Euclid fl. C. 300 B.C.
Euclid is often referred to as the "father of geometry" due to the concepts he explored in Elements of Geometry, his most famous and influential work. It has been noted by critic G. C. Evans (1958) that "with the single exception of the Bible, no work has been more widely studied or edited." Euclid also wrote other mathematical treatises, two of which, like Elements, are extant. In addition to purely mathematical works, he also wrote on the mathematical nature of vision and on the use of spherical geometry in relation to astronomy, and he is believed to have written on the mathematical components of music.
Little information is available about Euclid's life; his birthplace and birth and death dates are unknown. Based on references made by other classical writers, scholars can only conclude that Euclid flourished circa 300 B.C. It is probable that Euclid received his mathematical training in Athens from students of Plato. Additionally, it is believed that Euclid served as the first mathematics professor at the University of Alexandria and that he founded the Alexandrian School of Mathematics.
Elements is universally regarded as Euclid's greatest work. Written in thirteen books and containing 465 propositions, it superseded all other works on the subject. While others before Euclid had made efforts to identify mathematical "elements" (the leading theorems which are widely and generally used in a subject), Euclid's selection and organization of these elements is one of his primary accomplishments. Much of the material from Books I, II, and IV of Elements was probably developed by the early Pythagoreans, and the material in Book X most likely originated with Theaetetus.
In addition to Elements, Euclid's other two extant purely geometrical works are Data and On Divisions
(of Figures). Data focuses on plane geometry and facilitates the process of analysis with which higher geometry is concerned. In On Divisions Euclid discusses the divisions of such figures as circles and rectilinear figures, as well as the resulting ratios. Euclid's lost works, which further study various aspects of elementary geometry and geometrical analysis, include Pseudaria, Porisms, Conica, and Surface-loci.
Euclid wrote several works in which mathematical principles are applied to other fields. In Phaenomena, the geometry of the sphere is applied to astronomy for the purpose of examining problems related to the rising and setting of stars and of circular arcs in the "celestial sphere." Optica applies geometrical analysis to establish a theory of vision based on the concept of the emission of "visual rays" from the eye. This idea of emission—that the eye apprehends what it sees—is contrasted with the concept of intromission, in which the eye receives what is in the plane of vision. Similarly, Catoptrica examines visual phenomena caused by reflected visual rays or rays of light. Euclid's authorship of Catoptrica is debated among critics. Elements of Music is attributed to Euclid by Proclus (410-485 A.D.); the commentary of Proclus on Euclid is one of the primary sources of information on the history of Greek geometry. While there is no extant copy of Elements of Music, two musical treatises originally attributed to Euclid still exist. One of these, Introductio harmonica, has been proven to be the work of Cleonides; the other treatise, Sectio canonis, is believed by some critics to be the work of Euclid, while others doubt this attribution. The latter work focuses on the mathematical analysis of music and examines how musical sound may be construed numerically.
There exists no original version of Elements and no copy which can be dated to Euclid's time. A revision of the work prepared by Theon of Alexandria, who lived seven hundred years after Euclid's time, became the basis for all Greek editions of the text until the nineteenth century. By the end of the tenth century, several Islamic translations and commentaries had been compiled. The Middle Ages saw numerous Arabic, Near Eastern, Greek, and Latin commentaries as well. The first printed edition of Elements appeared in Latin in 1482; French and German translations were published in the sixteenth century. The first complete English translation was completed by Sir Henry Billingsley and appeared in London in 1570. In 1808, a tenth-century copy of an edition which predated Theon's was found by François Peyrard. It was not until J. L. Heiberg reconstructed the text using most available manuscripts, including Theon's and the manuscript discovered by Peyrard, that the first critical edition of Elements was published (1883-88). As Elements was translated, disseminated, and used in medieval mathematical curricula, Euclid's other extant works followed paths similar to that of their predecessor in terms of translation and distribution.
G. C. Evans has noted that from its first appearance, Elements "was accorded the highest respect." Additionally, most critics agree that Elements was so successful when it first appeared that other early efforts at establishing or collecting mathematical "elements" soon disappeared. While the validity or utility of individual propositions in Elements has been questioned by some scholars, the overall significance of the work has increased with the passage of time. Modern mathematical scholars and historians now investigate such topics as the relationship between Elements and Greek logic and whether the work truly developed geometry on an axiomatic basis. A. Seidenberg remarked on the "assumption" that Elements is an example of the use and development of the axiomatic method, a form of analysis in which one begins from a set of assumed "common notions" which need not be proved. While Evans has explained Euclid's development of this method, as well as some of the "logical short-comings" that exist within Elements, Seidenberg has argued that Euclid did not use or develop an axiomatic method. Furthermore, Seidenberg has asserted that the content of Elements suggests that its author felt it unacceptable to make any geometrical assumptions whatsoever. Ian Mueller has used Elements to examine the relationship between Greek logic and Greek mathematics. Mueller has found no evidence to suggest that Euclid was aware of Aristotle's syllogisms (a form of logical argument) or that he understood such a basic principle of logic that an argument's validity is dependent upon its form.
Euclid's Optica, which is also known by the title of its Latin translation, De visu, has received much critical attention as well. As David C. Lindberg has pointed out, Optica is often faulted for its strictly mathematical approach to vision which ignores "every physical and psychological aspect of the problem of vision." Lindberg has also discussed the critique of Optica made by Alkindi (d. 873), whom Lindberg identifies as the first great philosopher of the Islamic world. Alkindi agrees with most of Euclid's theory, but takes issue with the nature of the "visual cone" as conceived by Euclid. Like Lindberg, Wilfred Theisen has traced the influence of Optica. Discussing the medieval significance of the work, he has noted that it "became one of the standard texts for the teaching of the mathematical arts in the thirteenth century."
Of the remaining extant works attributed to Euclid, Catoptrica and Sectio canonis have both received substantial critical attention as scholars continue to debate their authorship. Sir Thomas Heath has briefly discussed the dubious nature of the attribution of Catoptrica to Euclid, while Ken'ichi Takahashi (1992) challenged critics who argue that the work is not Euclid's. Alan C. Bowen has attacked critics who judge Sectio canonis as non-Euclidean. To critics who argue that Sectio canonis differs too much in style and structure from the rest of Euclid's work to be his, Bowen has pointed out that they are ignoring the variety of logical structure and language throughout the Euclidean canon. To those who maintain that the inconsistencies within the text make the work unlikely to be Euclid's, Bowen has replied that such inconsistencies represent a failure in scholarship, not a problem within the text itself. While such controversies continue to be disputed, the lasting significance of Euclid's Elements remains clear and modern high school students are still taught the principles of Euclidian geometry.
†Catoptrica [On Reflections']
Elements of Geometry
On Divisions (of Figures)
†Sectio Canonis [Division of the Canon]
*The dates of Euclid's treatises are unknown.
†Authorship of these works is an issue of debate among scholars.
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Principal English Translations
The Thirteen Books of Euclid's "Elements" [translated from the text of J. L. Heiberg by Sir Thomas L. Heath] 1926
The Medieval Latin Translation of the Data of Euclid [translated by Shuntaro Ito; English translation of the medieval Latin version of Data] 1980
Euclid's Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy [translated by J. L. Berggren and R. S. D. Thomas] 1996
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SOURCE: "The History of Geometry to the Time of Euclid: Commentary on Euclid's 'Elements I,'" in A Source Book In Greek Science, edited by Morris R. Cohen and I. E. Drabkin, translated by I. E. Drabkin, McGraw-Hill Book Company, Inc., 1948, pp. 33-85.
[In the following excerpt—with translation by Drabkin and notes by Cohen and Drabkin, Proclus offers a brief overview of geometry, from that of the ancient Egyptians up to that of Euclid's Elements.]
We must next speak of the origin of geometry in the present world cycle. For, as the remarkable Aristotle tells us, the same ideas have repeatedly come to men at various periods of the universe. It is not, he goes on to say, in our time or in the time of those known to us that the sciences have first arisen, but they have appeared and again disappeared, and will continue to appear and to disappear, in various cycles, of which the number both past and future is countless. But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered. It owed its discovery to the practice of land measurement. For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person's land to disappear.1 Furthermore, it should occasion no surprise that the discovery both...
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SOURCE: A History of Greek Mathematics, Clarendon Press, 1921, 446 p.
[In the following excerpt, Heath discusses the significance and content of several of Euclid's lesser-known works.]
…Most closely connected with the Elements as dealing with plane geometry, the subject-matter of Books I-VI, is the Data, which is accessible in the Heiberg-Menge edition of the Greek text, and also in the translation annexed by Simson to his edition of the Elements (although this translation is based on an inferior text). The book was regarded as important enough to be included in the Treasury of Analysis … as known to Pappus, and Pappus gives a description of it; the description shows that there were differences between Pappus's text and ours, for, though Propositions 1-62 correspond to the description, as also do Propositions 87-94 relating to circles at the end of the book, the intervening propositions do not exactly agree, the differences, however, affecting the distribution and numbering of the propositions rather than their substance. The book begins with definitions of the senses in which things are said to be given. Things such as areas, straight lines, angles and ratios are said to be 'given in magnitude when we can make others equal to them' (Defs. 1-2). Rectilineal figures are 'given in species' when their angles are...
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SOURCE: The Thirteen Books of Euclid's Elements, translated by Sir Thomas L. Heath, revised edition, Cambridge at the University Press, 1926, 432 p.
[In the following introductory chapters to the translated text of Elements, Heath offers an overview of Euclid's life; provides a brief survey of his writings; and reviews early commentary on Elements.]
Euclid and the Traditions about Him
As in the case of the other great mathematicians of Greece, so in Euclid's case, we have only the most meagre particulars of the life and personality of the man.
Most of what we have is contained in the passage of Proclus' summary relating to him, which is as follows1:
"Not much younger than these (sc. Hermotimus of Colophon and Philippus of Medma) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry4. He is then younger than the pupils of Plato...
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SOURCE: "The Time of Euclid," in Introduction to the History of Science, 1927. Reprint by Robert E. Krieger Publishing Company, 1975, pp. 149-64.
[In the following excerpt, first published in 1927 and reprinted in 1975, Sarton offers a brief overview of the scientific developments taking place during the first half of the third century B.C., the time in which Euclid flourished.]
I. Survey of Science in First Half of Third Century B.C.
1. The period which we are now going to consider is widely different from the previous one. In the fourth century Athens was the greatest intellectual center of the world; by the beginning of the third century that center had already moved to Alexandria. Strictly speaking, the new period does not coincide with the first half of the third century, but began a little earlier, some time after the advent of the Ptolemies in Egypt. It can not have begun much earlier: Alexandria was founded only in 332, and the brilliant civilization which blossomed there can not have started at once—the material preparation required some time; we may assume that the activity of one generation was largely devoted to it. Strangely enough, we do not know the exact dates of three of the great men of this period—Euclid, Herophilos, and Manethon; we know only that they flourished under the first Ptolemy (Ptolemaeos Soter), who ruled Egypt from 323 to 285. Thus it is...
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SOURCE: "Euclid's Elements" in An Introduction to the Foundations and Fundamental Concepts of Mathematics, Holt, Rinehart and Winston, 1958, pp. 30-57.
[In the following excerpt, Eves and Newsom review the formal nature and significance of Elements, arguing that the work offers the earliest extensive development of the axiomatic method, and that the impact of this form of analysis on the development of mathematics has been tremendous.]
The Importance and Formal Nature of Euclid's Elements
The earliest extensively developed example of the use of the axiomatic method that has come down to us is the very remarkable and historically important Elements of Euclid. The production of this treatise is generally regarded as the first great landmark in the history of mathematical thought and organization, and its subsequent influence on scientific thinking can hardly be overstated.
Of Euclid himself, however, disappointingly little is known. It is from Proclus' Commentary on Euclid, Book I, that we obtain our most satisfying information about Euclid. He writes,
Euclid, who put together the Elements, collected many of the theorems of Eudoxus. He perfected many of the theorems of Theaetetus, and also brought to irrefragable demonstration the things which were only somewhat loosely proved by his...
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SOURCE: "The Survival of Babylonian Methods in the Exact Sciences of Antiquity and Middle Ages," in Proceedings of the American Philosophical Society, Vol. 107, No. 6, December, 1963, pp. 528-35.
[In the following essay, Neugebauer examines the influence of Babylonian mathematical methods on the development of Greek mathematics. Neugebauer states that while a large part of the information in Euclid's Elements had been known for more than a millennium, "mathematics in a modern sense" began with Euclid's addition of general mathematical proof.]
Among the many parallels between our own times and the Roman imperial period could be mentioned the readiness to ascribe to the "Chaldeans" discoveries whenever their actual origin was no longer known. The basis for such assignments is usually the same: ignorance of the original cuneiform sources, excusable in antiquity but less so in modern times. Given this situation, it seems to me equally important to establish what we can say today about knowledge which the Babylonians did have and to distinguish this clearly from methods and procedures which they did not have. In other words, it seems to me that it is high time that an effort is made to eliminate historical clichés, both for the Mesopotamian civilizations and their heirs, and to apply common sense to the fragmentary but solid information obtained from the study of the original sources...
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SOURCE: "Alkindi's Critique of Euclid's Theory of Vision," in Isis, Vol. 62, No. 214, December, 1971, pp. 469-89.
[In the following essay, Lindberg presents an analysis of Euclid's Optica by Alkindi (d. 873), an early Islamic philosopher. Lindberg states that Alkindi "placed himself firmly on the side of Euclid" in many respects, but that the philosopher disagreed with Euclid on the nature of the "visual cone, " one aspect of the mathematician 's theory of vision.]
Alkindi, undoubtedly the first great philosopher of the Islamic world, was a leader in the endeavor to communicate Greek philosophy to Islam.1 Not only did he participate in the translating activity of the ninth century, but he also attempted to integrate Greek philosophy with Mu'tazilite theology and thus, in Walzer's phrase, "to naturalise Greek philosophy in the Islamic world."2 Alkindi's deep respect for ancient thought is revealed in the preface to one of his works on metaphysics:
It is fitting then to acknowledge the utmost gratitude to all those who have contributed even a little to truth not to speak of all those who have contributed much. If they had not lived, it would have been impossible for us, despite all our zeal, during the whole of our lifetime, to assemble these principles of truth which form the basis of the final inferences of our research. The assembling of...
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SOURCE: "Greek Mathematics and Greek Logic," in Ancient Logic and Its Modern Interpretations, edited by John Corcoran, D. Reidel Publishing Company, 1974, pp. 35-70.
[In the following essay, delivered as a paper in 1972 and published in 1974, Mueller examines the nature of Euclidean reasoning (as evidenced in Elements), and its relationship to Aristotle's syllogistic logic (a type of logical argument). Mueller concludes that Euclid demonstrates no awareness of syllogistic logic or of the basic concept of logic—that is, that an argument's validity depends on its form.]
By 'logic' I mean 'the analysis of argument or proof in terms of form'. The two main examples of Greek logic are, then, Aristotle's syllogistic developed in the first twenty-two chapters of the Prior Analytics and Stoic propositional logic as reconstructed in the twentieth century. The topic I shall consider in this paper is the relation between Greek logic in this sense and Greek mathematics. I have resolved the topic into two questions: (1) To what extent do the principles of Greek logic derive from the forms of proof characteristic of Greek mathematics? and (2) To what extent do the Greek mathematicians show an awareness of Greek logic?
Before answering these questions it is necessary to clear up two preliminaries. The first is chronological. The Prior...
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SOURCE: "Did Euclid's Elements, Book I, Develop Geometry Axiomatically?," in Archive for History of Exact Sciences, Vol. 14, 1974/1975, pp. 263-95.
[In the following essay, Seidenberg challenges the assumption that Euclid, in Elements, developed geometry on an axiomatic basis. Seidenberg argues that, by insisting on this assumption, the work is viewed "from a false perspective" and its accomplishments are thus displayed "in a bad light."]
Historians are fond of repeating that Euclid developed geometry on an axiomatic basis, but the wonder is that any mathematician who has looked at The Elements would agree with this. Anyone who looks at The Elements with modern hindsight sees that something is wrong; but we have all been told in our childhood that Euclid had the axiomatic method, so the usual reaction is to speak of "gaps". This word is hardly right, though, if there was nothing there in the first place.
Could it be that, by insisting on the axiomatic basis, we are viewing The Elements from a false perspective and seeing its accomplishments in a bad light? This is precisely what I intend to prove.
The Greeks of Euclid's time1 had the axiomatic method; Aristotle's description of it can be considered a close approximation to our own. Or better yet, one may consider Eudoxus' theory of magnitude as presented in Book V of The...
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SOURCE: "Euclid's Optics in the Medieval Curriculum," in Archives Internationales D'Histoire des Sciences, Vol. 32, 1982, pp. 159-76.
[In the following excerpt, Theisen discusses the impact of Euclid's Optica on Western scholars in the twelfth and thirteenth centuries and maintains that by the thirteenth century, a "firm tradition " of the critical analysis of Euclid's text was established.]
Defending the utility of a liberal education, John Henry Newman stressed the advantages of learning " … to think and to reason and to compare and to discriminate and to analyze …"1. Newman's words are an apt description of one of the chief aims of the medieval curriculum, the formation of a discriminating, critical mind. Although it is true that medieval students were concerned primarily with studying the texts of great writers like Euclid, Aristotle and Boethius, they were encouraged, indeed generally constrained, to work their way through a text in a critical fashion. The medieval student did not simply hear a lecture without exerting his own mind; a proof of any proposition was not merely presented, elucidated and passively accepted. The proof was often presented as a test, a statement to be challenged and contested.
This insistence on taking a critical approach to texts is apparent in the glosses to the Latin text of Euclid's Optica, Liber de visu [A...
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SOURCE: Introduction to The Euclidean "Division of the Canon," University of Nebraska Press, 1991, pp. 1-108.
[In the following essay, Barbera examines the evidence and scholarly opinion surrounding the issue of the authorship of Sectio Canonis, concluding that "it would be bold to assert definitely" that Euclid is or is not the author.]
The Division of the Canon … is an ancient Pythagorean treatise on the relationship between mathematical principles and acoustical truths. Composed largely in the style of Euclid's Elements of Geometry, the Division is handed down in three distinct traditions: (1) a semi-independent version in Greek, which is attributed to Euclid or to Cleonides; (2) a Greek version contained in the fifth chapter of Porphyry's commentary on Ptolemy's Harmonics; and (3) a Latin version comprising the first two chapters of the fourth book of Boethius's De institutione musica. Of the three traditions, the semi-independent version is the longest. The other two versions present portions of the long version as well as some significant textual variants.
The long version consists of an introduction, a series of mathematical propositions, a series of acoustical propositions, a passage devoted to the enharmonic genus, and a division of the canon. The Introduction is philosophical in character, connecting the existence of sound to...
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SOURCE: "What Euclid Meant: On the Use of Evidence in Studying Ancient Mathematics," in Science and Philosophy in Classical Greece, translated by Alan C. Bowen, Garland Publishing, Inc., 1991, pp. 119-63.
[In the following essay, Knorr explores, through Elements, the role of authorial meaning in critical analysis and argues that mathematical historians often make the mistake of reading ancient texts in "the context of modern notions. "]
For most historians of mathematics the principal data are documents—records of past thoughts preserved in writing. It follows that the interpretation of documents is central to the methodology of historians and, hence, that discussions of the principles of interpretation can be brought to bear on efforts in this field.
As a specialist in mathematical history, I have found that my colleagues in the areas of literary studies tend to register surprise at the thought that mathematical texts are subject to interpretation, even as they take for granted that all literary texts require interpretation. Moreover, I would anticipate that associates in the disciplines of mathematics and the physical sciences would be surprised—perhaps appalled—at the suggestion that the understanding of technical documents could be illuminated through the insights of theorists of literary criticism. Somehow, the patent universality of mathematical discourse might...
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SOURCE: "Euclid's Sectio canonis and the History of Pythagoreanism," in Science and Philosophy in Classical Greece, translated by Alan C. Bowen, Garland Publishing, Inc., 1991, pp. 164-87.
[In the following essay, Bowen discusses the content of, and issues surrounding, Sectio Canonis. Bowen addresses the question of authorship and responds to critical arguments on this topic, maintaining that the work is Euclid's. Bowen also contends that the belief that the work is Pythagorean may be as "ill-founded" as the authorship debate.]
The treatise which has come down to us as the Sectio canonis or Division of the Canon consists in an introduction of thirty-three lines [Menge 1916, 158.1-160.4] and twenty interconnected demonstrations articulated in roughly the same way as those in Euclid's Elements [cf. Jan 1895, 115-116].1 Beyond this most everything is in dispute. To begin, scholars debate the authorship of the Sectio. Those who deny or qualify the thesis that it derives from Euclid usually proceed by comparing it to treatises more commonly acknowledged to be Euclid's, and by pointing out supposed inconsistencies in the Sectio itself which are presumed inappropriate for a mathematician of Euclid's stature [cf., e.g., Menge 1916, xxxviii-xxxix]. None of the arguments, however, are particularly persuasive. In the first place, the critics tend to ignore...
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SOURCE: Introduction to Euclid's "Phaenomena": A Translation and Study of a Hellenistic Treatise in Spherical Astronomy, Garland Publishing, Inc., 1996, pp. 1-18.
[In the following essay, Berggren and Thomas discuss the objectives and content of Phaenomena, suggesting that Euclid's application of spherics to questions of astronomy implies that some study of spherics and astronomy had been done before. While there is no evidence of this, the critics state that perhaps, as in the case of Elements, the appearance and success of Phaenomena resulted in the disappearance of earlier texts on the subject.]
The purpose and strategy of the Phaenomena
The Phaenomena is a geometrical treatment of some fundamental problems related to the risings and settings of stars and of important circular arcs on the celestial sphere. In fact, just over half its theorems (the last ten) are devoted to one of these problems, that of determining the length of daylight on a given day at a given locality, the two data on which the length of daylight obviously depends. Euclid's is the earliest extant treatise dealing with this particular question. Theodosius (late second century BC) and Menelaus (end of the first century AD) also wrote treatises bearing on this problem: indeed, Neugebauer has described its solution as 'one of the major goals' of their spherical...
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Baron, Margaret E. "Greek Mathematics." In The Origins of the Infinitesimal Calculus, pp. 11-59. Oxford: Pergamon Press, 1969.
Offers a detailed survey of the early Greek mathematicians, including Euclid, and their mathematical developments, theories, and influence.
Brownson, C. D. "Euclid's Optics and Its Compatibility with Linear Perspective." Archive for History of Exact Sciences 24 (1981): 164-94.
Challenges the "influential tradition" which contends that Optics conflicts with linear perspectives and argues that Euclid's system and that of linear perspective take similar approaches to closely related questions.
Clagett, Marshall. "The Greek and Arabic Forerunners of Medieval Statics." In The Science of Mechanics in the Middle Ages, pp. 3-68. Madison: University of Wisconsin Press, 1961.
Discusses, in the first portion of this chapter, the contents of the mechanical treatise On the Balance, originally attributed to Euclid, and addresses the controversy surrounding this attribution.
Grant, Edward, trans. "The Definitions of Book V of Euclid's Elements in a Thirteenth Century Version, and Commentary," by Campanus of Novara. In A Source Book in Medieval Science, edited by Edward...
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