# History and philosophy of Greek Mathematics

Symposium organized by by David Fowler & Eberhard Knobloch, from 20 – 26 July, 1998.

**Participants **

Lennart Berggren (Simon Fraser University Canada), Alan Bowen (Institute for Research in Philosophy and Science, Princeton), Lis Brack-Bernsen (Regensburg University Germany) Jean Christianidis (University of Athens), Achmed Djebbar (Université de Paris-Sud, Orsay), David Fowler (University of Warwick), Jens Hoyrup (Roskilde University), Alexander Jones (University of Toronto), Vassilis Karasmanis (European Cultural Centre of Delphi), Eberhard Knobloch (Technical University of Berlin), Henry Mendell (California State University at Los Angeles), Reviel Netz (Gonville and Caius College, Cambridge), Eleanor Robson (The Oriental Institute, Oxford), Sabine Rommevaux (CNRS, université de Lille-III), Ken Saito (Osaka Prefecture University), Ivo H.L. Schneider (Universität der Bundeswehr, München), Jacques Sesiano (Ecole polytechnique fédérale de Lausanne), Christian-Marinus Taisbak (University of Copenhague), Sabetai Unguru (Tel-Aviv University), Bernard Vitrac (Centre Louis-Germet, CNRS, Paris), Hans-Joachim Waschkies (Kiel University, Germany).

**Summary**

This highly successful Fourth International Colloquium on the History and Philosophy of Greek Mathematics was the latest in a series of biennial meetings of a group who meet to present papers and discuss a range of topics on Mesopotamian, Egyptian, and Greek mathematics and its transmission. At each meeting, new young members have been invited to join the group and, this time, the contributions of Reviel Netz, Eleanor Robson, and Sabine Rommevaux added further zest to the meeting.

The facilities for formal and informal sessions at the Fondation are outstanding: meals, in particular, often lasted well beyond their scheduled time as animated discussions took place on all aspects of our field. The group was cloistered from the outside world and liberated from the need to organise details of everyday lite which, combined with the facilities of the estate, contributed to the intensity of the session. The staff provided discreet and total support for our activity.

One wholly unexpected event of the session — some said even more memorable than the conference itself ! — was an impromptu concert given by Yo-Yo Ma who, as we only discovered in the middle of the session, was staying at Barjeantane. So warm and harmonious was this event that a quartet including Mr Ma, along with Veronika Hagen (viola) and Rainer Schmidt (violin) of the Hagen Quartet, who had just arrived to run a masterclass at Villecroze, and Paul Gulda (piano), arranged for an open session of music-making on the next evening, again in La Grande Maison.

Another item that was slipped into our very full programme was a talk by Pierre Léna and Yves Quéré on their project ‘La main à la pâte’, on Science Teaching in Elementary Schools. This generated a lot of interest and discussion, and was an example of the kind of interdisciplinary interchange that the Fondation provides.

The whole Colloquium was judged by the group to be exceptionally fruitful.

**Report**

**Len Berggren: ***The Development of Analysis and Synthesis from Ancient Greece to Medieval Islam*

In this paper, written jointly with Glen van Brummelen, we argue the following points:

- In view of the reversibility of many geometrical propositions, Pappus’s description of analysis and synthesis in the
*Mathematical Collection*vii does not show the confusion or contradictions that some writers have argued it does. Indeed we find the same dual descriptions of analysis in some of the Islamic writers. - The two parts of a typical synthesis of a geometric problem (construction and proof) correspond, in reverse order, to the two parts of a typical geometric analysis (transformation and resolution).
- In an analysis of a geometric problem, Euclid’s
*Data*served as a handbook of geometrical objects that could be constructed by the methods of the*Elements*from varions sets of givens. - Evidence from Archimedes’
*Sphere and Cylinder ii*suggests that one reason analyses were included in ancient mathematical treatises was to show that a particular assumption such as the constructibility of two mean proportionals — was necessary to the solution of the problem. - Ibrahim ibn Sinan’s classification of problems in his
*On Analysas and Synthesis*fits well with the types of analysis that one finds in mathematical works of his time (10th century AD). Typical of such ‘analyses by knowns’ (as we call them) are those of al-Kuhi in his*On Drawing Two Lines from a Point at a Known Angle.* - Ibrahim’s defense of analysis as containing, despite the charges of some critics, all that was needed to effect the synthesis, suggests that there was a tenth-century debate about the extent to which a synthesis should mirror an analysis.
- Evidence from the writings of al-Kuhi and Ibn al-Haytham suggests that these writers viewed the fonction of analysis as less that of showing that a particular object was constructible in a Euclidean sense to guarantee than showing, in Ibrahim’s par-lance, that the problem is `true’, i.e., it has one or finitely many solutions.
- Finally, we call attention to the word `known’ having replaced the Greek `given’ in medieval Islam, and suggest that it may have been due to the writings of a Hellenistic source, such as Diodorus, who described the given as
*gnorimon*and whose writings were known to the Islamic authors. Further, it appears that a definition of `known’ found in some Arabic authors may show the influence of Apollonius’s definition of `given’, as reported by Marinus.

**Alan C. Bowen: Reflections on an ancient calendar **

*(P. Hibeh 27)*- Hibeh 27 is a lengthy papyrus that comes in 16 fragments written in 2 hands probably before -240. In the first hand, we have some introductory remarks explaining the purpose of what follows and providing a general astronomical framework. In the second hand (with a few corrections, perhaps, in the first), we have a calendar in Greek that either records the month and day number in the Egyptian civil year when the Sun is in a new zodiacal sign or just gives the day number in the Egyptian month. Typically the risings and settings
^{.}of certain stars are listed, as are the changes in the weather and condition of the Nile, the length of daytime and nighttime in equinoctial hours, and the religions festivals for that day.

As shown in this paper, the calendar of P. Hibeh 27 is arithmetically structured by its scheme for the length of daytime and nighttime throughout a year of 365 days. It turns out, however, that this scheme is a clever adaptation of a Babylonian linear zigzag scheme defining the length of daytime over the course of a year of 360 days. This linear zigzag scheme is found in such documents as MUL.APIN which dates from mid-7th century BC.

The presentation concludes with a brief review of the problems involved in dating the composition of this papyrus and a few suggestions concerning its possible use.

**Lis Brack-Bernsen: Time intervals in Babylonian astronomy**

During the two last centuries BC, Babylonian astronomers were able to calculate the times between sunset and moonrise on the days around opposition (full moon). These time intervals are easy to observe but, because they depend on four different astronomical variables, they are very difficult to calculate. How did the Babylonians arrive at their calculational schemes? How were the basic parameters determined from observations?

Concentrating on the movement of the moon relative to the sun, we found that some general concepts are present in all known astronomical cuneiform texts. In the astrological series Enuma Anu Enlil (probably dating from old Babylonian times but compiled towards the end of the second millennium BC), tablet XIV contains numerical schemes concerning the moon. According to these tables, the daily moonrise around opposition was delayed by an amount equal to one fifteenth of the night. The Assyrian astronomical compendium MUL.APIN dating from 1000 BC contains similar tables which show us that until then the daily change of visibility of the moon was determined by the length of the night.

The Text TU 11 from Uruk (written in cuneiform around 200 BC) is a collection of ancient and younger prediction rules. Through examples, calculated in Section 19 of the tablet, it can be demonstrated how the rising time of the invisible moon was found on the days around conjunction. The ancient astronomers started with the time between the last visible moonrise and the sunrise before conjunction. From this time interval they subtracted the daily retardation of the moon and found thus the time intervals between moonrise and sunrise for the following days on which the moon was not visible. But this estimate of the retardation was quise inaccurate.

Later, the Babylonians found a better estimate for the daily retardation of the moon. On the evening just before opposition, the moon rises some time before sunset. This time interval was called ME. On the next evening (after opposition), the moon rises a time interval GE after sunset. The sum ME+GE is the retardation of the moonrise, measured on the day of opposition. Analogously, the time between moonset and sunrise were observed on the two mornings around opposition. The sum SU+NA of these time intervals is the retardation of moonset on the day of opposition. Around conjunction, the moon is invisible, hence its retardation cannot be observed.

The Goal-Year Tables collect astronomical data to be used for predictions. The lunar phenomena recorded on them start with the sums SU+NA and ME+GE and continue with eclipses and the time intervals mentioned above between setting and rising of sun and moon. An analysis of computer-simulated lunar data showed that the daily retardations of the moonrise (and -set), i.e., ME+GE and SU+NA, repeat themselves after one Saros. This fact was discovered by the Babylonians and used for predicting lunar phenomena. ME for a full moon can be found as the ME observed one Saros earlier plus one third of ME+GE (the daily retardation), while GE is one third of ME+GE smaller than GE measured one Saros earlier. TU 11 contains a short remark showing that this method was indeed known to the Babylonians. Similarly, the retardation of the moonset was used to predict the finie of moonset in comparison to sunrise on the days around opposition.

The sum of the daily retardations of moonrise ME+GE and moonset SU+NA gives a good measure for the velocity of the moon relative to the sun at opposition. ft turns out that this sum SU+NA+ME+GE defines a function which has the same period, amplitude and phase as the Babylonian column which served as a basis for calculating the lunar velocity. We are convinced that it was a skilled handling of these time intervals which enabled the Babylonians to calculate the lunar velocity.

**Jean Christianidis : Le Diophante byzantin : un aspect du rôle d’intermédiaire de la science byzantine**

La contribution des Byzantins — dans la mesure où elle a jamais existé — à l’augmentation du corps des connaissances mathématiques, telles qu’elles avaient été léguées par les oeuvres classiques d’Euclide, d’Archimède, d’Apollonios, de Diophante, etc., est de mince importance. Toutefois, on a reconnu comme très important le rôle que le monde byzantin a joué pour la conservation de l’héritage scientifique ancien, et sa transmission tant à l’Est (au monde islamique) qu’a l’Ouest. Il y a pourtant une dimension de ce rôle d’intermédiaire qui est insuffisamment estimée dans la recherche historique. Nous parlons de la tradition scoliastique et, en particulier, de la contribution des scoliastes byzantins à l’éclaircissement de certains points de la pensée scientifique ancienne qui, dans les textes originaux, ne sont pas suffisamment élucidés et sont, pour cette raison, susceptibles de multiples interprétations. Telle est le cas de la pensée diophantienne.

L’inventaire des byzantins qui, à différents titres — du simple collationneur de manuscrits au lecteur, à l’éditeur ou au commentateur — se sont occupés à l’oeuvre diophantienne, montre que Diophante était connu des cercles lettrés de Byzance __déjà. au__VIII’ siècle, son influence étant plus visible pendant les derniers siècles de l’Empire, notamment avec Georges Pachymère (1241 – environ 1310) et Maxime Planude (1255-1305 ou 1260-1310). Le commentaire de ce dernier marque le plus haut niveau de l’intérêt des Byzantins pour l’oeuvre diophantienne et s’avère éminemment utile pour la révélation de la méthode employée par le mathématicien alexandrin au cours de la résolution des problèmes indéterminés de ses *Arithmétiques.*

On sait que les procédés de résolution de Diophante ont fait l’objet de nombreuses discussions, les positions extrêmes étant, l’une que Diophante avait à sa disposition une méthode bien déterminée, et l’autre que chaque problème recevait une résolution différente. Cette question reçoit aujourd’hui une nouvelle lumière grâce au commentaire de Planude sur le problème Il 8 des *Arithmétiques, *commentaire qui nous offre une interprétation permettant de mettre en évidence l’unicité et la généralité des procédés de Diophante et les met en relation avec la théorie des proportions numériques, une caractéristique majeure des mathématiques grecques.

**Achmed Djebbar, Sabine Rommevaux, Bernard Vitrac : ***L’histoire du texte des ***Eléments ***d’Euclide : le débat Heiberg-Klamroth “revisité”*

Editer un texte antique suppose d’en trouver les manuscrits conservés, de recenser leurs variantes pour détecter les filiations éventuelles entre copies. Le but ultime est de se rapprocher, autant que faire se peut, de l’autographe original. En dehors des manuscrits portant une version du texte qu’il veut éditer, ou une partie de ce texte, voire des papyri anciens quand ils existent — on appelle cet ensemble tradition directe — , le philologue a recours à d’autres textes : les citations par d’autres auteurs, les commentaires quand ils existent, les traductions anciennes… qui constituent ce qu’on appelle habituellement la tradition indirecte. Les *Eléments *d’Euclide sont incontestablement le texte mathématique grec ancien dont la tradition indirecte est la plus riche, et cela n’est pas sans rapport avec le succès du traité durant une longue période, qui va de l’Antiquité tardive à la fin du XVIe siècle. Ainsi, les *Eléments *font partie des premiers textes mathématiques traduits en arabe au début du D.Ce siècle, puis, au MF, de l’arabe au latin.

Au début des années 1880, un débat a opposé deux éminents savants, M. Klamroth et J.-L. Heiberg, à propos de la valeur comparée de la tradition des manuscrits grecs et des dif férentes traductions arabes. Celles-ci présentent en effet certaines caractéristiques communes qu’elles ne partagent pas avec le grec, et qui paraissent trop importantes pour pouvoir être simplement expliquées par les corruptions successives dues aux fautes mécaniques de la copie (orthographe, mauvaise lecture, saut du même au même…). Deux explications, au moins, se présentent :

- Les traducteurs médiévaux ont pris de grandes libertés avec le texte, qu’ils n’ont pas hésité à adapter à leurs besoins.
- Leurs versions s’appuient sur des modèles grecs sensiblement différents de ceux que nous connaissons.

Klamroth et Heiberg ont défendu deux points de vue opposés, le premier soutenant l’idée d’une plus grande “pureté” de la tradition arabe, le second concluant, a contrario, que l’autorité de la tradition arabe est moindre que celle des manuscrits grecs. En 1996, dans son dernier grand article, le regretté Wilbur Knorr a repris la question, en s’appuyant sur les récentes éditions des versions arabo-latines. Il a montré la faiblesse de plusieurs arguments de Heiberg et s’est rallié au point de vue de Klamroth. Son analyse était cependant limitée à une portion des Livres dits stéréométriques (ou de géométrie “dans l’espace”) des *Eléments.*

Notre groupe de travail a entrepris d’étendre l’enquête à l’ensemble de l’ouvrage et surtout de prendre en compte les versions arabes encore inédites à ce jour, accessibles seulement sous forme manuscrite. Nos premières conclusions, pour provisoires qu’elles soient, nous conduisent à remettre en cause le caractère lacunaire et homogène des traditions médiévales arabes et arabo-latines qu’admettaient nos prédécesseurs, et à dissocier le problème de la fidélité des traducteurs de celui de la qualité des modèles grecs utilisés par les traducteurs arabes. A partir de l’exemple du livre X et des Définitions du Livre V (l’un des plus célèbres du traité), nous avons essayé de montrer que la transmission des *Eléments *de l’Antiquité au Moyen Age est un phénomène complexe dans lequel il faut distinguer, d’une part, les variantes provoquées par la multiplicité des traducteurs, la contamination ultérieure et réciproque des différentes versions médiévales, et, d’autre part, le fait qu’aussi bien les manuscrits grecs byzantins du DCe siècle que les modèles utilisés par les traducteurs médiévaux ont été réalisés, semble-t-il, à partir de plusieurs éditions grecques divergentes. De fait, les versions médiévales ne sont peut-être pas meilleures, mais elles s’avèrent davantage discriminantes, d’où l’intérêt de les étudier, pour elles-mêmes, mais aussi pour renouveler l’histoire du texte grec des *Eléments.*

**David Fowler: Greek arithmetic and some of its implications**

My underlying thesis is that the kind of basic arithmetic that is taught in schools and used in everyday calculations has profound implications on the way that mathematics develops, even up to the most sophisticated levels. If we examine the whole range of explicit Greek evidence — school texts, commercial documents, and scientific treatises, up to the few calculations that we find in Archimedes, Aristarchus, and those parts of Ptolemy that corne from the Greek tradition — what we find is unambiguous: the Greek procedures are the same as those found throughout corresponding Egyptian material, that is they use suros of ‘parts’, also called ‘unit fractions’ or *‘quantièmes’. *The calendar P. Hibeh i 27, discussed by Alan Bowen, provides a very clear and informative illustration of this and other features of numerical practice. A distinction should be made between geometrical manipulations with fines (where, for instance, multiplications and divisions are not part of the calculations) and arithmetic with numbers: Archimedes’ *Measurement of a Circle *should be considered with this in mind.

Two small illustrations of the effects of this proposal are to put into question the almost universal interpretation of Euclid’s *Elements, *Book VII, in which ratios of numbers are treated as a mathematical way of dealing with common fractions, and to lead to an examination of just precisely what is meant by asserting that a pivotal event in early Greek mathematics was the discovery of `irrational’ numbers.

**Jens Hoyrup: A Babylonian miscellany: some fruits picked from a book in progress**

The talk covered three topics, drawn from a book manuscript on Old Babylonian `algebra’ and its kin.

- Several of the Old Babylonian texts
*(c.*1800 to 1600 BC) show an interest in a generalized geometry, where non-linear magnitudes (square areas, the volume of cubes) occur as the Sicles’ of rectangles. This suggests that the knowledge underlying Hero’s formula’ for the area of a triangle goes back at least in part to the first half of the second millennium BC — but it tells us absolutely nothing about the further elaboration of channels of transmission. - Several tables of technical constants and problem texts both from the Old Babylonian and the Seleucid epoch (third to second century BC) give one fair and one very good approximation to the square root of Neugebauer and Sachs pointed out that both values follow from an iteration procedure which however, as shown by Fowler and Robson, is not easily implemented. It turns out that both values also follow from the side-and-diagonal number algorithm known from Theon of Byzantium, and argued by Leonardo Fibonacci by a diagram that seems to have Babylonian origins. It is therefore a possibility that the approximation was Pound in this way.

- The Old Babylonian period was preceded by ‘Ur III’, a period in which scribes were central cogs in a highly bureaucratized economy, and appear to have had very little autonomy. Analysis of the terminology of the early Old Babylonian mathematical texts reveals that a number of terms with roots in practical computation are invariably written with Sumerian word signs. Mathematical operations may be written thus, or in phonetic Babylonian. All terms that correspond to the formulation of PROBLEMS, however, are invariably written phonetically; this regards questions (what’, ‘how much’ …), announcement of results (`you see’, ‘n comes up’ …), and logical operators ‘when’, Since’). In earlier Sumerian mathematics such terms ARE present. Their disappearance suggests that the very use of problems in school teaching vanished during Ur III; this would amount to a very strong confirmation of the disappearance of scribal autonomy.

**Alexander Jones: ***Babylonian Mathematical Astronomy in Some Greek Papyri*

The corpus of Greek astronomical papyri has grown considerably with the discovery of more than a hundred fragments from Oxyrhynchus (modem Bahnasa, Egypt). These texts are in the course of publication (A. Jones, *Astronomical Papyri from Oxyrhynchus, *2 vols., Memoirs of the American Philosophical Society). Among the information that the new papyri yield is an insight into Greek knowledge, use, and understanding of the methods of Babylonian astronomy in the period from the second half of the first century BC to the fourth century of our era, as illustrated by the following papyri:

– P. Oxy. 4139 is a small scrap preserving only the last words of several lins of a text on lunar theory. The subject malter is similar to that of Ptolemy, *Almagest *IV 2, a discussion of periods containing whole numbers of lunar (synodic) months and of anomalistic months. But whereas Ptolemy ascribes to Hipparchus a period relation that is actually based on the Babylonian System B lunar model, the papyrus stases a period relation that we can recognize as that of die Babylonian System A. Moreover, the papyrus goes on to refer to the ‘people of Uruk’, one of the two sites (along with Babylon) from which cuneiform tablets of Babylonian mathematical astronomy have survived.

– P. Oxy. 4152-4161 are tables of dates and longitudes of the planets’ first and last visibilities, acronychal risings, and stations. The basis of computation of these phenomena turns out to be precisely the planetary models of the Babylonian texts. Babylonian methodology is modified in a minor respect by the adaptation of the model to the Egyptian calendar, in which the fundamental unit is the day, in place of the Babylonian calendar, in which the fundamental unit is the lunar month. A more significant extension is the invention of tables that use arithmetical sequences (e.g. second order) to bridge the intervals between phenomena, permitting the ready calculation of the longitude of a planet on an arbitrary date.

– P. Oxy. 4136 is a procedure text setting out rides for calculating partial totals of a linear zigzag function representing the moon’s variable daily progress in longitude. The text is expressed in a manner strikingly reminiscent of Babylonian mathematical and astronomical procedure texts, although it cannot be a translation of a Babylonian text since the function in question is itself a product of Greek astronomy. Underlying the cules is an understanding of the mathematical properties of a linear zigzag function quite unexpected in an ancient writer (whether Babylonian or Greek).

**Vassilis Karasmanis: De Lineis Insecabilibus 968B5-22**

This small pseudo-Aristotelian treatise is written by one of the immediate followers of Aristotle. At that time, some people (probably Xenocrates and some of his followers) argued that not only malter but also space, time, and mathematical entities (fines, surfaces, etc.) are discontinuous and constituted by atoms. Aristotle, in his *Physics *(296a17) refers to them saying that it is not a hard task to destroy the theory of indivisible fines. So he may have suggested to some pupil to write up the subject.

The Aristotelian author of this treatise starts by referring to five arguments of the supporters of indivisible fines, and after that tries to refute them. The firth argument sets out to prove that geometrical objects are not continuous but are made up of small indivisible parts, thus undermining the whole science of geometry. This argument is the most interesting, ingenious and the most difficult to refute. The expositor of the argument seems to be very well acquainted with mathematics. The definition of incommensurable magnitudes and the technical mathematical terms are exactly the same as in the tenth book of Euclid’s *Elements. *The structure of the argument reminds us of mathematical demonstrations.

The text of the treatise is corrupted in many places, especially in the exposition of the above argument, and one of the problems of my paper is the appropriate reading of the text and its translation. There is also the logical formulation of the argument and its interpretation. My interpretation of the argument ends with a paradox: I want to emphasise its importance, demonstrate the probable ways to escape the paradox, and show that scholars who worked on this treatise have not paid the proper attention to it.

**Eberhard Knobloch: Archimeclism in the 17th century**

Though 17th-century mathematicians still adhered to a glorification of Archimedes, quite a few authors distinguished between his extraordinary inventiveness, his discernment, and his obscure method of proving and teaching. The lecture concentrated on Paul Guldin and Gottfried Wilhelm Leibniz. While Guldin elaborated a new proof method relying on Aristotelian concepts and notions, Leibniz referred to Archimedes in order to justify his method of dealing with infinitely small and great quantities. Guldin claimed to be a perfect demonstrator because he wanted to substitute his own ostensive demonstrations for Archimedes’s indirect demonstrations. But the realization of this program suffered from severe deficiencies. Leibniz did what Guldin did not: he proved the principles of his proof method. Like Guldin, he underlined the universality of his quadratures and rejected the reproach of obscurity. Infinitely small and infinitely great quantities are well defined notions. He justified his approach with the aid of Archimedes. But his method was, as he said in his *Arithmetical quadrature of the circle, etc. *(ed. Eberhard Knobloch, 1993), more direct and more appropriate for the art of invention. He generalized the Archimedian approach by avoiding the double procedure of inscriptions and circumscriptions.

**Henry Mendell: The trouble with Eudoxus: textual and interpretive problems for bis astronomical theory, its motivation, and its demise**

Eudoxus and his school were the first astronomers to propose geometrical models of celestial motions: systems of homocentric, uniformly rotating spheres, where each outer sphere carries the next one within it. After a period of about 120 years during which the work of Schiaparelli has formed the basis of our interpretation of the mathematical astronomy of Eudoxus, Callippus, and the school of Cyzicus, there have been several recent articles challenging many of the assumptions of his work (Heglmeier, Yavetz, Mendell, and Bowen), opening up a vast array of interpretations. We can distinguish three areas of difficulties: the astronomical phenomena to be explained, the mathematics of the theory of homocentric spheres, and the ways in which the mathematics were or were not to be applied to the phenomena. These prompt a fourth, more basic issue: how are we to evaluate our principal sources, Aristotle and Simplicius? Moreover, is there any new evidence which bears on these issues? If we cannot trust Aristotle, then we may retire from the issue entirely. The difficulty is Simplicius, who wrote about 860 years after the death of Aristotle.

After evaluating the possible fines of transmission, I conclude that Simplicius either used Sosigenes’ *On Unwinders *or a lemma of Alexander of Aphrodisias *(c. *300), who used his teacher Sosigenes’ work, and that Sosigenes most likely used the *History of Astronomy *of Aristotle’s contemporary, Eudemus. I introduce some intuitive philological principles, such as that if we find that text A does things completely different from what we would suspect if A were in the tradition of B, then A is not in that tradition, and thus *lectio indocti doctior potior *(a very learned reading of a unlearned man must come from another source, and so deserves our attention). Simplicius is a cautious reporter (as we can see from his distinguishing the testimonium of Eudemus on the requirements for Eudoxan astronomy from the testimonium of Sosigenes on Plato’s alleged setting of the requirements at *In de caelo, *488), but he is not as learned as his material (as we can see from his argument that homocentric spheres cannot explain anomaly at *In de caelo *507-9, which may go back to Sosigenes). I argue, therefore, that the text at 493-7 is practically a quotation from Eudemus, but that every other relevant passage except for those traditionally attributed to Eudemus is based in a later source.

Yet, if we look carefully at this passage in isolation, we find some startling things. The solar model is the opposite of what we expect given Sosigenes’ terminology, but it determines the mathematics and observational motivation, except for the periods of the second and third spheres and their angle (which we may derive from other sources) . Departing from the work of the last 170 years, a plausible lunar model can be reconstructed from the text, except that it is not what we expect from a source influenced by Hipparchus or Ptolemy. Here everything except the periods of the second and third sphere is provided. The planetary theory is full of surprises. We may restore by reasonable emendation the synodic period of Mars to 860 or 840 days, but there is no trace of the basic cule that for a given amount of time, the synodic cycles plus the zodiacal cycles equal the solar cycles. Moreover, the only phenomenon ascribed to the third and fourth spheres is latitudinal motion, but this could not have been the primary purpose of the model given that a simpler model would do as well. However, it seems that Simplicius would have had a difficult time seeing this. I argue *(pave *Yavetz) that at least Sosigenes understood the motion of the fourth sphere to be relative to the third and not the second sphere, and that the planet must be on the equator of motion, but also that this is a strange assumption. The resulting curve is Schiaparelli’s hippopede or horse-fetter (with some strange difficulties on the way in the representation of horse fetters in Greek art and the method of horse training, called the ‘fetter’, but with some weak confirmation from Dercyllides, *c. *end of 1st cent. BC). Moreover, for a given angle of inclination of the third and fourth spheres, there are four possible hippopede motions, for which the text fully determines one, although some questions remain as to its latitude to the ecliptic. Yet it does not provide the angle of inclination. This too is notable.

Hence, although the mathematical model is fully determined, we have no idea of the phenomena to be modelled, and retrogradation is but one candidate among several, such as invisibility times, the time between last morning and first evening, etc. However, although it is clearly more likely that horizon phenomena motivated early astronomers, retrogradation can also be determined from horizon phenomena in several ways. Hence we are faced with a wide variety of horizon phenomena requiring anomalistic motion.

When we turn to Callippus, we know much less, but there is one peculiarity of the testimonium in Simplicius: the motivation for the addition of two spheres to the solar model is the variation in seasons attributed to Meton and Euctemon, and not the seasons attributed to Callippus by the *Ars Eudoxi. *Moreover, we need also to look at Eudemus’ attribution of this inequality to Thales. Using Theon of Smyrna, we may speculate on the solar model, but not much more than that. These oddities serve to strengthen the plausibility of our sources. However, a recently edited text of Epicurus from the last decade of the fourth century BC suggests the importance of horizon phenomena, at least for the sun. It also indicates that we should not suppose that the models were numerical calcula-don devices, but were rather the basis for physical models, and that these physical models were to explain celestial positions. This too is surprising and adds profoundly to our understanding.

**Reviel Netz: Why did Greek mathematicians publish their analyses?**

Some Greek mathematical propositions contain not only the regular presentation (which may be called a Synthesis’), but also a further unit, which may be called an ‘analysis’. This has been presented and studied in the past as part of the context of discovery of Greek mathematical results. However, the fact remains that this is also written down, and the question is, why? What purpose do analyses serve *in the context of presentation? *Having put forward the question, I offer a brief conjecture: starting from the role of analysis for problems; showing the more competitive nature of different solutions to the saure problem; and arguing that an analysis could serve as a competitive tool in support of a new solution to the problem, especially by creating the illusion that a certain solution is not only possible but, in a sense, necessary. It is also argued that, regardless of such specific conjectures, once our attention has been shifted to the presentational function of the analysis, it should be recognized how little evidence in fact we have for its purported role for discovery. The general moral is simple: texts tell of presentation, not of discovery.

**Reviel Netz and Eleanor Robson: Greek and Babylonian Formulaic Language**

While it is recognized that both Greek and Babylonian mathematics are written in a regimented, specialized language, hardly any attempt has yet been made to characterize the nature of those uses of languages. We offer an analysis of mathematical texts, based on the idea of `formulae’ (as developed in Homeric studies), with a special emphasis on the way in which larger structures are composed out of such basic formulae. We develop in detail two cases in Greek and Babylonian mathematics, pointing out the underlying similarities and differences. Part of the differences have to do with the role of numerical values in Babylonian mathematics. More fundamentally, the Babylonian text is much less articulated, so that larger-scale structures based on atomic formulae do not lead to complex, rich systems, allowed by the Greek mode of combination of formulae.

**Eleanor Robson: Wrong mathematics from Nippur**

We know of several hundred school mathematical problems from early second millennium Mesopotamia, but so far only one of them could be traced to the ancient scribal centre of Nippur. This year I discovered a few more in the collection of the University Museum, Philadelphia, two of which I present in this talk.

The first tablet, CBS 11618, contains two problems on finding the volume of a cube-shaped hole from its sides, and then working back from the volume to find the sides of the cube again. This second problem is also known from an eighteenth century school tablet from the small town of Eshnunna, some two hundred miles north of Nippur. Both exemplars use the same numerical data: the sides of the hole are 1/2 rod, or around 3 metres, long, and its volume therefore 1/8 cubic rod (thus neatly avoiding the need to find difficult cubes and cube roots). The solution is complicated slightly by the fact that the Mesopotamian unit of volume was not a cube of length 1 rod, but a block measuring 1 rod square horizontally by 1 cubit (= 1/12 rod) height. Therefore, to solve the problem successfully the cube on the length must be multiplied by a factor of 12 to convert it to the correct units. While the Eshnunna example does this, the Nippur ones do not.

The second Nippur tablet, 3N-T 117, contains just two fines, giving the base 60 reciprocal of 4 26 40 (4 4/9) erroneously as 2 13 20 (2 2/9) instead of 13 30 (13 1/2).

I suggest possible reasons for these elementary blunders and request further suggestions from the audience.

**Sabine Rommevaux : La proportionnalité numérique dans le livre VII des Eléments de Campanus**

Nous ne savons que peu de choses concernant Campanus. Cet ecclésiastique, né en Italie dans le premier quart du MF siècle, bénéficia de la protection du pape Urbain IV jusqu’en 1263. Il fut ensuite, à Paris, aumônier du pape Nicolas IV, puis du pape Boniface VIII. Il passa probablement ses dernières années au couvent de Viterbe, en Italie, où il mourut en 1296. Il** **apparaît dans certains écrits avec le titre “Magister Campanus Novariensis”, sans que l’on sache à quelle université il aurait été rattaché.

Son édition des *Eléments *d’Euclide date des environs de 1260. Elle fut éditée dès 1482 à Venise et connut un grand succès à la Renaissance. Ce n’est pas une traduction, mais une recension, c’est-à-dire une réécriture, une adaptation. Le cadre en est la traduction latine faite au XII’ siècle probablement par Robert de Chester à partir d’une traduction arabe d’un texte grec. Cette version latine fut très diffusée au Moyen Age et continuellement retravaillée par les étudiants, ce qui rend très complexe l’histoire de ce texte. Cette version est caractérisée par des preuves souvent résumées et réduites à de simples indications des constructions et des numéros des propositions à utiliser.

Le livre VII des *Eléments, *sur lequel nous avons porté notre étude, est le premier des livres arithmétiques. Il concerne essentiellement la proportionnalité numérique. Les historiens y ont décelé un certain nombre de lacunes : absences de postulats et axiomes propres à l’arithmétique, notions non ou mal définies, propriétés non démontrées, démonstrations incomplètes. Par ailleurs, la version latine utilisée par Campanus a des caractéristiques propres, comme l’absence des définitions des notions fondamentales de la proportionnalité que sont les notions de partie, parties et multiple, et l’utilisation de propositions du livre V, qui concernent la proportionnalité des grandeurs dont les nombres sont exclus.

Campanus, excellent mathématicien, a porté son attention sur la structure logique du traité euclidien tel qu’il l’a reçu via la traduction latine de Robert. Ainsi, il rejette explicitement l’usage du livre V dans les livres arithmétiques, notant qu’ils n’ont pas le même objet et sont fondés sur des principes différents. Par ailleurs, il ajoute des axiomes et des définitions, complète les preuves des propositions, démontrant au passage les propriétés fondamentales de la proportionnalité numérique non explicitées par Euclide. Pour tout cela, il puise ce qui lui manque dans *l’Arithmétique *de Jordanus, remarquable synthèse entre l’arithmétique euclidienne et l’arithmétique de Nicomaque transmise par Boèce.

Notre étude a mis à jour le travail de Campanus sur le texte euclidien, caractérisant son projet et montrant comment il utilisait ses sources.

**Ken Saito: Compounded Ratio Revisited**

Compounded ratio is a simple concept for us: the ratio of A to C is said to be compounded from that of A to B and B to C, and this is what Greek geometers understood. One can also associate compounding to multiplication, since A/C = A/B.B/C, and this kind of understanding can be found in later Greek authors.

A thorough examination of the use of compounded ratio in the first four books of Apollonius’ *Conics *reveals that:

- The terms concerning the compounding and de-compounding of ratios are common to the terms of addition and subtraction, so that compounding was not conceived as a multiplication at least in the context of geometrical arguments (but not so much even as addition or as any operation as is explained below).
- When the term `compounded’ is used for a ratio, it is always stated that a ratio `is compounded’ (from other ratios), or a magnitude ‘have a ratio compounded’ (from other ratios) to another magnitude. It never occurs that one compounds two ratios to produce a new ratio. This suggests that there did not exist an operation of (compounding; but only compounded’ ratio, a complicated ratio to be expressed by simpler ratios. In short, Greek geometry was not very much concerned with abstract operations of magnitudes or ratios.

- The most frequent type of use of compounded ratio in Apollonius (e.g. in I-11) can be replaced by another argument using only more fondamental technique. This is in fact what Euclid does in
*Data*70. I suggest that repeated use of this technique may have led Euclid (or someone else) to introduce the term and concept of compounded ratio, formulating the proof of VI-23 in the*Elements.*This conjecture would explain almost all of the idiosyncrasy about compounded ratio in the*Elements.*

What emerges from ail these arguments, and what seems to be more important than each assertion or suggestion I made, is the need to consider the role of problem-solving techniques in the development of Greek mathematics.

**Ivo Schneider: The influence of Greek mathematics on the content and fate of the mathematical production of German reckoningmasters**

In the seventeenth century we can witness a change in the way of dealing with Greek mathematics. In the first half, Greek mathematics was still used as kind of a quarry for finding new problems and their solutions, which eventually reached a new level of abstraction. Alter the middle of the seventeenth century, interest in Greek mathematics concerned mainly editorial and interpretive work, for example Edmund Halley tried to reconstruct in the early eighteenth century the lost books of Apollonius’ *Conica, *but this reconstruction had nothing to do with his other mathematical activities. At about the same time we can observe this shift in the treatment of Greek mathematics: the mathematical activity of the reckoningmasters declined to a degree that they ceased to be creative mathematicians.

My claim is that these two events are not coincidental but in a way are interrelated: at least the availability of Greek mathematics from the middle of the sixteenth century seriously influenced the productivity of the professional reckoningmasters. I propose to distinguish four phases for this impact of Greek mathematics:

- Greek mathematics, especially the works of Euclid, Archimedes and Apollonios, raised new expectations and demands of potential clients of the reckoningmasters from the late sixteenth century onwards.
- Greek mathematics offered an alternative model to the mathematical style of the reckoningmasters. In contrast to the presentations of the reckoningmasters, Greek mathematics also encouraged independent creative mathematical activities amongst the philomaths.
- Greek mathematics also influenced the mathematical production of professional mathematicians, who tried to satisfy the new demands of their clients within the limits given by their interests.
- The inherent conflict between the economically conditioned interests of the mathematical professionals and the new research interests of the amateurs could not be solved.

The growing mathematical production of the mathematical amateurs eventually outdated that of the mathematical practitioners and reckoningmasters. The new model of the creative economically disinterested amateur was shaped by Descartes and his *Géométrie *from 1637.

**Jacques Sesiano : Les tables de multiplication grecques des entiers**

Le premier outil de travail dont on a besoin pour le calcul est la table de multiplication. Pour les Grecs, qui utilisaient trente-six symboles (pour chacune des neuf unités, dizaines, centaines et milliers), une table de calcul complète devait donc contenir 36^{2}=1 296 produits, ou 666 si l’on supprime les répétitions dues à la commutativité du produit, ou encore 1 369, respectivement 703, si l’on convient d’introduire encore comme facteur la première myriade.

Alors que les papyri grecs n’ont conservé que des fragments des tables de multiplication, leur forme générale nous est connue par des sources ultérieures qui en avaient conservé l’usage. Lune de ces sources est arménienne, une autre byzantine, la troisième étant, elle, arabe, mais décrit les tables qui étaient alors en usage chez les Coptes. Il apparaît que les papy-ri confirment pleinement les descriptions ainsi apportées et que les tables de multiplication étaient de trois types :

- Table commune (selon la source arabo-copte). Chacun des 36 chiffres de 2 à 10 000 est multiplié par les chiffres 1, 2, …, 10 ; il y aura donc 360 produits. Les papyri montrent que ce type de tables était d’usage courant, et étudié en particulier par les écoliers. Les documents coptes sont également abondants.
- Table étendue (selon les deux sources, arménienne et byzantine). La table précédente est augmentée des produits des mêmes 36 chiffres par 20, 30, …, 90, 100, 200, …, 900, 1000, 2 000, …,9 000, 10 000. Elle comprend donc en tout 1 369 produits.

– Autre table étendue (selon la source arabo-copte). L’arrangement des produits est ici différent. Chacun des 36 chiffres est multiplié par la tétrade que forme chaque unité des quatre ordres, se terminant en général (de manière partiellement répétitive) par la tétrade 10, 100, 1000, 10 000 ; la table complète contiendra donc 1440 produits. Les témoignages conservés de l’existence de la première forme de ces tables étendues sont rares. Ils sont en revanche nombreux pour la deuxième forme, tant en copte qu’en grec. Il apparaît que cette table était destinée à l’usage professionnel plutôt que scolaire.

**Christian-Marinus Taisbak: ***Are the foundations of Euclid’s ***Data ***sound or rotten?*

Most propositions in the *Data *are tools for geometric analysis, raising few problems of interpretation. Normally a theorem states that if some items are `given’, some other items are also `given’. Obviously `given’ is used not only of the ‘input’ to a proposition, but also of its ‘output’, that is what is proved.

Proposition 5 says that if a magnitude has a given ratio to some part of itself, it will also have a given ratio to the remainder. By taking that proposition to pieces, **I **aim in my talk to isolate the different meanings of `given’ (including the sense of laken’) , and to illustrate how `givenness’ is proved by importing extraneous items (hidden correlates’) in order to apply propositions from Euclid’s *Elements.*

The analysis eventually concentrates on Definitions 1 and 2, and Propositions 1 and 2, which say:

Definition 1. ‘Given’ in magnitude is said of figures and lines and angles for which we can provide equals.

Definition 2. A ratio is said to be given for which we can provide the same.

Datum 1. The ratio of given magnitudes to one another is given.

Datum 2. If a given magnitude has a given ratio to some other magnitude, the other is also given in magnitude.

Even if it is possible to agree on what ‘provide’ might mean, it remains a question whether such statements as Propositions 1 and 2 can be proved, respecting Aristotle’s daim that a statement should be proved by statements that are simpler than the one to be proved. When the *Data *in 1 and 2 use such heavy ammunition as the alternative metamorphosis of *Elements *V.16, circularity is an eminent risk. So I am afraid that the Foundations of the *Data *are not so sound as was to be expected if its author was Euclid of Alexandria. But then, the House is in places a very clever building, much used and admired by Arabic and later mathematicians, so perhaps it really does not matter.

**Christian-Marinus Taisbak: A reconstructed analysis of Apollonius, Conics 127 (whicb bas troubled some), and a piece of heuristics: how Euclid (or some Greek) may have bit on the idea of Elements X111.10 and its proof**

If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the sum of the squares on the side of the hexagon and on that of the decagon inscribed in the same circle. I call the method used in my analysis `Splitting a square’, recalling the Pythagorean Theorem, *Elements *1.47, which has inspired my analysis.

**Sabetai Unguru and Michael Fried: Apollonius, Conica V, and Evolutes**

The *Conica *is perhaps the greatest extant Hellenistic mathematical text. lis interpretation by historians of Greek mathematics has typically been along analytical (algebraic) fines. The treatise is, however, purely geometric in character, and the geometry involved is Greek, i.e., synthetic geometry. Book V of the *Conica *is the most extended of the eight books comprising the treatise (only seven of which are extant, four in the original Greek, the other three in an Arabic translation, Book VIII being lost) and it has received special attention due to its being perceived as a book dealing with *normals, evolutes, *and *envelopes, *all modem mathematical entities crucial in the study of curves. And yet there is nothing to substantiate such an interpretation except the ahistorical ability of the modem mathematician to apply to the understanding of ancient texts techniques and concepts foreign to the Greek mathematician. It is against such anachronistic interpretations that our communication stands up, showing their inappropriateness and the distortion of the text to which they lead, arguing, on the contrary, for the genuineness of the Apollonian conception of Book V as just a book about maximum and minimum lines.

**Hans-Joachim Waschkies: Mathematics in the Aristotelian treatise Perî atômoon granunoôn.**

According to ancient sources, a treatise *On indivisible fines *is ascribed both to Aristotle and Theorphrastus. The ancient commentators already did not know whether the preserved *Perî atômoon grammoôn *is the one written by Aristotle, but in any case it is a source from the second half of the 4th century BC, which is of great interest for the historian of Greek mathematics. The reason is that the mathematical knowledge of its author gives hints for the relative or even the absolute chronology of some of the mathematics summed up in Euclid’s *Elements. *There is circumstantial evidence that Greek geometers denoted points by special names like middle *(kéntron), *end *(eschaton) *or extremity *(peras) *until the general terms *stigma** *and *sämeion, *of which eventually only *sämeion *survived, were introduced to denote points in general in the time of Plato. The author of *Peri atômoon grammoôn *uses *stigma**. *On the other hand he says that irrational lines from Book X of the *Elements, *part of which is said to be written in the first quarter of the 4th century, were a topic of conversation in his day. So it is advisable to look for propositions of Book X in which the term ‘point’ is explicitly used. There are surprisingly few of them among the 115 propositions summed up there: X.17 (commensurability of special pairs of lines) ; X.42-47 (theorems of uniqueness) ; X.54, 91, 93, 95 (theorems connected with the Lesser *(ellatoon), *the Greater *(meîzoon) *and their family), X.97, X.99 and X.100 (construction of special complex irrational fines). This seems to indicate that the blocks from Book X, which are not only connected by the use of *sämeion *in them but also differentiated by the mathematical topics treated in them, are lace additions to an old core of Book X. As in the *Physics *and the *On coming to be and passing *away of Aristotle there is ample evidence in the treatise *Peri atômoon grammoôn *that Euclid’s definitions of the point (1.2), the fine (1.2) the surface (I.5) and the solid (X1.1) were introduced *en bloc *to show that it is impossible to add up points to form a fine, fines to form a surface and surfaces to form a solid. These problems are clearly connected with the so-called proofs by the method of exhaustion introduced by Eudoxos of Cnidus *(c. *39-337 BC), and therefore the relevant passages from die treatise *Peri atômoon grammoôn *offer another opportunity to date parts of the *Elements. *These examples indicate that the writings of ancient authors who themselves were not mathematicians, are a still insufficiently exploited source for dating Greek mathematical theories.