given elements (points, lines, or circles). In Neyseis he sought the position of a line verging toward a given point and such that a marked segment of it lies exactly between given lines or circles. In Topoi epipedoi circles were produced as loci satisfying stated conditions, several of these being equivalent to expressions now familiar in analytic geometry.

It is significant that these last three works were restricted to planar constructions—that is, ones requiring only circles and straight lines. Pappus classifies problems in three categories: In addition to the planar, he names the solid (solvable by conics) and the linear (solvable by special curves, such as certain curves of third order, or others, such as spirals, now termed “transcendental,” composed of coordinated circular and rectilinear motions). For Pappus, this scheme is normative; a planar solution, if known, is preferable to a solid one, and, similarly, a solid solution to a linear. For example, the problems of circle quadrature, cube duplication, and angle trisection can be solved by linear curves, but the last two can also be solved by conics and so are classed as solid.

Historians often misinterpret this classification as a restriction on solutions, as if the ancients accepted only the planar constructions. To the contrary, geometers throughout antiquity so fully explored all forms of construction as to belie any such restriction. Presumably, in his three books on planar constructions, Apollonius sought to specify as completely as possible the domain of such constructions rather than to eliminate those of the solid or linear type. In any event, from works before Apollonius there is no evidence at all of a normative conception of problem-solving methods.

There survive isolated reports of Apollonian studies bearing on the regular solids, the cylindrical spiral, irrationals, circle measurement, the arithmetic of large numbers, and other topics. For the most part, little is known of these efforts, and their significance was slight in comparison with his treatises on geometric constructions.

Ptolemy reports in Mathēmatikē suntaxis (c. 150 c.e.; Almagest) that Apollonius made a significant contribution to astronomical theory by establishing the geometric condition for a planet to appear stationary relative to the fixed stars. Since, according to Ptolemy, he proved this condition for both the epicyclic and the eccentric models of planetary motion, Apollonius seems to have had some major responsibility for the introduction of these basic models. Apollonius studied only the geometric properties of these models, however, for the project of adapting them to actual planetary data became a concern only for astronomers such as Hipparchus a few decades later in the second century b.c.e.

Significance

If Apollonius of Perga did indeed institute the eccentric and epicyclic models for planetary motion, as seems likely, he merits the appellation assigned to him by historian Otto Neugebauer: “the founder of Greek mathematical astronomy.” These geometric devices, when adjusted to observational data and made suitable for numerical computation, became the basis of the sophisticated Greek system of astronomy. Through its codification by Ptolemy in the Almagest, this system flourished among Arabic and Hindu astronomers in the Middle Ages and Latin astronomers in the West through the sixteenth century. Although Nicolaus Copernicus (1473-1543) made the significant change of replacing Ptolemy’s geocentric arrangement with a heliocentric one, even he retained the basic geometric methods of the older system. Only with Johannes Kepler (1571-1630), who was first to substitute elliptical orbits for the configurations of circles in the Ptolemaic-Copernican scheme, can one speak of a clear break with the mathematical methods of ancient astronomy.

Apollonius’s work in geometry fared quite differently. The fields of conics and advanced geometric constructions he so fully explored came to a virtual dead end soon after his time. The complexity of this subject, proliferating in special cases and lacking convenient notations (such as the algebraic forms, for example, of modern analytic geometry that first appeared only with François Viète, René Descartes, and Pierre de Fermat in the late sixteenth and the seventeenth centuries), must have discouraged further research among geometers in the second century b.c.e.

In later antiquity, interest in Apollonius’s work revived: Pappus and Hypatia of Alexandria (fourth to early fifth century c.e.) and Eutocius (sixth century) produced commentaries on the Conics. Their work did not extend the field in any significant way beyond what Apollonius had done, but it proved critical for the later history of conic theory, by ensuring the survival of Apollonius’s writing. When the Conics was translated into Arabic in the ninth century, Arabic geometers entered this field; they approached the study of Apollonius with considerable inventiveness, often devising new forms of proofs, or contributing new results where the texts at their disposal were incomplete. Alhazen (early eleventh century), for example, attempted a restoration of Apollonius’s lost book 8.

In the early modern period, after the publication of the translations of Apollonius and Pappus by Federigo Commandino in 1588-1589, the study of advanced geometry received new impetus in the West. Several distinguished mathematicians in this period (François Viète, Willebrord Snel, Pierre de Fermat, Edmond Halley, and others) tried their hand at restoring lost analytic works of Apollonius. The entirely new field of projective geometry emerged from the conic researches of Gérard Desargues and Blaise Pascal in the seventeenth century. Thus, the creation of the modern field of geometry owes much to the stimulus of the Conics and the associated treatises of Apollonius.

Bibliography

Apollonius. On Cutting Off a Ratio. Translated by Edward Macierowski. Fairfield, Conn.: Golden Hind Press, 1987. This translation is literal and provisional; a full critical edition is being prepared by Macierowski.

Apollonius. Treatise on Conic Sections. Translated and edited by Thomas Little Heath. Cambridge, England: Cambridge University Press, 1896. Translation in modern notation, with extensive commentary. Heath surveys the older history of conics, including efforts by Euclid and Archimedes, and then summarizes the characteristic terminology and methods used by Apollonius. A synopsis appears in Heath’s History of Greek Mathematics (Oxford, England: Clarendon Press, 1921), together with ample discussions of the lost Apollonian treatises described by Pappus.

Fried, Michael N. Apollonius of Perga’s “Conica”: Text, Context, Subtext. Boston: E. J. Brill, 2001. A scholarly analysis. Bibliographic references, index.

Hogemdijk, J. P. Ibn al-Haytham’s Completion of the “Conics.” New York: Springer-Verlag, 1984. This edition of the Arabic text of Alhazen’s restoration of the lost book 8 of the Conics is accompanied by a literal English translation, a mathematical summary in modern notation, and discussions of the Greek and Arabic traditions of Apollonius’s work.

Knorr, W. R. Ancient Tradition of Geometric Problems. Cambridge, Mass.: Birkhauser Boston, 1986. A survey of Greek geometric methods from the pre-Euclidean period to late antiquity. Chapter 7 is devoted to the work of Apollonius, including his Conics and lost analytic writings.

Neugebauer, Otto. A History of Ancient Mathematical Astronomy. New York: Springer-Verlag, 1975. The section on Apollonius in this work provides a detailed technical account of his contributions to ancient astronomy.

Pappus of Alexandria. Book 7 of the “Collection.” Translated by A. Jones. New York: Springer-Verlag, 1986. A critical edition of Pappus’s Greek text (collated with the former edition of F. Hultsch in volume 2 of Pappi Collectionis Quae Supersunt, 1877), with English translation and commentary. Pappus’s book preserves highly valuable information on Apollonius’s lost works on geometric construction. Jones surveys in detail Pappus’s evidence of the lost works and modern efforts to reconstruct them.

Pappus of Alexandria, ed. Apollonius: Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banu Musa. New York: Springer-Verlag, 1990. This is the first literal English translation of this work ever to be published. Based on all known manuscripts, it includes the Arabic text with a full critical apparatus, an accurate English translation, and a commentary to elucidate both mathematical and historical difficulties.

Waerden, Bartel Leendert van der. Science Awakening. Translated by Arnold Dresden. 4th ed. Princeton Junction, N.J.: Scholar’s Bookshelf, 1988. In this highly readable survey of ancient mathematics, Waerden includes a useful synopsis of the geometric work of Apollonius.