Euclid

Greek geometer

• Born: c. 330 b.c.e.
• Birthplace: Probably Greece
• Died: c. 270 b.c.e.
• Place of death: Alexandria, Egypt

Euclid took the geometry known in his day and presented it in a logical system. His work on geometry became the standard textbook on the subject down to modern times.

Early Life

Little is known about Euclid (YOO-klihd), and even the city of his birth is a mystery. Medieval authors often called him Euclid of Megara, but they were confusing him with an earlier philosopher, Eucleides of Megara, who was an associate of Socrates and Plato. It is virtually certain that Euclid came from Greece proper and probable that he received advanced education in the Academy, the school founded by Plato in Athens. By the time Euclid arrived there, Plato and the first generation of his students had already died, but the Academy was the outstanding mathematical school of the time. The followers of Aristotle in the Lyceum included no great mathematicians. The majority of the geometers who instructed Euclid were adherents of the Academy.

Euclid traveled to Alexandria and was appointed to the faculty of the Museum, the great research institution that was being organized under the patronage of Ptolemy Soter, who ruled Egypt from 323 to 283. Ptolemy, a boyhood friend of Euclid and then a lieutenant of Alexander the Great, had seized Egypt soon after the conqueror’s death, become the successor of the pharaohs, and managed to make his capital, Alexandria, an intellectual center of the Hellenistic Age that outshone the waning light of Athens. Euclid presumably became the librarian, or head, of the Museum at some point in his life. He had many students, and although their names are not recorded, they carried on the tradition of his approach to mathematics. His influence can still be identified among those who followed in the closing years of the third century b.c.e. He was thus a member of the first generation of Alexandrian scholars, along with Demetrius of Phalerum and Strato of Lampsacus.

Two famous remarks are attributed to Euclid by ancient authors. On being asked by Ptolemy if there was any easier way to learn the subject than by struggling through the proofs in Euclid’s work the Stoicheia (c. 300 b.c.e.; The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara, 1570, commonly known as the Elements), Euclid replied that there is no “royal road” to geometry. Then when a student asked him if geometry would help him get a job, he ordered his slave to give the student a coin, “since he has to make a profit from what he learns.” In spite of this rejoinder, his usual temperament is described as gentle and benign, open, and attentive to his students.

Life’s Work

Euclid’s reputation rests on his greatest work, the Elements, consisting of thirteen books of his own and two spurious books added later by Hypsicles of Alexandria and others. This work is a systematic explication of geometry in which each brief and elegant demonstration rests on the axioms and postulates given previously. It embraces and systematizes the achievements of earlier mathematicians. Books 1 and 2 discuss the straight line, triangles, and parallelograms; books 3 and 4 examine the circle and the inscription and circumscription of triangles and regular polygons; and books 5 and 6 explain the theory of proportion and areas. Books 7, 8, and 9 introduce the reader to arithmetic and the theory of rational numbers, while book 10 treats the difficult subject of irrational numbers. The remaining three books investigate elementary solid geometry and conclude with the five regular solids (tetrahedron, cube, octahedron, dedecahedron, and icosahedron). It should be noted that the Elements discusses several problems that later came to belong to the field of algebra, but Euclid treated them in geometric terms.

The genius of the Elements lies in the beauty and compelling logic of its arrangement and presentation, not in its new discoveries. Still, Euclid showed originality in his development of a new proof for the Pythagorean theorem as well as his convincing demonstration of many principles that had been advanced less satisfactorily by others. The postulate that only one parallel to a line can be drawn through any point external to the line is Euclid’s invention. He found this assumption necessary in his system but was unable to develop a formal proof for it. Modern mathematicians have maintained that no such proof is possible, so Euclid may be excused for not providing one.

Other works by Euclid are extant in Greek. Ta dedomena (c. 300-270 b.c.e.; Data in Euclid’s Elements of Geometry, 1661) is another work of elementary geometry and includes ninety-four propositions. The Optika (c. 300-270 b.c.e.; The Optics of Euclid, 1945), by treating rays of light as straight lines, makes its subject a branch of geometry. Spherical geometry is represented by the Phainomena (c. 300-270 b.c.e.; Euclid’s Phaenomena, 1996), which is an astronomical text based in part on a work of Autolycus of Pitane, a slightly older contemporary. Euclid wrote on music, but the extant Katatomē kanonos (known by its Latin title, Sectio canonis) is at best a reworking by some later, inferior writer of a genuine text by Euclid, containing no more of his actual words than some excerpts. Discovered in Arabic translation was Peri diaireseon biblion (c. 300-270 b.c.e.; On Divisions of Figures, 1915), for which the proofs of only four of the propositions survive.

Also discovered have been the names of several lost books by Euclid on advanced geometry: The Pseudaria (fallacies) exposed fallacies in geometrical reasoning, and Konika (conics) laid some of the groundwork for the later book of the same title by Apollonius of Perga. There was a discussion of the relationships of points on surfaces titled Topoi pros epiphaneia (surface loci), and Porismata (porisms), a work of higher geometry, treated a kind of proposition intermediate between a theorem and a problem.

In addition to the last two books of the Elements, there are works bearing Euclid’s name that are not genuinely his. These include the Katoptrica (catoptrica), a later work on optics, and Eisagōgē armonikē (Introduction to Harmony), which is actually by Cleonides, a student of Aristoxenus. None of Euclid’s reputation, however, depends on these writings falsely attributed to him.

Significance

Euclid left as his legacy the standard textbook in geometry. There is no other ancient work of science that needs so little revision to make it current, although many modern mathematicians, beginning with Nikolay Lobachevski and Bernhard Riemann and including Albert Einstein, have developed non-Euclidean systems in reaction to the Elements, thus doing it a kind of honor. The influence of Euclid on later scientists such as Archimedes, Apollonius of Perga, Galileo Galilei, Sir Isaac Newton, and Christiaan Huygens was immense. Eratosthenes used his theorems to measure with surprising accuracy the size of the sphere of Earth, and Aristarchus attempted less successfully, but in fine Euclidean style, to establish the sizes and distances of the moon and the sun.

Other Hellenistic mathematicians, such as Hero of Alexandria, Pappus, Simplicius, and, most important, Proclus, produced commentaries on the Elements. Theon of Alexandria, father of the famous woman philosopher and mathematician Hypatia, introduced a new edition of the Elements in the fourth century c.e. The sixth century Italian Boethius is said to have translated the Elements into Latin, but that version is not extant. Many translations were made by early medieval Arabic scholars, beginning with one made for Harun al-Rashid near 800 c.e. by al-Hajjaj ibn Yusuf ibn Matar. Athelhard of Bath made the first surviving Latin translation from an Arabic text about 1120 c.e. The first printed version, a Latin translation by the thirteenth century scholar Johannes Campanus, appeared in 1482 in Venice. Bartolomeo Zamberti was the first to translate the Elements into Latin directly from the Greek, rather than Arabic, in 1505. The first English translation, printed in 1570, was done by Sir Henry Billingsley, later the lord mayor of London. The total number of editions of Euclid’s Elements has been estimated to be more than a thousand, making it one of the most often translated and printed books in history and certainly the most successful textbook ever written.

Bibliography

Euclid. The Thirteen Books of Euclid’s “Elements.” Translated by Thomas Little Heath. 3 vols. 1925. Reprint. New York: Dover, 1956. This English translation contains extensive commentary on Euclid’s Elements. This admirable work supersedes all previous translations. It contains a full introduction, 151 pages in length, touching on all the major problems.

Fraser, P. M. Ptolemaic Alexandria. 3 vols. Oxford, England: Clarendon Press, 1972. Has a useful section on the intellectual background and influences of Euclid but is primarily valuable in providing a study of the cultural setting of Alexandria in Euclid’s day.

Heath, Thomas Little. From Thales to Euclid. Vol. 1 in A History of Greek Mathematics. New York: Dover, 1981. Places Euclid in the context of the development of ancient mathematics. A thoroughly dependable treatment.

Mlodinow, Leonard. Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace. New York: Free Press, 2001. In this history of geometry, reason, and abstraction, Euclid is represented as a major figure.

Mueller, Ian. Philosophy of Mathematics and Deductive Structure in Euclid’s “Elements.” Cambridge, Mass.: MIT Press, 1981. A study of the Greek concepts of mathematics found in the Elements, emphasizing philosophical, foundational, and logical rather than historical questions, although the latter are not totally neglected. Attention is directed to Euclid’s work, not that of his predecessors. This monograph requires mathematical literacy, and the general reader may find it overly technical.

Reid, Constance. A Long Way from Euclid. New York: Thomas Y. Crowell, 1963. An explanation of how modern mathematical thought has progressed beyond Euclid, written for those whose introduction to mathematics consisted mainly of studying the Elements. Accessible to the general reader, this study takes Euclid as its starting point and shows that he did not provide the reader with all the answers, or even all the questions, with which mathematicians concern themselves.

Szabo, Arpad. The Beginnings of Greek Mathematics. Translated by A. M. Ungar. Boston: D. Reidel, 1978. Places Euclid within the context of the development of the Greek mathematical tradition.