Euclid's Elements

The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid.

The Elements is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus, and Theaetetus, the Elements is a collection in 13 books of definitions, postulates, geometric constructions, and theorems with their proofs that covers plane and solid Euclidean geometry, elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the construction of regular polygons and polyhedra.

Often referred to as the most successful textbook ever written, the Elements has continued to be used for introductory geometry. It was translated into Arabic and Latin in the medieval period, where it exerted a great deal of influence on mathematics in the medieval Islamic world and in Western Europe, and has proven instrumental in the development of logic and modern science, where its logical rigor was not surpassed until the 19th century.

Euclid's Elements is the oldest extant large-scale deductive treatment of mathematics. Proclus, a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians, including Eudoxus, Hippocrates of Chios, Thales, and Theaetetus, while other theorems are mentioned by Plato and Aristotle. It is difficult to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics. The Elements version available today also includes "post-Euclidean" mathematics, probably added later by later editors such as the mathematician Theon of Alexandria in the 4th century. The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps" and the historian Serafina Cuomo described it as a "reservoir of results". Despite this, historian Michalis Sialaros opines that its "remarkably tight structure" suggests that Euclid wrote the Elements himself rather than merely editing together the works of others.

The detailed attribution of parts of the Elements to specific mathematicians is still the subject of scholarly debate. According to W. W. Rouse Ball, Pythagoras was probably the source for most of books I and II, Hippocrates of Chios for book III, and Eudoxus of Cnidus for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. Wilbur Knorr ascribes the origin of the material in Books I, III, and VI of the Elements to the time of Hippocrates of Chios, and of the material in books II, IV, X, and XIII to the later period of Theodorus of Cyrene, Theaetetus, and Eudoxos. However, this suggested history has been criticized by van der Waerden, who believed that books I through IV were largely due to the much earlier Pythagorean school.

Other similar works are also reported to have been written by Hippocrates of Chios, Theudius of Magnesia, and Leon, but are now lost.

The Elements does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry (books I–VI), basic number theory (books VII–X) and solid geometry (books XI–XIII)—though book V (on proportions) and X (on incommensurability) do not exactly fit this scheme. The heart of the text is the theorems scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια). The postulates (that is, axioms) and common notions occur only in book I. Close study of Proclus suggests that older versions of the Elements may have followed the same distinctions but with different terminology, instead calling each definition a "hypothesis" (ύπόΘεςιζ) and the common notions "axioms" (άξιώμα). The second group consists of propositions, presented alongside mathematical proofs and diagrams. It is unknown whether Euclid intended the Elements as a textbook, despite its wide subsequent use as one. As a whole, the authorial voice remains general and impersonal.

Book I of the Elements is foundational for the entire text. It begins with a series of 20 definitions for basic geometric concepts such as points, lines, angles and various regular polygons. Euclid then presents 10 assumptions (see table, right), grouped into five postulates and five common notions. These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system. The common notions exclusively concern the comparison of magnitudes, the sizes of geometric objects. In modern mathematics these magnitudes would be treated as real numbers measuring arc length, angle, or area, and compared numerically, but Euclid instead found ways of comparing the magnitude of shapes using geometric operations, without interpreting these magnitudes as numbers. While the first four postulates are relatively straightforward, the fifth is not. It is known as the parallel postulate, and the question of its independence from the other four postulates became the focus of a long line of research leading to the development of non-Euclidean geometry.