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Born: 350-325 B.C., place unknown

Died: C. 275 B.d., probably in Alexandria, Egypt

Major Work: The Elements (Stoicheiai)

Major Ideas

A rigorous, systematic treatment of mathematics requires the statement of all assumptions and the proof of all propositions by means of a uniform methodology.

All mathematical quantities can be expressed by geometrical figures, either lines, areas, or solids.

Physical events can ;be modeled using mathematical expressions.

Space is infinite in extent, but not infinitely divisible.

Euclid was the author of history's longest-lived textbook, the Elements, which forms to this day the basis of instruction in plane geometry, the chief mathematical achievement of ancient Greek civilization. In addition to the familiar theorems of geometry, the, thirteen books (long chapters) of the Elements also cover number theory, incommensurable or irrational quantities, solid geometry and the properties of polyhedra (regular solids, from the tetrahedron--a four-sided solid--to the icosahedron, the twenty-sided solid), and a type of "geometrical algebra."

Nothing definite can be said about Euclid's life, although two anecdotes circulated in antiquity. One relates that the king of Egypt, Ptolemy, who was the first of the Greek dynasty that ruled Egypt from 323 B.C. to 30 B.C., asked Euclid if there was any easier way to geometry than that of the Elements. Euclid replied, "There is no royal road to geometry," meaning that there is no specially smoothed and easy road to learning. The second anecdote records Euclid's answer to a student who asked him, "How shall I benefit if I learn these theorems?" Euclid told his assistant to give the student an obol (a small coin), "Since he must profit from whatever he learns."

From these stories we can deduce that Euclid taught geometry in Alexandria around 300 B.C., during the reign of Ptolemy I. He may have been born or lived as a youth in Athens, and he may have studied at Plato's Academy, since that institution was the center of mathematical research in the fourth century B.C. At any rate, he spent his later life in Alexandria, where Ptolemy was setting up the scholarly institute known as the Museum (Center for the Muses), and taught geometry there. The famous mathematician Archimedes (born c. 287 B.C.) studied with some of Euclid's students, not with Euclid himself, so we may be sure that Euclid had died before the mid-third century B.C..

Euclid can best be seen as a talented collector and systematizer of all then-existing mathematical knowledge. He collected the work of the earlier mathematicians Eudoxus, Theaetetus, and Pythagoras, and presented this work in a uniform format, with many additions of his own. The success of his systematization insured that geometry and geometrical proofs would dominate mathematics until modern times. Even a work as late as Newton's Principia (first edition 1687) has the same outward appearance as the Elements, with geometrical diagrams and proofs filling each page, even though Newton had by then developed the calculus, which would have better served his purpose.

The Elements

Since the Elements describes a complete mathematical universe, Euclid begins with the basic task: to define this universe. To do so, he lists the definitions, the postulates, and the axioms on which this universe is based. He lists twenty-three definitions; for example,

1. A point is that which has no length.

2. A line is a length without breadth. down to

23. Parallel straight lines are straight lines which, being in the same plane and being extended indefinitely in both directions, do not meet one another in either direction.

This last definition establishes that the universe of Euclidean geometry is infinite. It would not be true in closed-space geometries.

Next Euclid lists five postulates, that is, statements that are not proven, but simply demonstrated. For example, postulate 1 says:

1. It is possible to draw a straight line from any point to any other point.

The most controversial of these postulates is 5:

5. If a straight line falls on two straight lines in such a way as to make the interior angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, will meet on that side on which the angles less than two right angles occur.

(In non-Euclidean, curved-space geometries this postulate does not hold true.)

As his final preliminary step, Euclid lists five axioms, or common notions, which must be accepted as the basis of this geometry, but which cannot be proven. For example, axiom 1 says:

1. Things which are equal to the same thing are also equal to each other.

These are all common-sense observations about normal space, but they are difficult to prove in any rigorous sense. If other axioms are adopted, non-Eudidean geometry results.

After listing the definitions, postulates, and axioms, Euclid begins his thirteen books of propositions, which may be theorems (in which something is proven) or problems (in which something is constructed). Every proposition, unless it is abbreviated, exhibits the same method of proof:

1. the statement of the problem;

2. the setting-out of the materials for the proof--usually a geometric figure;

3. the definition or statement of the conditions under which the proposition may be satisfied;

4. the construction or drawing of the figure;

5. the proof;

6. the conclusion, which usually ends with Q.E.D., the Latin for "what had to be proven."

Take as an example of Euclid's procedure his proof of the Pythagorean theorem (book 1, proposition 47). The statement is as follows:

In right triangles, the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

Then follows the setting-out:

Let ABC be a right triangle having the angle BAC right.

Then the definition:

I say that the square on BC is equal to the squares on BA and AC.

Next the construction, which in this proposition is two pages long.

Finally, the proof:

Therefore the whole square, BDEC, is equal to the two squares, GB and HC ... Therefore the square on the side BC is equal to the squares on the sides BA and AC. and the conclusion, which repeats the original statement, Q.E.D.

The Elements is the first such systematic treatment of the entire mathematical universe, beginning with fundamental definitions and employing a standard method of proof throughout. By no means did Euclid consider this book an elementary text, and he gives no hint as to how the truth of these propositions was discovered or why they are presented in this particular sequence. The propositions have leaped like Athena, fully formed from the brow of the mathematician. Thomas Hobbes, when he first read the example given above is said to have exclaimed, "By God, this is impossible!" Such is the impression of the majesty of the propositions and the inexorability of their march through the text.

Books 1 through 4 of the Elements deal with the geometry of points, lines, areas, and rectilinear and circular figures. Books 5 and 6 deal with ratios and proportions, a topic first treated by the mathematician Eudoxus a century earlier. For Euclid, a ratio is a relationship according to size of two magnitudes, whether numbers, lengths, or areas. For example, 6 has a certain ratio to 10, in modern terms, 6/10. Two pairs of magnitudes that have the same ratio are proportional. For example, 12 has the same ratio to 20 as 6 has to 10; therefore the two pairs, 6 and 10, 12 and 20, are proportional to each other. The manipulation of ratios and proportions is fundamental to Greek mathematics and to a large extent takes over the function of modern algebra. For example, in book 6, proposition 13, finding of a mean proportional between two lines, using geometrical figures, is the equivalent of finding the square root of the length of one of the lines. It is precisely this combination of geometry with ratios and propo rtion to solve problems for which modern mathematics uses arithmetic or algebra that gives ancient Greek mathematics its peculiar character.

In book 5, definition 4, Euclid states that any magnitude/quantity can, when multiplied by a factor, exceed any other magnitude. This means that infinitely small or infinitely large magnitudes are impossible. (For instance, one could not find a number bigger than an infinitely large number, if such an infinitely large number could exist--but it cannot, according to definition 4.)

Books 7 through 9. of the Elements deal with "arithmetic" land numbers, beginning with the usual definitions: the (to us) familiar--even, odd, prime, composite (nonprime), squares, cubes; and the unfamiliar--the even-times-even numbers (products of two even numbers, for example 12 = 6 X 2), the even-times-odd numbers (the product of an even and an odd number, for example, 18 = 6 x 3), perfect numbers (which are equal to the sum of their factors, for example, 6 = 3 + 2 + 1). It must be remembered that "arithmetic" in ancient texts means the study of numbers and their properties, not the techniques of calculation, which was called "logistic." Much of the material in books 7-9 is attributable to the semilegendary Pythagoras and his school.

In the course of his long book 10, often considered his masterpiece, Euclid develops the theory of incommensurability, a topic that had previously been treated by the mathematician Theaetetus. Two magnitudes are commensurable if they have a common measure/factor that divides into each an exact integral number of times. Incommensurable magnitudes are those which have no such common measure: "If the lesser of two quantities is continually subtracted from the greater, and the remainder never measures (is a factor of) the quantity which precedes it, then the quantities will be incommensurable" (book 10, proposition 2). In modem terms, incommensurable quantities are those which cannot be expressed by a common fraction (such as 1/3) and which, if put in decimal notation, are expressed by an infinite decimal. The most commonly cited example of incommensurability is the hypotenuse of a right triangle with sides equal to one. By the Pythagorean theorem, the hypotenuse of this triangle is [1.sup.2] + [1.sup.2] = [ p.2], therefore the hypotenuse is [square root of ]2, quantity that is incommensurable with the sides. (In modern notation, this means that [square root of]2 is an infinite decimal. Euclid's reliance on geometrical means of expression means that he avoids the problem of how to represent incommensurable quantities; he does not, for example, have to find a numerical approximation to [square root of]2.) The theorems of book 10 were closely studied by the developers of algebra, Paciuolo, Cardano, and Stevin.

Books 11 through 13 deal with solid geometry: the construction and relationships between circumscribed and enclosed figures, the volumes of pyramids and other solid figures, and the construction of the five "Platonic" solids, the regular tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. These figures were given quasi-mystical importance by the school of Plato. In these books Euclid employs the method of exhaustion: by taking smaller and smaller areas, he shows that the area of a circle can he "exhausted," that is, reduced to an amount smaller than any given quantity. Less than a century later, Archimedes refined this method to find an approximation to [pi] = 3.14+.

Euclid wrote other books, the most significant of which is the Optics, a treatise on sight and perspective. It assumes that vision is produced by rays that are emitted from the eyes and "touch" the object that is seen. These rays are straight and their behavior can be modeled by the straight lines of plane geometry. For example, things that are farther away are seen less distinctly because fewer of these eye-rays hit them; the rays diverge like lines from a point. This book on optics was the only treatment of the topic until Ptolemy wrote his Optics in the second century A.D.

Further Reading

Euclid. The Thirteen Books of the Elements. Translated with introduction and commentary by Sir Thomas L. Heath. Cambridge University Press, 1926. Reprint, New York: Dover Publications, 1956. This is the essential work for a study of Euclid: It includes not only a translation but also an extensive commentary on the text with many citations from the history of mathematics.

Greenberg, Marvin. Euclidean and Non-Euclidean Geometries: Development and History. San Francisco: W. H. Freeman, 1980. Greenberg outlines the essence of Euclidean geometry and shows how the hyperbolic and elliptic, non-Euclidean, geometries have developed in modem times.

Heath, Sir Thomas L. A History of Greek Mathematics. Oxford: Clarendon Press, 1921. Reprint, New York: Dover Publications, 1981. The best overall history of Greek mathematics, this text puts Euclid in context.

Knorr, Wilbur R. The Evolution of the Euclidean Elements. Dordrecht and Boston: D. Reidel, 1975. Not a text for beginners, this book discusses pre-Euclidean theories, particularly the development of the concept of incommensurability.
COPYRIGHT 1999 COPYRIGHT 1992 Ian P. McGreal
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Publication:Great Thinkers of the Western World
Article Type:Biography
Geographic Code:4EUGR
Date:Jan 1, 1999
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