A Quick Trip through the Elements

To get an idea of what's in the Elements, here are a few highlights in the order that they appear:

Book I on basic plane geometry

• Def. I.23, definition of parallel lines, one of many definitions in Book I
• Post. I.5, the parallel postulate
• Common notions, the axioms for magnitudes
• Prop. I.1, the first proposition which shows how to construct an equilateral triangle
• The congruence theorems for triangles: Prop. I.4, side-angle-side, Prop. I.8, side-side-side, and Prop. I.26, angle-side-angle
• Propositions on isosceles triangles: Prop. I.5, equal angles imply equal sides, and the converse, Prop. I.6, equal sides imply equal angles
• Prop. I.9 and Prop. I.10, constructions to bisect angles and line segments
• Prop. I.11 and Prop. I.12, constructions to draw perpendicular lines
• Prop. I.16, an exterior angle of a triangle is greater than either of the opposite interior angles (compare I.32 below)
• Prop. I.29, about angles made when a line crosses two parallel lines
• Prop. I.20, the triangle inequality (the sum of two sides is greater than the third)
• Prop. I.22, to construct a triangle with given sides
• Prop. I.32, an exterior angle of a triangle is the sum of the two opposite interior angles; the sum of the three interior angles equals two right angles.
• On application of areas: Prop. I.42 to find a parallelogram equal in area to any given triangle, and Prop. I.45 to find a parallelogram equal in area to any given polygon
• Prop. I.47, the Pythagorean theorem and its converse Prop. I.48

Book II on geometric algebra

• Prop. II.4, a geometric version of the algebraic identity (x + y)² = x² + 2xy + y²
• Prop. II.5, a sample proposition showing how to factor the difference of two squares
• Prop. II.6, a geometric version to solve the quadratic equation (b – x)x = c
• Prop. II.11, construction to cut a line in the golden ratio
• Prop. II.12 and Prop. II.13, a pre-trigonometry version of the the law of cosines
• Prop. II.14, a final proposition on application of areas—to find a square equal in area to any given polygon

Book III on circles and angles

• Prop. III.1, how to find the center of a circle
• Prop. III.17, how to draw a line tangent to a circle
• Propositions on angles in circles: Prop. III.20, Prop. III.21, and Prop. III.22
• Prop. III.31, Thales' theorem that an angle inscribed in a semicircle is right, and similar statements giving acute and obtuse angles
• Prop. III.35, when two chords are drawn through a point inside a circle, then the product of the two segments of one chord equals the product of the two segments of the other chord
• Prop. III.36, if from a point outside a circle both a tangent and a secant are drawn, then the square of the tangent is the product of the whole secant and the external segment of the secant, and the converse in Prop. III.37

Book IV on constructions of regular polygons

• Inscribing and circumscribing circles and arbitrary triangles Prop. IV.2, Prop. IV.3, Prop. IV.4, and Prop. IV.5,
• Prop. IV.10, how to construct a particular triangle needed for regular pentagons
• Prop. IV.11, how to construct a regular pentagon
• Prop. IV.15, how to construct a regular hexagon
• Prop. IV.16, how to construct a regular 15-gon

Book V on Eudoxus’ abstract theory of ratio and proportion, abstract algebra

• Def. V.3, the definition and nature of ratio
• Def. V.5 and V.6, the definition of proportion (equality of ratios)
• Def. V.9, the definition of duplicate proportion (the square of a ratio)
• Prop. V.2, distributivity of multiplication over addition
• Prop. V.3, associativity of multiplication of whole numbers
• Prop. V.11, transitivity of equality of ratios
• Prop. V.16, alternate proportions
• Prop. V.22, ratios ex aequali

Book VI on similar figures and geometric proportions

• Def. VI.1, definition of similar figures
• Prop. VI.1, areas of triangles (also parallelograms) of the same height are proportional to their bases
• Prop. VI.2, a line parallel to the base of a triangle cuts the sides proportionally
• Propositions on similar triangles: Prop. VI.4, Prop. VI.5,
• Prop. VI.6, side-angle-side similarity theorem
• Prop. VI.9, to cut a line into a given number of equal segments
• Prop. VI.10, to cut a line into a specified ratio
• Constructions of fourth proportionals Prop. VI.12, and mean proportionals Prop. VI.13,
• Prop. VI.16, if four lines are proportional, w : x = y:z, then the rectangle contained by the extremes, w by z, has the same area as the rectangle contained by the means, x by y
• Prop. VI.19, on areas of similar triangles
• Prop. VI.25, on application of areas
• Prop. VI.31, a generalization of the Pythagorean theorem to figures other than squares

Book VII on basic number theory

• Def. VII.11, definition of prime number
• Prop. VII.12, the Euclidean algorithm for finding greatest common divisors
• Several basic properties of numbers, such as Prop. VII.16, commutativity of multiplication of numbers, mn = nm.
• Prop. VII.29, if a prime number doesn’t divide a number, then it is relatively prime to it
• Prop. VII.30, if a prime number divides a product of two numbers, then it divides one of them
• Prop. VII.34, on constructing least common multiples

Book VIII on continued proportions (geometric progressions) in number theory

• Prop. VIII.2 and Prop. VIII.4, on finding continued proportions of numbers
• Many propositions on squares and cubes, such as Prop. VIII.22, if three numbers are in continued proportion, and the first is square, then the third is also square

Book IX on number theory

• Prop. IX.14, a partial version of the fundamental theorem of arithmetic that says no prime number can divide a product of other prime numbers
• Prop. IX.20, there are infinitely many prime numbers
• Several propositions on even and odd numbers, such as Prop. IX.23 which says that if you add an odd number of odd numbers together, then the sum is odd
• Prop. IX.35, how to get the sum of a geometric progression
• Prop. IX.36, on perfect numbers

Book X on classification of irrational magnitudes

• Def. X.1, definition of commensurable magnitudes
• Prop. X.1, a principle of exhaustion
• Prop. X.2, a characterization of incommensurable magnitudes
• Prop. X.9, commensurability in square as opposed to commensurability in length
• Prop. X.12, transitivity of commensurability
• Prop. X.26, the difference between two medial areas is irrational
• Lemma 1 for Prop. X.29, to find two square numbers whose sum is also a square

Book XI on basic solid geometry

• Def. XI.14, definition of a sphere
• Def. XI.25 through 28, definitions of regular polygons
• Prop. XI.3, the intersection of two planes is a straight line
• Prop. XI.6, two lines perpendicular to a plane are parallel
• Constructions to draw lines perpendicular to planes: Prop. XI.11 and Prop. XI.12
• Prop. XI.14, two planes perpendicular to the same line are parallel
• Prop. XI.23, how to construct solid angles
• Several propositions on volumes of parallelopipeds, such as Prop. XI.32
• Prop. XI.39 on volumes of prisms

Book XII on measurement of solids

• Prop. XII.2, areas of circle are proportional to the squares on their diameters
• Prop. XII.6 and Prop. XII.7, a triangular prism can be divided into three pyramids of equal volume, hence, the volume of a pyramid is one third of that of the prism with the same base and same height
• Prop. XII.10, the volume of a cone is one third of that of the cylindar with the same base and same height
• Prop. XII.11, volumes of cones and cylinders are proportional to their heights
• Prop. XII.18, on volumes of spheres

Book XIII on constructing regular polyhedra

• Prop. XIII.9, on hexagons and decagons inscribed in a circle, and the golden ratio
• Prop. XIII.10, on hexagons and decagons inscribed in a circle, and the golden ratio
• Prop. XIII.11, when a pentagon, hexagon, and decagon are inscribed in a circle, the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon
• Constructions of regular polyhedra XIII.13, XIII.15, XIII.14, XIII.16, and XIII.17
• These five are shown to be the only regular solids in proposition XIII.18.