Abstracts of Talks
Euclid's Elements in the Islamic world (8th-19th centuries)
In my paper, I will briefly outline the current research history of the Arabic transmission of Euclid's Elements and its new results. The focus of my paper will, however, be on other, so far less studied aspects of the transmission of the Elements in the Islamic world. I will survey features such as the remarkable variability of the linguistic format of the text, the introduction of Greek commentaries into the main body of the Elements, the substantial differences between the transmission of the Elements in Arabic and that in Persian and the scholarly as well as broader cultural relevance of the text and its transmitters.
Interpreting Euclid - early and late
Euclid's Book XIII is concerned with constructing the regular polyhedra, a complicated business. One of his constructions,that of the icosahedron, is in particular notoriously difficult. It is therefore ironic that the earliest fragments of Greek geometry, found on Elephantine Island in the German expedition of 1906, are concerned with it. I will show photographs of thesefragments and explain what they and Euclid's construction are all about with the help of some simple computer animations. I will also show photographs of the graphical tradition of this theme.
At the end I'll comment on the role of computer graphics in explaining Euclid at various levels of education.
Rethinking the elements – two thousand years of reflections on the foundations of geometry
What is geometry about? The answer given in Euclid's Elements is that it is about shapes, triangles, rectangles, circles and the like, presented in a systematic way that proceeds rigorously from a small number of explicit assumptions. A more modern answer would be that (metrical) geometry is about concepts such as length and angle, and the contexts where those terms make sense. The conceptual basis for Euclid's Elements is not always explicit, and when it has been found obscure there have been interesting disagreements about what needed to be done to clarify the matter. Some of these debates will be discussed, most, but not all, concerning the so-called problem of parallels.
The mathematical legacy of Euclid's Elements
In this talk I will discuss the mathematical content of the Elements, together with various questions that arise upon reading Euclid, and the later developments stimulated by them. These include the axiomatic foundations of geometry, the discovery of non-Euclidean geometries, the development of the real number system, and connections between geometry and modern algebra and analysis.
Euclid in Chinese... and in Manchu
The Jesuits who entered China in the late 16th century put science in the service of evangelisation in their missionary work. Euclid's Elements of Geometry, in the Latin edition done by Clavius, was among the first works translated into Chinese as a result of their collaboration with some Chinese scholars. The reception of this translation, the Jihe yuanben (1607) during the 17th century, parallels the history of the Elements in Europe at the time: the work underwent several rewritings as the context of study of geometry changed. This culminated with the imperial appropriation of mathematics during the reign of Kangxi (1662-1722), the second emperor of the Qing dynasty (1644-1911), for whom Euclidean geometry was rewritten, first in Manchu, then in Chinese.
From Euclid to Arethas
The oldest complete, or even nearly complete, surviving manuscripts of Euclid's Elements, including Bodleian d'Orville 301, were made in the ninth and tenth centuries of our era; Euclid is usually supposed to have written it in the early third century B.C. This talk will survey what we currently believe was the history of the book and its use in the Greek-speaking world during its first twelve centuries starting with the elusive Euclid himself.
Euclid's Elements in Hebrew
As far as Medieval Hebrew mathematics are concerned, Euclid's Elements is the most translated work, the most commented upon and the most widely distributed. Limiting oneself strictly to the various versions of this treatise, one can point to more than thirty extant manuscripts. I have identified four different versions of the text in Hebrew (translated from Arabic sources), the most widely circulated version being that of Moses in Tibbon, completed in 1270 in Provence. To this corpus should be added two adaptations of the Euclidean text written in Arabic and translated into Hebrew: one of them is Avicenna's (Ibn Sina)'Foundations of geometry', a well-known shortened Arabic version of Euclid's Elements.
Who started the Euclid myth?
The Euclid myth is described by Davis and Hersh as the view that "the books of Euclid contain truths about the universe which are clear and indubitable." The claim of indubitability (or certainty) plays an important role in sixteenth-century discussions of mathematics in which the influence of Proclus' commentary on the Elements is apparent and in later texts which seem to descend in some way from those discussions. The assertion that mathematics is certain can be traced back to Averroes and probably to earlier Islamic philosophers. I argue that there is no good ground for assigning the propagation of the Euclid myth to either the author of the Elements or to Plato or Aristotle or even to Proclus. An important historical factor in its propagation was a misunderstanding of the Greek word for exactness or precision, akribeia, although doubtless other factors (e.g., changes in the character of mathematics, the rise of modern skepticism) were involved as well.
Clay mathematics: Euclid's Babylonian counterparts
So canonical has the Elements become that it is easy to forget that it did not belong to the only mathematical tradition of antiquity. Euclid's contemporaries in Hellenistic Babylon, who wrote on clay, were heirs to a mathematical tradition at least as ancient as Euclid's is for us today. Using a mixture of archaeological evidence and published and unpublished cuneiform tablets from ancient Iraq I shall explore the lives, works, and motives of these long-forgotten individuals the better to understand the extraordinary nature of Euclid's achievement in the context of his time.
New technologies for the study of Euclid's Elements
This presentation will explore some of the contributions that can be made by advanced information technology to the study of Euclid's Elements, focusing on three applications in particular. (1) The linking of electronic versions of the text of the Elements to online manuscript images. (2) The use of linguistic technology for automatic morphological analysis, discovery of technical terminology, and generation of multilingual lexica in the analysis of parallel versions of the Elements in different languages. (3) The mapping of deductive structures and their visual representation. All software to be demonstrated is in the public domain and available for download at http://archimedes.fas.harvard.edu; in addition, participants will have the opportunity to obtain guidance in the use of these tools for individual research purposes.
The Heiberg Edition of Euclid's Elements : an incorrect text or a false history of the text?
During the last century, the prevailing history of the text of Euclid's Elements is that which Heiberg proposed when he wrote what is, still now, the latest critical edition of the Greek text. Since the nineteen-nineties, this history has been challenged. I will return to this debate, in particular to the contributions and the limits of the Arabic and Arabo-Latin medieval translations. What can we learn from them about the Greek text ?
The Elements: the transmission of the Greek text
The aim of this lecture is to throw light on how Euclid's major work the Elements was handed down through the long period in which hand-written copies were the only means of preserving a text. It happens that among the various manuscript copies in the collections of the Bodleian Library there is one of exceptional interest written early in the Middle Ages (A.D. 888). I shall concentrate on this MS, with a reading of a chosen proposition as shown on the handout. I shall add a few remarks on other aspects of the transmission, in particular the occasional use of Arabic numerals by medieval scribes who added annotations to the margins of their copies.
The achievements and limitations of the theory of proportion in Euclid's Elements Book V
Book V is an exposition of the work of Eudoxus, and is considered to be the greatest achievement of Euclidean geometry. It is the first example of abstract algebra, and enabled ratios to be defined and used rigorously in proofs (without the availability of real numbers). For example, a:b = c:d, where
However, a serious criticism of Euclid's treatment is that, by using the additive structure of magnitudes, he was unable to define the ratio of two ratios. This prevented the Greeks from being able to (i) define cross-ratios and develop projective geometry; (ii) define products and develop group theory; and (iii) define acceleration and develop dynamics.
In the lecture, we shall introduce a new axiom for magnitudes (related to the Archimedean axiom) that can be used instead of the additive structure to prove all the propositions of Book V not involving addition or subtraction. This enables us to develop the theory without appealing to the additive structure, and hence to define ratios of ratios.