Elements

The classic treatise in geometry written by Euclid and used as a textbook for more than 1,000 years in western Europe. The Elements, which went through more than 2,000 editions and consisted of 465 propositions, are divided into 13 ``books'' (an archaic word for ``chapters'').

 Book Contents 1 Triangles 2 Rectangles 3 Circles 4 Polygons 5 proportion 6 Similarity 7-10 Number Theory 11 solid geometry 12 Pyramids 13 Platonic Solids

The elements started with 23 definitions, five Postulates, and five ``common notions,'' and systematically built the rest of plane and solid geometry upon this foundation. The five Euclid's Postulates are

1. It is possible to draw a straight Line from any Point to another Point.

2. It is possible to produce a finite straight Line continuously in a straight Line.

3. It is possible to describe a Circle with any Center and Radius.

4. All Right Angles are equal to one another.

5. If a straight Line falling on two straight Lines makes the interior Angles on the same side less than two Right Angles, the straight Lines (if extended indefinitely) meet on the side on which the Angles which are less than two Right Angles lie.
(Dunham 1990). Euclid's fifth postulate is known as the Parallel Postulate. After more than two millennia of study, this Postulate was found to be independent of the others. In fact, equally valid Non-Euclidean Geometries were found to be possible by changing the assumption of this Postulate. Unfortunately, Euclid's postulates were not rigorously complete and left a large number of gaps. Hilbert needed a total of 20 postulates to construct a logically complete geometry.

References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, 6th ed. Dublin: Hodges, Figgis, & Co., 1892.

Dixon, R. Mathographics. New York: Dover, pp. 26-27, 1991.

Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 30-83, 1990.

Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 1: Books I and II. New York: Dover, 1956.

Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 2: Books III-IX. New York: Dover, 1956.

Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 3: Books X-XIII. New York: Dover, 1956.

Joyce, D. E. ``Euclid's Elements.'' http://aleph0.clarku.edu/~djoyce/java/elements/elements.html