# High school Euclid paper

text of paper
Endnotes
Bibliography
bottom of page

This is the original version of my Euclid paper, done for a Survey of Math class at Bellaire High School (Bellaire, Texas). The teacher of that class, Ms. Dianne Resnick, also taught statistics (and still does, as far as I know). Her statistics class was what got me interested in statistics in the first place. (In case you missed it, I'm now a graduate student in the Statistics Department at Texas A&M University.)

## Euclid and his Elements

One of the most influential mathematicians of ancient Greece, Euclid flourished around 300 B.C. Not much is known about the life of Euclid. One story which reveals something about Euclid's character concerns a pupil who had just completed his first lesson in geometry. The pupil asked what he would get from learning geometry. So Euclid told his slave to give the pupil a coin so he would be gaining something from his studies. Included in the many works of Euclid is Data, concerning the solution of problems through geometric analysis, On Divisions (of Figures), the Optics, the Phenomena, a treatise on spherical geometry for astronomers, several lost works on higher geometry, and the Elements, a thirteen volume textbook on geometry. 

The Elements, which surely became a classic soon after its publication, eventually became the most influential textbook in the history of civilization. In fact, it has been said that apart from the Bible, the Elements is the most widely read and studied book in the world.  It has also been said that the Greeks used to post over the doors of their schools the inscription: ``Let no one come to our school who has not learned the Elements of Euclid.''  Probably every great Western mathematician to arise in the last two thousand years has studied Euclid's Elements.

In writing the Elements Euclid collected and extended many of the ideas of other Greek mathematicians before him. The Elements is basically a chain of 465 propositions encompassing most of the geometry, number theory, and geometric algebra of the Greeks up to that time.  Book I contains twenty-three definitions, five common notions (axioms), five postulates, and forty-eight propositions of plane geometry.

The definitions of Book I include those of points, lines, planes, angles, circles, triangles, quadrilaterals, and parallel lines.

The five postulates may be translated into the following:

1. Two points determine a straight line.
2. A line segment extended infinitely in both directions produces a straight line.
3. A circle is determined by a center and distance.
4. All right angles are equal to one another.
5. If a straight line falling an two straight lines forms interior angles on the same side less than 180 degrees, the two straight lines, if produced indefinitely, will meet on that side.

The last of these, commonly known as the ``parallel postulate,'' is by far the most important of the five. Through manipulation, the following statement may be derived: ``The sum of the angles in a triangle is equal to 180 degrees.'' Changing ``equal to'' to ``less than'' or ``greater than'' results in entirely different geometries -- non-Euclidean geometries. In spherical geometry, for example, this would read: ``The sum of the angles in a triangle is greater than 180 degrees.'' In hyperbolic geometry it would read: ``The sum of the angles in a triangle is less than 180 degrees.'' Hyperbolic geometry was invented by the Russian mathematician Nicolai Ivanovitch Lobachevsky. 

Postulates, by definition, are not and cannot be proven. However, some mathematicians have claimed that postulate four can be proven;  and many have believed that postulate five, partly because of its length and complexity, can be proven.  Lobachevsky's geometry grew out of his unsuccessful attempts to prove Euclid's parallel postulate.  Zeno of Sidon in the first century B.C. believed that Euclid's list of postulates was incomplete. He claimed that one must postulate that two distinct straight lines cannot have a segment in common. Without this, he claimed, some of the propositions in Book I are fallacious. 

Unlike the specialized nature of the postulates, the five common notions, or axioms, were essentially taken to be universal truths in all of mathematics and the sciences. The fifth axiom breaks down when exposed to the concept of infinite sets. For example, the set of all integers is not larger than the set of all even integers. 

The final section of Book I includes the forty-eight postulates. Included in these are the familiar results on triangles, such as proposition 5 [that the angles at the base of an isosceles triangle are equal], as well as the four congruence theorems for triangles: side-angle-side (prop. 4), side-side-side (prop. 8), angle-side-angle (prop. 26), and side-angle-angle (prop. 26, also). The last two propositions are the Pythagorean theorem and its converse.

### Endnotes


Heath, Sir Thomas, trans., The Thirteen Books of Euclid's Elements, Great Books of the Western World 11 (N.p.: Encyclopaedia Britannica, Inc., 1952) ix.

Faber, Richard L., Foundations of Euclidean and Non-Euclidean Geometry, Pure and Applied Mathematics: A Program of Monographs, Textbooks, and Lecture Notes 73 (New York: Marcel Dekker, Inc., 1983) 60.

Heath, x.

Faber, 60.

Faber, 162.

Smith, David Eugene, History of Mathematics, Volume II: Special Topics of Elementary Mathematics (Boston: Ginn and Company, 1925) 281.

Blumenthal, Leonard M., A Modern View of Geometry, A Series of Books in Mathematics (San Francisco: W. H. Freeman and Company, 1961) 4-5.

Faber, 162.

Smith, 281.

Blumenthal, 25.

### Bibliography

Blumenthal, Leonard M.
A Modern View of geometry. A Series of Books in Mathematics. San Francisco: W. H. Freeman and Company, 1961.
Faber, Richard L.
Foundations of Euclidean and Non-Euclidean Geometry. Pure and Applied Mathematics: A Program of Monographs, Textbooks, and Lecture Notes 73. New York: Marcel Dekker, Inc., 1983.
Heath, Sir Thomas, trans.
The Thirteen Books of Euclid's Elements. Great Books of the Western World 11. N.p.: Encyclopaedia Britannica, Inc., 1952.
Smith, David Euqene.
History of Mathematics, Volume II: Special Topics of Elementary Mathematics. Boston: Ginn and Company, 1925.