Book:Euclid/The Elements

Subject Matter

 * Geometry
 * Number Theory

Contents

 * Book I: Straight Line Geometry
 * Definitions
 * Postulates and Common Notions
 * Proposition 1: Construction of Equilateral Triangle
 * Proposition 2: Construction of an Equal Straight Line
 * Proposition 3: Construction of Equal Straight Lines from Unequal
 * Proposition 4: Triangle Side-Angle-Side Equality
 * Proposition 5: Isosceles Triangles have Two Equal Angles
 * Proposition 6: Triangle with Two Equal Angles is Isosceles
 * Proposition 7: Two Lines Meet at Unique Point
 * Proposition 8: Triangle Side-Side-Side Equality
 * Proposition 9: Bisection of an Angle
 * Proposition 10: Bisection of a Straight Line
 * Proposition 11: Construction of a Perpendicular
 * Proposition 12: Perpendicular through a Given Point
 * Proposition 13: Two Angles on a Straight Line make Two Right Angles
 * Proposition 14: Two Angles making Two Right Angles make a Straight Line
 * Proposition 15: Two Straight Lines make Equal Opposite Angles (Vertical Angle Theorem)
 * Proposition 16: External Angle of Triangle Greater than Internal Opposite
 * Proposition 17: Two Angles of Triangle Less than Two Right Angles
 * Proposition 18: Greater Side of Triangle Subtends Greater Angle
 * Proposition 19: Greater Angle of Triangle Subtended by Greater Side
 * Proposition 20: Sum of Two Sides of Triangle Greater than Third Side
 * Proposition 21: Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides
 * Proposition 22: Construction of Triangle from Given Lengths
 * Proposition 23: Construction of an Equal Angle
 * Proposition 24: Hinge Theorem
 * Proposition 25: Converse Hinge Theorem
 * Proposition 26: Triangle Angle-Side-Angle and Side-Angle-Angle Equality
 * Proposition 27: Equal Alternate Interior Angles Implies Parallel
 * Proposition 28: Equal Corresponding Angles or Supplementary Interior Angles Implies Parallel
 * Proposition 29: Parallel Implies Equal Alternate Interior Angles, Corresponding Angles, and Supplementary Interior Angles
 * Proposition 30: Parallelism is Transitive
 * Proposition 31: Construction of a Parallel
 * Proposition 32: Sum of Angles of Triangle Equals Two Right Angles
 * Proposition 33: Lines Joining Equal and Parallel Straight Lines
 * Proposition 34: Opposite Sides and Angles of Parallelogram are Equal
 * Proposition 35: Parallelograms with Same Base and Same Height have Equal Area
 * Proposition 36: Parallelograms with Equal Base and Same Height have Equal Area
 * Proposition 37: Triangles with Same Base and Same Height have Equal Area
 * Proposition 38: Triangles with Equal Base and Same Height have Equal Area
 * Proposition 39: Equal Sized Triangles on Same Base are Same Height
 * Proposition 40: Equal Sized Triangles on Equal Base are Same Height
 * Proposition 41: Parallelogram on Same Base as Triangle has Twice its Area
 * Proposition 42: Construction of Parallelogram Equal to Triangle in Given Angle
 * Proposition 43: Complements of Parallelograms are Equal
 * Proposition 44: Construction of Parallelogram on Given Line Equal to Triangle in Given Angle
 * Proposition 45: Construction of Parallelogram in Given Angle Equal to Given Polygon
 * Proposition 46: Construction of Square on Given Straight Line
 * Proposition 47: Pythagoras's Theorem
 * Proposition 48: Square equals Sum of Squares implies Right Triangle


 * Book II: Geometrical Algebra
 * Definitions
 * Proposition 1: Real Multiplication Distributes over Addition
 * Proposition 2: Square is Sum of Two Rectangles
 * Proposition 3: Rectangle is Sum of Square and Rectangle
 * Proposition 4: Square of Sum
 * Proposition 5: Difference of Two Squares
 * Proposition 6: Square of Sum less Square
 * Proposition 7: Square of Difference
 * Proposition 8: Square of Sum with Double
 * Propositions 9 and 10: Sum of Squares of Sum and Difference
 * Proposition 11: Construction of Square Equal to Rectangle
 * Proposition 12: Relative Sizes of Sides of Obtuse Triangle
 * Proposition 13: Relative Sizes of Sides of Acute Triangle
 * Proposition 14: Construction of Square Equal to Given Polygon


 * Book III: Circles
 * Definitions
 * Proposition 1: Finding Center of Circle
 * Proposition 2: Chord Lies Inside its Circle
 * Proposition 3: Conditions for Diameter to be Perpendicular Bisector
 * Proposition 4: Chords Do Not Bisect Each Other
 * Proposition 5: Intersecting Circles Have Different Centers
 * Proposition 6: Touching Circles Have Different Centers
 * Proposition 7: Relative Lengths of Lines Inside Circle
 * Proposition 8: Relative Lengths of Lines Outside Circle
 * Proposition 9: Condition for Point to be Center of Circle
 * Proposition 10: Two Circles Have At Most Two Points of Intersection
 * Proposition 11: Line Joining Centers of Two Circles Touching Internally
 * Proposition 12: Line Joining Centers of Two Circles Touching Externally


 * Book IV: Circles: Inscription and Circumscription
 * Definitions


 * Book V: Theory of Proportions
 * Book VI: Theory of Proportions as applied to Plane Geometry
 * Book VII: Number Theory
 * Book VIII: Theory of Proportions as applied to Number Theory


 * Book IX: Further Number Theory: Infinitude of Prime Numbers, Geometric Series, Perfect Numbers
 * Proposition 20: For any finite set of prime numbers, there exists a prime number not in that set (Euclid's Theorem)
 * Proposition 36: Theorem of Even Perfect Numbers (first part)


 * Book X: Irrational Numbers, steps towards Calculus
 * Book XI: Spatial Geometry
 * Book XII: Cones, Pyramids and Cylinders
 * Book XIII: The Five Platonic Solids

Notable Translations and Editions

 * c. 364: Theon of Alexandria


 * 1620: Henry Briggs (the first 6 books)


 * 1908 (2nd edition 1925): Sir Thomas L. Heath - 3 volumes:
 * Vol. 1: Books I and II (ISBN 0-486-60088-2)
 * Vol. 2: Books II - IX (ISBN 0-486-60089-0)
 * Vol. 3: Books X - XIII (ISBN 0-486-60090-4)

Online

 * Java -version.