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 Written in thirteen books and containing 465 propositions, it superseded all other works on the subject.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