For textual studies relating to the ancient mathematical corpus the efforts by the Danish philologist, 1. L. Heiberg (1854-1928), are especially significant. Beginning with his doctoral dissertation, Quaestiones Archimedeae (Copen hagen, 1879), Heiberg produced an astonishing series of editions and critical studies that remain the foundation of scholarship on Greek mathematical 4 science. For comprehensiveness and accuracy, his editions are exemplary. In his textual studies, as also in the prolegomena to his editions, he carefully described the extant evidence, organized the manuscripts into stemmata, and drew out the implications for the state of the text. 5 With regard to his Archimedean work, Heiberg sometimes betrayed signs of the philologist's occupational disease - the tendency to rewrite a text deemed on subjective grounds to be unworthy. 6 But he did so less often than his prominent 7 contemporaries, and not as to detract appreciably from the value of his editions. In examining textual questions bearing on the Archimedean corpus, he attempted to exploit as much as possible evidence from the ancient commentators, and in some instances from the medieval translations. It is here that opportunities abound for new work, extending, and in some instances superseding, Heiberg's findings. For at his time the availability of the medieval materials was limited. In recent years Marshall Clagett has completed a mammoth critical edition of the medieval Latin tradition of Archimedes,8 while the bibliographical instruments for the Arabic tradition are in good order thanks to the work of Fuat Sezgin."

practice, some of which is translated into the standard forms of public discourse, in publication, and then retranslated by readers and adapted again to local practice at self-selected other sites. Less may be left implicit, and additional personal and contextual information is carried, by the "informal" methods of communication which mediate local projects and international publication. But both methods of communication are screens as well as conduits of information. History and Background of the Volume When the planning of this volume began in the spring of 1977, it seemed a natural part of the mandate for the Yearbook. There had also been a number of more specific calls for deeper studies of research in social and historical context (3). These calls can be seen as giving permission and legitimacy to ask questions otherwise seen as irrelevant, or even disrespectful, and as attempts to develop new perspectives from which to ask and to answer them. The implied and expressed irreverence toward traditions and institutions of great respect may have prolonged this process of initial apologetics. In any case, in May 1977 the theme of 'The Social Process of Scientific Investigation' was proposed to the Editorial Board for Volume IV as "the heart of the subject. " That is, the ethnographic and detailed historical study of actual scientific activity and thinking at or close to the work site.

practice, some of which is translated into the standard forms of public discourse, in publication, and then retranslated by readers and adapted again to local practice at self-selected other sites. Less may be left implicit, and additional personal and contextual information is carried, by the "informal" methods of communication which mediate local projects and international publication. But both methods of communication are screens as well as conduits of information. History and Background of the Volume When the planning of this volume began in the spring of 1977, it seemed a natural part of the mandate for the Yearbook. There had also been a number of more specific calls for deeper studies of research in social and historical context (3). These calls can be seen as giving permission and legitimacy to ask questions otherwise seen as irrelevant, or even disrespectful, and as attempts to develop new perspectives from which to ask and to answer them. The implied and expressed irreverence toward traditions and institutions of great respect may have prolonged this process of initial apologetics. In any case, in May 1977 the theme of 'The Social Process of Scientific Investigation' was proposed to the Editorial Board for Volume IV as "the heart of the subject. " That is, the ethnographic and detailed historical study of actual scientific activity and thinking at or close to the work site.

The present work has three principal objectives: (1) to fix the chronology of the development of the pre-Euclidean theory of incommensurable magnitudes beginning from the first discoveries by fifth-century Pythago reans, advancing through the achievements of Theodorus of Cyrene, Theaetetus, Archytas and Eudoxus, and culminating in the formal theory of Elements X; (2) to correlate the stages of this developing theory with the evolution of the Elements as a whole; and (3) to establish that the high standards of rigor characteristic of this evolution were intrinsic to the mathematicians' work. In this third point, we wish to counterbalance a prevalent thesis that the impulse toward mathematical rigor was purely a response to the dialecticians' critique of foundations; on the contrary, we shall see that not until Eudoxus does there appear work which may be described as purely foundational in its intent. Through the examination of these problems, the present work will either alter or set in a new light virtually every standard thesis about the fourth-century Greek geometry. I. THE PRE-EUCLIDEAN THEORY OF INCOMMENSURABLE MAGNITUDES The Euclidean theory of incommensurable magnitudes, as preserved in Book X of the Elements, is a synthetic masterwork. Yet there are detect able seams in its structure, seams revealed both through terminology and through the historical clues provided by the neo-Platonist commentator Proclus."

The present work has three principal objectives: (1) to fix the chronology of the development of the pre-Euclidean theory of incommensurable magnitudes beginning from the first discoveries by fifth-century Pythago reans, advancing through the achievements of Theodorus of Cyrene, Theaetetus, Archytas and Eudoxus, and culminating in the formal theory of Elements X; (2) to correlate the stages of this developing theory with the evolution of the Elements as a whole; and (3) to establish that the high standards of rigor characteristic of this evolution were intrinsic to the mathematicians' work. In this third point, we wish to counterbalance a prevalent thesis that the impulse toward mathematical rigor was purely a response to the dialecticians' critique of foundations; on the contrary, we shall see that not until Eudoxus does there appear work which may be described as purely foundational in its intent. Through the examination of these problems, the present work will either alter or set in a new light virtually every standard thesis about the fourth-century Greek geometry. I. THE PRE-EUCLIDEAN THEORY OF INCOMMENSURABLE MAGNITUDES The Euclidean theory of incommensurable magnitudes, as preserved in Book X of the Elements, is a synthetic masterwork. Yet there are detect able seams in its structure, seams revealed both through terminology and through the historical clues provided by the neo-Platonist commentator Proclus."