Proclus Diadochus (essay date c. 485 A.D.)

(Classical and Medieval Literature Criticism)

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 excerptwith 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...

(The entire section is 2639 words.)

Sir Thomas Heath (essay date 1921)

(Classical and Medieval Literature Criticism)

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.]

The Data

…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...

(The entire section is 7988 words.)

Sir Thomas Heath (essay date 1926)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 22257 words.)

George Sarton (essay date 1927)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 1148 words.)

Howard Eves and Carroll V. Newsom (essay date 1958)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 9072 words.)

O. Neugebauer (essay date 1963)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 5320 words.)

David C. Lindberg (essay date 1971)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 12359 words.)

Ian Mueller (essay date 1972)

(Classical and Medieval Literature Criticism)

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 logicthat is, that an argument's validity depends on its form.]

Introduction

By 'logic' I mean 'the analysis of argument or proof in...

(The entire section is 14035 words.)

A. Seidenberg (essay date 1974-75)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 16760 words.)

Wilfred Theisen (essay date 1982)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 5633 words.)

André Barbera (essay date 1991)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 12203 words.)

Wilbur R. Knorr (essay date 1991)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 20923 words.)

Alan C. Bowen (essay date 1991)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 10728 words.)

J. L. Berggren and R. S. D. Thomas (essay date 1996)

(Classical and Medieval Literature Criticism)

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...

(The entire section is 7005 words.)