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.
Books I to VI: Plane geometry Book I from the proof of the
Pythagorean theorem, in the colored version used by Byrne's 1847 edition. The proof shows that the black and yellow areas are equal, as are the red and blue areas. 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
pons asinorum, 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 from Book II Proposition 11, with an arc added to the traditional diagram. The construction finds the midpoint of side of square , intersects line at with a circle of radius , and constructs a second square , whose vertex is the subdivision point. 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.
Christopher Zeeman has argued that Book V's focus on the behavior of ratios under the addition of magnitudes, and its consequent failure to define ratios of ratios, was a flaw that prevented the Greeks from finding certain important concepts such as the
cross ratio (central to
projective geometry).
Book VI Book VI uses the theory of ratios from Book V in the context of plane geometry, especially the construction and recognition of
similar figures. It is built almost entirely of its first proposition: "Triangles and parallelograms which are under the same height are to one another as their bases". That is, if two triangles have the same height, the ratio of their areas is the same as the ratio of lengths of their two base segments (and analogously for two parallelograms of the same height). This proposition provides a connection between ratios of lengths and ratios of areas. Proposition 25 constructs, from any two
polygons, a third polygon similar to the first and with the same area as the second.
Plutarch attributes this construction to Pythagoras, calling it "more subtle and more scientific" than the Pythagorean theorem. The famous ancient Greek problem of
doubling the cube, now known impossible with compass and straightedge, is a special case of the analogous 3d problem of constructing a figure with a specified shape and volume. The book ends as it begins, by connecting two types of ratios: ratios of angles, and ratios of circular arc lengths, in proposition 33.
Books VII to X: Number theory Number theory, the theory of the arithmetic of
natural numbers, is covered by books VII to X. Book VII begins with a set of 22 definitions for
parity (whether a number is even or odd),
prime numbers, and other arithmetic-related concepts. The first of these definitions is for the unit (in modern terms, the number one), while the second states that "a number is a multitude composed of units"; this is generally interpreted to mean that, for Euclid, one is not a number, and the natural numbers begin at two.
Book VII Book VII deals with elementary number theory, and includes 39 propositions, which can be loosely divided into: the
Euclidean algorithm, a method for determining whether numbers are
relatively prime and for finding the
greatest common divisor (1–4), fractions (5–10), the theory of proportions for numbers (11–19), prime and relatively prime numbers and the theory of greatest common divisors, (20–32), and
least common multiples (33–39).
Book VIII The topic of Book VIII is
geometric progressions. For Euclid, these were defined by the property of being in continued proportion (each two consecutive magnitudes have the same ratio) rather than, as in modern treatments, by
exponentiation (the ith term of the progression has the form ax^i for constants a and x). This allowed Euclid to avoid multiplication of more than two values, but led to some awkward proofs of facts that exponential notation would make obvious. The first part of Book VIII (propositions 1 through 10) deals with the construction and existence of geometric progressions of integers in general, and the
divisibility of members of a geometric progression by each other. Propositions 11 to 27 deal with
square numbers and
cube numbers in geometric progressions, and the relation between these special progressions and the elements two or three steps apart in an arbitrary geometric progression.
Book IX After continuing the investigations of Book VIII on squares and cubes in geometric progressions, Book IX applies the results of the preceding two books and gives the
infinitude of prime numbers (Euclid's theorem, proposition 20), the formula for the sum of a
finite geometric series (proposition 35) and a construction using this sum for even
perfect numbers (proposition 36). Here, a number is perfect if it equals the sum of its
proper divisors, as for instance 28 = 1 + 2 + 4 + 7 + 14.
Alhazen conjectured , and in the 18th century
Leonhard Euler proved, that this construction generates
all even perfect numbers. This result is the
Euclid–Euler theorem.
Book X Of the
Elements, book X is by far the largest and most complex, dealing with (in modern terms)
irrational numbers in the context of magnitudes. Proposition 9 (as restated in modern terms) proves the irrationality of the square roots of all non-square integers such as \sqrt2, the
square root of 2. A lemma to Proposition 29 gives
Euclid's formula for producing all fundamental
Pythagorean triples. Additionally, this book classifies irrational lengths into thirteen disjoint categories, related to their construction by various combinations of other lengths that are integers and their square roots. However,
Wilbur Knorr warns that "The student who approaches Euclid's Book X in the hope that its length and obscurity conceal mathematical treasures is likely to be disappointed. ... the mathematical ideas are few." Rather than treating magnitudes as
real numbers and asking whether these are
rational numbers, Euclid handles this material in terms of the
commensurability of lengths or areas: whether two line segments or two rectangles can both be measured by an integer number of copies of a common subunit. His classification of lengths as rational or irrational differs from the modern meaning: for Euclid, a line segment is rational when the square on its side has a rational area. That is, for Euclid, a length such as \sqrt2 that is the square root of a rational area is itself rational. This book is connected to a short passage in
Plato's dialogue
Theaetetus among
Socrates,
Theodorus of Cyrene, and
Theaetetus, a younger mathematician. This passage discusses a proof by Theodorus that the non-square integers from 3 to 17 have irrational square roots (after the much earlier discovery of the irrationality of \sqrt2), the generalization of this result to all non-square integers by Theaetetus, and a partial classification of the irrational numbers (with fewer than 13 classes).
Books XI to XIII: Solid geometry , foundational components of
solid geometry which feature in Books 11–13 The final three books primarily discuss
solid geometry. By introducing a list of 37 definitions, Book XI contextualizes the next two. Although its foundational character resembles Book I, unlike Book I it features no axiomatic system or postulates.
Book XI Book XI generalizes the results of book VI to solid figures: perpendicularity, parallelism, volumes, and similarity of
parallelepipeds (polyhedra with three pairs of parallel faces). The three sections of Book XI include content on: solid geometry (1–19), solid angles (20–23), and parallelepipeds (24–37).
Book XII Book XII studies the volumes of
cones,
pyramids, and
cylinders in detail by using the
method of exhaustion, a precursor to
integration, and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing that the volume of a
sphere is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids.
Book XIII Book XIII constructs the five
Platonic solids (regular polyhedra) inscribed in a sphere, compares the ratios of their edges to the radius of the sphere, and concludes the
Elements by proving that these are the only regular polyhedra.
Apocryphal books Two additional books, that were not written by Euclid, Books XIV and XV, have been transmitted in the manuscripts of the
Elements: • Book XIV was likely written by
Hypsicles, following a treatise by
Apollonius of Perga. It continues the study in Book XIII of the Platonic solids and their circumscribed spheres. It concludes that, for a dodecahedron and icosahedron inscribed in a common sphere, the ratio of their surface areas and the ratio of their volumes are equal, both being\sqrt{\frac{10}{3(5-\sqrt{5})}} = \sqrt{\frac{5+\sqrt{5}}{6}}. • Book XV may have been written by a student of
Isidore of Miletus. It also studies the Platonic solids; it inscribes some of them within each other, counts their edges and vertices (without however finding
Euler's formula V-E+F=2 relating these counts to each other), and computes the
dihedral angles between their faces. The practice of adding to the works of famous authors, exemplified by these books, was not unusual in ancient Greek mathematics. ==Euclid's method and style of presentation==