Euclid's Elements is by far the most famous mathematical work of classical antiquity, and also has the distinction of being the world's oldest continuously used mathematical textbook. Little is known about the author, beyond the fact that he lived in Alexandria around 300 BCE. The main subjects of the work are geometry, proportion, and number theory.
Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms.
Euclid is known to almost every high school student as the author of The Elements, the long studied text on geometry and number theory. No other book except the Bible has been so widely translated and circulated. From the time it was written it was regarded as an extraordinary work and was studied by all mathematicians, even the greatest mathematician of antiquity -- Archimedes, and so it has been through the 23 centuries that have followed. It is unquestionably the best mathematics text ever written and is likely to remain so into the distant future.
The Elements-- Basic facts
written about 2300 years ago,
no copies extant,
a few potsherds dating from 225 BC contain notes about some propositions,
Many new editions were issued (e.g. Theon of Alexandria, tex2html_wrap_inline379 cent. AD)
Earliest copy dates from 888AD -- in Oxford
Style: no examples, no motivations, no calculation, no witty remarks, no introduction, no preamble --- nothing but theorems and their proofs.
Book 1 contains Euclid's 10 axioms, which 5 named postulates—including the parallel postulate—and 5 named axioms, and the basic propositions of geometry: the pons asinorum, the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" have the same area.
The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight".
Since 1482, there have been more than a thousand editions of Euclid's Elements printed. It has been the standard source for geometry for millennia. It is only in recent decades that we have started to separate geometry from Euclid.
Euclid's indirect demonstrations always end with a contradiction to the immediate hypothesis; but as the propositions to which he applies the method are so extremely elementary, this could scarcely happen other wise, as, so far, deductions would be made from the hypothesis by direct steps.
Euclid's reasoning is said to be Deductive, because by a connected chain of argument it deduces new truths from truths already proved or admitted. Thus each proposition, though in one sense complete in itself, is derived from the Postulates, Axioms, or former propositions, and itself leads up to a subsequent proposition.
In these three books he defines prime numbers, develops many properties of divisibility, presents the Euclidean algorithm for finding the greatest common divisor of two integers, shows how to find an even perfect number from (what is now called) a Mersenne prime, proves that there are infinitely many primes, and states a version of the fundamental theorem of arithmetic.