Euclid: Elements

Please note: The above is only an image. To obtain the stand-alone program that presents the ancient text and see what else is included there, go to the downloading page.
 

Euclid is the author of The Elements, the definitive work on classical geometry, which today is named after him: we call it “Euclidean geometry”, to distinguish it from other, non-Euclidean geometries that were developed in the 19th century.

Nearly nothing is known of Euclid, except that he lived in Alexandria, Egypt, during the reign of Ptolemy I (323 BC – 283 BC). He was active at the great Library of Alexandria, and may have studied at Plato’s Academy in Athens, Greece.

The Elements consist of 13 books, some of which deal with fields of mathematics that today we call algebra and number theory, but the ancient Greeks viewed these subjects always from the perspective of geometry. The Elements treat such important topics in planar geometry as: the Pythagorean theorem, equality of triangles, angles and their relation to triangles, triangles & angles inscribed in circles, tangents, circumscribed circles, polygons, Thales’ theorem, the golden ratio, and many more. In algebra and number theory, topics include divisibility, prime numbers, proof of the infinity of primes, greatest common divisor, least common multiple, prime factorization, perfect numbers, geometric sequences, sums of geometric series, irrational numbers, and more.

Finally, in spacial geometry, topics include perpendicularity and parallelism in three dimensions, areas and volumes of parallelepipeds, cones, pyramids, cylinders, prisms, and the sphere, the regular (Platonic) solids inscribed in a sphere, and more.

Besides being a foundation of geometrical and number-theoretical knowledge, another, most important contribution of the Elements is that it exemplified the axiomatic method and logical deduction (proving conclusions from premises), which became part of the Western subconscious thought in later times. When today we say “Can you prove it?”, what we mean is, ideally, to start from some undeniable assumptions (e.g., hard facts), and, making only logical deductions, prove the consequent proposition in the same way as propositions are proven in The Elements. In practice, of course, this can almost never be done in a manner that employs mathematical rigor; but it is the idealization of this process that was inculcated in the minds of Western scholars and thinkers, in large part due to Euclid’s Elements.

Euclid’s proofs are not error-free, because in some cases he makes use of propositions that appear “obvious”, but which he never stated (or proved) explicitly. This, however, is unavoidable, as anyone who has worked in automated theorem proving knows: if one wants to give a complete proof of anything but the simplest statements, the true number of propositions on which the proof must rest explodes to proportions hardly manageable by the human mind — hence only machines can keep track of some proofs in their entirety. Euclid used shortcuts, as every schoolteacher who teaches geometry inevitably must do.

The author of the present web-page has translated the introduction of The Elements into English and Modern Greek; specifically, the definitions and axions. The translation can be found in the program Classics Reader (downloadable and installable in Windows).