Article abstract: Greek geometer. 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.
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.
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...
(The entire section is 1674 words.)