Euclid’s Elements, one of the most influential mathematics texts of all time, set the standard for logical mathematical thought throughout Europe and the Middle East.

The city of Alexandria was founded by Alexander the Great in 332 b.c.e. After Alexander’s death in 323 b.c.e., his empire was divided between his generals. Alexandria came under the power of Ptolemy Soter, who founded a great library and school there. The first of what was to become a long line of mathematicians at that school was Euclid. Euclid

The only sources of information on Euclid’s life are commentaries in mathematics texts, which discuss his mathematics more than his personal life. The main sources are the commentaries of Pappus, which were written in the third century c.e., and the commentaries of Proclus, which were written in the fifth century c.e., significantly later than Euclid’s lifetime. What can be said about Euclid’s life is that he flourished about 300 b.c.e. and probably studied mathematics in Athens at the school founded by Plato. What is certain is that he wrote the Stoicheia (compiled c. 300 b.c.e.; Elements, 1570), an elementary introduction in thirteen books to all of Greek geometry as it was known at the time.

Euclid’s great task was not in the creation of that geometry, although some of the proofs of the theorems in the Elements are thought to be Euclid’s. Rather, his accomplishment is in the collection, categorization, and simplification of the contemporary knowledge of geometry. Euclid’s achievement is twofold.

First, he collected the entire corpus of ancient Greek geometry and arranged it in a logical fashion. Each theorem in the Elements, including those of other authors, is proved using theorems that precede it in the text. Thus the theory is built up, theorem by theorem, on a solid logical foundation.

The second, and perhaps more important, of Euclid’s accomplishments is the statement of the five axioms forming the logical basis of the entire work. An axiom is an unprovable statement, one that is simply accepted as true and is then used as the basis of a mathematical theory. Euclid’s genius lay in recognizing that all of Greek geometry flowed from five simple axioms: (1) a straight line can be drawn between any two points; (2) a straight line can be extended indefinitely; (3) given any center and any radius, a circle can be drawn; (4) all right angles are equal to each other; and (5) if a line intersects two other lines, and if the sum of the interior angles made on one side of the first line is less than two right angles, then the other two lines, when extended, meet on that side of the first line. All the theorems in the Elements are logically based on just these five axioms.

Special note must be made of the fifth axiom. It is often called the Parallel Postulate, because in fact, it is a statement about parallel lines: If a line happens to intersect two others, and the sum of the interior angles on neither side of the first line is less than two right angles, then the other two lines do not meet; in other words, they are parallel. This axiom stands out from the others. The first four are all simply stated and quickly understood and believed; the fifth takes some time to state and to understand, and it caused much anxiety among mathematicians, ancient and modern. Many tried to prove the fifth postulate from the other four, to no avail. Finally, in the nineteenth century, it was shown that the fifth could not be proved from the other four. In fact, one can replace the fifth axiom with certain other axioms and obtain “non-Euclidean” geometries.

Euclid.

(Library of Congress)

The geometry of Euclid covers more than the modern definition of geometry. In fact, it covers a great variety of mathematical subjects from a modern perspective. For instance, Euclid’s geometry does deal with plane and solid figures, but it also deals with the application of these figures to many other problems. Plane figures such as rectangles, triangles, and circles are treated in books 1, 3, and 4 of the Elements. These books cover modern geometry. In books 2 and 6, geometric methods are used to solve what today are considered algebraic problems, such as solving linear and quadratic equations. The geometry of ratios of magnitudes, covered in book 5, is in today’s terminology the study of rational numbers; book 10 covers the geometry of magnitudes that are not in a simple ratio, or are incommensurable, which is the study of irrational numbers. The geometry in books 7, 8, and 9 is used to do what is now called number theory, including divisibility of one whole number by another, factoring whole numbers, and treatment of prime numbers. Solid figures also appear prominently in Euclid’s geometry, in books 11, 12, and 13. These books hold theorems from the most basic facts about solid figures up to the fact that there are only five regular solid figures all of whose sides are a given regular planar figure. These five figures are known as the Platonic solids.

Euclid is thought to have written nine works besides the Elements. These other works deal with some more specialized areas of geometry. Data (compiled c. 300 b.c.e.; English translation, 1751) is a text that deals further with plane geometry, expanding on books 1 through 6 of the Elements. Peri Diairéson biblion (compiled c. 300 b.c.e.; On Divisions of Figures, 1915) treats the taking of a single plane figure and dividing it according to a rule; for example, dividing a triangle into a quadrilateral and a triangle of certain areas, or dividing a figure bounded by two straight lines and the arc of a circle into equal parts. Parts of these two works are extant. The rest of Euclid’s works are known only because they are mentioned by other mathematicians or historians. Euclid produced a work called the Pseudaria, or the “book of fallacies.” In it he gives examples of common errors and misgivings in geometry, with the idea that later geometers could avoid these mistakes. The Porisms are described by Pappus as “neither theorems nor problems, but . . . a species occupying a sort of intermediate position.” An example of this is the task of finding the center of a circle: it can be stated as a problem, or as a theorem proving that the point found is actually the center. Euclid also wrote a work entitled Conics, which deals with conic sections, or the shapes obtained when one slices a cone. Surface-Loci deals with figures drawn on surfaces other than a plane, for example, triangles drawn on a sphere. Euclid also produced works in applied mathematics: Phaenomena, dealing with astronomy; Optics, Caltropics, or the theory of mirrors; and the Elements of Music.

The influence of Euclid’s Elements has been felt across the ages. From the rise of the Roman Empire through the early medieval period, the value of the kind of abstract intellectual thought embodied in the Elements was largely ignored in Europe. The Arabian world, however, had inherited Greek intellectualism and continued to study geometry, copying and translating the Elements and adding to geometry and mathematics in general. In the eleventh and twelfth centuries c.e., this intellectual thought was reintroduced to Europe as a result of the Crusades and the Moorish invasion of Spain. Euclid was translated into Latin, first from Arabian copies, then from older Greek copies. The rise of the universities in Europe introduced many to Euclid’s Elements, and European intellectuals began to add to mathematical knowledge. The geometry of Euclid is studied to this very day. Although much has been added to mathematical thought in the twenty-three centuries years since the writing of the Elements, that text remains one of the most published and most revered of mathematical treatises.

• Artmann, Benno. Euclid: The Creation of Mathematics. New York: Springer Verlag, 1999. This book is, in essence, a close reading of the Elements, walking readers through each book and providing sample proofs, interspersed with chapters of more general commentary.
• Euclid. Elements. 300 b.c.e. Translated with commentary by Thomas L. Heath. 1908. Reprint. New York: Dover Publications, 1956. English translations of the oldest extant Greek sources, with copious commentary.
• Fauvel, John, and Jeremy Grey, eds. The History of Mathematics: A Reader. 1987. Reprint. Washington, D.C.: The Mathematical Association of America, 1997. Excerpts on commentaries on the Elements, from Pappus to modern commentators. Bibliography and index.
• Grattan-Guinness, Ivor. The Norton History of the Mathematical Sciences. New York: W. W. Norton, 1999. A history of mathematics from the earliest beginnings to the early twentieth century. Bibliography and index.
• Heath, Sir Thomas L. A History of Greek Mathematics: From Thales to Euclid. 1921. Reprint. New York: Dover Publications, 1981. A history specifically looking at Greek mathematics, by one of the best authors in the field. Index.
• Mlodinow, Leonard. Euclid’s Window: A History of Geometry from Parallel Lines to Hyperspace. New York: Touchstone, 2002. This history of geometry, written for a general audience, is divided into five sections, the first of which is devoted to Euclid.

Apollonius of Perga; Archimedes; Diophantus; Euclid; Eudoxus of Cnidus; Hero of Alexandria; Hypatia; Pappus; Pythagoras; Thales of Miletus. Stoicheia (Euclid)