The Elements ( ) is a mathematical treatise written 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.
Background 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's theorems, perfecting many of Theaetetus's, 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to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics.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.Review of The Evolution of the Euclidean Elements by Bartel Leendert van der Waerden (1976), Historia Mathematica 3 (4): 497–499, .
Other similar works are also reported to have been written by Hippocrates of Chios, Theudius of Magnesia, and Leon, but are now lost.
Contents + Summary Contents of Euclid's Elements (Heath edition) Book I II III IV V VI VII VIII IX X XI XII XIII Totals Definitions 23 2 11 7 18 4 22 – – 16 28 – – 131 Postulates 5 – – – – – – – – – – – – 5 Common Notions 5 – – – – – – – – – – – – 5 Propositions 48 14 37 16 25 33 39 27 36 115 39 18 18 465 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" ( or ), "postulate" (), or a "common notion" (). 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" () and each common notion an "axiom" (). 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. + Euclid's postulates and s Postulates Let the following be postulated: 1 To draw a straight line from any point to any point. 2 To produce a finite straight line continuously in a straight line 3 To describe a circle with any centre and distance 4 That all right angles are equal to one another 5 That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles Common notions 1 Things which are equal to the same thing are also equal to one another 2 If equals be added to equals, the wholes are equal 3 If equals be subtracted from equals, the remainders are equal 4 Things which coincide with one another are equal to one another 5 The whole is greater than the part
Books I to VI: Plane geometry
Book I
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.
Book I also includes 48 propositions, which can be loosely divided into: basic theorems and constructions of plane geometry and triangle congruence (1–26), parallel lines (27–34), the area of triangles and parallelograms (35–45), and the Pythagorean theorem and its converse (46–48).
Proposition 5, that the base angles of an isosceles triangle are equal, became known in the Middle Ages as the , or bridge of asses, separating the mathematicians who could prove it from the fools who could not. Papyrus Oxyrhynchus 29, a 3rd-century AD papyrus, contains fragments of propositions 8–11 and 14–25. The last two propositions of Book I comprise the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate". The figure for the Pythagorean theorem has itself become well known under multiple names: the Bride's Chair, the windmill, or the peacock's tail.
Book II
The second book focuses on area, measured through quadrature, meaning the construction of a square of equal area to a given figure. It includes a geometric precursor of the law of cosines, and culminates in the quadrature of arbitrary rectangles. In the late 19th and 20th centuries, Book II was interpreted by some mathematical historians to establish a "geometric algebra", an expression of algebraic manipulation of linear and quadratic equations in terms of geometric concepts of length and area, centered on the quadratic case of the binomial theorem. This interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later. Nevertheless, taken as statements about geometry, many of the propositions in this book are superfluous to modern mathematics, as they can be subsumed by the use of algebra.
Proposition 11 of Book II subdivides a given line segment into extreme and mean proportions, now called the golden ratio. It is the first of several propositions involving this ratio: It is later used in Book IV to construct a golden triangle and regular pentagon and in Book XIII to construct the regular dodecahedron and regular icosahedron, and studied as a ratio in Book VI Proposition 30.
Book III Book III begins with a list of 11 definitions, and follows with 37 propositions that deal with circles and their properties. Proposition 1 is on finding the center of a circle. Propositions 2 through 15 concern chords, and intersecting and tangent circles. Tangent lines to circles are the subjects of propositions 16 through 19. Next are propositions on inscribed angles (20 through 22), and on chords, arcs, and angles (23 through 30), including the inscribed angle theorem relating inscribed to central angles as proposition 20. Propositions 31 through 34 concern angles in circles, including Thales's theorem that an angle inscribed in a semicircle is a right angle (part of proposition 31). The remaining propositions, 35 through 37, concern intersecting chords and tangents; proposition 35 is the intersecting chords theorem, and proposition 36 is the tangent–secant theorem.
Book IV Book IV treats four problems systematically for different polygons: inscribing a polygon within a circle, circumscribing a polygon about a circle, inscribing a circle within a polygon, and circumscribing a circle about a polygon. These problems are solved in sequence for triangles and then for constructible regular polygons (i.e., those that have a straightedge and compass construction) with 4, 5, 6, and 15 sides.
Book V Book V, which is independent of the previous four books, concerns ratios of magnitudes (intuitively, how much bigger or smaller one shape is relative to another) and the comparison of ratios. Heath and other translators have formulated its first six propositions in symbolic algebra, as forms of the distributive law of multiplication over division and the associative law for multiplication. However, Leo Corry argues that this is anachronistic and misleading, because Euclid did not treat magnitudes as numbers, nor taking a ratio as a binary operation from numbers to numbers.
Much of Book V was probably ascertained from earlier mathematicians, perhaps Eudoxus, although certain propositions, such as V.16, dealing with "alternation" (if a : b :: c : d, then a : c :: b : d) likely predate Eudoxus.