ProofWiki:Books/Euclid/The Elements
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Euclid: The Elements
Published c. 300 B.C.E.
Subject Matter
Contents
Book I: Straight Line Geometry
- 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
- 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
- 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
- Proposition 13: Circles Touch at One Point at Most
- Proposition 14: Equal Chords in Circle
- Proposition 15: Relative Lengths of Chords of Circles
- Proposition 16: Line at Right Angles to Diameter of Circle
- Proposition 17: Construction of Tangent from Point to Circle
- Proposition 18: Radius at Right Angle to Tangent
- Proposition 19: Right Angle to Tangent to Circle goes through Center
- Proposition 20: Inscribed Angle Theorem
- Proposition 21: Angles in Same Segment of Circle are Equal
- Proposition 22: Opposite Angles of Cyclic Quadrilateral
- Proposition 23: Segment on Given Base Unique
- Proposition 24: Similar Segments on Equal Bases are Equal
- Proposition 25: Construction of Circle from Segment
- Proposition 26: Equal Angles in Equal Circles
- Proposition 27: Angles on Equal Arcs are Equal
- Proposition 28: Straight Lines Cut Off Equal Arcs in Equal Circles
- Proposition 29: Equal Arcs of Circles Subtended by Equal Straight Lines
- Proposition 30: Bisection of an Arc
- Proposition 31: Relative Sizes of Angles in Segments
- Proposition 32: Angles made by Chord with Tangent
- Proposition 33: Construction of Segment on Given Line Admitting a Given Angle
- Proposition 34: Construction of Segment on Given Circle Admitting a Given Angle
- Proposition 35: Intersecting Chord Theorem
- Proposition 36: Tangent Secant Theorem
- Proposition 37: Converse of Tangent Secant Theorem
Book IV: Circles: Inscription and Circumscription
- Book IV: Circles: Inscription and Circumscription
- Definitions
- Proposition 1: Fitting a Chord Into a Circle
- Proposition 2: Inscribing in Circle a Triangle Equiangular with Given
- Proposition 3: Circumscribing about Circle a Triangle Equiangular with Given
- Proposition 4: Inscribing a Circle in a Triangle
- Proposition 5: Circumscribing a Circle about a Triangle
- Proposition 6: Inscribing a Square in a Circle
- Proposition 7: Circumscribing a Square about a Circle
- Proposition 8: Inscribing a Circle in a Square
- Proposition 9: Circumscribing a Circle about a Square
- Proposition 10: Construction of Isosceles Triangle whose Base Angle is Twice Apex
- Proposition 11: Inscribing a Regular Pentagon in a Circle
- Proposition 12: Circumscribing a Regular Pentagon about a Circle
- Proposition 13: Inscribing a Circle in a Regular Pentagon
- Proposition 14: Circumscribing a Circle about a Regular Pentagon
- Proposition 15: Inscribing a Regular Hexagon in a Circle
- Proposition 16: Inscribing a Regular 15-gon in a Circle
Book V: Theory of Proportions
- Book V: Theory of Proportions
- Definitions
- Proposition 1: Multiplication of Numbers Distributes over Addition (1)
- Proposition 2: Multiplication of Numbers Distributes over Addition (2)
- Proposition 3: Multiplication of Numbers is Associative
- Proposition 4: Multiples of Terms in Equal Ratios
- Propositions 5 and 6: Multiplication of Numbers Distributes over Subtraction
- Proposition 7: Ratios of Equal Magnitudes
- Proposition 8: Relative Sizes of Ratios on Unequal Magnitudes
- Proposition 9: Magnitudes with Same Ratios are Equal
- Proposition 10: Relative Sizes of Magnitudes on Unequal Ratios
- Proposition 11: Equality of Ratios is Transitive
- Proposition 12: Sum of Components of Equal Ratios
- Proposition 13: Relative Sizes of Proportional Magnitudes
- Proposition 14: Relative Sizes of Components of Ratios
- Proposition 15: Ratio Equals its Multiples
- Proposition 16: Proportional Magnitudes are Proportional Alternately
- Proposition 17: Magnitudes Proportional Compounded are Proportional Separated
- Proposition 18: Magnitudes Proportional Separated are Proportional Compounded
- Proposition 19: Proportional Magnitudes have Proportional Remainders
- Proposition 20: Relative Sizes of Successive Ratios
- Proposition 21: Relative Sizes of Elements in Perturbed Proportion
- Proposition 22: Equality of Ratios Ex Aequali
- Proposition 23: Equality of Ratios in Perturbed Proportion
- Proposition 24: Sum of Antecedents of Proportion
- Proposition 25: Sum of Antecedent and Consequent of Proportion
Book VI: Theory of Proportions as applied to Plane Geometry
- Book VI: Theory of Proportions as applied to Plane Geometry
- Definitions
- Proposition 1: Areas of Triangles and Parallelograms Proportional to Base
- Proposition 2: Parallel Line in Triangle Cuts Sides Proportionally
- Proposition 3: Angle Bisector Theorem
- Proposition 4: Equiangular Triangles are Similar
- Proposition 5: Triangles with Proportional Sides are Similar
- Proposition 6: Triangles with One Equal Angle and Two Sides Proportional are Similar
- Proposition 7: Triangles with One Equal Angle and Two Other Sides Proportional are Similar
- Proposition 8: Perpendicular in Right-Angled Triangle makes two Similar Triangles
- Proposition 9: Construction of a Part of a Line
- Proposition 10: Construction of Similarly Cut Straight Line
- Proposition 11: Construction of Third Proportional Straight Line
- Proposition 12: Construction of Fourth Proportional Straight Line
- Proposition 13: Construction of Mean Proportional
- Proposition 14: Sides of Equiangular Parallelograms are Reciprocally Proportional
- Proposition 15: Sides of Equiangular Triangles are Reciprocally Proportional
- Proposition 16: Rectangles Contained by Proportional Straight Lines
- Proposition 17: Rectangles Contained by Three Proportional Straight Lines
- Proposition 18: Construction of Similar Polygon
- Proposition 19: Ratio of Areas of Similar Triangles
- Proposition 20: Similar Polygons Composed of Similar Triangles
- Proposition 21: Similarity of Polygons is Equivalence
- Proposition 22: Similar Figures on Proportional Straight Lines
- Proposition 23: Ratio of Areas of Equiangular Parallelograms
- Proposition 24: Parallelograms About Diameter are Similar
- Proposition 25: Construction of Figure Similar to One and Equal to Another
- Proposition 26: Parallelogram Similar and in Same Angle has Same Diameter
- Proposition 27: Similar Parallelogram on Half a Straight Line
- Proposition 28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram
- Proposition 29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram
- Proposition 30: Construction of Golden Section
- Proposition 31: Similar Figures on Sides of Right-Angled Triangle
- Proposition 32: Triangles with Two Sides Parallel and Equal
- Proposition 33: Angles in Circles have Same Ratio as Arcs
Book VII: Number Theory
- Book VII: Number Theory
- Definitions
- Proposition 1: Sufficient Condition for Coprimality
- Proposition 2: Greatest Common Divisor of Two Numbers (Euclidean Algorithm)
- Proposition 3: Greatest Common Divisor of Three Numbers
- Proposition 4: Natural Number Divisor or Multiple of Divisor of Another
- Proposition 5: Divisors Obey Distributive Law
- Proposition 6: Multiples of Divisors Obey Distributive Law
- Proposition 7: Subtraction of Divisors Obeys Distributive Law
- Proposition 8: Subtraction of Multiples of Divisors Obeys Distributive Law
- Proposition 9: Alternate Ratios of Equal Fractions
- Proposition 10: Multiples of Alternate Ratios of Equal Fractions
- Proposition 11: Proportional Numbers have Proportional Differences
- Proposition 12: Ratios of Numbers is Distributive over Addition
- Proposition 13: Proportional Numbers are Proportional Alternately
- Proposition 14: Proportion of Numbers is Transitive
- Proposition 15: Alternate Ratios of Multiples
- Proposition 16: Natural Number Multiplication is Commutative
- Proposition 17: Multiples of Ratios of Numbers
- Proposition 18: Ratios of Multiples of Numbers
- Proposition 19: Relation of Ratios to Products
- Proposition 20: Ratios of Fractions in Lowest Terms
- Proposition 21: Numbers in Fractions in Lowest Terms are Coprime
Book VIII: Theory of Proportions as applied to Number Theory
Book IX: Further Number Theory: Infinitude of Prime Numbers, Geometric Series, Perfect Numbers
Book X: Irrational Numbers, steps towards Calculus
Book XI: Spatial Geometry
Book XII: Cones, Pyramids and Cylinders
Book XIII: The Five Platonic Solids
The So-Called Book XIV
Notable Translations and Editions
- c. 364: Theon of Alexandria
- 1620: Henry Briggs (the first 6 books)
- 1908: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements - 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 [1].
