Introduction¹

This paper is based on a talk given at the International Workshop: History of Mathematics in the Last 25 Years: New Departures, New Questions, New Ideas. Tel-Aviv and Jerusalem, May 14-17, 2001. Much of it was adapted also from Fried and Unguru, Apollonius of Perga’s Conica: Text, Context, Subtext; however, it develops and clarifies the analogy-proportion-image relationship in ways not found in that book.

Although this paper is on a second-century BC mathematician, I would have the reader begin by finding and placing before him a copy of Rembrandt’s well known Aristotle Contemplating a Bust of Homer (1653). Any copy will do. The reason for such an odd request is this. Rembrandt portrayed in the dress of his own time²

an ancient subject contemplating his ancient object just as we in the history of mathematics have in past years viewed ancient mathematicians through the eyes of moderns and endowed them likewise with modern dress. His painting, therefore, serves well as an image for the state of history of Greek mathematics as it was until some 25 years ago when Sabetai Unguru wrote his landmark article “On the Need to Rewrite the History of Greek Mathematics” and as a reminder of the difficulties we still face in trying to see the world as ancient thinkers saw it themselves. In the present paper, I shall consider a certain aspect of Apollonius of Perga’s Conica with these difficulties and Unguru’s article always tacitly in the background.

But first, I wish to return for a moment to Aristotle Contemplating a Bust of Homer. Using the painting as an image, I have set up an obvious analogy. The painting, of course, suggests its own analogies, for instance that between the richly dressed philosopher, with his expression of humility and dreamy resignation, standing beside the bust of the unadorned artist, the poet Homer, and philosophy next to poetry, or, even, the wealthy Antonio Ruffo, who apparently commissioned the painting, beside the outwardly poor, but inwardly rich, painter, Rembrandt.³

One need not look far to find such analogies, for images like that provided by Rembrandt invariably embody them. Indeed, it is really analogies that make images images—that ability to say this is to this as that is to that. Now, the word “analogy” derives from a Greek word analogia whose basic meaning is “proportion,” and that, principally, in the mathematical sense. The use of analogia, of proportion, is ever-present in Greek mathematics, and its theory, needless to say, is one of its great achievements. But the use of analogy, in the usual sense of making an image, is also present in Greek mathematical work, including, as I shall show, in the work of Apollonius.

A striking instance from Apollonius can be found in the analogies between propositions concerning conic sections in the Conica and propositions concerning circles such as those in the third book of Euclid’s Elements. Consider, for example, Conica II.26: “If in an ellipse or circumference of a circle two straight lines not through the center cut each other, then they do not bisect each other,” and Elements III.4: “If in a circle two straight lines cut one another which are not through the center, they do not bisect one another”; or Conica IV.25: “A section of a cone does not cut a section of a cone or circumference of a circle at more than four points,” and Elements III.10: “A circle does not cut a circle at more points than two.” These propositions, however, are not those I wish to emphasize in this paper.⁴

For the point of this paper is not to speak about proportion as opposed to analogy or analogy as opposed to proportion, rather, it is to show how image, analogy, and proportion can be united. To this end, I shall concentrate on Book VII of Apollonius’ Conica, the book Apollonius himself described as containing “many wonderful and beautiful things on the topic of diameters and the figures constructed on them.” In particular, I shall look at what Apollonius called the homologue, which he developed and used extensively throughout Book VII. My main point is this: that the homologue was not just an abstract component in an abstract proportion, rather, the homologue was meant to be seen, and the proportion, or analogia, by which it was defined was the vehicle of its becoming a genuine visual image.

The Latus Rectum

To understand what the homologue is, we ought to review what is the orthia pleura or latus rectum (to use the more common Latin translation) of the figure or eidos of a conic section, say, of a hyperbola.

As in proposition I.11 for the case of the parabola and in I.13 for the ellipse, in proposition I.12, the latus rectum of a hyperbola is defined in the course of defining the section itself, and, like the section, it is defined by means of the cone. It is a line ZΛ (see fig. 1) such that the transverse diameter ΘZ has the same ratio to ZΛ that the square on AK, the line from the vertex to the base of the cone drawn parallel to the diameter, has to the rectangle contained by BK and KΓ, the segments AK cuts off from the diameter of the base circle. Its primary use is revealed in the symptoma of the hyperbola, namely, that if MN is an ordinate (i.e., if MN is a line drawn from the section to the diameter parallel to the common section of the cutting plane and the base of the cone) then the square on MN is equal to the parallelogram “…applied to ZΛ, having ZN as breadth, and exceeding by a figure ΛΞ similar to the rectangle contained by ΘZ and ZΛ.”⁵

Very quickly and without preparation the rectangle contained by the transverse diameter ΘZ and the latus rectum ZΛ becomes the “figure (

)” of the hyperbola. Thus, in I.14 Apollonius refers to the transverse side as “…the transverse side of the figure (

) [emphasis added], that between the vertices of the sections.…” As the upright side of the “figure,” the latus rectum has a true concrete geometric identity. Indeed, what sense has its name, “upright side (

),” and what use is its being perpendicular to the diameter, its very “uprightness,” unless it is understood as the side of a rectangle? Given the section, that is, given the cone and given the plane cutting off the particular hyperbola, the sides of this rectangle, the “figure,” are naturally given in ratio. The ratio of the sides of the “figure” comes up one way or another all through the Conica. For example, in I.21 where it is proven that in a hyperbola and an ellipse the square on a segment drawn ordinatewise is to the rectangle contained by the segments cut off from the transverse diameter as the latus rectum is to the transverse diameter (in fig. 2, sq.ZH : rect.AH,BH :: AΓ : AB :: latus rectum:transverse diameter).

A more telling example is V.9 in which we are shown that if EΓ along the axis of a hyperbola is greater than half the latus rectum of the hyperbola (see fig. 3), and HZ:ZE is the same ratio as the ratio of transverse diameter to the latus rectum, where H is the center of the hyperbola, then if ZΘ is perpendicular to EΓ, EΘ will be a minimum line (i.e. no line drawn from E to the section will be shorter than EΘ).

In this example, as in others like it, one might view the line ZE as the counterpart of the latus rectum, where the latter is still understood as the side of the “figure” though the figure itself is “invisible.”⁶

What is the evidence for this? How do we know, in other words, that Apollonius truly intends for us to think of the latus rectum with its fullest geometrical significance when we are faced with a line like ZE in V.9? It is here we must begin and look at the homologue in Book VII, so to this we shall now turn.

The Homologue

The homologue makes its debut in propositions VII.2, 3 (VII.2 in the case of the hyperbola and VII.3 in the case of the ellipse); afterwards it appears over and over again in Book VII.⁷

Proposition VII.2 states that “If the axis in a hyperbola is extended in a straight line so that the part of it falling outside of the section is the transverse diameter, and a line is cut off adjoining one of the ends of the transverse diameter such that the transverse diameter is divided into two parts in the ratio of the transverse diameter to the latus rectum, while the line cut off corresponds to the latus rectum, and a line is drawn from that end of the transverse diameter which is the end of the line which was cut off to the section, in any position,⁸ and, from the place where [that line] meets it [the section], a perpendicular is drawn to the axis, then the ratio of the square on the line drawn from the end of the transverse diameter to the rectangle contained by the two lines between the foot of the perpendicular and the two ends of the line which was cut off is equal to the ratio of the transverse diameter to the excess of it [the transverse diameter] over the line which was cut off.” Thus:

“Let the hyperbola be the section with extended axis AΓE [fig. 4], and let the figure of the section be ΓΔ. Let line AΘ be cut off from line AΓ, and let

We draw from point A to the section an arbitrary line, AB, and draw line BE perpendicular to the axis.

That the proposition itself serves as counterpart to I.21, I shall leave aside,¹¹

for I want to concentrate only on the construction of the homologue. The homologue, of course, is the line AΘ; it is the line constructed by dividing the transverse axis AΓ at a point Θ such that ΓΘ : ΘA :: transverse axis: latus rectum. Clearly, in a given hyperbola (or ellipse) it is a fixed line, since it is produced by dividing another fixed line, the transverse axis, in a fixed ratio. Although this is an obvious point, it should not be overlooked. That the homologue is a fixed line justifies its being given a name. Plainly, a line such as ZE in V.9 does not have a special name because it is not fixed – ZE is determined only to the extent that E is determined, and the only condition on E is that it be taken farther from the vertex than half the latus rectum; other than that, E is arbitrary, and, to that extent, so is ZE.

Since it has a name and, therefore, also a definition, the homologue can now be associated with other line segments in the Conica that also have a name and definition, and which are intimately associated with the equally defined conic sections—lines such as the transverse diameter (diametros plagia), the upright diameter (diametros orthia), and the upright side, and, of course, the latus rectum (pleura orthia). Obviously, ΘA has the particular name, homologue, because in the proportion, ΓΘ :: ΘA :: transverse axis: latus rectum, ΘA is made to correspond to the latus rectum, and is, therefore, in this sense, homologous to the latus rectum.

But is this the only reason why the homologue earns its name? Does it function in a way that is homologous to the latus rectum? Does Apollonius want us to think of the homologue, in other words, as truly homologous to the latus rectum? A comparison between VII.2,3 and VII.1 gives a first hint that the answer is yes. Indeed, the very purpose of VII.1 seems to be that it present an analogous counterpart to VII.2,3, perhaps as a preparation for those propositions.¹²

For VII.1 shows that if AB is a parabola (see fig. 5), AΓ its axis, AΔ its latus rectum, then with AB being any line from the vertex to the section, sq.AB = rect.ΔΓ,AΓ. So, to make the comparison, in VII.1 the ratio of the square on AB to a rectangle whose sides differ by the latus rectum is always the same (namely, the ratio borne by two equal magnitudes), while in VII.2,3 the ratio of the square on AB to a rectangle whose sides differ by the homologue is always the same.

The proof of VII.2, moreover, makes it clear that we are to relate the homologue, ΘA its defining proportion ΓΘ : ΘA :: transverse axis: latus rectum notwithstanding, directly to the latus rectum in its capacity as the upright side of the “figure.” It makes this clear by presenting us with the “figure” ΓΔ explicitly (and, with that, the diagrams accompanying VII.2,3 have drawn in them the whole rectangle whose sides are the transverse axis and latus rectum, that is, the whole “figure”), rather than only with the latus rectum, AΔ.¹³

Indeed, Apollonius does not say initially that AΔ is the latus rectum, but it is clear that AΔ is the latus rectum because we know where it is in the “figure.” With the “figure” before us, the homologue can then be defined in terms of the parts of this “figure”: “ΓΘ : ΘA = ΓA : AΔ (which is the latus rectum),” and, therefore, the homologue is set in direct comparison to the latus rectum as the upright side of the “figure.” One must go further and say that besides making this comparison, there is no other reason Apollonius presents us with the “figure,” for ΓΔ is never used in the proof as a figure. This contrast with VI.12, for example, where the “figure” does play a part as a figure in the demonstration that “Hyperbolas and ellipses in which the figures constructed on their axes are similar are also [themselves] similar; and if the sections are similar, then the figures constructed on their axes are similar.”

From what has been said up until now, one can see that the homologue is analogous to the latus rectum (with respect to the axis) in two senses. First, it is analogous in the strict sense that it corresponds to the latus rectum in a proportion, an analogia. But it is also analogous to the latus rectum in the sense that the homologue is assimilated to the latus rectum, that it is an image of latus rectum. Four facts support this:

1. that in a given hyperbola (or ellipse) it is a fixed line;
2. that it has a name;
3. that it functions in VII.2,3 in a way analogous to the latus rectum of the parabola in VII.1;
4. that it is put in direct comparison with the latus rectum in the latter’s fullest geometrical manifestation, namely, as the upright side of the “figure.”

Obviously, none of these facts has anything to do with the logical demands of the book; the proof of VII.2,3 is not strengthened or weakened by Apollonius’ calling AΘ the homologue. These other aspects of the homologue give us, rather, a way of thinking of the line, AΘ, of relating it to the geometrical significance of the latus rectum.

Images and Analogies

Naturally, just as the homologue corresponds to the latus rectum with respect to the axis, the remaining part of the axis corresponds to the axis itself. How the homologue induces other correspondences with other lines, or, we might say, creates new images of other lines, can be seen in a less trivial way in VII.6,7, which brings us also to one of the main concerns of the book, the conjugate diameters.

Proposition VII.6 states: “If there are constructed on the extension of the axis of a hyperbola two lines adjacent to the two ends of the axis which is the transverse diameter, each of them being equal to the line which we called ‘homologue,’ and placed as it [the homologue] is placed, and two conjugate diameters from among the diameters of the section are drawn, and, from the vertex of the section, a line is drawn parallel to the ‘erect’ diameter of the two to cut the section, and from the place where it intersects it a perpendicular is drawn to the axis: then the ratio of the transverse of the two conjugate diameters to the erect one is equal in square to the ratio of the line between the foot of the perpendicular and the end of the more remote of the two homologues to the line between the foot of the perpendicular and the end of the nearer of the two homologues; and the ratio of the transverse diameter to the parameter of the lines drawn to it [the transverse diameter] parallel to the second diameter, which [parameter] is its latus rectum, is, in length, equal to the ratio of the two lines which we mentioned previously to each other in length.” So, let the transverse axis of the hyperbola be AΓ with center Θ (see fig. 6), AN, ΓΞ be the homologues (i.e., ΓN : NA :: transverse axis : latus rectum with respect to the axis :: AΞ : ΞΓ), and ZH and BK be conjugate diameters (the latter being the transverse diameter and the former the “erect” diameter). Then, if line AΛ is drawn parallel to line ZH,¹⁴

and ΛM is drawn perpendicular to the axis, Apollonius proves that sq.BK : sq.ZH :: ΞM : MN and, from that, that BK : lr(BK) is also the same ratio as ΞM : MN.¹⁵

Proposition VII.7 is the corresponding proposition for the ellipse, and the construction is identical except that the points Ξ and N, there, lie on the extension of the axis (see fig. 7).¹⁶

For the present discussion, what is important to stress about the proofs of VII.6,7 is that while Apollonius provides a detailed argument for the first claim, that sq.BK : sq.ZH is the same ratio as ΞM : MN, he merely states the second, that BK : lr(BK) is also the same ratio as ΞM : MN, with hardly any argument at all. The second claim, in fact, requires little more than stating since the proportion BK : ZH :: ZH : lr(BK) is provided by the very definition of the second or conjugate diameter. The question is, then, why does Apollonius emphasize such an obvious conclusion in the enunciation? The reason, I believe, is this, that the purpose of these two propositions is to explain the significance of the line segments ΞM and MN, namely, that they provide a derivative image of a diameter and its conjugate, being in the same ratio as that between the conjugate diameters in square, and that they provide an image of a diameter and its latus rectum. This may be why Apollonius remarks at the end of VII.7, as an illustration, that from what was proven in the proposition, “…it will be proven that, if the perpendicular falling from point Λ on the axis passes through center Θ, then diameter KB will be equal to diameter ZH, because MΞ = MN.” In other words, it is obvious in this instance that KB will equal ZH because their images can be seen to be equal.

This interpretation of VII.6,7 helps resolve some puzzling aspects of the preceding proposition on the parabola. Proposition VII.5 states that “If there is a parabola, and one of its diameters is drawn in it, and from the vertex of that diameter a perpendicular is drawn to the axis, then the parameter of the lines drawn from the section to the diameter parallel to the tangent drawn from the vertex of the diameter, [i.e.],¹⁷

when figures are applied to that line, [those ordinates] are equal in square to them—and that line is its [the diameter’s] latus rectum—is equal to the latus rectum of the axis increased by four times the amount cut off from it by the perpendicular from the axis adjacent to the vertex of the section.” Thus, if AB is a parabola (see fig. 8), AH its axis, BI a diameter (therefore, BI is parallel to AH), AΓ the latus rectum with respect to AH, and BZ a perpendicular from B to AH, then the latus rectum with respect to BI is equal to the latus rectum with respect to AH together with four times the segment AZ. So, what VII.5 says is clear enough. What is puzzling about it is, first of all, where it is in Book VII.

There are three propositions in Book VII dedicated to the parabola, VII.1, 5, and 32.¹⁸

Of those, VII.1 and VII.32 evidently prepare the reader for the propositions immediately following them; this we have already seen in the case of VII.1. But what is the connection between VII.5 and VII.6,7? On the face of it, there is none. Moreover, the only use Apollonius makes of VII.5 is in VII.32. This leads Toomer to conclude that VII.5 is a lemma for VII.32 (see Toomer 1990, lxxi). But if this is the case why does VII.5 not appear immediately before VII.32? Why is VII.5 VII.5 and not VII.31? We must conclude that, despite the seeming unrelatedness of VII.5 and VII.6,7 (and, for that matter, the propositions following VII.7), VII.5 does somehow set the stage for the propositions following it; indeed, I believe it anticipates the image-making function of VII.6,7 which I described above. The evidence for this is not in the statement of VII.5, but in its proof. A slightly paraphrased version of Apollonius’ argument runs as follows: Let EA be drawn perpendicular to the axis (and therefore tangent to the parabola at A) and BI be extended to E (see fig. 8). Let BΔ be tangent to the parabola at B and let BH be drawn perpendicular to BΔ. Then by similar triangles we have,

BΘ : BE :: ΔH : BΔ(:: 2ΔH : 2BΔ).

And therefore, by Conica I.49, we have,

BΘ : BE :: latus rectum with respect to BI : 2BΔ.¹⁹

• Therefore, 1/2 lr. (BI) = ΔH (or lr.(BI) = 2ΔH)
• But by Conica I.35 (and because ΔBH is a right triangle), it follows that,
• sq.BZ = rect.ΔZ,ZH = rect.2AZ,ZH
• And, by the symptoma of the parabola (given in Conica I.11), we have,
• sq. BZ = rect. (lr.(AZ)), AZ
• Therefore, rect. AΓ, AZ = rect.2AZ,ZH (= rect.AZ,2ZH)
• Therefore, AΓ = 2ZH
• And 4AZ = 2ΔZ
• Therefore, lr.(BI) = 2ΔH = 2ZH + 2ΔZ = lr.(AZ) + 4AZ.

Now, proposition VII.5 can be proved just as easily without drawing the line BH as follows (fig. 9):

• As before, BΘ : BE :: ΔB : ΔZ : lr.(BI) : 2BΔ by similar triangles and by Conica I.49.
• Therefore, 2sq.ΔB = rect.ΔZ,lr.(BI) = 2rect.AZ,lr.(BI) by Conica I.35.
• Therefore, sq.ΔB = rect.AZ,lr.(BI).
• But sq.BZ = sq.ΔB − sq.2AZ
• Therefore, sq.BZ = rect.AZ,lr.(BI) − 4sq.AZ
•    = rect.AZ,(lr.(BI) − 4AZ)
• But sq.BZ = rect.AZ, lr.(AZ)
• So, lr.(BI) − 4AZ = lr.AZ or lr.(BI) = lr.(AZ) + 4AZ.

So, why does Apollonius choose to prove VII.5 as he does? In particular, why does he choose a proof that requires the extra line BH? The reason, I believe, is that by means of the line BH in Apollonius’ proof, and the subsequent relations ΔA= 1/2 lr.(BI), ZH = 1/2 lr.(AZ), and ΔZ= 2AZ, the lines of interest, lr(BI), lr(AZ), AZ, can all be seen or have visible traces in the diagram, and these traces are set along the axis just as they are in VII.6,7; in the alternative proof none of these lines, except for AZ (and lr.(AZ), but only because we are told to draw it!) can be seen in diagram—there is no image of them in the proof. So, to answer the question with which we began this discussion, Apollonius places VII.5 before VII.6,7, and not before VII.32, as the deductive structure of the book would suggest, because Apollonius can prove VII.5 in a way that will anticipate the setting up of images of latera recta and other lines which characterizes VII.6,7.

Let us now return to VII.6,7 and review what images, or, since they are such because of their place in various proportions, what analogues are contained in construction for those propositions. I shall list below only the correspondences for the hyperbola, that is, for the case of VII.6; for the ellipse the correspondences are similar. To make things easier to grasp, moreover, I shall write Tr for the transverse axis, Lr for its latus rectum, D for an arbitrary diameter, lr(D) for its latus rectum, d for its conjugate diameter or second diameter (similarly, I shall write tr for the second or “erect axis”). Moreover, I shall write Im(Tr), Im(Lr), Im(d), and so on, for the images of the transverse axis, latus rectum, diameter.

1) ΓΞ and AN   Im(Lr)

2) AΞ and ΓN   Im(Tr)

Observe, by I.15 and the definition of the second diameter after I.16, it follows that if D is any diameter and d its conjugate then D : lr(D) :: lr(d) : d,²¹

so that, ΓN : NA :: AΞ : ΞΓ :: D : lr(D) :: lr(d) : d. Thus, ΓΞ and AN can be taken, alternatively, as analogues for the second axis, and AΞ and ΓN for the latus rectum of the figure on that axis:²²

3) ΓΞ and AN   Im(tr)

4) AΞ and ΓN   Im(lr(tr))

5) AΓ   Im(Tr + Lr)

6) NΞ   Im(Tr − Lr) or Im(Lr-Tr)

(In the ellipse, it is the other way around, that is, NΞ is the analogue for the sum of the axis and its latus rectum, while AΓ is the analogue for the difference between the axis and its latus rectum.)

7) MΞ   Im(D)

8) MN;   Im(lr(D))

In three propositions, propositions VII.8,9,10,²³

there appears another line segment drawn along the axis, ΞI, which is the mean proportional between MΞ and NM, and so it can be taken as an analogue to the conjugate diameter, since the conjugate diameter is the mean proportional between the diameter and its latus rectum. Thus, also:

9) ΞI;   Im(d)

Although the stated goal of VII.8 is to prove “that the ratio of the square on AΓ (which is the transverse diameter) to the square on BK and ZH, the two conjugate [diameters], when they are joined together in a straight line, is equal to the ratio of rect.NΓ,MΞ to the square on a line equal to line MΞ plus the line which is equal in square to rect.MN,MΞ),” the main work of the proof and the fact which is used widely in the subsequent propositions is the proportion sq.AΓ : sq.BK :: rect.NΓ,ΞM : sq.MΞ. Since, from (7) above, sq.BK corresponds to sq.MΞ, so that, from the proportion, the analogia, we have the further correspondence:

10) rect.NΓ,ΞM;   Im(sq.Tr)

The way these images make propositions in Book VII visually evident can be seen in the fifteen corollaries between propositions VII.31 and VII.32. The fifth of these corollaries, for example, states that “if there is a hyperbola in which the transverse diameter of the sides of the figure of the section constructed on the axis is greater than the latus rectum, then the transverse diameter of [each of] the figures of the section constructed on the other diameters is greater than its latus rectum.” Relying on VII.21, which states that “If there is a hyperbola, and its transverse axis is greater than its erect axis, then the transverse diameter of each pair of conjugate diameters among its other diameters is greater than the erect diameter of [that pair]; and the ratio of the greater axis to the lesser axis is greater than the ratio of the transverse diameter to the erect diameter among the other conjugate diameters; and the ratio of a transverse diameter nearer to the greater axis to the erect diameter conjugate with it is greater than the ratio of a transverse diameter farther [from that axis] to the erect diameter conjugate with it,” one can construct a rigorous proof for the fifth corollary: By the definition of the second diameter of a hyperbola given after Conica I.16,²⁴

we have Tr : tr :: tr : lr(Tr) and D : d :: d : lr(D) or sq.Tr : sq.tr :: Tr : lr(Tr) and sq.D : sq.d :: D : lr(D). So, given Tr > lr(Tr), it follows that sq.Tr > sq.tr and, hence, also Tr > tr. This means that D > d, since, by VII.21, if Tr > tr then also D > d. Therefore, sq.D > sq.d, from which it follows that D > lr(D).

But did Apollonius intend for us to reason in this way? It is not impossible that he did. To be sure, there are enough places in the Conica where Apollonius leaves the details of an argument to the reader—and often the details are longer than the argument that I have just produced above. On the other hand, there are also enough instances where Apollonius does supply such straightforward arguments to make me believe that he expected these corollaries to be truly immediate, by which I mean he expected the results to be seen rather than actually proved. To be specific, when Apollonius says at the start of this list of corollaries, “Thus it has been shown from the preceding propositions…,” I believe he means that the results, literally, appear in the diagrams for VII.6,7 (which are virtually repeated in every proposition between VII.6,7 and VII.20) when one keeps in mind the images set up in the preceding proposition. Thus, in the corollary we have been discussing, he expects the reader to see that if the transverse axis is greater than its latus rectum, AΞ will be greater than ΞΓ, that is, Ξ will be farther from A than N is. Therefore, MΞ will be greater than NΞ. Thus, it will be evident to Apollonius’s reader that the transverse diameter, whose image is NΞ, will be greater than the latus rectum of that transverse diameter, whose image is NΞ.