Euclid Essay - Critical Essays

Euclid

Introduction

Euclid fl. C. 300 B.C.

Greek mathematician.

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.

Biography

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.

Major Works

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 Written in thirteen books and containing 465 propositions, it superseded all other works on the subject.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.

Textual History

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.

Critical Reception

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.

*Principal Works

Mathematical Treatises

Catoptrica [On Reflections']

Conica [Conics]

Data

Elements of Geometry

On Divisions (of Figures)

Optica [Optics]

Phaenomena

Porisms

Pseudaria [Fallacies]

Sectio Canonis [Division of the Canon]

Surface-loci

*The dates of Euclid's treatises are unknown.

†Authorship of these works is an issue of debate among scholars.

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

(The entire section is 64 words.)

Criticism

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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.)