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The Elements

The Elements is the classic treatise in geometry written by Euclid  and used as a textbook for more than 1,000 years in western Europe.  An Arabic version appeared at the end of the eighth century, the first printed version was produced in 1482, and subsequently, The Elements has gone through more than 2,000 editions.  It consists of 465 propositions, and is divided into 13 books (i.e., chapters).  

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 site, <http://aleph0.clarku.edu/~djoyce/java/elements/elements.html>, provides a fairly comprehensive treatment, where the author notes, among other things:  “Euclid’s Elements form one of the most beautiful and influential works of science in the history of humankind.  Its beauty lies in its logical development of geometry and other branches of mathematics.  It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied for twenty four centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many of the modern languages.”  

The Elements starts with 23 definitions, five postulates, and five “common notions”, and then systematically builds 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.  

Euclid’s  fifth postulate is known as the parallel postulate -- although it’s actually defining lines which are not parallel.  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.  

But, of course, he had the advantage of 24 centuries of intervening work on which to base his more modern efforts.  (Plus which Hilbert didn’t have to learn Greek!)  

In general, however, Euclid’s The Elements and the study of geometry in general has the unique advantages to encouraging the development of deductive reasoning and logic -- the kind that would have made Dr. Spock of Star Trek fame proud.  

But before you begin your Geometry and Greek lessons, feel free to link via:  


Sacred Geometry          Platonic Solids

Forward to:

Golden Mean         Golden Mean Mathematics         The Golden Spiral



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