Euclid of Alexandria
325 BC – 265 BC

Euclid, according to this source, is the most prominent mathematician of ancient times. One of his greatest works was his dissertation on mathematics The Elements. He is known as the leading mathematics teacher of all time; he is still teaching us today. It is thought that Euclid must have studied in Plato’s Academy in order to have learned the geometry of Eudoxus and Theaetetus.

The Elements was a compilation of knowledge which is still incorporated into today’s teachings. Although he may not have proved all of the results in this work, he did organize the material using earlier textbooks. He begins this dissertation with a number of definitions and five postulates. The first three postulates deal with construction; they assume the existence of points, lines and circles. The existence of other geometrical objects is deduced from these assumptions. For example, the first postulate states that “it is possible to draw a straight line between any two points.” The last two postulates are of a different nature. The fourth states that “all right angles are equal.” The last states that “one and only one line can be drawn through a point parallel to a given line.” The decision made by Euclid to make this statement a postulate is what led to Euclidean geometry. (In the 19th century, this postulate was dropped to result in the study of non-Euclidean geometry.)

Euclid is also responsible for certain axioms, which he refers to as “common notions”. These are general assumptions that allow mathematicians to proceed with their deductive reasoning. For example, one axiom states that “things which are equal to the same thing are equal to each other.”

The Elements is divided into 13 books. The first six address plane geometry. The first two deals with basic properties of triangles, parallels, parallelograms, rectangles, and squares. The third addresses properties of the circle, and the fourth deals with problems about circles. Book five looks at proportion and book six looks at applications to book five. In the next three books, Euclid addresses number theory. In particular, book seven, explains the Euclidean algorithm, which we have all learned to use in order to find the greatest common divisor of any two positive integers. The tenth book appreciates the theory of irrational numbers. The last three books deal with three-dimensional geometry. For example, the last book discusses the properties of the five regular polyhedra. In this, there is a proof that there are only five regular polyhedra.

We credit Euclid with the axiomatic method. This is a method of proving that results are correct. There are two requirements that must be met to consider a proof as correct:

1.) Certain statements, known now as axioms or postulates, must be accepted without further justification.

2.) There should be an agreement on how and when a statement follows logically from another.

So, Euclid singled out a few simple postulates and deduced from them 465 propositions, containing all the geometric knowledge of his time.
Euclid’s geometry was based on five fundamental axioms, or postulates, the first four of which have always been readily accepted by mathematicians. The fifth, the parallel postulate, was highly controversial. In fact, consideration of alternatives to this axiom resulted in non-Euclidean geometries. The axioms are as follows:

Euclid’s Postulate I. For every point P and for every point Q not equal to P there exists a unique line l that passes through P and Q.

Euclid’s Postulate II. For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE.

Euclid’s Postulate III. For every point O and every point A not equal to O there exists a circle with center O and radius OA.

Euclid’s Postulate IV. All right angles are congruent to each other.

The Euclidean Parallel Postulate. For every line l and for every point P that does not lie on l there exists and unique line m through P that is parallel to l.

References:

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries:

Development and History. 3rd ed. New York: W.H. Freeman and Company,
1993. 6-19.

Historical Topics for the Mathematics Classroom. Washington D.C.:

National Council of Teachers of Mathematics, 1969.

http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Thales.html (March 2005)

http://geometryalgorithms.com/history.htm (March 2005)

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