Euclid's Elements, Quick Trip
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Date: 0101-01-01T00:00:00+00:00

Euclid’s Elements, Quick Trip

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 + 2xy2

+ 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 areasto 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’s 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

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.

Table of Contents

Copyright Š 2002.

http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce

Dept. Math. & Comp. Sci.

Clark University Select bookBook IBook IIBook IIIBook IVBook VBook VIBook VIIBook VIIIBook IXBook XBook XIBook XIIBook XIII Select topicIntroductionTable of ContentsGeometry appletAbout the textEuclidWeb referencesA quick tripSubject index These pages are located at
http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.

Table of Contents

- Book I on basic plane geometry
- Book II on geometric algebra
- Book III on circles and angles
- Book VII on basic number theory
- Book IX on number theory
- Book XII on measurement of solids
- Book XIII on constructing regular polyhedra