Chapter 15

Apollonius of Perga

From what we have already seen of Greek geometry, we can understand how the study of the conic sections came to seem important. From commentators like Pappus, we know of treatises on the subject by Aristaeus, a contemporary of Euclid who is said to have written a book on Solid Loci, and by Euclid himself. We have also mentioned that Archimedes studied the conic sections. The only extensive treatise devoted just to conic sections that has survived, however, is that of Apollonius. Until recently, there was no adequate and accessible study of the whole treatise in English. The version most accessible was that of Heath, who said in his preface that writing his translation involved “the substitution of a new and uniform notation, the condensation of some propositions, the combination of two or more into one, some slight re-arrangements of order for the purpose of bringing together kindred propositions in cases where their separation was rather a matter of accident than indicative of design, and so on.” He also replaced Apollonius' purely synthetic arguments with analytic arguments, based on the algebraic notation we are familiar with. All this labor has no doubt made Apollonius more readable. On the other hand, Apollonius' work is no longer current research; and from the historian's point of view, this kind of tinkering with the text only makes it harder to place the work in proper perspective. Nevertheless, one can fully understand the decision to use symbolic notation, since the mathematical language in which the original was couched was the cumbersome metric-free “synthetic” approach of Euclid in which the basic tools are lines and circles, and all relations must be reduced to proportions proved using something equivalent to the Eudoxan definition. A 1952 translation by R. Catesby Taliaferro of the first three books was included in the Great Books of the Western World series; it unfortunately went to the other extreme from the Heath translation and preserved the full obscurity of Apollonius' original exposition. A translation of Books 5–7 (Toomer, 1990) at least made that portion of the work available to those like the present author, who could not read the Arabic in which the only extant manuscripts are written. Fortunately, all these gaps have now been filled in a thorough study of the entire work (Fried and Unguru, 2001).

In contrast to his older contemporary Archimedes, Apollonius remains a rather obscure figure. His dates can be determined from the commentary written on the Conics by Eutocius. Eutocius says that Apollonius lived in the time of the king Ptolemy Euergetes and defends him against a charge by Archimedes' biographer Heracleides that Apollonius plagiarized results of Archimedes. Eutocius' information places Apollonius in the second half of the third century BCE, perhaps a generation or so younger than Archimedes.

Pappus says that as a young man Apollonius studied at Alexandria, where he made the acquaintance of a certain Eudemus (probably not the student of Aristotle whose history of mathematics was used by Proclus). It is probably this Eudemus to whom Apollonius addresses himself in the preface to Book 1 of his treatise. From Apollonius' own words we know that he had been in Alexandria and in Perga, which had a library that rivaled the one in Alexandria. Eutocius reports an earlier writer, Geminus by name, as saying that Apollonius was called “the great geometer” by his contemporaries. He was highly esteemed as a mathematician by later mathematicians, as the quotations from his works by Ptolemy and Pappus attest. In Book 12 of the Almagest, Ptolemy attributes to Apollonius a geometric construction for locating the point at which a planet begins to undergo retrograde motion. From these later mathematicians we know the names of several works by Apollonius and have some idea of their contents. However, except for a few fragments that exist in Arabic translation, only two of his works survive to this day, and for them we are indebted to the Islamic mathematicians who continued to work on the problems that Apollonius considered important. Our present knowledge of Apollonius' Cutting Off of a Ratio, which contains geometric problems solvable by the methods of application with defect and excess, is based on an Arabic manuscript, no Greek manuscripts having survived. Of the eight books of Apollonius' Conics, only seven have survived in Arabic and only four in Greek. The astronomer Edmund Halley (1656–1743) published a Latin edition of all seven books in 1710. Halley also produced what Fried and Unguru (2001, p. 295) call “a reasonable and intelligent partial restoration” of Book 8, based on Apollonius' preface to Book 7, which he explains contains certain lemmas needed to prove what is in Book 8. As many people have pointed out, that statement does not necessarily cover the entire contents of Book 8; hence the use of the word partial by Fried and Unguru.

15.1 History of the Conics

The evolution of the Conics was reported by Pappus five centuries after they were written in Book 7 of his Collection.

By supplementing Euclid's four books on the conics and adding four others Apollonius produced eight books on the conics. Aristaeus. . . and all those before Apollonius, called the three conic curves sections of acute-angled, right-angled, and obtuse-angled cones. But since all three curves can be produced by cutting any of these three cones, as Apollonius seems to have objected, [noting] that some others before him had discovered that what was called a section of an acute-angled cone could also be [a section of] a right-or obtuse-angled cone. . .changing the nomenclature, he named the so-called acute section an ellipse, the right section a parabola, and the obtuse section a hyperbola.

In a preface addressed to the aforementioned Eudemus, Apollonius lists the important results of his work: the description of the sections, the properties of the figures relating to their diameters, axes, and asymptotes, things necessary for analyzing problems to see what data permit a solution, and the three-and four-line locus. He continues:

The third book contains many remarkable theorems of use for the construction of solid loci and for distinguishing when problems have a solution, of which the greatest part and the most beautiful are new. And when we had grasped these, we knew that the three-line and four-line locus had not been constructed by Euclid, but only a chance part of it and that not very happily. For it was not possible for this construction to be completed without the additional things found by us.

We have space to discuss only the definition and construction of the conic sections and the four-line locus problem, which Apollonius mentions in the passage just quoted.

15.2 Contents of the Conics

The earlier use of conic sections had been restricted to cutting cones with a plane perpendicular to a generator. As we saw in our earlier discussion, this kind of section is easy to analyze and convenient in the applications for which it was intended. In fact, only one kind of hyperbola, the rectangular, is needed for duplicating the cube and trisecting the angle. The properties of a general section of a general cone were not discussed. Also, it was considered a demerit that the properties of these plane curves had to be derived from three-dimensional figures. Apollonius set out to remove these gaps in the theory.

First it was necessary to define a cone as the figure generated by moving a line around a circle while one of its points, called the apex and lying outside the plane of the circle, remains fixed. Next, it was necessary to classify all the sections of a cone that happen to be circles. Obviously, those sections include all sections by planes parallel to the plane of the generating circle (Book 1, Proposition 4). Surprisingly, there is another class of sections that are also circles, called subcontrary sections. Once the circles are excluded, the remaining sections must be parabolas, hyperbolas, and ellipses.

We shall give some details of Apollonius' construction of the ellipse and then briefly indicate how the same procedure applies to the other conic sections. Consider the section of a cone shown in Fig. 15.1, made by a plane cutting all the generators of the cone on the same side of its apex. This condition is equivalent to saying that the cutting intersects both sides of the axial triangle (see Fig 6 of Chapter 11). Apollonius proved that there is a certain line (EH in the figure), which he called the [up]right side (or perpendicular side or vertical side), now known by its Latin name latus rectum, such that the square on the ordinate from any point of the section to its axis equals the rectangle applied to the latus rectum with width equal to the abscissa and whose defect on the latus rectum is similar to the rectangle formed by the axis and the latus rectum. He gave a rule, too complicated to go into here, for constructing the latus rectum. This line characterized the shape of the curve. Because of its connection with the problem of application with defect, he called the resulting conic section an ellipse. Similar connections with the problems of application and application with excess, respectively, arise in Apollonius' construction of the parabola and hyperbola. These connections motivated the names he gave to these curves.

Figure 15.1 Apollonius' construction of the ellipse with latus rectum p = EH. Given x = EM and y = LM, these two lines are connected by the relation . Since OH is proportional to EM, this says y² = px − kx², where k is the slope of the diagonal of the rectangle whose sides are the axis EF and the latus rectum EH.

In Fig. 15.1, where the latus rectum is the line EH, the locus condition¹ is that the square on the ordinate LM equals the rectangle on EO and EM, that is, . The reason for the term ellipse is that the rectangle applied to the latus rectum with area equal to the square on the ordinate and width equal to the abscissa leaves a defect of prescribed shape (the shape of the rectangle whose sides are the axis and the latus rectum) on the remainder of the latus rectum.

In one sense, this locus definition for an ellipse is not far removed from what we now think of as the equation of the ellipse, but that small gap was unbridgeable in Apollonius' time. We shall digress briefly to “translate” this language to its modern algebraic equivalent, again warning the reader that Apollonius was certainly not thinking of the figure this way. If we write LM = y and EM = x in Fig. 15.1 (so that we are essentially taking rectangular coordinates with origin at E), we see that Apollonius is claiming that y² = x · EO. Now, however, EO = EH − OH, and EH is constant, while OH is directly proportional to EM, that is, to x. Specifically, the ratio of OH to EM is the same as the ratio of the latus rectum EH to the axis EF. It follows that an ellipse is uniquely determined by the latus rectum and its major axis. Thus, if we write OH = kx—a crucial step that Apollonius could not take, since he did not have the concept of a dimensionless constant of proportionality—and denote the latus rectum EH by p, we find that Apollonius' locus condition can be stated as the equation y² = px − kx². Here k is the slope of the dashed line HF. By completing the square on x, transposing terms, and dividing by the constant term, we can bring this equation into what we now call the standard form for an ellipse with center at (a, 0):

where a = p/(2k) and . In this notation, the latus rectum p is 2b²/a. Apollonius, however, did not have the concept of an equation nor the symbolic algebraic notation we now use, and if he had known about these things, he would still have lacked the letter k used above as a constant of proportionality. These “missing” pieces gave his work on conics a ponderous character with which most mathematicians today have little patience. That is why Heath's translation of the Conics looks more like a textbook of analytic geometry than an ancient Greek treatise translated into English.

Apollonius' constructions of the parabola and hyperbola also depend on the latus rectum. A parabola is completely determined by its latus rectum and the locus condition that the square on the ordinate equals the rectangle on the abscissa and the latus rectum (y² = px). In other words, the rectangle on the latus rectum whose width is the abscissa is exactly equal to the square on the ordinate, with no excess or defect. For a hyperbola, y² = px + kx², so that the hyperbola is determined by the latus rectum p and the constant k, which is the negative of the slope of the dashed line in the right-hand drawing in Fig. 15.2. In this case, the rectangle having a side along the latus rectum, width equal to the abscissa, and area equal to the square on the ordinate has length that exceeds the latus rectum, creating an “excess” rectangle whose shape is the same for all points on the hyperbola. The now-standard form for this equation is

and the latus rectum p is once again 2b²/a.

Figure 15.2 Left: The parabola with latus rectum p. The condition for a point to be on the locus is that the square on its ordinate (y) equal the rectangle on its abscissa (x) and the latus rectum (p), that is y² = px. Right: A hyperbola with latus rectum p. Here y² = px + kx², where the oblique dashed line has slope −k.

Apollonius was the first to take account of the fact that a plane whose intersection with a cone is a hyperbola must cut both nappes of the cone. He regarded the two branches as two hyperbolas, referring to them as “opposites” and using the term hyperbola for either branch. For the hyperbola, Apollonius proved the existence of asymptotes—that is, a pair of lines through the center that never meet the hyperbola but such that any line through the center passing into the region containing the hyperbola does meet the hyperbola. The word asymptote means literally not falling together—that is, not intersecting. For the hyperbola shown on the right-hand side of Fig. 15.2, the asymptotes are the two lines .

With these new characterizations of the three conic sections, it becomes possible to discard the cone itself. Once the latus rectum and the shape of the excess or defect (measured in our terms by the constant of proportionality that we denoted by k) are given, the locus condition defining the curve is determined. It makes no reference to anything outside the plane of the curve itself. The original cone is like the scaffolding around a building, which is removed after the construction is complete. With these curves now defined as plane loci, their properties can then be developed using Euclid's plane geometry. Apollonius proceeds to do so.

15.2.1 Properties of the Conic Sections

Books 1 and 2 of the Conics are occupied with finding the proportions among line segments cut off by chords and tangents in conic sections, the analogs of results on circles in Books 3 and 4 of the Elements. These constructions involve finding the tangents to the curves satisfying various supplementary conditions such as being parallel to a given line. Fried and Unguru (2001, Chapter 7) argue that these analogies probably guided Apollonius in his choice of material.

15.3 Foci and the Three-and Four-line Locus

We are nowadays accustomed to constructing the conic sections using the focus–directrix property, so that it comes as a surprise that the original expert on the subject does not seem to recognize the importance of the foci. He never mentions the focus of a parabola, and for the ellipse and hyperbola he refers to these points only as “the points arising out of the application.” The “application” he has in mind is explained in Book 3. Propositions 48 and 52 of Book 3 read as follows:

(Proposition 48) If in an ellipse a rectangle equal to the fourth part of the figure is applied from both sides to the major axis and deficient by a square figure, and from the points resulting from the application straight lines are drawn to the ellipse, the lines will make equal angles with the tangent at that point.

(Proposition 52) If in an ellipse a rectangle equal to the fourth part of the figure is applied from both sides to the major axis and deficient by a square figure, and from the points resulting from the application straight lines are drawn to the ellipse, the two lines will be equal to the axis.

The “figure” referred to is the rectangle whose sides are the major axis of the ellipse and the latus rectum. In Fig. 15.3 the points F₁ and F₂ must be chosen on the major axis AB so that the rectangle on AF₁ and F₁B and the rectangle on AF₂ and BF₂ both equal one-fourth of the area of the rectangle formed by the whole axis AB and the latus rectum p.

Proposition 48 expresses the focal property of these two points: A light ray emanating from one will be reflected to the other. Proposition 52 is the string property that characterizes the ellipse as the locus of points such that the sum of the distances to the foci is constant. These are just two of the theorems Apollonius called “strange and beautiful.” Apollonius makes little use of these properties, however, and does not discuss the use of the string property to draw an ellipse.

A very influential part of the Conics consists of Propositions 54–56 of Book 3, which contain the theorems that Apollonius claimed (in his cover letter) would provide a solution to the three-and four-line locus problems. Both in their own time and because of their subsequent influence during the seventeenth century (when analytic geometry was being created), the three-and four-line locus problems have been of great importance for the development of mathematics. These propositions involve the proportions among pieces of chords inscribed in a conic section. Three propositions are needed because the hyperbola requires two separate statements according as the points involved lie on the same or opposite branches of the hyperbola.

We limit ourselves to stating the four-line locus problem and illustrating it. The data for the problem are four lines, which for definiteness we suppose to intersect two at a time, and four given angles, one corresponding to each line. The problem requires the locus of points P such that if lines are drawn from P to the four lines, each making the corresponding angle with the given line (for simplicity all shown as right angles in Fig. 15.4), the rectangle on two of the lines will have a constant ratio to the rectangle on the other two. The solution is in general a pair of conics.

The origin of this kind of problem may lie in the problem of two mean proportionals, which was solved by drawing fixed reference lines and finding the loci of points satisfying a condition resembling the condition here. In that problem, the square on the line drawn perpendicular to one reference line equals the rectangle on a fixed line and the line drawn to the other reference line. The commentary on this problem by Pappus, who mentioned that Apollonius had left a great deal unfinished in this area, inspired Fermat and Descartes to take up the implied challenge and solve the problem completely. Descartes offered his success in solving the locus problem to any number of lines as proof of the value of his analytic method in geometry.

Figure 15.3 Focal properties of an ellipse.

Figure 15.4 The four-line locus. If a point moves so that the product of its distances to two lines bears a constant ratio to the product of its distances to two other lines, it must move in a conic. In this illustration, two conics satisfy the condition: one an ellipse, the other a hyperbola.

Problems and Questions

Mathematical Problems

15.1 The string property of an ellipse is illustrated in Fig. 15.3. It implies that all broken lines starting at one focus, going to any point on the ellipse, and then going to the other focus have the same total length. Taking for granted that an ellipse is a convex figure, you may assume that the tangent to an ellipse at any point lies entirely outside the ellipse, except for the point of tangency itself. Use this fact to prove the reflection property of the ellipse. You will need to establish that if two points D and E are on the same side of a given line MN, then the shortest path from D to a point Q on the line and thence to E is the one for which the lines DQ and QE make equal angles with the line MN. This theorem is illustrated in Fig. 15.5.

Figure 15.5 Shortest path meeting a line.

15.2 Show from Apollonius' definition of the foci that the product of the distances from each focus to the ends of the major axis of an ellipse equals the square on half of the minor axis.

15.3 We have seen that the three-and four-line locus problems have conic sections as their solutions. State and solve the two-line locus problem. You may use modern analytic geometry and assume that the two lines are the x axis and the line y = ax. The locus is the set of points whose distances to these two lines have a given ratio. What curve is this? (The distance from a point to the line whose equation is ax + by = c is .)

Historical Questions

15.4 How much of the treatise on conics by Apollonius has been preserved, and in what form?

15.5 On what basis can we conjecture what was in the missing Book 8 of the Conics?

15.6 Why did Apollonius rename the conic sections?

Questions for Reflection

15.7 As we have seen, Apollonius was aware of the string property of ellipses, yet he did not mention that this property could be used to draw an ellipse. Do you think that he did not notice this fact, or did he omit to mention it because he considered it unimportant, or for some other reason?

15.8 Is the apparent generality of Apollonius' statement of the three-line locus problem, in which arbitrary angles can be prescribed at which lines are drawn from the locus to the fixed lines, really more general than the particular case in which all the angles are right angles? Observe that the ratio of a line from a point P to line l making a fixed angle θ with the line l bears a constant ratio to the line segment from P perpendicular to l. How would a particular locus problem be altered if recast in terms of the perpendicular distances to the same lines?

15.9 A circle can be regarded as a special case of an ellipse. What is the latus rectum of a circle? (Consider the expression given for the latus rectum in terms of the semi-axes of the ellipse.)

Note

1. The Latin word locus is the equivalent of the Greek word tópos, from which our word topology comes. Both mean place. The Greek mathematicians had to imagine a cone generated by a line with one of its points fixed moving around a circle. A locus was thought of as the path followed by a moving point. Modern mathematics has replaced the concept of a locus by the concept of a set, meaning the points satisfying a certain condition. This concept is a static one, not the kinematic picture imagined by the Greeks. But it is more realistic, since a set may be disconnected and hence difficult to picture as the path of a moving point.