**1. The evolution of the Euclidean elements. A study of the theory of incommensurable magnitudes and its significance for early Greek geometry (1975).**

**1.1. From the Introduction.**

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 Pythagoreans, advancing through the achievements of Theodorus of Cyrene, Theaetetus, Archytas and Eudoxus, and culminating in the formal theory of *Elements*, Book 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.

**1.2. Review by: David H Fowler.**

*The Mathematical Gazette* **60** (413) (1976), 229.

This book studies the history of the Greek theory of irrationals, which culminated, with Theaetetus and Eudoxus, in the geometric constructions and classification of ratios (medial, binomial, apotome, bimedial, etc.) in Euclid's Elements Book X. It also includes a reassessment of some aspects of Pythagorean arithmetic and the geometric algebra of Book II. The explicit view adopted is that a study of incommensurable magnitudes was of interest in its own right to Greek geometers; Eudoxus' later foundational innovations in Book V do not concern the author, and so he deals with him only incidentally and then mainly in connection with his contribution to the older theory of irrationals. This subject-matter( Boyer describes Book X as "the most admired and the most feared" of the Elements), the impeccably scholarly treatment, and the uncompromising attitudes the author takes towards the responsibilities of a historian, make the book difficult but rewarding reading. ... Much of the book sets out to challenge and replace currently held beliefs, and so it is right that the evidence and arguments are presented in detail. Unfortunately this will restrict the readership and even price it out of reach of many libraries. But anybody with a special interest in Greek mathematics, or who has been dissatisfied with the tentative accounts of pre-Euclidean mathematics given in more general histories, should make a special effort to study this book.

**1.3. Review by: Bartel L van der Waerden.**

*Historia Mathematica* **3** (4) (1976), 497-499.

The main part of this book is excellent. It is based on a thorough study of the testimonies and a sound mathematical and philological analysis. It presents a consistent, plausible picture of the development of the theory of incommensurable magnitudes from Theodorus to Euclid. On the other hand, in the introduction and in some other parts, the author exposes his views on the origin of the first four books of Euclid. In particular, he ascribes Book 2 to Theodorus of Cyrene. With these views I do not agree.

**1.4. Review by: Sabetai Unguru.**

*Isis* **68** (2) (1977), 314-316.

The unexpected discovery of incommensurable magnitudes by the Pythagoreans marks a turning point in the history of mathematics. It occurred, according to Wilbur Knorr's analysis of the sources, sometime between 430 and 410 B.C. in the context of studies of the side and diagonal of the square. It did not lead to any real crisis among working mathematicians who kept on doing what they have been doing before the discovery. Only later did Plato (and his associates) call for a rigorisation of mathematical practice. Within the philosophical domain of the Pythagorean school, however, the impact of the discovery was such as to lead to modifications of the earlier naive views on number atomism. Wilbur Knorr (W.K.) speculates that the discovery of incommensurability might have taken place at Thebes, where Lysis, Philolaus, and Eurytus congregated at different times, and that from there it was carried over to Athens, perhaps by Hippasus of Metapontum, who taught the discovery to Theodorus. ... Knorr advances his own interpretation and reconstruction of the mathematical proofs involved in the work of Theodorus and Theaetetus. By adhering closely to the Platonic text (Theaetetus, 147C-148B), and by refusing to allow foreign, unacceptable mathematical techniques to encroach on his interpretation and dictate its character, Knorr tries to avoid the historical pitfalls of the solutions offered by his predecessors, all of which he finds wanting in various respects and degrees: all these solutions remain at a (variable) distance from the text and all had recourse to historically inadmissible procedures, be they approximating techniques, algebraic reconstructions, or unwarranted appeal to anthyphairesis. ... It is a well-told story, in which the ancient and modern sources are fully mastered and handled competently and insightfully. Wilbur Knorr is alert to the need to deal with his materials in an historically acceptable way, he shuns as a rule anachronisms in interpretation, and he always surveys critically previous interpretations before amending or dismissing them. This is a remarkable accomplishment by any reasonable standards. It is a book to be taken seriously by students of ancient Greek mathematics.

**1.5. Review by: Maurice Caveing.**

*Revue d'histoire des sciences* **29** (2) (1976), 180-183.

The issue of incommensurable magnitudes was of great importance in the science and Greek philosophy prior to Euclid. However, in the absence of pre-Euclidean treaties that did not survive, we can make history in confronting systematically indirect evidence for traces of ancient treatments that can be located in Euclid himself. This book uses this perspective for the following three problems: what were the mathematical techniques used at various stages of the theory? What was its significance for pre-Euclidean geometry and for the ordering of the 'Elements'? What does its development give to the contemporary development of logic and thinking of philosophers of science?

**2.**Ancient Sources of the Medieval Tradition of Mechanics: Greek, Arabic, and Latin studies of the balance (1982).

**2.1. From the Introduction.**

The medieval science of weights owed an extraordinary debt to the production of a single work, a treatise on the balance, *Kitab al-Qarastun, *by the 9th-century mathematician-astronomer, Thabit ibn Qurra. It retained a prominent place within the theoretical section on mechanics in the rich compendium compiled by al-Khazinl, *Kitab Mizan al-Hikma, *two centuries later. Beginning from the 12th century, it exercised a major influence on mechanical studies in the Latin West, through the translation as the *Liber Karastonis *made by Gerard of Cremona. Four centuries later, writings on mechanics still clearly betrayed their provenance through elaborations and commentaries on this work. Thabit's writing is the centre of interest in the present study. Through a close examination of the Arabic and Latin versions now extant I propose to reveal the pattern of their complex interrelation and to discover the nature of the connections between this work and similar writings from this period. In the second part of this study I will take up the problem of the character of the dependence of these works on the earlier Greek tradition of mechanics.

**2.2. Review by: Gian Carlo Garfagnini.**

http://www.mirabileweb.it/mel/-ancient-sources-of-the-medieval-tradition-of-mech/377084

At the centre of Knorr's study is the debt that the medieval science of the balance had towards the work of Thabit ibn Qurra, known as 'Liber Karastonis' and translated by Gerard of Cremona. Knorr's research also involves other works that, in the wake of Vitruvius (De architectura) and Aristotle (Mechanica), have dealt with the problem of scale. Among the ancient authors who are discussed and studied, are fragments of Euclid, Archimedes, Menelaus, Philon of Byzantium, Heron of Alexandria, Pappus and Eutocius; among the medieval, especially the Arabic versions of Greek authors and the 'Liber de canonio', an anonymous treatise of the XII/XIII century, the Latin version of an original Greek text which is lost. The eight appendices which conclude the volume examine particular aspects of traditional Arab and Latin versions of 'Liber Karastonis' with particular attention to their interrelations.

**3.**The ancient tradition of geometric problems (1986).

**3.1. From the Preface.**

Within the ancient geometry, a geometric "problem" seeks the construction of a figure corresponding to a specific description. The solution to any problem requires for its completion an appeal to the constructions in other problems already solved, and in turn will be applied to the solutions of yet others. In effect, then, the corpus of solved problems forms an ordered sequence in which each problem can be reduced to those preceding. The implications of this simple conception struck me, as I was completing a paper on Apollonius' construction of the hyperbola (1980; published in Centaurus, 1982), for it served to unify a diverse range of geometric materials I had then been collecting for some five years. That the ancient problem-solving effort took on such a structure is much what one would expect, given the prominence of the role of "analysis" for discovering and proving solutions; for this method seeks in each case to reduce the stated problem to others already solved. We possess one ancient work, Euclid's Data. which organizes the materials of elementary geometry in this manner; but there survives none which attempts the same for the more advanced field. The Conics of Apollonius, for instance, is by its own account ancillary to this effort, providing the essential introduction to the theory of conics through which the so-called "solid" problems might be solved; but it undertakes the actual solution of such problems only to a very limited extent. This salient omission from the extant record thus defines the project of the present work: to exploit the materials extant from Archimedes, Apollonius, Pappus and others in order to retrieve a sense of the nature and development of this ancient tradition of analysis. The present effort is conceived as an exploratory essay, intended to reveal the opportunities which the evidence available to us provides for an interpretation of the ancient field. A definitive survey overreaches its scope, however, and may ultimately be unattainable in view of the gaps in our documentation. Similarly, an exhaustive survey of the immense secondary literature on the history of geometric constructions could not be attempted here. The sheer bulk of this literature has made omissions inevitable, but I have endeavoured to include references to those contributions which seemed to me historically and technically stimulating, as well as directly pertinent to my special objectives.

**3.2. From the Introduction.**

The problems of cube duplication, angle trisection and circle quadrature have long attracted major interest among scholars of ancient Greek geometry. One reason is that the ancient literature on these problems constitutes an unusually well-documented area of ancient mathematics, spanning the whole course of antiquity from the pre-Euclidean period through the Hellenistic, Graeco-Roman and Byzantine periods, even into the Arabic Middle Ages. Throughout, these problems were affiliated with researches in the more advanced fields of geometry A historical survey of these studies could thus be hoped to provide insight of value into the development of the techniques of geometry in antiquity: What were the sources of the interest in these problems? How did their study relate to prior efforts on the same or associated problems? How did the results of these studies contribute to the development of new geometric techniques for the solution of geometric problems? What were the precise specifications for the construction of solutions, and did these conditions themselves change with the introduction of new techniques and concepts? Ultimately, did the ancient geometers and philosophers view the quest for solutions as having succeeded? Unfortunately, only a small part of this project now permits its presentation in the form of a straightforward narrative of how things happened. The extant technical literature has large gaps, so that we cannot claim to know all the solutions to these problems which the ancients worked out, nor even, in some cases, the geometers responsible for solutions which have survived. More difficult still is the assessment of motive, since technical documents rarely provide direct insight into the reasons which led geometers to take up specific problems and treat them in specific ways. For this we are often compelled to resort to testimonies in nonmathematical writings. But we then face new difficulties, in that their authors may not fully comprehend the technical issues and, at any rate, will have their own literary or philosophical concerns. The latter will invariably discourage them from presenting the technical materials in the clear and detailed manner we would prefer.

**3.3. Review by: Ivor Bulmer-Thomas.**

*The Classical Review, New Series* **39** (2) (1989), 364-365.

The basic idea behind this study of problem-solving in ancient Greek geometry by Dr Wilbur Knorr, Associate Professor of Stanford University, is that 'the corpus of solved problems forms an ordered sequence in which each problem can be reduced to those preceding'. There is one ancient work, Euclid's Data, which organizes the materials of elementary geometry in such a manner as to illustrate this principle, but nothing surviving attempts the same for the more advanced field. Knorr's aim in this work is to exploit the materials extant for Archimedes, Apollonius, Pappus and others in order to retrieve a sense of the nature and development of this ancient tradition of analysis. There were three classic problems which fascinated the Greeks for centuries and provoked many solutions - the duplication of the cube, the squaring of the circle and the trisection of an angle. The first illustrates Knorr's thesis perfectly, for the construction of a cube double the size of a given cube was reduced by Hippocrates of Chios to the finding of two mean proportionals between two lines, one double the other, and thereafter was always pursued in that form. Knorr's study of these problems is the most detailed yet offered, and as he investigates every by-way it virtually amounts to a history of Greek geometry within his chronological limits.

**3.4. Review by: Thomas Drucker.**

*Isis* **82** (4) (1991), 718-720.

One of the standard topics in Greek mathematics is the unsolvability of three problems: the duplication of the cube, the trisection of an angle, and the squaring of the circle. Other historians have recognized that ancient treatments of these topics were adorned with mythological touches, but in general the assessment of work on these problems in programmatic statements (such as by Pappus) has been taken at face value. Knorr proceeds instead by working his way exhaustively through the surviving texts on the subject from earliest Greek times through Plato and Euclid to the high points of Archimedes and Apollonius. He is looking at technical work rather than philosophy, but this is not the result of philosophical insensitivity. Knorr argues that recent scholarship has looked at ancient mathematics through anachronistic philosophical lenses. To see the response to the inability to solve the classical problems as akin to the foundational crisis in mathematics in the 1930s may have support from commentators, but Knorr argues that there is no sign of crisis in the practitioners.

**3.5. Review by: Derek Thomas Whiteside.**

*The British Journal for the History of Science* **23** (3) (1990), 373-375.

The three traditional problems of Greek classical mathematics are of course the duplication of the cube (more generally the finding of two mean proportionals between two given quantities or magnitudes), the trisection of an arbitrary angle, and above all, the quadrature of the circle. ... To gain all insight possible into the generality as well as particular instances of [Greek geometrical] analysis is Knorr's brief in this book; and he vigorously and forthrightly pursues it. Only bloodless professionals will carp, jeering at what they conceive to be his lack of rigorous scholarship. In his full-blooded, gutsy onrush he often ventures far out onto many a shaky branch. I do not propose to make tedious listings of what I conceive to be the times he seems almost like some Tom Cat to saw himself off from the main tree. It is enough for me, and I hope it will be for everyone, that all such weaknesses are much more than made up for by the freshness of his re-interpretations, the penetration of his insights, and the sturdy common sense of his suggested explanations and conjectures: all these founded on a deep knowledge of his subject. What he urges must be weighed by scholars before judgment is passed. I will not presume to anticipate.

**3.6. Review by: Maurice Caveing.**

Revue d'histoire des sciences 44 (3/4), La diffusion des sciences au XVIIIe siècle (1991), 487-489.

The book follows, through the history of antiquity, the evolution of three classical geometry problems: the duplication of the cube, the trisection of the angle and squaring the circle. The material examined is not different from that described in Heath (A History of Greek Mathematics (1960)); he works from sources preserved in Arabic. The author has also provided a companion volume, containing only the textual studies, which also includes some medieval Latin texts (Wilbur Knorr, Textual Studies in Ancient and Medieval Geometry (1989)). Heath follows the route of chronology, based on the methods of scholarly criticism; he often replaces demonstrations by modern equivalents and does not always justify its reconstructions. Knorr wants to make this book historically well enchained from a mathematical point of view. He attempts to reconstruct the "analyses" of the problems, to find motives and methods, "lines of thought" which should allow enable accounting for "syntheses", proposing unsuspected connections and changes to the chronology; it also gives some importance to the "personal" relationships between mathematicians, often subject anecdotes classical historiography from secondary or even apocryphal sources.

**3.7. Review by: Colin R Fletcher.**

*Mathematical Reviews*, MR0884893 **(88e:01010).**

This latest work of the author forms a comprehensive survey of the available literature on Greek problem-solving in geometry. The author says that his effort is conceived as an "exploratory essay", but the noun is hardly adequate whilst the adjective is a gross understatement. However, he does tell us that a second volume, on textual issues, is in preparation. In Greek geometry, a problem and a theorem are distinct objects. The latter asserts a property which a given configuration can be proved to possess; the former seeks the construction of some geometric entity which has a certain relationship to a given configuration. The construction may not exist for the general case and extra conditions (the diorisms) must be applied. Very early on, the trick was learnt of transforming one problem into another. Hippocrates (c. 400 B.C.) showed that the cube could be duplicated if two mean proportionals could be found between the side and its double. Thus, in one direction one had the synthesis, and in the other, analysis. Throughout the book, the author shows how a lost analysis can be reassembled to produce an extant synthesis. So a sequence of constructions is built up. The author concentrates on the three classical problems of the Greeks, cube duplication, angle trisection and circle quadrature, mainly because of the sources which have survived. Nevertheless he leaves no ivy leaf unturned in his investigation into the other Greek geometrical problems and their solutions. But he goes further. Where did the impetus for these problems come from? Was it from inside or outside geometry? And what did the Greeks mean by "a solution"? Why did they continue to work on the problems once they had a solution? Was this something to do with the classification of problems into planar, solid and linear? And what was this ridiculous classification anyway? Or had this continual effort something to do with "mechanical" solutions, as opposed to "geometrical" or "demonstrative" ones? Is a circle not mechanical simply because Euclid postulated its existence? And talking of the circle, what was the special significance of ruler and compass constructions? The author examines the whole fascinating field in an attempt to produce a unified whole. From Hippocrates through Eudoxus, Archytas and Menaechmus, to Archimedes and Apollonius, and finishing with the later commentators, he puts forward his views on the classical problems, the three and four line locus, the cissoid, conchoids, quadratrix, spirals, hippopede. The book is essential reading for anyone wishing to understand the complexities, technical, philosophical, textual and chronological, of the geometric problems of the Greeks.

**4.**Textual studies in ancient and medieval geometry (1989).

**4.1. From the Introduction.**

To the classical scholar, unlike the ancient physician, time is the enemy, for time - far too much of it - has afflicted the documentary record of antiquity with all manner of ills. Indeed, the metaphor of disease pervades the language of textual criticism: one speaks of the "corruption" that texts suffer in the course of their transmission, even as the purity of one textual form is "contaminated" or "infected" by defects from another. In the words of a distinguished guide to classical scholarship.

At times this goal is unattainable, and the text must be diagnosed as "incurable." But to the extent it is possible, the philologist, like a physician, takes on the role of "restorer" of texts. This description captures, I think, an essential aspect of the philologist's task, as commonly conceived within the field. In effect, the aim of philology is defined in terms of its product, the establishment of textual prototypes. This aim was already evident among the Renaissance humanists, but only in the 19th century were the methods of textual analysis refined to attain it, by systematizing procedures and minimizing the role of subjective judgments. Basically, through the critical examination of the extant manuscripts of a given work, one determines those that appear the most trustworthy (not necessarily the earliest), hence reflective of the oldest accessible form of the text. Through their collation one seeks to construct a text that is the closest possible approximation to the original form, but, where the evidence is questionable, to identify among the variants those most likely to be candidates for the original reading. At the same time, one seeks to identify passages likely to be later accretions - interpolations and other editorial changes - that in the course of a text's transmission have come to spoil its pristine form.The business of textual criticism is in a sense to reverse this process, to follow back the treads of transmission and try to restore the texts as closely as possible to the form which they originally had.

**4.2. Review by: Thomas Drucker.**

*Isis* **82** (4) (1991), 718-720.

In Studies Knorr turns to the detailed examination of the textual survival of some of the material discussed in 'The ancient tradition of geometric problems'. As a result, he is looking more at commentators and translators than at the original mathematicians ... Knorr ranges chronologically from late antiquity to medieval Europe and deals with texts in Greek, Arabic, Hebrew, and Latin. All these languages are represented by both transcriptions and facsimile reproductions, which bring home some of the difficulties involved in even starting the task of textual criticism. ... The task to which Knorr dedicates most of his energy is the detection of lines of ancestry and resemblance between surviving texts on the duplication, trisection, and squaring problems. He displays a familiarity with recent and older scholarship in these areas and takes advantage of earlier workers in the field (most notably Marshall Clagett on Archimedes) without being tied to their conclusions. Knorr subjects hypotheses as well as texts to detailed examination (such as doubts on the authenticity of Eutocius's text of Eratosthenes), but also proposes hypotheses of his own as he goes along. As he notes of the study of ancient science, "Given the fragmentary nature of the surviving evidence, the security of one's hypotheses is almost inversely proportional to their content. The price of being always correct would be to venture no hypotheses at all, and that is surely too high a price"

**4.3. Review by: Ivor Bulmer-Thomas.**

*The Classical Review, New Series* **41** (1) (1991), 210-212.

In this substantial and beautifully produced volume Wilbur R. Knorr, who is Associate Professor in the Programme in History of Science at Stanford University, takes the reader into the workshop where he composed The Ancient Tradition of Geometric Problems and other works. In the main it is a minute examination of the writings concerned with the three classic problems of ancient Greek mathematicians - the doubling of the cube, the trisection of an angle, and the squaring of the circle, and not merely the ancient Greek writings but the Arabic, Latin and Hebrew translations. Many manuscript renderings are here reproduced, saving scholars the need to go to Istanbul or the Escorial. As such the volume may be regarded as an apparatus criticus on the grand scale. But there is still more in it. References in these studies are expanded in essays on such subjects as the anonymous treatise on isoperimetric figures and the competence of Hypatia as a mathematician. K. has a thoroughly independent mind and has no hesitation in challenging received opinions even when propounded by such recognized scholars as Wilamowitz, Heiberg and Heath. The documentation is exhaustive, and it may safely be predicted that scholars will be dipping into the work for many years to come.

**4.4. Review by: George Molland.**

*Speculum* **68** (1) (1993), 189-191.

Like Umberto Eco's novel Foucault's Pendulum, this book is concerned in a complex way with a complex tradition. In part 1 (pp. 9-245) special attention is paid to treatments within the Greek cultural ambit of the mathematical problems of duplicating the cube and trisecting the angle. Pappus of Alexandria and Eutocius of Ascalon figure prominently, both for their roles as transmitters and for their own extant writings. In this connection Knorr discusses the genuineness of the ascription of two texts to Eutocius. Part 2 (pp. 247-372) moves on to the treatment of the issues in Arabic circles. Part 3 (pp. 373-816) brings in the third of the famous triad of geometrical problems, namely, the quadrature of the circle. It is entitled "The Textual Tradition of Archimedes' Dimension of the Circle" and traces the fortuna of this short work (or rather the Archimedean prototype from which, Knorr holds, the extant Greek text derives at some remove) in Greek, Arabic, and medieval Latin, although ranging rather more widely than the part's title would suggest. ... Eco's tradition was occultist and secretive, and hence by its very nature murky. Knorr's concern is with the supposedly highly rational field of mathematics, but his tradition is also murky and hard to assimilate and assess. This is partly due to the nature of the materials and partly to Knorr's own methods of research and presentation. A major problem is the paucity of the extant evidence and the need to postulate the previous existence of numerous texts and versions of texts, which have since been lost through the fragilities of transmission of knowledge in a manuscript culture. ... I have no wish to argue that Knorr's conclusions are wrong but only that his arguments are insufficient to rule out other viable hypotheses. ... I shall value Knorr's book: not for its providing an overall definitive picture of the tradition that it treats, but for its presentation of evidence and arguments that must be taken into account in particular detailed pieces of research.

**4.5. Review by: Jan P Hogendijk.**

*Mathematical Reviews*, MR1029522 **(91c:01008).**

This book is devoted to the ancient and medieval tradition of three great problems of Greek geometry: the duplication of the cube, the trisection of the angle and the quadrature of the circle. The book is a complement to the author's previous work *The ancient tradition of geometric problems* [Birkhäuser Boston, Boston, MA, 1986]. The purpose of the book under review is, in the words of the author, to give "a redaction history of a specific set of ancient mathematical works", in other words, "a body of specimens of the alternations such texts underwent in the course of transmission". For, the author says, "by sorting out the strands of tradition, we will aim to document how certain mathematical texts have been studied and edited, and through these cases to understand better the process of textual transmission". The book is in three parts. In Part I, entitled "Ancient texts on geometric problems", Knorr presents translations with detailed commentaries of all extant Greek texts on the duplication of the cube, the construction of two mean proportionals between two given lines and the trisection of an angle. ... Part II, "Arabic geometrical texts and their ancient sources", contains discussions, translations, editions and facsimiles of texts on the construction of two mean proportionals ... Part III, which occupies more than half of the book, is devoted to "the textual tradition of Archimedes' *Dimension of the circle*". This short text was, in the words of Knorr, "Archimedes' most accessible work and for that reason, the one most studied in the schools of late antiquity. It was thus bound to suffer most modification in the light of the instructional aim of editors." Knorr discusses numerous Greek, Arabic, Latin and Hebrew versions of this work and of related texts. He argues that the extant Greek text of the *Dimension of the circle* is an abstract of a much larger Archimedean original, which he characterizes as "a body of properties on the circle comparable in format and scope to Archimedes' writings on the sphere". Knorr's book is long and technical, but it is useful as a reference work for research in ancient and medieval geometry. The book contains a wealth of textual material and many interesting speculations, which cannot all be taken as established facts, but which may well stimulate further discussion on the transmission of mathematical texts in antiquity and the Middle Ages.