Apollonius of Perga

(b. second half of third century b.c.; d. early second century b.c.)

mathematical sciences.

Very little is known of the life of Apollonius. The surviving references from antiquity are meager and in part untrustworthy. He is said to have been born at Perga (Greek Πέργη), a small Greek city in southern Asia Minor, when Ptolemy Euergetes was king of Egypt (i.e., between 246 and 221 b.c.)¹ and to have become famous for his astronomical studies in the time of Ptolemy Philopator, who reigned from 221 to 205 b.c.² Little credence can be attached to the statement in Pappus that he studied for a long time with the pupils of Euclid in Alexandria.³ The best evidence for his life is contained in his own prefaces to the various books of his Conics. From these it is clear that he was for some time domiciled at Alexandria and that he visited Pergamun and Ephesus.

The prefaces of the first three books are addressed to one Eudemus of Pergamum. Since the Preface to Book II states that he is sending the book by the hands of his son Apollonius, he must have been of mature age at the time of its composition.⁴ We are told in the Preface to Book IV that Eudemus is now dead;⁵ this and the remaining books are addressed to one Attalus. The latter is commonly identified with King Attalus I of Pergamum (reigned 241–197 b.c.); but it is highly unlikely that Apollonius would have negelected current etiquette so grossly as to omit the title of “King” (βασιλ∊ύς) when addressing the monarch, and Attalus was a common name among those of Macedonian descent. However, a chronological inference can be made from a passage in the Preface to Book II, where Apollonius says,“... and if Philonides the geometer, whom I introduced to you in Ephesus, should happen to visit the neighborhood of Pergamum, give him a copy [of this book].”⁶ Philonides, as we learn from a fragmentary biography preserved on a papyrus and from two inscriptions, was an Epicurean mathematician and philosopher who was personally known to the Seleucid kings Antiochus IV Epiphanes (reigned 175–163 b.c.) and Demetrius I Soter (162–150 b.c.). Eudemus was the first teacher of Philonides. Thus the introduction of the young Philonides to Eudemus probably took place early in the second century b.c. The Conics were composed about the same time. Since Apollonius was then old enough to have a grown son, it is reasonable to accept the birth date given by Eutocius and to place the period of Apollonius’ activity in the late third and early second centuries b.c. This first well with the internal evidence which his works provide on his relationship to Archimedes (who died an old man in 212–211 b.c.); Apollonius appears at times to be developing and improving on ideas that were originally conceived by Archimedes (for examples see p. 189). It is true that Apollonius does, however, refer to Conon, an older (?) contemporary and correspondent of Archimedes as a predecessor in the theory of conic sections.⁷

Of Apollonius’ numerous works in a number of different mathematical fields, only two survive, although we have a good idea of the content of several others from the account of them in the encyclopedic work of Pappus (fourth century a.d.). But it is impossible to establish any kind of relative chronology for his works or to trace the development of his ideas. The sole chronological datum is that already established, that the Conics in the form that we have them are the work of his mature years. Thus the order in which his works are treated here is an arbitrary one.

The work on which Apollonius’ modern fame rests, the Conics (κωνικά), was originally in eight books. Books I–IV survive in the orginal Greek, Books V–VII only in Arabic translation. Book VIII is lost, but some idea of its contents can be gained from the lemmas to it given by Pappus.⁸ Apollonius recounts the genesis of his Conics in the Preface to Book I⁹: he had originally composed a treatise on conic sections in eight books at the instance of one Naucrates, a geometer, who was visiting him in Alexandria; this had been composed rather hurriedly because Naucrates was about to sail. Apollonius now takes the opportunity to write a revised version. It is this revised version that constitutes the Conics as we know it.

In order to estimate properly Apollonius’ achievement in the Conics, it is necessary to know what stage the study of the subject had reached before him. Unfortunately, since his work became the classic textbook on the subject, its predecessors failed to survive the Byzantine era. We know of them only from the scattered reports of later writers. It is certain, however, that investigation into the mathematical properties of conic sections had begun in the Greek world at least as early as the middle of the fourth century b.c., and that by 300 b.c. or soon after, textbooks on the subject had been written (we hear of such by Aristaeus and by Euclid). Our best evidence for the content of these textbooks comes from the works of Archimedes. Many of these are concerned with problems involving conic sections, mostly of a very specialized nature; but Archimedes makes use of a number of more elementary propositions in the theory of conics, which he states without proof. We may assume that these propositions were already well known. On occasion Archimedes actually states that such and such a proposition is proved “in the Elements of Conics” (ʾεν τoις κωνικoις στoιϰείoις).¹⁰ Let us leave aside the question of what work(s) he is referring to by this title; it is clear that in his time there was already in existence a corpus of elementary theorems on conic sections. Drawing mainly on the works of Archimedes, we can characterize the approach to the theory of conics before Apollonius as follows.

The three curves now known as parabola, hyperbola, and ellipse were obtained by cutting a right circular cone by a plane at right angles to a generator of the cone. According to whether the cone has a right angle, an obtuse angle, or an acute angle at its vertex, the resultant section is respectively a parabola, a hyperbola, or an ellipse. These sections were therefore named by the earlier Greek investigators “section of a right-angled cone,” “section of an obtuse-angled cone,” and “section of an acute-angled cone,” respectively; those appellations are still given to them by Archimedes (although we know that he was well aware that they can be generated by methods other than the above). With the above method of generation, it is possible to characterize each of the curves by what is known in Greek as a σύμπτωμα, i.e., a constant relationship between certain magnitudes which vary according to the position of an arbitrary point taken on the curve (this corresponds to the equation of the curve in modern terms). For the parabola (see Figure 1), for an arbitrary point K, KL² = 2 AZ • ZL (for suggested proofs of this and the σύμπτωμα of hyperbola and ellipse, see Dijksterhuis, Archimedes, pp.57–59, whom I follow closely here). In algebraic notation, if KL = y, ZL = x, 2 AZ = p, we get the characteristic equation of the parabola y² = px. Archimedes frequently uses this relationship in the parabola and calls the parameter p “the double of the distance to the axis” (ʿα διπλασία τας μέxρι τού ʾάξονος)¹¹ exactly describing 2 ZA in Figure 1 (“axis” refers to the axis of the cone). For the hyperbola and ellipse the following σύμπτωμα can be derived (see Figures 2 and 3):

in algebraic notation, if KL = y, ZL = x₁, PL = x₂ 2 ZF = p, PZ = a,

This is found in Archimedes in the form equivalent to

It is to be noted that in this system ZL always lies on the axis of the section and that KL is always at tight angles to it. In other words, it is a system of “orthogonal conjection.”

Apollonius’ approuch is radically differnt. He generates all three curves from the double oblique circular cone, as follows: in Figures 4,5, and 6 ZDE is the cutting plane. We now cut the cone with another plane orthogonal to the first and passing through the axis of the cone; the is known as the axial triangle (ABG); the latter must intersect the base of the cone in a diameter (BG) orthogonal to the line in which

the cutting plane intersects it (or its extension); it intersects the cutting plane in a straight line ZH. Then, if we neglect the trivial cases where the cutting plane generates a circle, a straight line, a pair of straight lines, or a point, there are three possibilities:

(a) The line ZH in which the cutting plane intersects the axial triangle intersects only one of the two sides of the axial triangle, AB, AG; i.e., it is parallel to the other side (Figure 4).

(b) ZH intersects one side of the axial triangle below the vertex A and the other (extended) above it (Figure 5).

(c) ZH intersects both sides of the axial triangle below A (Figure 6).

In all three cases, for an arbitrary point K on the curve.

Furthermore, in case (a),

If we now construct a line length Θ such that Θ = BG² • AZ/AB • AG, it follows from (1) and (2) that KL² = LZ • Θ. Since none of its constituent parts is dependent on the position of K, Θ is a constant. In algebraic terms, if KL = y, LZ = x, and Θ = p, then y² = px. In cases (b) and (c),

If we now construct a line length Ξ such that Ξ = BS • SG • ZT/AS², it follows from (1) and (3) that

Thus

and

In algebraic terms, if KL = y, LZ = x, Ξ = p, and ZT = a,

The advantage of such formulation of the σύμπτẃματα of the curves from the point of view of classical Greek

geometry is that now all three curves can be determined by the method of “application of areas,” which is the Euclidean way of geometrically formulating problems that we usually express algebraically by equations of second degree. For instance, Euclid (VI, 28) propounds the problem “To a given straight line to apply a parallelogram equal to a given area and falling short of it by a parallelogrammic figure similar to a given one.” (See Figure 7, where for simplicity rectangles have been substituted for parallelograms.) Then the problem is to apply to a line of given length b a rectangle of given area A and side x such that the rectangle falls short of the rectangle bx by a rectangle similar to another with sides c, d. This is equivalent to solving the equation

Compare the similar problem Euclid VI, 29: “To a given straight line to apply a parallelogram equal to a given area and exceeding it by a parallelogrammic figure similar to a given one.”

This method is used by Apollonius to express the σύμπτẃματα of the three curves, as follows (see Figures 8–10).

For case (a) (Figure 8), a rectangle of side x (equal to the abscissa) is applied (παραβάλλεται) to the line-length p (defined as above): this rectangle is equal to the square on the ordinate y. The section is accordingly called parabola (παραβολη, meaning “exact application”).

For case (b) (Figure 9), there is applied to p a rectangle, of side x, equal to y² and exceeding

ύπєρβάλλον) by a rectangle similar to p/a. The section is accordingly named hyperdola (ύπєρβολη), meaning “excess”).

For case (c) (Figure 10) there is applied to p a rectangle of side x. equal to y² anf falling short (ʾєλλєίπον) of p by a rectangle similar to p/a. The section is accordingly named ellipe (ʾєλλєιψς, meaning “falling short”).

This approach has several advantages over the older one. First, all three curves can be represented by the method of “application of areas” favored by classical Greek geometry (it has been appropriately termed “geometrical algebra” in recent times); the older approuch allowed this to be done only for the parabola. In modern terms, Apollonius refers the equation of all three curves to a coordinate system of which one axis isd a given diameter of the diameter. This brings us to a second advantage: Apllonius’ method of generating the curves immediately produces

oblique conjugation, whereas the older method produces orthogonal conjection. As we shall see, oblique conjugation was not entirely unknown to earlier geometers; but it is typical of Apollonius’ approach that he immediately develops the most general formulation. It is therefore a logical step, given this approuch, for Apollonius to prove (1,50 and the preceding propositions) that a σύμτπωμα equivalent to those derived above casn be established for any diameter of a conic and its ordicates: in modern terms, the coordinated of the curves can be transposed to any diameter and its tangent.

We cannot doubt that Apollonius’ approach to the generation and basic definition of the conic sections, as outlined above, was radically new. It is not easy to determine how much of the content of the Conics is new. It is likely that a good deal of the nomenclature that his work made standard was introduced by him; in particular, the terms “parabola,” “hyperbola,” and “ellipse” make sense only in terms of Apollonius’ method. To the parameter which we have called p he gave the name ʾορθία(“orthogonal side” [of a rectangle]), referring to its use in the “application”: this term survives in the modern latus rectum. He defines “diameter” as any line bisecting a system of parallel chords in a conic, in accordance with the new generality of his coordinate system: this differs from the old meaning of “diameter” of a conic section (exemplified in Archimedes), which is (in Apollonian and modern terminology) the axis. But though this new terminology reflects the new approach, it does not in itself exclude the possibility that many of Apollonius’ results in the Conics were already known to his predecessors. That this is true at least for the first four books is suggested by his own Preface to Book I. He says there:¹³

The first four books constitue an elementary introduction. The first contains the methods of generating the three sections and their basic properties (ανμπτώματα) developed more fully and more generally (καθóλνμάλλον) than in the writings of others; the second contains the properties of the diameter and axes of the sections, the asymptotes and other things...; the third contains many surprising theorems useful for the syntheses of solid loci and for determinations of the possibilities of solutions (διοριομούς); of the latter the greater part and the most beautiful are new. It was the discovery of these that made me aware that Euclid has not worked out the whole of the locus for three and four lines,¹⁴ but only a fortuitous part of it, and that not very happily; for it was not possible to complete the synthesis without my additional discoveries. The fourth book deals with how many ways the conic sections can meet one another and the circumference of the circle, and other additional matters, neither of which has been treated by my predecessors, namely in how many points a conic section or circumference of a circle can meet another. The remaining books are particular extensions (πєριονσοιαστικώτєρα); one of them [V] deals somewhat fully with minima and maxima, another [VI] with equal and similar conic sections, another [VII] with theorems concerning determinations (διοριστικών), another [VIII] with determinate conic problems.

From this one gets the impression that Books I-IV, apart from the subjects specifically singled out as original, are merely reworkings of the results of Apollonius’ predecessors. This is confirmed by the statement of Pappus, who says that Apollonius supplemented the four books of Euclid’s Conics (which Pappus May have known) and added four more books.¹⁵

Apollonius also claims to have worked out the methods of generating the sections and setting out their ανμπτώματα “more fully and more generally” than his predecessors. The description “more generally” is eminently justified by our comparison of the two methods. However, it is not clear to what “more fully” (ʾєπì πλєον) refers. Neugebauer suggests that Apollonius meant his introduction of conjugate hyperboals (conjugate diameters in ellipse and hyperbola are dealt with in I, 15–16).¹⁶ At least as probable is a more radical alternative, rejected by Neugebauer, that Apollonius is referring to his treatment of the two branches of the hyperbola as a unit (exemplified in I, 16 and frequently later). It is true that Apollonius applies the name “hyperhbola” only to a single branch of the curve (he refers to the two branches as the “opposite sections” [τομαì ʾαντικєíμєναι]); it is also true that in his own Preface to Book IV¹⁷ he reveals that at least Nicoteles among his predecessors had considered the two branches together; but Apollonius’ very definition of a conic surface¹⁸ as the surface on both sides of the vertex is significant in this context; and it is unlikely that any of his predecessors had systematically developed the theory of both branches of the hyperbola. Here again, then, we may reasonably regard Apollonius as an innovator in his method. But we are not justified in assuming that any of the

results stated in the first four books were unknown before Apollonius, except where he specifically states this, In this part of the work we must see him rather as organizing the results of his predecessors, consisting in part of haphazard and disconnected sets of theorems, into an exposition ordered rationally according to his own very general method. His mastery is such that it seems impossible to separate difffernt sources (as one can, for instance, in the comparable work of Euclid on elementary geometry).

Nevertheless, we may suspect that Archimedes, could he have read Books I-IV of the Conies, would have found few results in them that were not already familiar to him (although he might well have been surprised by the order and manual connection of the theorems). The predecessors of Archimedes were already aware that section could be generated by methods other than that described on p. 180. Euclid states that an ellips can be produced by cutting a cylinder by a plane not parallel to the base. ¹⁹ Archimedes himself certainly knew that there were many different ways of generating the sections from a cone. The bewst proof of this is De Conoidibus et sphaerodibus VII-IX, in which it is shown that for any ellipse it is possible to find an oblique circular cone from which that ellipse can be generated. Furthermore, it is certain that the essential properties of the oblique conjugation of at least the parabola were known to Apollonius’ predecessors; for that is the essence of propositions I-III of Archimedes’ Quadrature of the Parabola, which he states are proved “in the Elements of Conics.”²⁰ In De Conoidibus et sphaeroidibus III, Archimedes states “If two tangents be drawn from the same point to any conic, and two chords be drawn inside the section parallel to the two tangents and intersecting one another, the product of the two parts of each chord [formed by the intersection] will have the same ratio to one another as the squares on the tangents … this is proved in the Elements of Conics.”²¹ It is plausible to interpret this, with Dijksterhuis, as treatment of all three sections in oblique conjugation.²²

It is probable then that much of the contents of Books I-IV was already known before Apollonius. Conversely, Apollonius did not include in the “elementary introduction” of Books I-IV al theorems on conies known to his predecessors. For example, in a parabola the subnormal to any tangent formed on the diameter (HZ in Figure 11) is constant and equal to half the parameter p. This is assumed without proof by Archimedes²³ and by Diocles (perhaps a contemporary of Apollonius) in his proof of the focal property of the parabola in his work On Burning Mirrors.²⁴ We can therefore be sure that it was a well–known theorem in the Elements of conics. Yet in Apollonius it can be found only by combining the results of propositions 13 and 27 of Book V, one of the “particular extensions.”

If Apollonius omitted some of his predecessors’ results from the elementary section, we must not be surprised if he omitted altogether some results with which he was perfectly familiar: his aim was not to compile an encyclopedia of all possible theorems on conic sections, but to write a systematic textbook on the “elements” and to add some more advanced theory which he happened to have elaborated. The question has often been raised in modern times why there is no mention in the Conics of the focus of the parabola. The focal properties of hyperbola and ellipse are treated in III, 45–52: Apollonius proves, inter alia, that the focal distances at any point make equal angles with the tangent at that point and that their sum (for the ellipse) or difference (for the hyperbola) is constant. There is no mention of directrix, and from the Conics we might conclude that Apollonius was totally ignorant of the focus-directrix property of conic sections. However, it happens that Pappus proves at length that if a point moves in such a way that the ratio of its distance from a fixed point and its orthogonal distance from a fixed straight line is constant, then the locus of that point is a conic section; and that according as the ratio is equal to, greater than, or less than unity, the section will be respectively a parabola, a hyperbola, or an ellipse.²⁵ This amounts to the generation of the sections from focus and directrix. Pappus gives this proof as a lemma to Euclid’s (lost) book On Surface Loci; hence, it has been plausibly concluded that the proposition was there stated without proof by Euclid.²⁶ If that is so, here is a whole topic in the theory of conics that must have been completely familiar to Apollonius, yet which he omits altogether. Thus the lack of any mention of the focus of the parabola in the Conics is not an argument for Apollonius’ ignorance of it. I agree with those who argue on a priori grounds that he must have known of it.²⁷ Since he very probably dealt with it in his work(s) on burning mirrors (see p.189) there was all the more justification for omitting it from the Conics. In any case, we now have a proof of it, by Diocles, very close to the time of Apollonius. Since Diocles further informs us that a parabolic burning mirror was constructed by Dositheus, who corresponded with Archimedes, it is highly probable that the focal property of the parabola was well known before Apollonius.

For a detailed summary of the contents of the Conics, the reader is referred to the works of Zeuthen and Heath listed in the Bibliography. Here we will only supplement Apollonius’ own description quoted above by noting that Book III deals with theorems on the rectangles contained by the segments of intersecting chords of a conic (an extension to conics of that proved by Euclid for chords in a circle), with the harmonic properties of pole and polar (to use the modern terms: there are no equivalent ancient ones), with focal properties (discussed above), and finally with propositions relevant to the locus for three and four lines (see n.14). Of Books V-VII, which are, to judge from Apollonius’ own account, largely original, Book V is that which has particularly evoked the admiration of modern mathematicians: it deals with normals to conics, when drawn as maximum and minimum straight lines from particular points or sets of points to the curve. Apollonius finally proves, in effect, that there exists on either side of the axis of a conic a series of points from which one can draw only one normal to the opposite side of the curve, and shows how to construct such points: these points form the curve known, in modern terms, as the evolute of the conic in question. Book VII is concerned mainly with propositions about inequalities between various functions of conjugate diameters. Book VIII is lost, but an attempt at restoration from Pappus’ lemmas to it was made by Halley in his edition of the Conics. If he is right, it contained problems concerning conjugate diameters whose functions (as “determined” in Book VII) have given values.

For a modern reader, the Conics is among the most difficult mathematical works of antiquity. Both form and content are far from tractable. The author’s rigorous rhetorical exposition is wearing for those used to modern symbolism. Unlike the works of Archimedes, the treatise does not immediately impress the reader with its mathematical brilliance. Apollonius has, in a way, suffered from his own success: his treatise became canonical and eliminated its predecessors, so that we cannot judge by direct comparison its superiority to them in mathematical rigor, consistency, and generality. But the work amply repays closer study; and the attention paid to it by some of the most eminent mathematicians of the seventeenth century (one need mention only Fermat, Newton, and Halley) reinforces the verdict of Apollonius’ contemporaries, who, according to Geminus, in admiration for his Conics gave him the title of The Great Geometer.²⁸

In Book VII of his mathematical thesaurus, Pappus includes summaries of and lemmas to six other works of Apollonius besides the Conics. Pappus’ account is sufficiently detailed to permit tentative reconstructions of these works, all but one of which are entirely lost. All belong to “higher geometry,” and all consisted of exhaustive discussion of the particular cases of one or a few general problems. The contrast with Apollonius’ approach in the Conics, where he strives for generality of treatment, is notable. A brief indication of the problem(s) discussed in these works follows.

(1) Cutting off of a Ratio (λóγον ʾαπομη), in two books, is the only surviving work of Apollonius apart from the Conics. However, it is preserved only in an Arabic version which, by comparison with Pappus’ summary, appears to be an adaptation rather than a literal translation. Pappus describes the general problem as follows; “To draw through a given point a straight line to cut off from two given straight lines two sections measured from given points on the two given lines so that the two sections cut off have a given ratio,”²⁹ Apollonius discusses particular cases before proceeding to the more general (e.g., in every case discussed in Book I the two given lines are supposed to be parallel) and solves every case by the classical method of “analysis” (in the Greek sense). That is, the problem is presumed solved, and from the solution is deduced some other condition that is easily constructible. Then, by “Synthesis” from this latter construction, the original condition is constructed. We may presume that Apollonius followed the same method in all six of these works, especially since Book VII of Pappus was named ʾαναλλóμєνος(“Field of Analysis”). In the Cutting off of a Ratio the problem was reduced to one of “application of an area.” Zeuthen³⁰ points out the relevance of this work to Conics III, 41: If one regards the theorem proved there as a method of drawing a tangent to a given point in a parabola by determining the intercepts it makes on two other tangents to the curve, that is exactly the problem discussed by Apollonius in this work. Although there is no mention in it of conic sections, the connection is surely not a fortuitous one. In fact, many of the problems discussed by Apollonius in the six works summarized in Book VII of Pappus can be reduced to problems connected with conics. (This helps to explain the great interest shown in this part of Pappus’ work by mathematicians of the sixteenth and seventeenth centuries.)

(2) Cutting off of an Area (ϰψρóον ʾαποτομη), in two books, has a general problem similar to that of the preceding work. But in this case the intercepts cut off from the two given lines must have a given product (in Greek terms, contain a given rectangle) instead of a given proportion.³¹ Here again Zeuthen has shown that Conics III, 42 and 43, which concern tangents drawn to ellipse and hyperbola, are equivalent to particular cases of the problem discussed by Apollonius in this work.³²

(3) Determinate Section (διωρισμένη ιομη) deals with with following general problem: Given four points— A, B, C, D—on a straight line l, to determine a point P on that line such that the ratio AP • CP / BP • DP has a given value.³³ Since this comparatively simple problem was discussed at some length by Apollonius, Zeuthen conjectured—plausibly—that he was concerned to find the limits of possibility of a solution for the various possible arrangements of the points (e.g., when two coincide).³⁴ We know from Pappus’ account that it dealt, among other things, with maxima and minima. Whether, as Zeuthen claims, the work amounted to “a complete theory of involution” cannot be decided on existing evidence. But it is a fact that the general problem is the same as determining the intersection of the line l and the conic that is the “locus for four lines,” the four lines passing through A, B, C, and D; and Apollonius must have known this. Here again, then, is a connection with the theory of conics.

(4) Tangencies (ʾєπαøαí), in two books, deals with the general problem characterized by Pappus³⁵ as follows: “Given three elements, either points, lines or circles (or a mixture), to draw a circle tangent to each of the three elements (or through them if they are points).” There are ten possible different combinations of elements, and Apollonius dealt with all eight that had not already been treated by Euclid. The particular case of drawing a circle to touch three given circles attracted the interest of Vieta and Newton, among others. Although one of Newton’s solutions³⁶ was obtained by the intersection of two hyperbolaones that can be reconstructed with some probability from Pappus’ accounts, and solutions to other cases can also be represented as problems in conics, Apollonius seems to have used only straight-edge and compass constructions throughout. Zeuthen provides a plausible solution to the three-circle problem reconstructed from Pappus’ lemmas to this work.³⁷

(5) Inclinations (νєύσєις), in two books, is described by Pappus on pages 670–672 of the Hultsch edition. In Greek geometry, a νєύσις problem is one that consists in placing a straight line of given length between two given lines (not necessarily straight) so that it is inclined (νєύєι) toward a given point. Pappus tells us that in this work Apollonius restricted himself to certain “plane” problems, i.e., ones that can be solved with straight–edge and compass alone. The particular problems treated by Apollonius can be reconstructed with some probability from Pappus’ account.

(6)Plane Loci (ιóποι ʾєπíπєδοι), in two books, is described on pages 660–670 of Pappus,. “Plane loci” in Greek terminology are loci that are either straight lines or circles. In this work, Apollonius investigated certain conditions that give rise to such plane loci. From them one can easily derive the equation for straight line and circle in Cartesian coordinates.³⁸

A number of other works by Apollonius in the field of pure mathematics are known to us from remarks by later writers, but detailed information about the contents is available for only one of these: a work described by Pappus in Books II of his Collectio.³⁹ Since the beginning of Pappus’ description is lost, the title of the work is unknown. It expounds a method of expressing very large numbers by what is in effect a place-value system with base 10,000. This way of overcoming the limitations of the Greek alphabetic numeral system, although ingenious, is not surprising, since Archimedes had already done the same thing in his Ψαμμèτης (or “Sand Reckoner”).⁴⁰ Archimedes’ base is 10,000². It is clear that Apollonius’ work was a refinement on the same idea, with detailed rules of the application of the system to practical calculation. Besides this we hear of works on the cylindrical helix (κοϰíας);⁴¹on the ratio between dodecahedron and eicosahedron inscribed in the same sphere;⁴²and a general treatise (καθóλον πραγμτєíα).⁴³It seems probable that the latter dealt with the foundations of geometry, and that to it are to be assigned the several rematks of Apollonius on that subject quoted by Proclus in his commentary on the first book of Euclid (see Friedlein’s Index).

Thus Apollonius’ activity covered all branches of geometry known in his time. He also extended the theory of irrationals developed in Book X of Euclid, for several sources mention a work of his on unordered irrationals (πєρí τών ʾατάκτων ʾαλóγων).⁴⁴ The only information as to the nature of this work comes from Pappus’ commentary on Euclid X, preserved in Arabic translation;⁴⁵ but the exact connotation of “unordered irrationals” remains obscure. Finally, Eutocius, in his commentary on Archimedes’ Measurement of a Circle,⁴⁶ informs us that in a work called ʾωκντóκτον, meaning “rapid hatching” or “quick delivery,” Apollonius calculated limits for π that were closer than Archimedes’ limits of 3–1/7 and 3–10/71. He does not tell us what Apollonius’ limits were; it is possible to derive closer limits merely by extending Archimedes’ method of inscribing and circumscribing regular 96-gons to polygons with an ever greater number of sides (as was frequently done in the sixteenth and seventeenth centuries).⁴⁷Very probably this was Apollonius’ procedure, but that cannot be proved.

In applied mathematics, Apollonius wrote at least one work on optics. The evidence comes from a late Greek mathematical work preserved only fragmentarily in a palimpsest (the “Bobbio Mathematical Fragment”). Unfortunately, the text is only partly legible at the crucial point,⁴⁸but it is clear that Apollonius wrote a work entitled On the Burning Mirror (πєρì τούς πνρ〈є〉íον), in which he showed to what points parallel rays striking a spherical mirror would be reflected. The same passage also appears to say that in another work, entitled To the Writers on Catoptrics(πρòς τούς κατοπτρικού), Apollonius proved that the supposition of older writers that such rays would be reflected to the center of sphericity is wrong. The relevance of his work on conics to the subject of burning mirrors is obvious. We may conjecture with confidence that Apollonius treated of parabolic as well as of spherical burning mirrors. But the whole history of this subject in antiquity is still wrapped in obscurity.

Several sources indicate that Apollonius was noted for his astronomical studies and publications. Ptolemaeus Chennus (see n. 2) made the statement that Apollonius was called Epsilon, because the shape of the Greek letter є is similar to that of the moon, to which Apollonius devoted his most careful study. This fatuous remark incidentally discloses some valuable information. “Hippolytus,” in a list of distances to various celestial bodies according to different authorities, says that Apollonius stated that the distance to the moon from the earth is 5,000,000 stades (roughly 600,000 miles).⁴⁹ But the only specific information about Apollonius’ astronomical studies is given by Ptolemy (fl. a.d. 140) in the Almagest.⁵⁰ While discussing the determination of the “station” of a planet (the point where it begins or ends its apparent retrogradation), he states that Apollonius proved the following theorem. In Figure 12, O is the observer (earth), the center of a circle on the circumference of which moves an epicycle, center C, with (angular) velocity v₁; the planet moves on the circumference of the epicycle about C with velocity v₂, and in the same sense as C moves about O. Then Apollonius’ theorem states that if a line OBAD is drawn from O to cut the circle at B and D, such that

B will be that point on the epicycle at which the planet is stationary. Ptolemy also indicates that Apol-lonius proved it both for the epicycle model and for an equivalent eccenter model (depicted in Figure 13; here the planet P moves on a circle, center M, eccentric to the earth O, such that OM/MP = CD/OC in Figure 12; M moves about O with speed [v₁ + v₂],

P about M with speed v₁). Even this much information is valuable, for it shows that Apollonius had already gone far in the application of geometrical models to explain planetary phenomena, and that he must have been acquainted with the equivalence of epicyclic and eccentric models (demonstrated by Ptolemy in Almagest III, 3); yet he was still operating with a simple epicycle/eccentric for the planets, although this would, for instance, entail that the length of the retrograde arc of a planet is constant, which is notoriously not the case. Neugebauer (see Bibliography) supposes, however, that the whole of the passage in which Ptolemy himself proves the above theorem is taken from Apollonius. That proof combines the two models of epicycle and eccenter in one by the ingenious device of using the same circle as both epicycle and eccenter; in other words, the epicycle model is transformed into the eccentric model by inversion on a circle. The procedure is worthy of Apollonius, and is indeed a particular case of the polepolar relationship treated in Conics III, 37. But Ptolemy (who of all ancient authors is most inclined to give credit where it is due) seems to introduce this device as his own,⁵¹ and to return to Apollonius only later.⁵² Fortunately, this uncertainty does not affect the main point: that Apollonius represents an important stage in the history of the adaptation of geometrical models to planetary theories. His real importance may have been much greater than we can ever know, since not only his astronomical works, but also those of his successor in the field, Hipparchus (fl. 130 b.c.), are lost.

It is not clear how far Appollonius applied his theoretical astronomical models to practical prediction (i.e., assigned sizes to the geometrical quantities and velocities). For the fact that he “calculated” the absolute distance of the moon need imply no more than imitation of the crude methods of Aristarchus of Samos (early third century b.c.); for “Hippolytus” also lists figures for distances in stades between the spheres of the heavenly bodies as given by Archimedes which cannot be reconciled with any rational astronomical

system.⁵³ We should not assume without evidence that Apollonius had any better basis for his lunar distance. There is, however, a passage in the astrologer Vettius Valens(fl. a.d. 160) that has been taken to show that Apollonius actually constructed solar and lunar tables.⁵⁴ The author says that he has used the tables of Hipparchus for the sun; of Sudines, Kidenas, and Apollonius for the moon; and also of Apollonius for both. But there is no certainty that “Apollonius” here refers to Apollonius of Perga. At least as likely is the suggestion of Kroll that it may be Apollonius of Mynda, who is known to us only from a passage of Seneca, from which it appears that he claimed to have studied with the “Chaldaeans” and that he was “very experienced in the examination of horoscopes.”⁵⁵ The Apollonius of the Vettius Valens passage is also associated with Babylonian names and practices.

Although the mathematical stature of Apollonius was recognized in antiquity, he had no worthy successor in pure mathematics. The first four books of his Conics became the standard treatise on the subject, and were duly provided with elementary commentaries and annotations by succeeding generations. We hear of such commentaries by Serenus (fourth century a.d.?) and Hypatia (d. a.d. 415). The commentary of Eutocius (early sixth century a.d.) survives, but it is entirely superficial. Of surviving writers, the only one with the mathematical ability to comprehend Apollonius’ results well enough to extend them significantly is Pappus (fl. a.d. 320), to whom we owe what knowledge we have of the range of Apollonius’ activity in this branch of mathematics. The general decline of interest in the subject in Byzantium is reflected in the fact that of all Apollonius’ works only Conics I-IV continued to be copied (because they were used as a textbook). A good deal more of his work passed into Islamic mathematics in Arabic translation, and resulted in several competent treatises on conics written in Arabic; but so far as is known, no major advances were made. (Ibn alHaytham discusses the focus of the parabola in his work on parabolic burning mirrors;⁵⁶ but this, too, may be ultimately dependent on Greek sources.) The first real impulse toward advances in mathematics given by study of the works of Apollonius occurred in Europe in the sixteenth and early seventeenth centuries. The Conics were important. but at least as fruitful were Pappus’ reports on the lost works, available in the excellent Latin translation by Commandino, published in 1588. (We must remember in this context that Books V-VII of the Conics were not generally available in Europe until 1661, ⁵⁷ too late to make a real impact on the subject.) The number of “restorations” of the lost works of Apollonius made in the late sixteenth and early seventeenth centuries, some by outstanding mathematicians(e.g., Vieta, whose Apollonius Gallus[1600] is a reconstruction of the Tangencies, and Fermat, who reconstructed the Plane Loci) attests to the lively interest that Pappus’ account excited. It is hard to overestimate the effect of Apollonius on the brilliant French mathematicians of the seventeenth century, Descartes, Mersenne, Fermat, and even Desargues and Pascal, despite their very different approach. Newton’s notorious predilection for the study of conics, using Apollonian methods, was not a chance personal taste. But after him the analytic methods invented by Descartes brought about a lack of interest in Apollonius which was general among creative mathematicians for most of the eighteenth century. It was not until Poncelet’s work in the early nineteenth century, picking up that of Desargues, Pascal, and la Hire, revived the study of projective geometry that the relevance of much of Apollonius’ work to some basic modern theory was realized. It is no accident that the most illuminating accounts of Apollonius’ geometrical work have been written by mathematicians who were themselves leading exponents of the revived “synthetic” geometry, Chasles and Zeuthen.

The contribution of Apollonius to the development of astronomy, although far less obvious to us now, may have been equally important but, unlike his geometrical work, it had an immediate effect on the progress of the subject. Hipparchus and Ptolemy absorbed his work and improved on it. The result, the Ptolemaic system, is one of the most impressive monuments of ancient science (and certainly the longest-lived), and Apollonius’ work contributed some of its essential parts.

NOTES

1. Eutocius, Commentary, Heiberg, II, 168, quoting one Heraclius.

2. Photius, Bibliotheca, p. 151b18 Bekker, quoting th dubious authority Ptolemaeus Chennus of the second century a.d.

3. Pappus, Collectio VII, Hultsch, p. 678

4. Heiberg, I, 192.

5.Ibid., II, 2.

6.Ibid., I, 192.

7.Ibid., Preface to Bk. IV, II, 2, 4.

8. Pappus, Collectio VII, Hultsch, p.990 ff.

9. Heiberg, 1, 2.

10.De quadratura parabolae III, Heiberg, II², 268; cf. De conoidibus et sphaeroidihus III, Heiherg, 1², 270.

11. E.g., De conoidibus et sphcteroidibus III, Heiherg, 1². 272.

12. for the ellipse, see. e.g. De conoidibus et sphaeroidihus VIII, Heiberg, I², 294, 22–26; for the hyperbola, ibid, XXV, Heiberg, I², 376, 19–23.

13. Heiberg, I, 2, 4.

14. In modern terms, the locus for four lines is the locus of a point whose distances x, y, z, u from four given straight lines, measured along a given axis, satisfy the equation xz/yu = constant. This locus is a conic. (The locus for three lines is just a particular case of the above: for the distances x, y, z from three lines, xz/y² = constant.) This is, in modern terms, an anharmonic ratio: it can be shown that the theorem that this locus is a conic is equivalent to some basic theorems of projective geometry. (See Michel Chalsles, Aperçu historique, pp. 58, 354 ff.)

15.Collectio VII, Hultsch, p. 672.

16. “Apollonius-Studien,” p. 219.

17. Heiberg, II, 2.

18.Ibid., I, 6.

19.Phaenomena, ed. H. Menge (Euclidis Opera Omnia VIII) (Leipzig, 1916), p. 6.

20. Heiberg, II², 266–268.

21.Ibid., I², 270

22. Dijksterhuis, Archimedes, pp. 66, n.1, 106.

23.De corporibus fluitantibus II, 4, Heiberg, II², 357.

24. Chester Beatty MS. Ar.5255, f.4v.

25.Collectio VII, Hultsch, pp. 1006–1014.

26. See, e.g., Zeuthen, Kegelschnitte, p. 367 ff.

27. For a method of proving the focal property of the parabola exactly parallel to Apollonius’ procedure for those of hyperbola and ellipse, see Neugebauer, “Apollonius-Studien,” pp. 241–242.

28. Eutocius, Commentary, Heiberg, II, 170.

29. Hultsch, p.640.

30.Kegelschnitte, p. 345.

31. See Pappus, ed. Hultsch, pp. 640–642.

32.Kegelschnitte, p. 345 ff.

33. Pappus, ed. Hultsch, pp. 642–644.

34.Kegelschnitte, p. 196 ff.

35. Hultsch, p. 644.

36. Principia, Bk., I, Lemma XVI (Motte-Cajori trans., pp. 72–73).

37.Kegelschnitte, p. 381 ff.

38. See T.L. Heath, A History of Greek Mathematics, II, 187–189.

39. Hultsch, p. 2ff.

40. Heiberg, II², 216ff.

41. Proclus, Commentary on Euclid, ed. Friedlein, p. 105.

42. “Euclid,” Bk, XIV, ed, Heiberg, V, 2:the problem is solved by the author of this part of the Elements, a man named Hypsicles(fl.ca. 150 b.c.), but we cannot tell exactly how much he owes to Apollonius.

43. See the commentary on Euclid’s Data by Marinus (fifth century a.d.), in Euclidis opera, ed. Heiberv-Menge, Vi, 234.

44. Proclus, op. cit., p. 74

45. Ed. Junge-Thomson, p. 219.

46.Archimedis opera, ed. Heiberg, III², 258.

47. See E.W. Hobson, Squaring the Circle (CAmbridge, 1913), pp. 26–28.

48.Mathematici Graec Minores, ed. Heiberg, p. 88.

49.Refutation of all Heresies IV, 8, ed, Wendland, III, 41.

50. XII, l, ed, Heiberg, II, 450 ff.

51.Ibid., pp. 451,22.

52.Ibid., pp. 456,9.

53.Refutation, ed, Wendland, pp. 41–42.

54.Anthologiae XI II, ed. Kroll, 354.

55.Questiones naturales VII, 4, I, ed. Oltramarde, II, 304.

56. Ed. Heiberg-Wiedemann, in Bibliotheca mathematica, 10 (1910), 201–237.

57. 1661 is the date of the publication at Florence of Abraham Ecchellensis’ unsatisfactory version. Some knowledge of it had trickled out before, for Mersenne mentions some of the propositions in a book published in 1644 (see Introduction, xlvi, of ver Eecke’s translation of the Conics).

BIBLIOGRAPHY

Ancient sources for Apollonius’ life include the Prefaces to Books I, II, IV, V, VI, and VII of the Conics (in editions of Heiberg and Halley); Eutocius, Commentary on Apollonius I (in Heiberg, II, 168, 170); Pappus, Collectio VII (Hultsch, p. 678); Photius, Bibliotheca, ed. Bekker (Berlin, 1824–1825), p. 151b18. The fragmentary papyrus containing the life of Philonides is edited by Wilhelm Crönert in “Der Epikureer Philonides,” in Sitzungsberichte der Königlich Preussichen Akademie der Wissenschaften zu Berlin, Jahrgang 1900.2, pp. 942–959. Crönert there points out the importance of this text for dating Apollonius. See further R. Phillippson, article “Philonides 5,” in Real-Encyclopädie, XX. I (Stuttgart, 1941), cols. 63ff. A convenient summary of the evidence is given by George Huxley in “Friends and Contemporaries of Apollonius of Perge,” in Greek, Roman and Byzantine Studies, 4 (1963), 100–103.

A critical text of books I-IV of the Conics (with Latin translation) and Eutocius’ commentary was published by J. L. Heiberg, Apollonii Pergaei quae Graece exstant cum commentar’s antiquis, 2 vols. (Leipzig, 1891–1893), Of the Arabic version, only part of Book V has been published, with German translation, by L. Nix, Das Fünfte Buch der Conica des Apollonius von Perga in der Arabischen Uebersetzung des Thabit ibn Corrah (Leipzig, 1889). For the rest of Books V-VII the basis is still Edmund Halley’s Latin translation from the Arabic in the first edition of the Greek text (Oxford, 1710). The most influential translation was Commandino’s Latin version of the first four books (Bologna. 1566). For other editions and early versions and a history of the text, see Heiberg, II, 1vii ff. The best modern translation is the French version of all seven books (from the Greek for I-IV and from Halley’s Latin for V-VII) by Paul ver Eecke, Les coniques d’Apollonius de Perge (Bruges, 1923; reprinted Paris, 1963); the introduction gives a good survey of the work of Apollonius. T. L. Heath’s Apollonius of Perga (Cambridge, 1896; reprinted 1961) is a free adaptation of the Conics rather than a translation. The fundamental modern work on Apollonius(and the ancient theory of conics in general) is H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886; reprinted Hildesheim, 1966), originally published in Danish. It is indispensable for anyone who wishes to make a serious effort to understand the methods underlying the Conics. The Introduction of Heath’s Apollonius is valuable for those who cannot read Zeuthen. A useful summary of the contents of the Conics is provided by T. L. Heath, A History of Greek Mathematics (Oxford, 1921), II, 126–175. O. Neugebauer’s “Apollonius-Studien,” in Quellen und Studien zur Geschichte der Mathematik, Abteilung B: Studien Band 2 (1933), pp. 215–253, a subtle analysis of some parts of the Conics attempts to trace certain “algebraic” procedures of Apollonius.

On the theory of conic sections before Apollonius, Zeuthen is again the best guide. On Archimedes in particular, J.L. Heiberg, “Die Kenntnisse des Archimedes über die Kegelschnitte,” in Zeitschrift für Mathematik und Physik, 25 (1880), Hist,-lit. Abt., 41–67, is a careful collection of the relevant passages. In English, an account of pre-Apollonian conic theory is provided by Heath, A History of Greek Mathematics, II, 110–126; and E. J. Dijksterhuis, Archimedes (Copenhagen, 1956), ch. 3, gives an illuminating comparison between the Apollonian and Archimedean approaches. Another relevant work is Diocles’ “On Burning Mirrors,” which is extant only in Arabic translation. The sole known manuscript is Chester Beauty Arabic no.5255, ff. 1–26, in the Chester Beauty Library, Dublin. An edition is being prepared by G. J. Toomer.

The Arabic text of Cutting off of a Ratio has never been printed. Halley printed a Latin version, together with a restoration of Cutting off of an Area, in Apollonii Pergaei De sectione rationis libri duo (Oxford, 1706); see also W. A. Diesterweg, Die Bücher des Apollonius von Perga De Sectione RAtionis (Berlin, 1824), adapted from Halley’s Latin.

Ancient texts giving information on lost mathematical works of Apollonius are the commentary of Proclus (fifth century A. D.) on Euclid Book I, edited by G. Friedlein, Procli Diadochi in primum Euclidis Elementorum librum (Leipzig, 1873); and The Commentary of Pappus on Book X of Euclid’s Elements, ed. G. Junge and W. Thomson (Cambridge, Mass., 1930); but the most important is in Book VII of Pappus’ Collection, ed. Fr. Hultsch, Pappi Alexandrini Collections Quae supersunt, 3 vols. (Berlin, 1876–1878). There is a good French translation of this work by P. ver Eecke, 2 vols. (Paris-Bruges, 1933). In modern times many attempts have been made at restoration of lost works of Apollonius on the basis of Pappus’ account. Here we mention only the following: for the Determinate Section, Willebrordus Snellius, Apollonius Batavus (Leyden, 1608), and Robert Simson, in Opera quaedam reliqua (Glasgow, 1776); for the Tangencies—apart from Vieta’s Apollonius Gallus (Paris, 1600)—J. Lawson, The Two Books of Apollonius Pergaeus Concerning Tangencies (Cambridge, 1764); for the Inclinations. Samuel Horsley, Apolllonii Pergaei inclinationum libri duo (Oxford, 1770); for the Plane Loci, Pierre de Fermat, Oeuvres, P. Tannery and C. H. Henry, eds., I (Paris, 1891), 3–51, and Robert Simson, Apollonii Pergaei locorum planorum libri II restituti (Glasgow, 1749).

For other restorations of all the above see the Introduction to ver Eecke’s translation of the Conics, pp. xxii-xxxiv. A good account of the probable contents of all six works is given by Heath, A History of Greek Mathematics, II, 175 ff. This is heavily dependent on Zeuthen’s Kegelschnitte; the Index to the 1966 reprint of the latter is the most convenient guide to Zeuthen’s scattered treatment of these lost works. F. Woepcke, “Essai d’une restitution de travaux perdus d’Apollonius sur les quantités irrationelles,” in Mémoires présentées à l“Académie des Sciences, 14 (Paris, 1856), 658–720, is devoted to the work on unordered irrationals; see also T. L. Heath, The Thirteen Books of Euclid’s Elements Translated, III (2nd ed., Cambridge, 1925), 255–259.

Ancient texts relevant to Apollonius’ astronomical works are “Hippolytus,” Refutationonmium haeresium, ed. P. Wendland, Hippolytus Werke III (Leipzig, 1916), IV 8–10; Vettius Valens, Anthologiarum libri, ed. W. Kroll (Berlin, 1908), IX, II; Seneca, Quaestiones naturales, ed. P. Oltramare, 2 vols. (Paris, 1961), VII, 4, 1; and especially Ptolemy, Almagest XII, i, ed. J.L. Heiberg, in Claudii Ptolemaei syntaxis mathematica, 2 vols. (Leipzig, 1898–1903).

For Apollonius’ astronomical work, see O. Neugebauer, “Apollonius’ Planetary Theory,” in Communications on Pure and Applied Mathematics,8 (1955), 641–648, and “The equivalence of Eccentric and Epicyclic Motion According to Apollonius,” in Scripta mathematica, 24 (1959), 5–21.

No detailed account of the influence of Apollonius on later mathematics exists. Much interesting information can be found in ver Eecke’s introduction to his translation. The best guide is Michel Chasles, AperCu historique sur l’origine et le développement des methodes en géométrie (Paris, 1837; reprinted 1875), a work which is also remarkable for its treatment of Apollonius in the light of nineteenth-century synthetic geometry.

G. J. Toomer

APOLLONIUS OF PERGA

(b. Perga, Asia Minor, second half of third century BCE.; d. early second century BCE)

mathematical sciences, geometry. For the original article on Apollonius see DSB, vol. 1.

The main advances in Apollonian scholarship since the original article come from the edition of texts extant only in Arabic translation. This article describes the recent researches, in the perspective of a new edition of the Greek text on the transmission of the Conics and a short account of Apollonius’s foundational endeavors.

A mistaken claim contained in the original article on Apollonius, namely, that in the pre-Apollonian approach to conic sections only the parabola could be represented by the method of “application of areas” (p. 184), was corrected by Gerald Toomer in his edition of Diocles’s On Burning Mirrors (1976).

Works and Fragments Extant in Arabic Translation. The masterly edition of the Arabic translation of books V–VII of the Conics provides scholars with an invaluable working tool. However, no relevant changes in the contents emerge when compared with Edmund Halley’s Latin translation.

Fragments recovered from Arabic sources, most notably Ibrâhîm ibn Sinân’s Selected Problems (SP) and al–Sijzî’s Geometrical Annotations (GA), can be related to proofs contained in Apollonius’ minor works, most of which were apparently translated into Arabic:

• Apollonius’s method of solution for two of the four problems treated in the Inclinations can be recovered from extensive fragments in GA.
• A proposition in SP solves a case of a particular locus problem. The proof is a synthesis and is similar to that of the lemma to Plane Loci II.1 given in Pappus’s Collectio VII.187. Therefore, the latter appears to rework original Apollonian material.
• An analysis (SP) and fragments related to the synthesis (GA) are reported to pertain to the locus problem known as the Circles of Apollonius (Plane Loci II.1). These texts confirm that the proof of the same proposition given by Eutocius in his commentary to the Conics was completely rewritten by him.
• Two lemmas to the Tangencies (GA), in form of an analysis followed by a very sketchy synthesis, coincide with lemmas to the same work in Pappus’s Collectio VII.167–174. The latter’s proofs are slightly different, more detailed, and contain also a complete synthesis. Al-Sijzî found the lemmas in his text of the Tangencies, but it is not said that they were contained in the original as well.

The Textual Tradition of the Conics. The Greek text of the Conics is available in the commented edition that the Neoplatonic scholar Eutocius prepared in the sixth century CE. Recent studies have thoroughly investigated aims and methods of this edition, trying to sift out possible additions to the original Apollonian text. Eutocius declares that he had found several divergent editions of the treatise; he chose the proofs to be retained in the main text following a criterion of mathematical clarity, and transcribed the others in his commentary. Eutocius’s criterion inclined him to give prominence in the constitution of his text to reworked propositions, coming from the preceding scholarly tradition. Secondary Arabic sources can be brought to bear on the subject, as well as the series of lemmas to the Conics that the fourth-century geometer Pappus proves in Collectio VII.233–311: It appears that the text of the Conics underwent conspicuous modifications both before and after Eutocius’s edition. Such modifications and those introduced by Eutocius’s editing, as can be reconstructed by a careful assessment of the available sources, include: the addition of some problems to book II; the addition of missing cases in single propositions; the rearrangement of the order of the propositions of the initial segments of books I and II; the partition of a single proposition into consecutive theorems or, inversely, fusion of several results into one single proposition, possibly with suppression of some of them; and the redrawing of some diagrams. A rather puzzling point is that some of the lemmas presented by Pappus do not find any application in the received text of the Conics.

Apollonius’s Foundational Endeavors. These endeavors are mainly attested by the fourth-century Neoplatonist Proclus, who ascribes to Apollonius a few arguments aiming at explaining or modifying notions and proofs in the Elements. They include:

• A clarification of the conception of line; an anonymous clarification of the conception of surface, very similar in structure to the clarification of the conception of line, also should be ascribed to Apollonius.
• A general definition of angle as “contraction of a surface or of a solid into one single point under an inflected line or surface.”
• A proof of Common Notion 1, resorting to the transitivity of the relation “occupying the same place as.”
• Alternative proofs of Elements I.10 (to find the midpoint of a segment), and of I.11, (to draw the perpendicular to a straight from a point on it). These proofs avoid resorting to I.1, as is done instead in the Elements. The alternative proof of Elements I.23 (to construct on a given straight an angle equal to a given one) applies III.27 and avoids resorting to I.22.

Two more proofs reported by Proclus are very likely Apollonian. They are an anonymous alternative proof of I.2 (to place a given segment at a given point as its extreme), that does not use I.1 and entails a less restrictive interpretation of Elements I.post.3; and the alternative proof to Elements I.5 (the angles at the basis of an isosceles triangle are equal) ascribed to Pappus. The latter proof cleverly employs superposition of the isosceles triangle with its mirror image; in his Prolegomena to Euclid’s Data, Marinus of Neapolis relates that in the General Treatise Apollonius introduced the term tetagmenon as a substitute for dedomenon.

Such interventions are guided by a concern with deductive economy, and aim at minimizing the number of notions and basic propositions employed. The rationale behind the alternative constructions appears to have been to avoid using Elements I.1 or, more generally, constructions of triangles; the proof of I.5 reduces to zero the number of auxiliary geometrical objects introduced. The proof of Common Notion 1 can be read as an attempt at coordinating the Elements and the Data, since in the latter the notion of “occupying a place” is the one to which the notion of “given in position” is reduced. Assuming the same notion as a basic one entails such a wider interpretation of Elements I.post.3 as permits to drastically simplify the proof of I.2, thereby greatly reducing the interest in a construction like I.1. The related notion of “superposition” is pivotal in the study of homeomeric lines, whose thorough study is very likely among Apollonius’s achievements. This was the only class of lines in Greek mathematics directly defined as a whole in terms of a characteristic property.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY APOLLONIUS OF PERGA

Conics, Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banû Mûsâ, 2 vols. Edited, translated, and with commentary by Gerald J. Toomer. Berlin: Springer-Verlag, 1990. The first critical edition of the books of the Conics which are extant only in Arabic.

Hogendijk, Jan P. “Arabic Traces of Lost Works of Apollonius.” Archive for History of Exact Sciences 35 (1986): 187–253. This fundamental contribution to the reconstruction of the lost analytical works of Apollonius contains a commented edition of the fragments extant in Arabic sources.

OTHER SOURCES

Decorps-Foulquier, Micheline. “Eutocius d’Ascalon éditeur du traité des Coniques d’Apollonios de Pergé et l’exigence de ‘clarté’: un exemple des pratiques exégétiques et critiques des héritiers de la science alexandrine.” In Sciences exactes et sciences appliquées à Alexandrie, edited by Gilbert Argoud and Jean-Yves Guillaumin. Saint-Étienne, France: Publications de l’Université de Saint-Étienne, 1998. The article discusses aims and methods of Eutocius’s commentary to the Conics.

———. “Sur les figures du traité des Coniques d’Apollonios de Pergé édité par Eutocius d’Ascalon.” Revue d’histoire des mathématiques 5 (1999): 61–82.

———. Recherches sur les Coniques d’Apollonios de Pergé et leurs commentateurs grecs. Paris: Klincksieck, 2000. This fundamental book is a detailed investigation of the modifications the Apollonian text has undergone during its transmission through late Antiquity and the Middle Ages.

———. “La tradition manuscrite du texte grec des Coniques d’Apollonios de Pergé (livres I–IV).” Revue d’Histoire des Textes 31 (2001): 61–116. The article reconsiders the whole manuscript tradition of the extant Greek text of the Conics. It confirms that the manuscript Vaticanus graecus 206 is the sole independent witness of the text. However, the relationships between its descendants are established on firmer basis than was made by the first editor, Johan L. Heiberg.

Diocles. Diocles On Burning Mirrors: The Arabic Translation of the Lost Greek Original. Edited, translated, and with commentary by G. J. Toomer. Berlin: Springer-Verlag, 1976. On pp. 3–17, a fuller treatment can be found of the pre-Apollonian theory of conic sections than the one offered on pp. 180–182 of the original article.

Euclide. Les Éléments: Traduction et commentaires par BernardVitrac, 4 vols. Vol. 3: Livre X. Paris: Presses Universitaires de France, 1998. On pp. 399–411, a compact reconstruction is presented of Apollonius’ work on irrational lines.

Federspiel, Michel. “Notes critiques sur le Livre I des Coniques d’Apollonius de Pergè.” Revue des Études Grecques 107 (1994): 203–218. This and the following articles by the same author throw considerable light on the peculiar mathematical language employed by Apollonius and, more generally, in Greek mathematics.

———. “Notes linguistiques et critiques sur le Livre II des Coniques d’Apollonius de Pergè (Première partie).” Revue des Études Grecques 112 (1999): 409–443.

———. “Notes linguistiques et critiques sur le Livre II des Coniques d’Apollonius de Pergè (Deuxième partie).” Revue des Études Grecques 113 (2000): 359–391.

———. “Notes linguistiques et critiques sur le Livre III des Coniques d’Apollonius de Pergè (Première partie).” Revue des Études Grecques 115 (2002): 110–148.

Fried, Michael N., and Sabetai Unguru. Apollonius of Perga'sConica: Text, Context, Subtext. Leiden: Brill, 2001. An overall assessment of the Conics that tries to explain and interpret the text without resorting to modern conceptions and notation. The book offers a remarkable overview of Apollonius’ masterly mathematical insight, pinpoints features of his geometrical approach to conic sections that were completely neglected by previous scholars, and firmly places his work in the Euclidean tradition.

Knorr, Wilbur R. “The Hyperbola-Construction in the Conics, Book II: Ancient Variations on a Theorem of Apollonius.” Centaurus 25 (1982): 253–291. The article shows that a theorem in the extant version of the Conics is a later addition.

———.The Ancient Tradition of Geometric Problems. Cambridge, MA: Birkhäuser, 1985. On pp. 293–338 of this book, a thorough analysis is developed of some among Apollonius’ most remarkable achievements.

Pappus of Alexandria. Book 7 of the Collection, 2 vols. Edited, translated, and with commentary by Alexander Jones. New York: Springer-Verlag, 1986. On pp. 510–546, an account is offered of Apollonius’s lost works. A translation of a part of the Arabic text of Apollonius’s Cutting off of a Ratio is given on pp. 606–619.

Saito, Ken. “Compounded Ratio in Euclid and Apollonius.” Historia Scientiarum 31 (1986): 25–59. A study of a mathematical tool widely employed by Apollonius.

Unguru, S. “A Very Early Acquaintance with Apollonius of Perga’s Treatise on Conic Sections in the Latin West.” Centaurus 20 (1976): 112–128. The author shows that a detailed knowledge of the Conics is exhibited in Witelo’s Perspectiva, surmising that this derives from a very early, otherwise unattested, Latin translation of the Apollonian treatise.

Fabio Acerbi

Apollonius of Perga

The Greek mathematician Apollonius of Perga (active 210 B.C.) was known as the "Great Geometer." He influenced the development of analytic geometry and substantially advanced mechanics, navigation, and astronomy.

Very little is known about the life of Apollonius, the last great mathematician of antiquity. He was born at Perga in Pamphylia, southern Asia Minor, during the reign (247-222 B.C.) of Ptolemy Euergetes, King of Egypt. When he was quite young, Apollonius went to study at the school in Alexandria established by Euclid.

Apollonius's fame in antiquity was based on his work on conics. His treatise on this subject consisted of eight books, of which seven have survived. Like most of the well-known Greek mathematicians, Apollonius was also a talented astronomer.

Apollonius had Euclid's great collection, the Elements, available and was thus able to draw upon the work of all previous major mathematicians. Also, Euclid's own work on conics, now lost, was a basis for Apollonius's further work.

Conics of Apollonius

The Conics was written book by book over a long period of time. The general preface to the work is given in Book I. Apollonius next outlines the contents of the eight books. The first four books are an "elementary introduction," that is, elementary in that they include those properties that are necessary to any further specialization. These books are thus an extension of the earlier conics by other mathematicians such as Euclid. Since most of these results were already well known, one might expect Apollonius's presentation to be more concise and to attempt a greater logic and generality. Beginning with Book V, more advanced topics are taken up. Book V is perhaps the best of the latter four.

Other Works

A number of other works by Apollonius are mentioned by ancient writers, but only one exists in its entirety today. The work, Cutting-off of a Ratio, was found in an Arabic version, and a Latin translation was published in 1706. It is concerned with the general problem: given two lines and a point on each of them, draw a line through a given point cutting off segments on the lines (measured from the fixed points on the lines) which have a given ratio to each other.

Another treatise, Cutting-off of an Area, was concerned with the same problem as the previous treatise except that the segments cut off were to contain a given rectangle or, in modern terms, have a given product.

Of a similar nature was the treatise On Determinate Sections. Here the general problem was: given a line with four points A, B, C, and D on it, determine a fifth point P on the line such that the product of lengths AP and CP is a given constant times the product BP and DP. The determination of point P is equivalent to solving a quadratic equation and is no great challenge. But the treatise apparently included more elaborate considerations.

The treatise On Contacts (or Tangencies) was devoted to the general problem: given three things (points, straight lines, or circles) in position, draw a circle which passes through the points (if any) and is tangent to the lines and circles (if any). For example, if two points and a line are given, then the problem would be to draw a circle through the two points and tangent to the given line. There are ten possibilities; two of them were already in Euclid's Elements. Six cases were treated in Book I of On Contacts, and Book II dealt with the remaining two, including the most difficult case of three circles. To draw a circle tangent to three given circles became known as the Apollonian problem.

Another treatise was On Plane Loci. Restorations of this have been attempted by many geometers. It was presumably concerned with straight lines and circles only and with the problem of showing, given certain conditions on a point, that the point must lie on a straight line or a circle.

A work in applied geometry, On the Burning-mirror, was probably about the properties of a mirror in the shape of a paraboloid of revolution. Even though the property is not mentioned by Apollonius in his treatise, he probably knew that light entering such a mirror parallel to its axis is reflected to a single point, its focal point.

Apollonius was also known as a great astronomer. In the Almagest, the great astronomical work by Ptolemy (2d century A.D.), Apollonius is mentioned as having proved two important theorems. These theorems, dealing with epicycles and eccentric circles, enabled the points on the planetary orbits to be determined where the planets, as seen from the earth, appeared stationary.

Further Reading

The standard English translation of Apollonius's principal work, with modern mathematical notation, is Thomas L. Heath, ed., Apollonius of Perga: Treatise on Conic Sections (1896). Apollonius's work is described and analyzed by Heath in A Manual of Greek Mathematics (1931) and by Bartel L. van der Waerden in Science Awakening (1950; trans. 1954). For Apollonius's place in the development of analytic geometry see Carl B. Boyer, History of Analytic Geometry (1956). □

Apollonius of Perga

Greek Geometer
262 b.c.e.–190 b.c.e.

Apollonius, known as "The Great Geometer" to his admirers was born in 262 b.c.e. in the Greek colonial town of Perga in Anatolia, a part of modern Turkey. Apparently Apollonius's intellectual ability was recognized early, and as a young man he attended the university in Alexandria, Egypt, where many of the great scholars of that time were gathered.

Apollonius's teachers had studied with Euclid (c. 330–c. 260 b.c.e.), who is regarded as the most outstanding mathematician of ancient times. Apollonius quickly gained a reputation for his thorough and creative approach to mathematics and was made a professor at the university.

Apollonius wrote a number of books on mathematics, especially geometry. He gathered, correlated, and summarized the mathematics of his predecessors. More importantly, he extended their work and made many creative and original contributions to the development of mathematics.

His best known work is contained in the eight volumes of Conics. Volumes I–IV survive in the original Greek, and volumes I–VII, like many other Greek intellectual works, survive in medieval Arabic translation. Volume VIII is lost but is known from references made to it by later Greek scholars.

Conics addressed the four types of curves that result when a solid cone is cut into sections by a plane : the circle, the ellipse , the hyperbola , and the parabola . Apollonius discovered and named the latter two curves. In Conics, he gives a thorough treatment of the theory of these curves and related matters, developing a total of 387 propositions.

Apollonius also demonstrated applications of the geometry of conic sections and curves to various mathematical problems. His work led to the separation of geometry into the two divisions of solid geometry and plane geometry .

In addition to Conics, Apollonius wrote at least eleven books, some in more than one volume. Of these, only one—Cutting Off a Ratio (also known as On Proportional Section )—survives in an Arabic translation. The others are known only by mention or discussion by other Greek mathematicians and authors.

In addition to his work in pure mathematics, Apollonius analyzed the properties of light and its reflection by curved mirrors, and he invented a sundial based on a conic section. Of particular importance was his application of geometry to astronomy. His use of elliptical orbits with eccentric and epicylic motion to explain the complex movement of planets, including their retrograde motion, was accepted until the time of Copernicus (1473–1543).

The work of Apollonius had an extensive effect on the subsequent development of mathematics and is still relevant today. Later mathematicians influenced by Apollonius's work include René Descartes (1596–1650) in the development of Cartesian mathematical science; Johannes Kepler (1571–1630) in his proposal of elliptical orbits for the planets; and Isaac Newton (1642–1727), who used conic sections in understanding the force of gravity.

see also Conic Sections; Descartes and His Coordinate System; Euclid and His Contributions; Newton, Sir Isaac.

J. William Moncrief

Bibliography

Mansfield, Jaap. Proegomena Mathematica: From Apollonius of Perga to Late Neoplatonism. Boston: Brill Academic Publishers, Inc., 1998.

Internet Resources

O'Connor, J. J., and E. F. Robertson. "Apollonius of Perga." School of Mathematics and Statistics, University of St Andrews, Scotland. January 1999. <http://wwwhistory.mcs.st-andrews.ac.uk/~history/Mathematicians/Apollonius.html>.

Apollonius of Perga

c. 262-c. 190 b.c.

Greek Mathematician

Though he is known as "The Great Geometer," even that title fails to do justice to Apollonius of Perga and his career. His Conics laid the foundations for Newtonian astronomy, ballistics, rocketry, and space science—all 2,000 or more years in the future when he wrote—with its discussion of conic sections, which describe the shape formed by the path of projectiles. Along the way, Apollonius developed his own counting system for large numbers, and put forth a new mathematical worldview that opened the way for the infinitesimal calculus many centuries later.

Born in the town of Perga in southern Asia Minor (now Turkey), Apollonius later studied Euclidean geometry in Alexandria. He also visited Pergamum and Ephesus, both important cities in Asia Minor. In addition to the Conics, he wrote a number of other works, all of which have been lost, but whose English titles include Quick Delivery, Vergings, Plane Loci, Cutting-Off of a Ratio, and Cutting-Off of an Area. Pappus (fl. c. a.d. 320), the principal source regarding these lost works, also summed up the material contained in them. Other ancient writers referred to lost writings of Pappus, such as his discussion of "burning mirrors" for military purposes, in which he disproved the claim that parallel rays of light could be focused on a spherical mirror.

By far the most influential of Apollonius's works, however, was the Conics, which consisted of eight books with some 400 theorems. In this great treatise, he set forth a new method for subdividing a cone to produce circles, and discussed ellipses, parabolas, and hyperbolas—shapes he was the first to identify and name. In place of the concentric spheres used by Eudoxus (c. 400-c. 350 b.c.), Apollonius presented epicircles, epicycles, and eccentrics, concepts that later influenced Ptolemy's (c. 100-170) cosmology. Even more significant was his departure from the Pythagorean tendency to avoid infinites and infinitesimals: by opening up mathematicians' minds to these extremes, Apollonius helped make possible the development of the infinitesimal calculus two millennia later.

In the first four volumes of his Conics, Apollonius examined notions of geometry passed down by Euclid (c. 325-c. 250 b.c.) and others, and maintained that he had made it possible for the first time to solve Euclidean problems such as finding the locus relative to three or four lines. The second half of the Conics discussed conic sections, and confronted problems such as that of finding a "normal" on a point along a curve.

The Conics also presented what became known as the "problem of Apollonius," which calls for the construction of a circle tangent to three given circles, and discussed a means of finding the point at which a planetary orbit took on an apparently retrograde motion. The most important factor in this monumental work, however, was not any one problem, but Apollonius's overall approach, which opened mathematicians' minds to the idea of deriving conic sections by approaching the cone from a variety of angles. By applying the latus transversum and latus erectum, lines perpendicular and intersecting, Apollonius prefigured the coordinate system later applied in analytic geometry.

Apollonius, whose work influenced mathematicians beginning with Hipparchus (fl. 146-127 b.c.) and Hypatia of Alexandria (c. 370-415), has continued to inspire thinkers throughout the ages. The last book of his Conics was lost, and among those who have attempted to recreate it were al-Haytham (Alhazen; 965-1039), Edmond Halley (1656-1742), and Pierre de Fermat (1601-1665). Even today, mathematicians are still examining the work of Apollonius, and finding in it applications to problems and situations they could scarcely have imagined.

JUDSON KNIGHT