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Elements


The classic treatise in geometry written by Euclid and used as a textbook for more than 1000 years in western Europe. An Arabic version The Elements appears at the end of the eighth century, and the first printed version was produced in 1482 (Tietze 1965, p. 8). The Elements, which went through more than 2000 editions and consisted of 465 propositions, are divided into 13 "books" (an archaic word for "chapters").

bookcontents
1triangles
2rectangles
3circles
4polygons
5proportion
6similarity
7-10number theory
11solid geometry
12pyramids
13Platonic 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.


See also

Euclidean, Parallel Postulate

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Densmore, D. (Ed.). Euclid's Elements: All Thirteen Books Complete in One Volume. The Thomas L. Heath Translation.anta Fe, NM: Green Lion Press, 2002.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://mathcs.clarku.edu/~djoyce/java/elements/elements.html.Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, pp. 8-9, 1965.

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Elements

Cite this as:

Weisstein, Eric W. "Elements." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Elements.html

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