# 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 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 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.