Introduction

The title of the present paper indicates that this work concerns the relationship between ancient rhetoric and ancient Greek mathematics. Such a title obviously raises a simple question: Is there such a relationship? The usual appreciation of ancient science and philosophy is at odds with such an idea. This appreciation is rooted in the pregnant categorization that ranks rhetoric and science at very different levels. While the philosophical tradition has criticized rhetoric as it was initially practiced (by fifth-century BC Sophists in particular), it has attributed to mathematics a close affinity to science in general. As far as sophistry and rhetoric are concerned, the locus classicus is Plato’s Gorgias, in which the initial discussion with Gorgias brings Socrates to make a sharp distinction between mere cookery (sophistry) and the true knowledge of the art of speaking. As for mathematics, it appears in the Republic as an intermediary stage necessary in the education of the best rulers of the City, again by opposition to the practitioners of a debased political oratory trained by the Sophists. These initial distinctions were further elaborated, particularly by Aristotle, and became usual standards in modern historiography.¹

Nevertheless, I want to argue that more recent trends in the history of Greek mathematics point, on the contrary, toward such a relationship, and my purpose is to explain why and how.

The idea of bringing together mathematics and rhetoric is not foreign to historians of pre-modern mathematics. Indeed, Giovanna Cifoletti’s recent studies on sixteenth-century French mathematicians have shown that the rise of algebra as a full discipline included in the mathematical curriculum during this period is deeply correlated to the way in which knowledge was reorganized in the frame of the humanist Encyclopaedia.²

Since the “rediscovery” of classical rhetoric was crucial to this reorganization, it appears that the fundamental shift that occurred in mathematics with the introduction of symbolic algebra is correlated to the no less fundamental shift that occurred in the way rhetorical concepts were taken up in order to reconsider the general organization of knowledge. This shift included a profound criticism of certain Aristotelian distinctions between scientific reasoning and dialectic, notably by Peletier and Ramus. In particular, the understanding of mathematical reasoning through rhetoric, which may seem surprising to us, had actually become a natural idea at the end of the sixteenth century (see Cifoletti 1992, introd. and ch. 5).

The development of late sixteenth-century mathematics also explains why so much attention was given to Greek texts that spoke about problems and analysis, namely the texts of Pappus, Diophantus, and Proclus. Indeed, problem, with the traditional meaning it had from medieval algebra, was progressively understood through, and finally confounded with, the rhetorical notion of quaestio. As for the Greek notion of analysis, the way it was understood in the sixteenth century was dependent partly on the aforementioned texts, and partly on the way it was progressively confounded with the rhetorical ars inveniendi and with symbolic algebra.³

Consequently, the study of this intricate process logically requires that we deepen our historical understanding of what ancient Greek analysis was in order to delineate the innovations and modifications that have created the modern understanding of it.⁴

In the endeavor to understand the nature of ancient analysis, there can be no question of using modern algebra, as Sabetai Unguru convincingly has shown in his 1975 groundbreaking article (Unguru 1975). Nevertheless, the previously mentioned approach to sixteenth-century mathematics suggests another possible way to explore this issue: through the study of ancient rhetoric. Since the “rediscovery” of ancient analysis is correlated to the “rediscovery” of ancient rhetoric, we may ask whether both were correlated from the outset, i.e., in Antiquity. After all, ancient analysis, understood in a loose sense, is meant to provide a concrete way to invent solutions for problems. But invention (heuresis) is a well-known part of ancient rhetorical theory.

I myself have devoted a detailed study to this question in the context of Late Antiquity (Bernard 2003a). I have shown that a difficult passage of Pappus’ Mathematical Collection can be better understood if one takes into account the techniques developed by orators and dialecticians that had become dominant in Late Antiquity. I am also preparing two other case studies: the first treats Eutocius’ commentary on Archimedes; the second examines Proclus’ commentary on Euclid.

My present purpose is not to dwell on these particular studies but rather to show how recent trends in the history of mathematics point toward ancient rhetoric. This objective parallels the general theme of the international workshop in Tel-Aviv in May 2001 organized by Leo Corry and Sabetai Unguru, which focused on recent trends in the history of mathematics.⁵

In other words, I want to defend the view that whereas there has not been, to my knowledge at least, any explicit comparison made between ancient rhetoric and Greek mathematics other than mine, this comparison stands behind some dilemmas of modern historiography. In other words, I believe that modern historians have been turning around this idea for some years without necessarily knowing it.

In the first part of this study, I will discuss the nature of the “dilemma” to which I have alluded. The issue at stake is the role that has been given (or denied) to mathematical practice in historical interpretations of Greek mathematics. I will illustrate this point by comparing the opposing views of Unguru and Knorr on this subject. In the second part, I will focus on the specific details of the controversy in order to show that the main question is the nature of ancient analysis. This will afford me the opportunity to list certain important questions related to this issue. In the third and concluding part, I will more fully develop how the recourse to ancient rhetoric could partly “solve” the previously described dilemma. Specifically, I will emphasize the reasons that make the choice of rhetoric as a historical context for Greek mathematics relevant and important for modern historians.

I

In a 1984 article on the recent history of Greek mathematics, Len Berggren remarks that interest in this field lies in the fact that “it is an area where there is still controversy over some of the main features and where issues of considerable historical importance are still unsettled” (Berggren 1984, 395).

This judgment has been true until now. Indeed, there are still debates about the correct understanding of quite basic features of Greek mathematics such as its synthetic or analytical character. The Tel-Aviv workshop actually reflected this atmosphere since many talks explicitly referred to Unguru’s strong contribution to this debate – specifically, the 1975 article On the Need to Rewrite the History of Greek Mathematics (Unguru 1975). More recently, Michael Fried and Sabetai Unguru have published the important study they were preparing on Apollonius’ Conics (Fried and Unguru 2001). The book presents itself as an explicit illustration of the 1975 thesis and eventuates from other articles Unguru has published in the past 25 years. It also presents itself as the counterpart of Zeuthen’s famous book about Apollonius’ work (Zeuthen 1886).⁶

Let us summarize Unguru’s statements about the way most of the history of Greek mathematics has been written. He argues that it has been spoiled, so to speak, by an abusive and illegitimate use of algebra or algebraic transcriptions of Greek mathematical works. Following Jacob Klein and Michael Mahoney (among others), Unguru says that the use of algebra in historical interpretations is not just a matter of external form that would leave the content of what is interpreted untouched. It also entails a basic misunderstanding of ancient methods because modern notation and conceptualization inherently carry a modern “intentionality” which is foreign to the ancient thought. The notion of “intentionality” is borrowed from Klein who follows Husserl on this point. To illustrate this general view, Unguru directs heavy attacks against the concept of “geometrical algebra,” first invented by Tannery and then adopted and extensively used by Zeuthen in his commentary on Apollonius. Unguru also argues in his study of Apollonius’ Conics that this idea of geometrical algebra, although it appeared in historiography in the nineteenth century, has its roots in the works of sixteenth- and seventeenth-century algebraists (Fried and Unguru 2001, 20ff). The idea of an art that the Greeks would have used but concealed in cloudy formulations leads us back at least to Descartes’ Regulæ.⁷

Unguru actually goes so far as to generally deny the legitimacy of any kind of mathematical approach to the truly historical study of ancient Greek mathematics. Mathematics and historical accounts are for him completely incompatible and he explains how so:

Let us contrast these views with those of Wilbur Knorr. Knorr is another major historian of Greek mathematics. Not only has he made an impressively large contribution in the past decades to the field of the history of Greek mathematics, he has also repeatedly proposed new and enlightening views about the chronology and the very nature of the Greek mathematical tradition. One obvious example can be found in his study on the evolution of Euclid’s Elements (Knorr 1975). Another can be seen in the emphasis he has put on the study of the late commentators for themselves (Knorr 1989, first chapters). The interesting fact about Knorr’s methodology is that his basic belief about Greek mathematics is diametrically opposed to Unguru’s. Namely, Knorr has taken up Viète’s and Descartes’ idea of a hidden art by which the Ancients made their discoveries, although the synthetic mode of exposition that is found most often in the texts testifies only indirectly about what this way of discovery was. This idea has become the major concept on which Knorr has built his principal work on the ancient tradition of geometric problems (Knorr [1986] 1993), as can be seen from the following passage taken from the first chapter:

Thus, just where Unguru sees the ultimate foundation of the special kind of anachronism made by most modern historians of mathematics up to the twentieth century, Knorr sees the best foundation of a faithful account of the Greek mathematical tradition. Accordingly, just as Unguru would not hesitate to dismiss most modern interpretations inspired by modern mathematics as the perfect antinomy of a historical study in the proper sense, Knorr, in his most polemical passages, does not hesitate to dismiss most modern historians like Heath, Tannery, or Szabó because they are not truly mathematicians but only philologists. Therefore, Knorr claims, they do not have even the beginnings of a true understanding of the mathematical tradition precisely because they do not belong to it (ibid., 87f).

It becomes obvious from this set of quotations that confronted with such opposite views, modern historians of mathematics face a true dilemma: should they use their own mathematical practice as a key tool for the understanding of ancient mathematical texts, or should they avoid it, on the contrary, as the worst (historical) sin they could commit?

In exploring this dilemma, I will refrain from resorting to any modern historians other than Unguru and Knorr. There are three reasons for this. First of all, Unguru and Knorr represent the extreme ends of the historiographical landscape concerning Greek mathematics and are therefore in some way representative of it. Secondly, they express their methodological claims very clearly. Lastly, I myself have been strongly influenced and inspired by both. One could object that focusing as I do on extreme and polemical positions implies getting involved in a tiring, old-fashioned, and therefore unproductive fray. But I cannot agree with this because if we leave aside the most polemical overtones of the conflicts that Knorr’s and Unguru’s positions have entailed, we must insist that there still remains a real dilemma. The polemic is not the core, but rather the most external manifestation of a deeper and real historical problem. Indeed, Unguru and Knorr both have successfully demonstrated, at least in my opinion, that they have made valuable contributions to the historical study of Greek mathematics. Their respective points of view on Greek mathematics are both possible and enlightening points of view. On the other hand, these views clearly seem incompatible. My view is that the study of rhetoric in Antiquity provides some possible keys to solve this dilemma in a way that does not seem like an unsatisfactory compromise. To understand this, we have to enter a little further into Knorr’s and Unguru’s arguments.

II

Taken on a large scale, the picture I have described above is faithful. Unguru dismisses any mathematical reading in historical studies as historically baseless since, he says, it is from the outset a source of anachronism. Knorr, on the contrary, rejects any attempt to avoid a mathematical involvement in historical interpretations of mathematical texts, for he deems the essential lines of development of Greek geometry must be judged from the inside.

Let us enter into the details of this controversy. The target of Unguru’s attacks centers on the question of algebra. The reason for this rejection is that, for Unguru, algebra and algebraic methods stand at the core of modern mathematical thinking.⁹

But on this point, Knorr clearly shares the same feelings as Unguru since he carefully avoids any interpretation that cannot be put into Greek form. Furthermore, he seeks a thorough correspondence between his reconstructed analyses and the actual synthetic presentation that is most often found in the texts. We have seen, for example, that he speaks in his book about the ancient traditions of geometric problems (ibid.), about “the essential, usually simple geometrical idea underlying each result.”¹⁰ To take an example from the thousands that can be found in Knorr’s work and especially in Knorr 1989, here is what Knorr writes in the introduction to his commentary on the beginning of the Third Book of Pappus’ Collection: This passage testifies that Knorr is at least sensitive to the objections to algebraic interpretations raised by Unguru. In this case, Knorr explicitly refuses to resort to algebra to build his commentary on the construction, even if in the criticism he makes about Pappus’ comments, he again resorts to some algebraic speculations (ibid. 79, n.12–13). In a way similar to Unguru, Knorr has distanced himself early on from Zeuthen’s concept of “geometrical algebra,” as the first chapter of his study of Euclid’s Elements makes clear: In fact, Knorr accepts the use of Zeuthen’s expression but with a much more restricted meaning. Thus, Knorr does not go as far as Unguru in his criticism of Zeuthen’s concept, but he nevertheless explains that neither Book II nor Book X of the Elements can be viewed as algebraic treatises (ibid., 10–11).

In summary, while taking into consideration the non-algebraic form to be given to his speculations, Knorr allows himself to propose reconstructed analyses (in the Greek sense) in order to show the way in which mathematical results were found, not just demonstrated, by Greek mathematicians. At the same time, he carefully avoids, as much as he can, giving any algebraic expression to these speculations. Now, insofar as he makes such a reconstruction, he is to some extent in line with the Cartesian idea that the Ancients must have had some kind of very simple art by which they found their results. Here, he strays from Viète and Descartes, who share the belief that the Ancients hid this art afterwards within cloudy formulations. The following quotation is the famous passage in which Descartes, speaking about the “true mathematics” of which we find remains in Pappus and Diophantus, explains why the Ancients hid their art by a certain injurious cunning (pernicioso quadam astutia):

It seems to me hard to deny that Knorr makes a valid point here since the idea of a different, yet correlated, way to that of demonstration is just the point that Archimedes makes in his famous Ephodos that Heiberg retrieved in a palimpsest. Archimedes speaks about a tropos, a way of doing things, by which you can eventually find certain results but not demonstrate them – although this tropos may also appear useful for demonstrations:

In a little exchange I had with Unguru on this point, he wrote to me that “quite often, Wilbur Knorr’s ‘analysis’ is a disguise for algebra” (personal communication). Here, he is applying to Knorr the exact argument he denied to other historians, namely that parts of Greek geometry are just “disguised algebra” or “algebra in geometrical garb” (Unguru 1975). To such a judgment, one can apply precisely the same criticism Unguru himself applied to the tenet of algebra disguised in Greek geometry: if we do not find any symbolic algebra in Knorr’s reconstruction, but only an analysis in the form that could be expected from a Greek mathematician, how can we maintain that it is not simply what it seems to be? Or shall we permit the general axiom that any modern historian making use of mathematics is disguising in geometrical garb his or her basic thoughts which can be only algebraic by nature? This clearly cannot be the case, since the basic problems underlying Unguru’s “suspicions” are more intricate. Jacob Klein summarizes them suggestively: This quotation highlights the fundamental and subtle idea that while Greek mathematics is not inaccessible to us, we are obviously and irremediably cut off from the “circumstances” that surrounded ancient science. Therefore, any access to ancient mathematics requires envisioning the missing circumstances in which it was created. Our access to it is, therefore, necessarily mediated through a partial historical reconstruction of these circumstances.

The debate concerning the ancient “way of discovery” in mathematics is obviously related to the historical questions surrounding ancient analysis. In this respect, an interesting touchstone is the series of modern commentaries made on Archimedes’ famous 4^th proposition of Book II of his treatise Sphere and Cylinder. It is the following proposition that is stated as a problem: to cut a given sphere in two segments that have a given ratio to one another (fig. 1).

III

To step out of the dilemmas described and analyzed in the previous sections, we have to return to the question of context. This question obviously lies at the core of Unguru’s main claim: Unguru wants modern historians of mathematics to avoid any recourse to modern mathematics in their interpretations. According to him, they must use instead only “Greek” tools. Correspondingly, he urges them to set Greek mathematics back into its “own” cultural context. As Unguru himself points out (Unguru 1975, 78, n.21), scholars like Árpád Szabó and Jacob Klein, both of whom Unguru utilizes extensively, have already adopted this cautious attitude. Jacob Klein, for example, is a very subtle minded and sharp analyst of Platonic, Peripatetic, and Neoplatonist philosophy. His study of Diophantus’ arithmetical work, for example, is grounded on the basic hypothesis that this work has to be set back into the context of the history of Platonism. More precisely, he claims that Diophantus’ work has to be seen as the “theoretical logistic” that is implicit in Plato and that later Neoplatonists could not adopt for reasons pertaining to their own philosophical bias. Thus, Diophantus would, so to speak, fill a gap in the philosophical landscape defined by Plato’s philosophy. Thus, for Klein, philosophy and its history play essential roles in the history of science. This belief can be discerned in many passages, for example in the following one:

Unguru alludes to this when he speaks about We find in both cases a fundamental assumption that is actually easily accepted: there is an organic relation between Greek philosophy and Greek mathematics. But Klein goes further and eventually sees mathematical texts as the expression of this or that philosophy. One clear example of this position is his analysis of the philosophical “shift” from the Platonic notion conceived by Aristotle on the question of the nature and measurement of numbers:

It is again interesting to compare the previously quoted views to Knorr’s since Knorr was profoundly opposed to Klein’s and Szabó’s perspective. This can already be witnessed in his study of Euclid’s Elements (Knorr 1975, 3–4), and after the polemical exchanges he had during the period around the 1981 Pisa Conference, he made his position very clear in different texts. For example, in the introduction and first chapter of his 1986 text, he strongly refutes the widespread idea that one has to look for “motivations” or “explanations” for the development of Greek mathematics in Greek philosophy:

One can even say that Knorr has emphasized the notion of problem precisely because he wanted to show the internal dynamic of this problem-solving activity that finds its motivation in the process of mathematical research itself and not in any external factor. Many of his studies do actually reflect this state of mind since his efforts tend to make clear the technical coherence of the geometrical tradition, sometimes at the cost of unwarranted hypotheses. This motivation is clearly expressed at the end of the first chapter of his study on geometric problems:

When Knorr resorted to the notion of problem to create his own reconstruction of the ancient tradition of Greek geometry, he was not aware of the fact that he was actually resorting to a central concept of ancient rhetoric.²⁵

On the other hand, scholars like Klein or Szabó, who were looking for a possible context for Greek mathematics, resorted to philosophy as the only available possibility. They were also unaware, though in a different way, that there is at least another possible context – that of ancient rhetoric. I would even be maliciously tempted to argue – although such debates quickly become sterile – that resorting to rhetoric or sophistry, in general, is more natural and justified than resorting to philosophy. Let us take, for example, the very notion of mathêma or “learning matter” according to Klein’s translation (Klein [1968] 1992, 51). This concept obviously stands at the very heart of Plato’s speculations on what learning means: the Meno dialogue is the locus classicus in this respect. But the notion of mathêmata stands even more obviously at the center of sophistic and rhetorical practice – not just speculations. This fact is usually overlooked because of some pregnant misunderstandings about sophistry that are probably due to the heavy shadow of Platonism on our historiography. The main purpose of sophistic techniques was much less to “persuade others” than to make others learn – or, in Greek words, to build mathemata in a very concrete sense.²⁶

Now, how can one concretely use the categories of ancient rhetorical practice in order to study ancient mathematical texts? To illustrate this point, I will come back to a case I have studied with much care: Pappus of Alexandria. Many historians tend to consider his Collection as characteristic of Late Antiquity, a period regarded as generally decadent. From this perspective, late antique authors are thought to have cared less for creative invention than for classification and inventory of the past “classical” texts.²⁷

Unfortunately, this view is at best partial and at worst a real misunderstanding of this period. Knorr had already acknowledged, although reluctantly, that Pappus demonstrates some sensitivity to ancient heuristic procedures (Knorr [1986] 1993, 360). But we should go much further: the period of Late Antiquity is characteristically obsessed with the idea of invention. The Latin word inventio is itself the translation of the Greek word heuresis. Let us reconsider the example of Pappus and the beginning of Book Three of his Collection.²⁸ What is it about? The pupil of Pandrosion, who was most probably a female competitor of Pappus (Jones 1986, 9, n.4), has suggested to the latter that he should examine an original geometrical construction, which is meant to solve the classical problem of inserting two means proportional by “plane means.” This construction is certainly new: Pappus, who is usually so eager to demonstrate the scope of his mathematical knowledge and even presents some solutions of the Delian problem as his own,²⁹ does not identify it as something known. Moreover, the criticism he sketches is precisely only sketched and again indicates that he is facing something new. We are thus in a context of research, to use a modern term, which nevertheless parallels the first words of Pappus’ foreword: This context is additionally confirmed by the type of criticism that Pappus elaborates: it is not apodictic (demonstrative in the logical sense) but rather epidictic (demonstrative in the usual sense). The concrete concern that Pappus has in mind is indeed to show his own value as an “instructor” in the technique of analysis. He thus does not reach any conclusion in a strictly logical sense, but a “rhetorical conclusion,” which is constituted in particular of new problems given in the form of lemmas (Coll. Math. I 48.19–52.30). Lastly, the effort made by Pandrosion’s pupil to propose an original construction is mainly encouraged by Pappus. In general, Pappus displays many marks of respect toward his interlocutors. The first of them is to justify the very fact that Pandrosion’s pupil has made a proposal. Pappus does it by referring to an ancient debate on the nature of mathematical investigations. It opposed those who held that such investigations were “problems” or “projections” to those who maintained that they were essentially “theoretic” or contemplative.³⁰ Pappus thus calls on the prestigious authority of (unnamed) Ancients to encourage the endeavor of his young competitor. But this is only an ad hoc erudite reference on the part of Pappus; the basic meaning of the word problem, as Pappus uses it in this passage, is rhetorical. It refers in general to a subject (hypothesis) of declamation cast (proballomenon, from proballein) by a public of listeners to the speaker, or more generally, set by circumstances, as when one has to defend a given case before a tribunal. In this particular case, it refers to the challenge set by Pandrosion’s pupil to Pappus: to examine a construction, which the pupil claims solves the Delian problem.³¹

Now, is the construction proposed by Pandrosion’s pupil an original one? Wilbur Knorr has shown, quite convincingly in my eyes, that it may be a clever variation on the construction of Eratosthenes’ mesolabe or “mean-taker” (Knorr 1989, 64–68). Eratosthenes’ device was well known in Pappus’ time: this is attested in particular by the description Pappus gives of it a few pages later (Coll. Math. I 56.18–58.22).³²

However, to conclude from this that the construction devised by the pupil of Pandrosion was not original would be a pure misunderstanding: the practice of invention, as it is known from ancient rhetoric, consists of building up an answer to a given challenge. The way to do it is to construct this answer, to the best of one’s ability, on the basis of topoi, that is to say, of a kind of “treasury” of arguments taken from classical literature. It is probably this kind of “treasury” to which Pappus alludes at the beginning of the seventh Book of his Collection, as he speaks about the analuomenos [topos], the “analytical topic.”³³ This is one of the texts taken from this “treasury” that Pappus uses himself to initiate the pupil of Pandrosion to the technique of analysis. This is also a kind of treasury that Pappus probably intends to constitute when he lists, after his criticism, a series of solutions to the Delian problem. We moderns tend to have a dramatic notion, so to speak, of invention – all the more so since we most often lack the rhetorical training that would give it a concrete content. But in antiquity, invention is still conceived of and practiced most often as a slight variation on a well-known basis of topoi. As such, to some extent it can be compared with modern jazz music, according to a striking comparison proposed by Henri-Irénée Marrou (Marrou 1948, 275). The kind of construction proposed by the pupil of Pandrosion is therefore perfectly inventive, provided one conceives of invention in the very way Pappus’ contemporaries understood and practiced it. In other words, there is a great disadvantage in not taking into account the (generally very subtle) categories elaborated by Greek rhetoricians in order to put into practice invention, which we now tend to relate to subjective “originality.”

The passage of Pappus I have discussed is of course a very isolated case. Nevertheless, I hold that it reveals powerful means to interpret other Greek mathematical texts, not only belonging to the same period (Eutocius, Proclus) but also of earlier periods. In other words, I claim that this kind of reading has a high degree of generality and can thus be applied to many other texts, but I cannot fully discuss and justify this claim here since it requires more detailed case studies.

Let us then come back to the thorny question I have mentioned concerning Greek analysis. It seems fitting to me to explore the field of ancient rhetoric in order to get not only the objective information that is often lacking in solely mathematical texts, but also a relevant point of view. By “relevant,” I mean, in perfect accordance to Unguru’s criteria, “non anachronistic.” As is well known, rhetorical practice is one of the most deep-seated phenomena pertaining to ancient Greek culture – and by that I am not only referring to something pregnant in the Greek and Roman ancient world, but also to something that lasted from the birth of ancient Greek civilization to the expiration of Greek pagan culture.³⁴

As a cultural fact, ancient rhetoric actually relates the notions of agôn and agonism that pervade all Greek culture (Lloyd [1996] 1999). In short, we hold here a huge corpus of texts which has been left unexplored till now by historians of mathematics, but which could nevertheless clear up many intricate debates about the way of discovery of the Ancients. Therefore, the study of ancient rhetoric is relevant to the aforementioned dilemma.

I must insist on an important point here. To make it clear, let us begin with a simple question: what is the core of ancient rhetorical practice? The first and foremost answer is: practice itself. Otherwise stated, a central key for understanding ancient rhetoric is that it is oriented toward an accomplishment (a performance), which is the so-called “declamation” or meletê.³⁵

For meletê, there are by nature no rules; one can only prepare for it. Hence, the important notion of progymnasta or preparatory exercises. We touch here on one of the most important ideas that Isocrates, the founder of the classical paideia according to Marrou, tries to define in his Antidosis : for the accomplishment which is the supreme aim of his paideia tôn logôn, there is no science (in the sense of epistêmê) that would enable one to cope in advance with all possible cases. This feature of Isocrates’s paideia is intimately related to the purpose of the rhetorical training which is to make others learn and practice rhetoric themselves. When one comes to the accomplishment, the meletê proper, Isocrates says one can no longer distinguish between “master” and “pupil” – at this stage one finds only competitors who are engaged in a common task.³⁶

If we keep in mind the essential function of the rhetorical exercise that I have quickly sketched,³⁷

we are able to understand that texts that have been “influenced” themselves by such practice possess a very particular form. To put it simply, the kind of text that someone trained in rhetoric would write is partially built in order to induce its reader to practice rhetoric. It therefore functions as a kind of trap for its reader or its listener (in the case of a speech). Mathematical texts, that is, texts that are mathemata in the true sense, “learning matters,” also share in this particular form. This view immediately raises a difficult question: how can one describe a text that is built to induce something, a text that for example leaves gaps to be filled by subsequent readers?³⁸ I do not have any definite answer to this subtle question, but one thing at least is sure: one cannot avoid this aspect once he or she has entered the “rhetorical dimension” of ancient texts. In other words, one cannot avoid the practical aspect lying behind or rather inside the form of the text. In this respect, I cannot agree with Unguru when he radically dismisses the Cartesian idea of an art carefully hidden in the ancient texts. That is to say, I cannot agree with his idea that The assumption Unguru speaks about may well be qualified “gratuitous” if it serves to justify an illegitimate resort to modern algebra. But taken separately, I think it represents for historians a fruitful assumption to which one should pay attention, provided one keeps an eye on ancient rhetorical practice. I think that the Cartesian idea, if it is taken as an historical idea, is not wrong in that it claims the Ancients had a hidden art, but rather in that the Ancients could have revealed it and turned it into an explicit method. The notion of secret as such plays an essential role in ancient science (this can be perceived in Archimedes’ introduction to his Ephodos).