conic sections
conic sections
- Michael N. Fried
Summary
The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated intensely in Greek mathematics. The most famous work on the subject was the Conics, in eight books by Apollonius of Perga, but conics were also studied earlier by Euclid and Archimedes, among others. Conic sections were important not only for purely mathematical endeavors such as the problem of doubling the cube, but also in other scientific matters such as burning mirrors and sundials. How the ancient theory of conics is to be understood also played a role in the general development of the historiography of Greek mathematics.
Keywords
Subjects
- Science, Technology, and Medicine
The term conic sections, familiar to all students in modern mathematics classrooms, is a direct translation of the Greek, tomai tōn kōnōn. For the Greek mathematicians, conic sections were, true to their name, those curves produced by cutting the surface of cone with a plane not containing the vertex of the cone. The exact cutting procedure—indeed, the definition of the cone itself—changed in significant ways from the end of the 4th to the beginning of the 2nd century bce, but by the end of that period, it was known that the three conic sections—the parabola, ellipse, and hyperbola—and the circle could be obtained by positioning the cutting plane in various ways with respect to any given cone. This was the discovery of Apollonius of Perga, who also gave us the names parabola, ellipse, and hyperbola.
Origin of the Study of Conic Sections
Why it occurred to Greek mathematicians to study the sections of a cone is not altogether clear. It may be that investigations concerning conic sections were inspired by considerations of cylindrical sections, such as one might see in cross-sections of columns broken at a slant. Conic and cylindrical sections are mentioned together in Euclid’s Phaenomena, where Euclid remarked, “If a cone or cylinder is cut by a plane not parallel to the base, the section produced will be a section of an acute-angled cone, which is similar to a shield.”1 However, since the section of a cylinder is described in terms of a section of a cone, it is likely that the investigation of conic sections preceded that of cylindrical sections, and not the other way about. In fact, the only known explicit treatment of the sections of a cylinder was very late, namely, that of Serenus (fl. 4th century ce), and the main theorem of Serenus’ work was that an oblique section of a cylinder is an ellipse, a theorem Serenus demonstrated precisely by showing how the same section can be produced by a cone.
Another possible source for interest in conic sections has to do with sundials. This was suggested by Neugebauer, who hypothesized a particular kind of sundial having a gnomon, the bar or post producing the shadow, daily aligned to point in the direction of the sun at noon and a shadow plane set perpendicular to the gnomon.2 By this construction, Neugebauer could explain one early way of defining conic sections. Unfortunately, as Neugebauer admitted, no such sundial had ever been found or reported. That said, the connection with sundials is reasonable. When the rays of the sun are intercepted by a point—say, the tip of a vertical gnomon—they will trace out a cone as the sun moves over the course of the day: the shadow on the ground, taken as a flat plane, must, therefore, trace out a conic section. Indeed, in the lower latitudes, where Greece is located, the shadow curve lies along a hyperbola, perhaps the oddest of all the conic sections.
Early sundials themselves were often conical. The tip of the gnomon would cast a shadow onto a conical surface whose axis pointed to the north celestial pole. The oldest known conical sundial was found in Heraclea ad Latmum, near modern Kapikiri in Turkey, and dates from approximately the end of the 3rd century bce.3 These would have been easier to produce than, say, a spherical or even cylindrical sundial, given that the profile could be measured by a taut string fixed at a point. More significantly, Vitruvius (On Architecture, Vitr. 9.8) associated conical sundials with Dionysodorus (early 2nd century bce), and Dionysodorus, according to Eutocius of Ascalon (c. 480–540 ce), used conic sections to complete a solution for Archimedes’ problem of cutting a sphere by a plane so that the ratio of the resulting volumes would be the same as a given ratio.4 The interest of Dionysodorus in both conical sundials and conic sections, leaves, of course, only a chicken and egg argument; however, such arguments do point to a close if not inseparable association between things.
The role of conic sections in problems, like that of Archimedes, has suggested yet another possibility, namely, that the investigations of conic sections arose in the context of more general problem-solving efforts, particularly those of Archytas (c. 400–350 bce), Menaechmus (c. 380–320 bce), and Aristaeus (c. 370–320 bce), especially related to the problem of finding four lines in continuous proportion or “doubling the cube.” For this problem, two lengths A and B are given (in the case of doubling the cube, B = 2A), and one must then find two lengths, M and N, so that A:M::M:N::N:B. The proportions, A:M::M:N (or M:N::N:B) implies the presence of a parabola, while the proportion A:M::N:B, a hyperbola: the intersection of these two curves, then, would provide the required lines. This, according to Eutocius, was how Menaechmus approached the problem and, accordingly, Menaechmus is given credit for an early theory of conic sections. Knorr, however, maintained that Menaechmus’s activities were connected more with problems of “application of area” than with conic sections as such and that a theory of conic sections developed afterwards, though he also admitted that his views rested only on inference.5 Nevertheless, even if Menaechmus’s work did not initiate the theory of conic sections, a close connection between the tradition of problem solving and conic section cannot be denied support, for that connection comes from the division of basic problem types noted by figures such as Pappus of Alexandria (c. 290–c. 350 ce). Pappus said that there are three types of problems, of which the second type, solid problems (sterea), are those that require specifically one or more of the sections of a cone.6 It can be said that this is the interpretation now most favored by scholars in the field.
Early Investigations of Conic Sections
Whatever its exact origin, one can say confidently that a systematic investigation of conic sections was well underway by the end of the 4th century bce. Euclid (fl. c. 300 bce) and Aristaeus, for example, certainly worked on conics and may have written treatises on the subject. More than once, Archimedes mentioned a certain work on the elements of conics (e.g., in Conoids and Spheroids, 3) and he may well have been referring to one or both of these treatises.
We are told by Eutocius that in this earlier period the different kinds of conic sections were associated each with a different cone. The cone was always a right-cone and most likely was conceived as a figure generated by rotating a right triangle about one of its legs, a description one finds in Euclid’s Elements (Book XI, defs.18–20). A plane containing the fixed leg of the triangle (the axis of the cone) cuts the cone in an isosceles triangle, called the axial triangle. With that, the conic sections were produced by a plane cutting the cone perpendicularly to the side of the axial triangle (see figure 1). Neugebauer’s hypothetical sundial, mentioned above, was designed specifically to explain this particular cutting procedure—the gnomon, accordingly, lay along one of the sides of the axial triangle, and the shadow plane set perpendicular to the gnomon, corresponded to the cutting plane.
Figure 1. The Pre-Apollonian procedure for cutting a cone and corresponding names for the conic sections.
Be that as it may, with this cutting procedure, three different conic sections arose according to whether the vertex angle of the axial triangle was acute, obtuse, or right. The sections were called, correspondingly: “a section of an acute-angled cone” (oxugōniou kōnou tomē), “a section of an obtuse-angled cone” (amblugōniou kōnou tomē), and, finally, “a section of a right-angled cone” (orthogōniou kōnou tomē).
These names are those used, for the most part, by Archimedes. Moreover, it is easy to see how the construction allows one to prove the basic properties of the conic sections which Archimedes employed so powerfully in works such as the Sphere and the Cylinder, Conoids and Spheroids and the Quadrature of the Parabola. For example, consider the principal property, or symptōma, for the section of the acute-angled cone, as it appears in Archimedes: let AB be the diameter of a section of an acute-angle cone (this can be shown to be the common section of the axial triangle and cutting plane), and let MN and PR be drawn perpendicularly to AB (“ordinate-wise”) from M and P on the section (see figure 2); then, we have:
sq.MN:sq.PR::rect.AN,NB:rect.AR,RB
Where, by sq.MN, we mean the square on the side MN and by rect.AN,NB we mean the rectangle whose sides are equal to AN and NB.
Figure 2. Ordinates drawn to the axis of the section of the acute-angled cone.
Draw planes EMF and GPH parallel to the base of the cone (see figure 3): they will cut circles from the cone—as would be true in any cone—and, like the cutting plane itself, EMF and GPH will also be perpendicular to the axial triangle. Thus, MN and PR are also perpendicular to EF and GH, respectively. So, by Elem. 3.35, it follows that sq.MN = rect. EN,NF and sq.PR = rect.GR,RH. But by similar triangles, EN:GR::AN:AR and NF:RH::NB:RB, from which we have, rect.EN,NF:rect.GR,RH::rect.AN,NB:rect.AR,RB, or sq.MN:sq.PR::rect.AN,NB:rect.AR,RB.
Figure 3. The chief property of the section of the acute-angle cone.
The analogous properties corresponding to the section of the right-angled and the section of the obtuse-angled cone can be demonstrated in a similar way. These are (see figure 4):
sq.MN:sq.PR::AN:AR for the section of the right-angled conesq.MN:sq.PR::rect.AN,NB:rect.AR,RB for the obtuse-angled cone
(B, in the latter, is the point at which the side of the axial triangle meets the extended diameter of the section).
Figure 4. Sections of the right and obtuse-angled cones.
It should be noted that the condition that the cutting plane be perpendicular to the side of the axial triangle, which Neugebauer wanted so badly to explain with his sundial, was not used in the argument above. By the end of the 3rd century bce, or at the start of the 2nd century, it must have become apparent that this mode of cutting the cone was overly restrictive, for the generalization of the procedure occurred about that time.
Apollonius of Perga’s Conics
The generalization of the procedure and, indeed, that of the cone itself was the achievement of Apollonius of Perga. Apollonius’s Conics, perhaps completed at the beginning of the 2nd century bce, was certainly the most extensive treatise on conic sections in antiquity and one of the summits of Greek mathematics altogether. In this work, which contained more than 330 propositions on conic sections, Apollonius redefined the cone and the mode of cutting it so that all three conic sections could be derived from a single cone and their principal properties (symptōmata) could be expressed in greater generality. In his commentary on the conics, Eutocius remarked that, according to Geminus (1st century bce), “Marveling at this, and on account of the marvelous theorems in conics proved by him, [Apollonius’s] contemporaries called him the ‘Great Geometer’.”7
The conic surface was defined by Apollonius to be the surface swept out by a line held fixed at one point while, at another point, it moves along the perimeter of a fixed circle (figure 5). The point held fixed is the vertex, and the line joining the vertex and the center of the fixed circle is the axis. The part of the surface between the vertex and the fixed circle is what Apollonius calls the cone; the circle is the base of the cone.
Figure 5. Generation of the double conic surface.
Apollonius’s definition of the conic surface and cone differed sharply from the right cone used previously in three ways. First, the conic surface comprises two sheets, one above and one below the vertex. Second, the surface extends indefinitely; unlike Euclid’s cone, which was generated as a bounded object ab initio, Apollonius’s cone was cut off from an unbounded surface. Third, the conic surface and cone are oblique, that is, the axis is not perpendicular to the base.
The definitions of the conic surface, vertex, axis, cone, and base of the cone are the first definitions of Book I of the Conics. These are followed by a series of definitions concerning curves in general, including the definition of a diameter and of chords drawn ordinate-wise. Given a curve in the plane, a diameter is a line that bisects all chords drawn parallel to some straight line; when chords are drawn this way, they are said to be drawn ordinate-wise. An example is shown in Figure 6, where AB is the diameter, and chords, such as PQ, drawn parallel to L, are bisected by AB. When the ordinate direction is perpendicular to the diameter, the diameter is called an axis. The endpoint of the diameter on the curve is called the vertex.
Figure 6. A diameter AB and chords PQ drawn ordinate-wise in a curve.
Apollonius also defined a diameter between two curves in a plane in a similar way. He called this a transverse diameter (diametros plagia) (see figure 7). These definitions prepare the way for diameters and ordinates in the conic sections; the transverse diameter, for example, anticipates the opposite sections, which have two branches and both diameters and transverse diameters.
Figure 7. Transverse diameter, AB, and ordinates PQ.
What are not defined in the opening definitions of Book I are the conic sections themselves. The definitions of the conic sections appear only after Apollonius has demonstrated ten propositions exploring the properties of the cone and conic surface, a kind of mini-treatise on the cone.8 Apollonius proved, for example, that a circle is produced not only by cutting a cone parallel to the base, but also by cutting it in a second way, which he called subcontrariwise (hupenantiōs) (Conics, I.5); this is a result true only for an oblique cone.
More significantly, proposition I.7 of the mini-treatise, gives us the new cutting procedure. Let ABC be any axial triangle, that is, as before, a triangle produced by a plane containing the axis of the cone, where A is the vertex of the cone and BC is the common section of ABC and the base of the cone. Let the cone be cut by a second plane—this is the plane producing the section—so that its common section with the base of the cone, the line L, is perpendicular to BC, possibly extended (see figure 8).
Figure 8. The cutting procedure.
The cutting procedure is designed to provide the section with a well-defined diameter, and this is what is actually proven in I.7. In Figure 8, the diameter is RS, the common section of the cutting plane and the axial triangle (true also in the Pre-Apollonian procedure), and the ordinate direction is given by the common section, L, of the cutting plane and base plane. At the end of this proposition, Apollonius remarked that the diameter need not be perpendicular to the ordinate direction, that is, that not every diameter is an axis. Later in the Conics, he showed that the ordinate direction is the same as the tangent at the vertex of the diameter; however, the fundamental explanation as to why there should be a diameter and ordinate direction at all was set out in terms of the cone and positions of the axial triangle, cutting plane, and base plane.
Finally, propositions 11–14 of Book I define the conic sections, provide their principal properties—the symptōmata—and bestow upon them the names we are familiar with, parabola, ellipse, and hyperbola. The parabola is obtained when the diameter (produced in the way prescribed by I.7) is parallel to one of the sides of the axial triangle; the ellipse is obtained when the diameter cuts both sides of the axial triangle on the same side of the vertex; and hyperbola is obtained when the diameter cuts the sides of the axial triangle above and below the vertex (figure 9). A hyperbola is produced on each sheet of the conic surface. In modern mathematics the hyperbola itself contains these two curves—two branches, as one would say today. For Apollonius, each branch was a separate hyperbola, and the two together formed a pair that he called the opposite sections (tomai antikeimenai) (figure 10). The opposite sections are described in the last of these initial propositions, Conics, I.14.
Figure 9. The conic sections according to Apollonius.
Figure 10. The opposite sections composed of two hyperbolas.
The names, parabola, ellipse, and hyperbola relate to the symptomata of the curves which Apollonius formulated in the language of “application of areas” (parabolē tōn xōriōn). Application of areas concerns a set of problems in which, roughly, one must apply a rectangle or parallelogram to a given line equal to a given area, or exceeding, or falling short of that area by another given area. Accordingly, in propositions I.11, 12, and 13 of Conics I, Apollonius proves that there is a line, PA, called the orthia or “upright side” (thus, latus rectum in Latin), defined in terms of the cone, such that (figure 11):
When the section LAM is a parabola, the square on MN is exactly equal to the rectangle contained by PA, AN (Conics, I.11).
When the section LAM is a hyperbola, the square on MN is equal to a rectangle greater than that contained by PA, AN; if the width of the rectangle is AN, then the excess will be a rectangle similar to the rectangle contained by PA, AB (Conics, I.12).
When the section LAM is an ellipse, the square on MN is equal to a rectangle less than that contained by PA, AN; if the width of the rectangle is AN, then the deficiency will be a rectangle similar to the rectangle contained by PA, AB (Conics I.13).
These are equivalent to the symptōmata given above for the sections of the right, acute, and obtuse-angled cones.
Figure 11. The orthia by which the symptōma are defined in Apollonius de Perge, Coniques.
Apollonius’s Conics comprised eight books, of which four are extant in Greek and three in Arabic. The final book has been lost. Curiously, the Conics contains no propositions relating to the areas contained one way or another by conic sections, which Archimedes investigated. Nevertheless, the eight books contain a mass of information about conic sections that underlines the richness of the Greek knowledge of these curves. It investigates, among other things, the properties and positions of diameters, conjugate diameters, tangents, and asymptotes (Books I-III); the number of points at which conic sections can intersect or touch (Book IV); shortest and longest lines drawn from a point and intercepted between the axis of a section and the section itself (Book V); similar and equal conic sections (Book VI); theorems connected to diorismoi (Book VII). Focal properties of the ellipse and hyperbola were considered by Apollonius in Book III of the Conics, though the focal property of the parabola does not appear; it does appear, however, in another work by a contemporary of Apollonius, Diocles (fl. c. 190 bce), On Burning Mirrors.
Historiographical Remark
The study of Greek mathematics in modern times was deeply influenced by a work that was largely an interpretation of Apollonius’s Conics. This was H. G. Zeuthen’s Die Lehre von den Kegelschnitten im Altertum (The Theory of Conic Sections in Antiquity) published in 1886.9 Zeuthen's position was that modern mathematical conceptions and especially its algebraic notation had the power to “...bring into light all those things concealed by the ancients’ exclusive attention to logical coherence.”10 He meant, among other things, their geometrical presentation. For him and others following him, such as Neugebauer, little was lost, and much was to be gained by representing conic sections and, generally, geometric relationships in the algebraic terms familiar to modern students of mathematics.11 Already in the 1930s, Jacob Klein's Greek Mathematical Thought and the Origin of Algebra though not specifically about the historiography of Greek mathematics, made it clear that the seemingly innocent use of symbolic representation, in fact, ignored deep conceptual differences between Greek and modern mathematics.12 Unguru, by contrast, directly attacked the then current historiographical approaches to Greek mathematics inspired by figures such Zeuthen and Neugebauer, especially their free use of algebra.13 Fried’s book on Apollonius’s Conics with Unguru was meant, subsequently, to be an answer to Zeuthen’s book on the Conics, arguing that Apollonius’s work on conics had a fundamentally geometric foundation.14 Though few historians today adopt the Zeuthen-Neugebauer position, there is a general acceptance of a problem-solving emphasis, such as that developed by Knorr; this, in a way, can be viewed as a compromise between a purely geometric approach and an algebraic analytic approach to conics. On the other hand, what problem solving really meant for Greek mathematics is still an issue being explored in the field.
Primary Texts
- Apollonius (Johan Ludvig Heiberg). Apollonii Pergaei quae Graece exstant cum commentariis antiquis. Editit et Latine interpretatus est I. L. Heiberg. 2 vols. Leipzig: Teubner, 1891,1893.
- Apollonius (Gerald J. Toomer). Apollonius Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banu Musa. 2 vols. Sources in the History of Mathematics and Physical Sciences 9. New York: Springer-Verlag, 1990.
- Apollonius, Roshdi Rashed, Micheline Decorps-Foulquier, and Michel Federspiel. Apollonius de Perge, Coniques: Texte grec et arabe etabli, traduit et commenté. Berlin: Walter de Gruyter, 2008–2010.
- Archimedes (Johan Ludvig Heiberg and E. S. Stamatis). Opera Omnia, 3 vol. Stuttgart, 1972.
- Euclid, Johan Ludvig Heiberg, and H. Menge. Euclidis Opera Omnia, 8 vol. Leipzig: Teubner, 1883.
- Eutocius (Johan Ludvig Heiberg). In Apollonium Commentaria: in Apollonius and Heiberg, Apollonii Pergaei quae Graece exstant, Vol. II, 168–361.
- Pappus d’Alexandrie Fridericus Hultsch. Collectionis quae supersunt, 3 vols. Berlin: Weidmann, 1876–1878.
Bibliography
- Decorps-Foulquier, Micheline. Recherches sur les Coniques d’Apollonios de Pergé et leurs commentateurs grecs. Histoire de la transmission des Livres grecs I–IV. Paris: Klincksieck, 2000,
- Dijksterhuis, Eduard Jan. Archimedes. Princeton, NJ: Princeton University Press, 1987. (Published originally by Ejnar Munksgaard (Copenhagen), 1956).
- Fried, Michael N., and Unguru, Sabetai. Apollonius of Perga’s Conica: Text, Context, and Subtext. Leiden: Brill, 2001.
- Heath, Sir Thomas. A History of Greek Mathematics, 2 vols. New York: Dover, 1981. (Originally published by Clarendon Press (Oxford), in 1921).
- Knorr, Wilbur Richard. The Ancient Tradition of Geometric Problems. Boston: Birkhäuser, 1986.
- Netz, Reviel. The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations. Cambridge, U.K.: Cambridge University Press, 2004.
Notes
1. Euclid, Johan Ludvig Heiberg, and Heinrich Menge, Euclidis Opera Omnia, vol. 8 (Leipzig: Teubner, 1883), 6.
2. Otto Neugebauer, “The Astronomical Origin of the Theory of Conic Sections,” Proceedings of the American Philosophical Society 92, no. 3 (1948): 136–138.
3. Robert Hannah, Time in Antiquity (London: Routledge, 2009), 91.
4. Reviel Netz, The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations (Cambridge, UK: Cambridge University Press, 2004), 29–39.
5. Wilbur R. Knorr, “Observations on the Early History of the Conics,” Centaurus 26, no. 1 (1982): 1–24.
6. Pappus and Fridericus Hultsch, Collectionis quae supersunt (Berlin: Weidmann, 1876–1878), 270.
7. Apollonius and Johan Ludvig Heiberg. Apollonii Pergaei quae Graece exstant cum commentariis antiquis. Editit et Latine interpretatus est I. L. Heiberg. 2 vols. (Leipzig: Teubner, 1891,1893), Vol. II, 170.
8. Michael N. Fried and Sabetai Unguru, Apollonius of Perga’s Conica: Text, Context, and Subtext (Leiden: Brill, 2001), 76
9. Hieronymus G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum (Hildesheim, Germany: Georg Olms, 1966). (Originally published in 1886.)
10. Zeuthen, Die Lehre von den Kegelschnitten, xii.
11. Neugebauer took a somewhat more extreme view than Zeuthen in thinking that an algebraic version of mathematical investigations based on “Babylonian algebra” was behind much of Greek mathematical thought in general and Apollonius’s work in particular. See David E. Rowe, “Otto Neugebauer’s Vision for Rewriting the History of Ancient Mathematics,” Anabases 18 (2013): 175–196.
12. Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, trans. Eva Brann (Cambridge, MA: MIT Press, 1968). Originally published in two parts: “Die griechische Logistik und die Entstehung der Algebra,” in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik (fasc. 1; Berlin: Springer, 1934), 18–105; and (fasc. 2; Berlin: Springer, 1936), 122–235).
13. Sabetai Unguru, “On the Need to Rewrite the History of Greek Mathematics,” Archive for History of Exact Sciences 15, no. 1 (1975): 67–114.
14. Michael N. Fried and Sabetai Unguru, Apollonius of Perga’s Conica.
