Euclid and His Contributions

Euclid was an ancient Greek mathematician from Alexandria who is best known for his major work, Elements. Although little is known about Euclid the man, he taught in a school that he founded in Alexandria, Egypt, around 300 b.c.e.

For his major study, Elements, Euclid collected the work of many mathematicians who preceded him. Among these were Hippocrates of Chios, Theudius, Theaetetus, and Eudoxus. Euclid's vital contribution was to gather, compile, organize, and rework the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean geometry.

In Euclid's method, deductions are made from premises or axioms. This deductive method, as modified by Aristotle, was the sole procedure used for demonstrating scientific certitude ("truth") until the seventeenth century.

At the time of its introduction, Elements was the most comprehensive and logically rigorous examination of the basic principles of geometry. It survived the eclipse of classical learning, which occurred with the fall of the Roman Empire, through Arabic translations. Elements was reintroduced to Europe in 1120 c.e. when Adelard of Bath translated an Arabic version into Latin. Over time, it became a standard textbook in many societies, including the United States, and remained widely used until the mid-nineteenth century. Much of the information in it still forms a part of many high school geometry curricula.

Axiomatic Systems

To understand Euclid's Elements, one must first understand the concept of an axiomatic system . Mathematics is often described as being based solely on logic, meaning that statements are accepted as fact only if they can be logically deduced from other statements known to be true.

What does it mean for a statement to be "known to be true?" Such a statement could, of course, be deduced from some other "known" statement. However, there must be some set of statements, called axioms, that are simply

assumed to be true. Without axioms, no chain of deductions could ever begin. Thus even mathematics begins with certain unproved assumptions.

Ideally, in any axiomatic system, the assumptions are of such a basic and intuitive nature that their truth can be accepted without qualms. Yet axioms must be strong enough, or true enough, that other basic statements can be proved from them.

Definitions are also part of an axiomatic system, as are undefined terms (certain words whose definitions must be assumed in order for other words to be defined based on them). Thus an axiomatic system consists of the following: a collection of undefined terms; a collection of definitions; a collection of axioms (also called postulates); and, finally, a collection of theorems . Theorems are statements that are proved by the logical conclusion of a combination of axioms, definitions, and undefined terms.

Euclid's Axioms

In the Elements, Euclid attempted to bring together the various geometric facts known in his day (including some that he discovered himself) in order to form an axiomatic system, in which these "facts" could be subjected to rigorous proof. His undefined terms were point, line, straight line, surface, and plane. (To Euclid, the word "line" meant any finite curve, and hence a "straight" line is what we would call a line segment.)

Euclid divided his axioms into two categories, calling the first five postulates and the next five "common notions." The distinction between postulates and common notions is that the postulates are geometric in character, whereas common notions were considered by Euclid to be true in general.

Euclid's axioms follow.

1. It is possible to draw a straight line from any point to any point.
2. It is possible to extend a finite straight line continuously in a straight line. (In modern terminology, this says that a line segment can be extended past either of its endpoints to form an arbitrarily large line segment.)
3. It is possible to create a circle with any center and distance (radius).
4. All right angles are equal to one another. (A right angle is, by Euclid's definition, "half" of a straight angle: that is, if a line segment has one of its endpoints on another line segment and divides the second segment into two angles that are equal to each other, the two equal angles are called right angles.)
5. If a straight line falling on (crossing) two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
6. Things which are equal to the same thing are equal to each other.
7. If equals are added to equals, the wholes (sums) are equal.
8. If equals are subtracted from equals, the remainders (differences) are equal.
9. Things that coincide with one another are equal to one another.
10. The whole is greater than the part.

It was Euclid's intent that all the remaining geometric statements in the Elements be logical consequences of these ten axioms.

In the two millennia that have followed the first publication of the Elements, logical gaps have been found in some of Euclid's arguments, and places have been identified where Euclid uses an assumption that he never explicitly states. However, although quite a few of his arguments have needed improvement, the great majority of his results are sound.

Euclid's Fifth Postulate

The axioms in Euclid's list do seem intuitively obvious, and the Elements itself is proof that they can, as a group, be used to prove a wide variety of important geometric facts. They also, with one exception, seem sufficiently basic to warrant axiom status—that is, they need not be proved by even more basic statements or assumptions. The one exception to this is the fifth postulate. It is considerably more complicated to state than any of the others and does not seem quite as basic.

Starting almost immediately after the publication of the Elements and continuing into the nineteenth century, mathematicians tried to demonstrate that Euclid's fifth postulate was unnecessary. That is, they attempted to upgrade the fifth postulate to a theorem by deducing it logically from the other nine. Many thought they had succeeded; invariably, however, some later mathematician would discover that in the course of his "proofs" he had unknowingly made some extra assumption, beyond the allowable set of postulates, that was in fact logically equivalent to the fifth postulate.

In the early nineteenth century, after more than 2,000 years of trying to prove Euclid's fifth postulate, mathematicians began to entertain the idea that perhaps it was not provable after all and that Euclid had been correct to make it an axiom. Not long after that, several mathematicians, working independently, realized that if the fifth postulate did not follow from the others, it should be possible to construct a logically consistent geometric system without it.

One of the many statements that were discovered to be equivalent to the fifth postulate (in the course of the many failed attempts to prove it) is "Given a straight line, and a point P not on that line, there exists at most one straight line passing through P that is parallel to the given line." The first "non-Euclidean" geometers took as axioms all the other nine postulates of Euclidean geometry but replaced the fifth postulate with the statement "There exists a straight line, and a point P not on that line, such that there are two straight lines passing through P that are parallel to the given line." That is, they replaced the fifth postulate with its negation and started exploring the geometric system that resulted.

Although this negated fifth postulate seems intuitively absurd, all our objections to it hinge on our pre-conceived notions of the meanings of the undefined terms "point" and "straight line." It has been proved that there is no logical incompatibility between the negated fifth postulate and the other postulates of Euclidean geometry; thus, non-Euclidean geometry is as logically consistent as Euclidean geometry.

The recognition of this fact—that there could be a mathematical system that seems to contradict our most fundamental intuitions of how geometric objects behave—led to great upheaval not only among mathematicians but also among scientists and philosophers, and led to a thorough and painstaking reconsideration of what was meant by words such as "prove," "know," and above all, "truth."

see also Postulates; Theorems and Proofs; Proof.

Naomi Klarreich and

J. William Moncrief

Bibliography

Heath, Sir Thomas L. The Thirteen Books of Euclid's Elements. 1908. Reprint, New York: Dover Publications, 1956.

Kline, Morris. Mathematical Thought from Ancient to Modern Times, vol. 1. New York: Oxford University Press, 1972.

Trudeau, Richard J. The Non-Euclidean Revolution. Boston: Birkhäuser, 1987.