# Book:Euclid/The Elements

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## Euclid:

#### The So-Called

## Euclid: *The Elements*

Published $\text {c. 300 B.C.E}$.

### Subject Matter

### Contents

#### Book $\text{I}$: Straight Line Geometry

- Proposition $1$: Construction of Equilateral Triangle
- Proposition $2$: Construction of Equal Straight Line
- Proposition $3$: Construction of Equal Straight Lines from Unequal
- Proposition $4$: Triangle Side-Angle-Side Equality
- Proposition $5$: Isosceles Triangle has 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 Angle
- Proposition $10$: Bisection of Straight Line
- Proposition $11$: Construction of Perpendicular Line
- Proposition $12$: Perpendicular through Given Point
- Proposition $13$: Two Angles on Straight Line make Two Right Angles
- Proposition $14$: Two Angles making Two Right Angles make Straight Line
- Proposition $15$: Two Straight Lines make Equal Opposite Angles
- 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 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 Angles implies Parallel Lines
- Proposition $28$: Equal Corresponding Angles or Supplementary Interior Angles implies Parallel Lines
- Proposition $29$: Parallelism implies Equal Alternate Angles, Corresponding Angles, and Supplementary Interior Angles
- Proposition $30$: Parallelism is Transitive Relation
- Proposition $31$: Construction of Parallel Line
- Proposition $32: \text{ Part } 1$: External Angle of Triangle equals Sum of other Internal Angles
- Proposition $32: \text{ Part } 2$: Sum of Angles of Triangle equals Two Right Angles
- Proposition $33$: Lines Joining Equal and Parallel Straight Lines are Parallel
- 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 have Same Height
- Proposition $40$: Equal Sized Triangles on Equal Base have 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 $\text{II}$: Geometrical Algebra

- Proposition $1$: Real Multiplication Distributes over Addition/Geometric Proof
- 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
- Proposition $9$: Sum of Squares of Sum and Difference (1)
- Proposition $10$: Sum of Squares of Sum and Difference (2)
- 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 $\text{III}$: Circles

- 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 of 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 sum to Two Right Angles
- 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 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 Given Angle
- Proposition $34$: Construction of Segment on Given Circle Admitting Given Angle
- Proposition $35$: Intersecting Chord Theorem
- Proposition $36$: Tangent Secant Theorem
- Proposition $37$: Converse of Tangent Secant Theorem

#### Book $\text{IV}$: Circles: Inscription and Circumscription

- Proposition $1$: Fitting Chord Into Circle
- Proposition $2$: Inscribing in Circle Triangle Equiangular with Given
- Proposition $3$: Circumscribing about Circle Triangle Equiangular with Given
- Proposition $4$: Inscribing Circle in Triangle
- Proposition $5$: Circumscribing Circle about Triangle
- Proposition $6$: Inscribing Square in Circle
- Proposition $7$: Circumscribing Square about Circle
- Proposition $8$: Inscribing Circle in Square
- Proposition $9$: Circumscribing Circle about Square
- Proposition $10$: Construction of Isosceles Triangle whose Base Angle is Twice Apex
- Proposition $11$: Inscribing Regular Pentagon in Circle
- Proposition $12$: Circumscribing Regular Pentagon about Circle
- Proposition $13$: Inscribing Circle in Regular Pentagon
- Proposition $14$: Circumscribing Circle about Regular Pentagon
- Proposition $15$: Inscribing Regular Hexagon in Circle
- Proposition $16$: Inscribing Regular 15-gon in Circle

#### Book $\text{V}$: Theory of Proportions

- Proposition $1$: Multiplication of Numbers is Left Distributive over Addition
- Proposition $2$: Multiplication of Numbers is Right Distributive over Addition
- Proposition $3$: Associative Law of Multiplication
- Proposition $4$: Multiples of Terms in Equal Ratios
- Proposition $5$: Multiplication of Real Numbers is Left Distributive over Subtraction
- Proposition $6$: Multiplication of Real Numbers is Right Distributive 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 $\text{VI}$: Theory of Proportions as applied to Plane Geometry

- 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 Part of 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 Equal and 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 are composed of Similar Triangles
- Proposition $21$: Similarity of Polygons is Equivalence Relation
- 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 $\text{VII}$: Number Theory

- 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 $21$: Coprime Numbers form Fraction in Lowest Terms
- Proposition $22$: Numbers forming Fraction in Lowest Terms are Coprime
- Proposition $23$: Divisor of One of Coprime Numbers is Coprime to Other
- Proposition $24$: Integer Coprime to all Factors is Coprime to Whole
- Proposition $25$: Square of Coprime Number is Coprime
- Proposition $26$: Product of Coprime Pairs is Coprime
- Proposition $27$: Powers of Coprime Numbers are Coprime
- Proposition $28$: Numbers are Coprime iff Sum is Coprime to Both
- Proposition $29$: Prime not Divisor implies Coprime
- Proposition $30$: Euclid's Lemma for Prime Divisors
- Proposition $31$: Composite Number has Prime Factor
- Proposition $32$: Natural Number is Prime or has Prime Factor
- Proposition $33$: Least Ratio of Numbers
- Proposition $34$: Existence of Lowest Common Multiple
- Proposition $35$: LCM Divides Common Multiple
- Proposition $36$: LCM of Three Numbers
- Proposition $37$: Integer Divided by Divisor is Integer
- Proposition $38$: Divisor is Reciprocal of Divisor of Integer
- Proposition $39$: Least Number with Three Given Fractions

#### Book $\text{VIII}$: Theory of Proportions as applied to Number Theory

- Proposition $1$: Geometric Sequence with Coprime Extremes is in Lowest Terms
- Proposition $2$: Construction of Geometric Sequence in Lowest Terms
- Proposition $3$: Geometric Sequence in Lowest Terms has Coprime Extremes
- Proposition $4$: Construction of Sequence of Numbers with Given Ratios
- Proposition $5$: Ratio of Products of Sides of Plane Numbers
- Proposition $6$: First Element of Geometric Sequence not dividing Second
- Proposition $7$: First Element of Geometric Sequence that divides Last also divides Second
- Proposition $8$: Geometric Sequences in Proportion have Same Number of Elements
- Proposition $9$: Elements of Geometric Sequence between Coprime Numbers
- Proposition $10$: Product of Geometric Sequences from One
- Proposition $11$: Between two Squares exists one Mean Proportional
- Proposition $12$: Between two Cubes exist two Mean Proportionals
- Proposition $13$: Powers of Elements of Geometric Sequence are in Geometric Sequence
- Proposition $14$: Number divides Number iff Square divides Square
- Proposition $15$: Number divides Number iff Cube divides Cube
- Proposition $16$: Number does not divide Number iff Square does not divide Square
- Proposition $17$: Number does not divide Number iff Cube does not divide Cube
- Proposition $18$: Between two Similar Plane Numbers exists one Mean Proportional
- Proposition $19$: Between two Similar Solid Numbers exist two Mean Proportionals
- Proposition $20$: Numbers between which exists one Mean Proportional are Similar Plane
- Proposition $21$: Numbers between which exist two Mean Proportionals are Similar Solid
- Proposition $22$: If First of Three Numbers in Geometric Sequence is Square then Third is Square
- Proposition $23$: If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube
- Proposition $24$: If Ratio of Square to Number is as between Two Squares then Number is Square
- Proposition $25$: If Ratio of Cube to Number is as between Two Cubes then Number is Cube
- Proposition $26$: Similar Plane Numbers have Same Ratio as between Two Squares
- Proposition $27$: Similar Solid Numbers have Same Ratio as between Two Cubes

#### Book $\text{IX}$: Further Number Theory: Infinitude of Prime Numbers, Geometric Series, Perfect Numbers

- Proposition $1$: Product of Similar Plane Numbers is Square
- Proposition $2$: Numbers whose Product is Square are Similar Plane Numbers
- Proposition $3$: Square of Cube Number is Cube
- Proposition $4$: Cube Number multiplied by Cube Number is Cube
- Proposition $5$: Number multiplied by Cube Number making Cube is itself Cube
- Proposition $6$: Number Squared making Cube is itself Cube
- Proposition $7$: Product of Composite Number with Number is Solid Number
- Proposition $8$: Elements of Geometric Sequence from One which are Powers of Number
- Proposition $9$: Elements of Geometric Sequence from One where First Element is Power of Number
- Proposition $10$: Elements of Geometric Sequence from One where First Element is not Power of Number
- Proposition $11$: Elements of Geometric Sequence from One which Divide Later Elements
- Proposition $12$: Elements of Geometric Sequence from One Divisible by Prime
- Proposition $13$: Divisibility of Elements of Geometric Sequence from One where First Element is Prime
- Proposition $14$: Expression for Integer as Product of Primes is Unique
- Proposition $15$: Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element
- Proposition $16$: Two Coprime Integers have no Third Integer Proportional
- Proposition $17$: Last Element of Geometric Sequence with Coprime Extremes has no Integer Proportional as First to Second
- Proposition $18$: Condition for Existence of Third Number Proportional to Two Numbers
- Proposition $19$: Condition for Existence of Fourth Number Proportional to Three Numbers
- Proposition $20$: For any finite set of prime numbers, there exists a prime number not in that set (Euclid's Theorem)
- Proposition $21$: Sum of Even Integers is Even
- Proposition $22$: Sum of Even Number of Odd Numbers is Even
- Proposition $23$: Sum of Odd Number of Odd Numbers is Odd
- Proposition $24$: Even Number minus Even Number is Even
- Proposition $25$: Even Number minus Odd Number is Odd
- Proposition $26$: Odd Number minus Odd Number is Even
- Proposition $27$: Odd Number minus Even Number is Odd
- Proposition $28$: Odd Number multiplied by Even Number is Even
- Proposition $29$: Odd Number multiplied by Odd Number is Odd
- Proposition $30$: Odd Divisor of Even Number also divides its Half
- Proposition $31$: Odd Number Coprime to Number is also Coprime to its Double
- Proposition $32$: Power of Two is Even-Times Even Only
- Proposition $33$: Number whose Half is Odd is Even-Times Odd
- Proposition $34$: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
- Proposition $35$: Sum of Geometric Sequence
- Proposition $36$: Theorem of Even Perfect Numbers (first part)

#### Book $\text{X}$: Irrational Numbers, steps towards Calculus

- Proposition $1$: Existence of Fraction of Number Smaller than Given
- Proposition $2$: Incommensurable Magnitudes do not Terminate in Euclid's Algorithm
- Proposition $3$: Greatest Common Measure of Commensurable Magnitudes
- Proposition $4$: Greatest Common Measure of Three Commensurable Magnitudes
- Proposition $5$: Ratio of Commensurable Magnitudes
- Proposition $6$: Magnitudes with Rational Ratio are Commensurable
- Proposition $7$: Incommensurable Magnitudes have Irrational Ratio
- Proposition $8$: Magnitudes with Irrational Ratio are Incommensurable
- Proposition $9$: Commensurability of Squares
- Proposition $10$: Construction of Incommensurable Lines
- Proposition $11$: Commensurability of Elements of Proportional Magnitudes
- Proposition $12$: Commensurability is Transitive Relation
- Proposition $13$: Commensurable Magnitudes are Incommensurable with Same Magnitude
- Proposition $14$: Commensurability of Squares on Proportional Straight Lines
- Proposition $15$: Commensurability of Sum of Commensurable Magnitudes
- Proposition $16$: Incommensurability of Sum of Incommensurable Magnitudes
- Proposition $17$: Condition for Commensurability of Roots of Quadratic Equation
- Proposition $18$: Condition for Incommensurability of Roots of Quadratic Equation
- Proposition $19$: Product of Rationally Expressible Numbers is Rational
- Proposition $20$: Quotient of Rationally Expressible Numbers is Rational
- Proposition $21$: Medial is Irrational
- Proposition $22$: Square on Medial Straight Line
- Proposition $23$: Straight Line Commensurable with Medial Straight Line is Medial
- Proposition $24$: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
- Proposition $25$: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
- Proposition $26$: Medial Area not greater than Medial Area by Rational Area
- Proposition $27$: Construction of Components of First Bimedial
- Proposition $28$: Construction of Components of Second Bimedial
- Proposition $29$: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater
- Proposition $30$: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Incommensurable with Greater
- Proposition $31$: Construction of Medial Straight Lines Commensurable in Square Only containing Rational Rectangle whose Square Differences Commensurable with Greater
- Proposition $32$: Construction of Medial Straight Lines Commensurable in Square Only containing Medial Rectangle whose Square Differences Commensurable with Greater
- Proposition $33$: Construction of Components of Major
- Proposition $34$: Construction of Components of Side of Rational plus Medial Area
- Proposition $35$: Construction of Components of Side of Sum of Medial Areas
- Proposition $36$: Binomial is Irrational
- Proposition $37$: First Bimedial is Irrational
- Proposition $38$: Second Bimedial is Irrational
- Proposition $39$: Major is Irrational
- Proposition $40$: Side of Rational plus Medial Area is Irrational
- Proposition $41$: Side of Sum of Medial Areas is Irrational
- Proposition $42$: Binomial Straight Line is Divisible into Terms Uniquely
- Proposition $43$: First Bimedial Straight Line is Divisible Uniquely
- Proposition $44$: Second Bimedial Straight Line is Divisible Uniquely
- Proposition $45$: Major Straight Line is Divisible Uniquely
- Proposition $46$: Side of Rational Plus Medial Area is Divisible Uniquely
- Proposition $47$: Side of Sum of Two Medial Areas is Divisible Uniquely

- Proposition $48$: Construction of First Binomial Straight Line
- Proposition $49$: Construction of Second Binomial Straight Line
- Proposition $50$: Construction of Third Binomial Straight Line
- Proposition $51$: Construction of Fourth Binomial Straight Line
- Proposition $52$: Construction of Fifth Binomial Straight Line
- Proposition $53$: Construction of Sixth Binomial Straight Line
- Proposition $54$: Root of Area contained by Rational Straight Line and First Binomial
- Proposition $55$: Root of Area contained by Rational Straight Line and Second Binomial
- Proposition $56$: Root of Area contained by Rational Straight Line and Third Binomial
- Proposition $57$: Root of Area contained by Rational Straight Line and Fourth Binomial
- Proposition $58$: Root of Area contained by Rational Straight Line and Fifth Binomial
- Proposition $59$: Root of Area contained by Rational Straight Line and Sixth Binomial
- Proposition $60$: Square on Binomial Straight Line applied to Rational Straight Line
- Proposition $61$: Square on First Bimedial Straight Line applied to Rational Straight Line
- Proposition $62$: Square on Second Bimedial Straight Line applied to Rational Straight Line
- Proposition $63$: Square on Major Straight Line applied to Rational Straight Line
- Proposition $64$: Square on Side of Rational plus Medial Area applied to Rational Straight Line
- Proposition $65$: Square on Side of Sum of two Medial Area applied to Rational Straight Line
- Proposition $66$: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
- Proposition $67$: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
- Proposition $68$: Straight Line Commensurable with Major Straight Line is Major
- Proposition $69$: Straight Line Commensurable with Side of Rational plus Medial Area
- Proposition $70$: Straight Line Commensurable with Side of Sum of two Medial Areas
- Proposition $71$: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
- Proposition $72$: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
- Proposition $73$: Apotome is Irrational
- Proposition $74$: First Apotome of Medial is Irrational
- Proposition $75$: Second Apotome of Medial is Irrational
- Proposition $76$: Minor is Irrational
- Proposition $77$: That which produces Medial Whole with Rational Area is Irrational
- Proposition $78$: That which produces Medial Whole with Medial Area is Irrational
- Proposition $79$: Construction of Apotome is Unique
- Proposition $80$: Construction of First Apotome of Medial is Unique
- Proposition $81$: Construction of Second Apotome of Medial is Unique
- Proposition $82$: Construction of Minor is Unique
- Proposition $83$: Construction of that which produces Medial Whole with Rational Area is Unique
- Proposition $84$: Construction of that which produces Medial Whole with Medial Area is Unique

- Proposition $85$: Construction of First Apotome
- Proposition $86$: Construction of Second Apotome
- Proposition $87$: Construction of Third Apotome
- Proposition $88$: Construction of Fourth Apotome
- Proposition $89$: Construction of Fifth Apotome
- Proposition $90$: Construction of Sixth Apotome
- Proposition $91$: Side of Area Contained by Rational Straight Line and First Apotome
- Proposition $92$: Side of Area Contained by Rational Straight Line and Second Apotome
- Proposition $93$: Side of Area Contained by Rational Straight Line and Third Apotome
- Proposition $94$: Side of Area Contained by Rational Straight Line and Fourth Apotome
- Proposition $95$: Side of Area Contained by Rational Straight Line and Fifth Apotome
- Proposition $96$: Side of Area Contained by Rational Straight Line and Sixth Apotome
- Proposition $97$: Square on Apotome applied to Rational Straight Line
- Proposition $98$: Square on First Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition $99$: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition $100$: Square on Minor Straight Line applied to Rational Straight Line
- Proposition $101$: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
- Proposition $102$: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
- Proposition $103$: Straight Line Commensurable with Apotome
- Proposition $104$: Straight Line Commensurable with Apotome of Medial Straight Line
- Proposition $105$: Straight Line Commensurable with Minor Straight Line
- Proposition $106$: Straight Line Commensurable with that which produces Medial Whole with Rational Area
- Proposition $107$: Straight Line Commensurable with that which produces Medial Whole with Medial Area
- Proposition $108$: Side of Remaining Area from Rational Area from which Medial Area Subtracted
- Proposition $109$: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted
- Proposition $110$: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
- Proposition $111$: Apotome not same with Binomial Straight Line
- Proposition $112$: Square on Rational Straight Line applied to Binomial Straight Line
- Proposition $113$: Square on Rational Straight Line applied to Apotome
- Proposition $114$: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio
- Proposition $115$: From Medial Straight Line arises Infinite Number of Irrational Straight Lines

#### Book $\text{XI}$: Spatial Geometry

- Proposition $1$: Straight Line cannot be in Two Planes
- Proposition $2$: Two Intersecting Straight Lines are in One Plane
- Proposition $3$: Common Section of Two Planes is Straight Line
- Proposition $4$: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane
- Proposition $5$: Three Intersecting Lines Perpendicular to Another Line are in One Plane
- Proposition $6$: Two Lines Perpendicular to Same Plane are Parallel
- Proposition $7$: Line joining Points on Parallel Lines is in Same Plane
- Proposition $8$: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane
- Proposition $9$: Lines Parallel to Same Line not in Same Plane are Parallel to each other
- Proposition $10$: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles
- Proposition $11$: Construction of Straight Line Perpendicular to Plane from point not on Plane
- Proposition $12$: Construction of Straight Line Perpendicular to Plane from point on Plane
- Proposition $13$: Straight Line Perpendicular to Plane from Point is Unique
- Proposition $14$: Planes Perpendicular to same Straight Line are Parallel
- Proposition $15$: Planes through Parallel Pairs of Meeting Lines are Parallel
- Proposition $16$: Common Sections of Parallel Planes with other Plane are Parallel
- Proposition $17$: Straight Lines cut in Same Ratio by Parallel Planes
- Proposition $18$: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane
- Proposition $19$: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane
- Proposition $20$: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
- Proposition $21$: Solid Angle contained by Plane Angles is Less than Four Right Angles
- Proposition $22$: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle
- Proposition $23$: Construction of Solid Angle from Three Plane Angles any Two of which are Greater than Other Angle
- Proposition $24$: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
- Proposition $25$: Parallelepiped cut by Plane Parallel to Opposite Planes
- Proposition $26$: Construction of Solid Angle equal to Given Solid Angle
- Proposition $27$: Construction of Parallelepiped Similar to Given Parallelepiped
- Proposition $28$: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected
- Proposition $29$: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume
- Proposition $30$: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume
- Proposition $31$: Parallelepipeds on Equal Bases and Same Height are Equal in Volume
- Proposition $32$: Parallelepipeds of Same Height have Volume Proportional to Bases
- Proposition $33$: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides
- Proposition $34$: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights
- Proposition $35$: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles
- Proposition $36$: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it formed on Mean
- Proposition $37$: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional
- Proposition $38$: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
- Proposition $39$: Prisms of equal Height with Parallelogram and Triangle as Base

#### Book $\text{XII}$: Cones, Pyramids and Cylinders

- Proposition $1$: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters
- Proposition $2$: Areas of Circles are as Squares on Diameters
- Proposition $3$: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition $4$: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition $5$: Sizes of Tetrahedra of Same Height are as Bases
- Proposition $6$: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases
- Proposition $7$: Prism on Triangular Base divided into Three Equal Tetrahedra
- Proposition $8$: Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides
- Proposition $9$: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition $10$: Volume of Cone is Third of Cylinder on Same Base and of Same Height
- Proposition $11$: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases
- Proposition $12$: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases
- Proposition $13$: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis
- Proposition $14$: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights
- Proposition $15$: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition $16$: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
- Proposition $17$: Construction of Polyhedron in Outer of Concentric Spheres
- Proposition $18$: Volumes of Spheres are in Triplicate Ratio of Diameters

#### Book $\text{XIII}$: The Five Platonic Solids

- Proposition $1$: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $2$: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $3$: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $4$: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $5$: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment
- Proposition $6$: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
- Proposition $7$: Equilateral Pentagon is Equiangular if Three Angles are Equal
- Proposition $8$: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio
- Proposition $9$: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio
- Proposition $10$: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in same Circle
- Proposition $11$: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
- Proposition $12$: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
- Proposition $13$: Construction of Regular Tetrahedron within Given Sphere
- Proposition $14$: Construction of Regular Octahedron within Given Sphere
- Proposition $15$: Construction of Cube within Given Sphere
- Proposition $16$: Construction of Regular Icosahedron within Given Sphere
- Proposition $17$: Construction of Regular Dodecahedron within Given Sphere
- Proposition $18$: Comparison of Sides of Five Platonic Figures

#### The So-Called *Book $\textit{XIV}$*

- Proposition $1$: Perpendicular from Center of Circle to Side of Inscribed Pentagon
- Proposition $2$: Circle Circumscribing Pentagon of Dodecahedron and Triangle of Icosahedron in Same Sphere
- Proposition $3$: Size of Surface of Regular Dodecahedron
- Proposition $4$: Size of Surface of Regular Icosahedron
- Proposition $5$: Ratio of Sizes of Surfaces of Regular Dodecahedron and Regular Icosahedron in Same Sphere
- Proposition $6$: Ratio of Sizes of Surfaces of Cube and Regular Icosahedron in Same Sphere
- Proposition $7$: Ratio of Lengths of Sides of Cube and Regular Icosahedron in Same Sphere
- Proposition $8$: Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere

## Translations and Editions

- c. 364: Theon of Alexandria

- c. 500: Boëthius, but this has not survived

- c. 1120: Adelard (or Athelhard) of Bath

- 1255 -- 1259: Campanus of Novara

### Early Latin translations

- 1482: First printed edition, based on the Campanus of Novara translation

- 1505: Bartolomeo Zamberti

- 1509: Luca Paciuolo

### Editions of the Greek text

- 1533: Simon Grynaeus the Elder

- 1536: Oronce Finé

- 1545: Angelo Caiani (severely edited and cut down, according to the whim of the editor)

- 1549: Joachim Camerarius

- 1550: Johannes Scheubel

- 1620: Henry Briggs (the first 6 books)

- 1654: Andrea Tacquet:
*Elementa Geometriae*(which included some work by Archimedes)

- 1655: Isaac Barrow (a simplified edition, which became a standard textbook)

- 1702: William Whiston:
*The Elements of Euclid, With Select Theorems out of Archimedes*(translation of*Elementa Geometriae*by Andrew Tacquet)- 1710: William Whiston:
*The Elements of Euclid, With Select Theorems out of Archimedes*(2nd ed.) (translation of*Elementa Geometriae*by Andrew Tacquet)

- 1710: William Whiston:

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements*(2nd ed.) -- 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]

## Critical View

*Euclid's**The Elements*is certainly one of the greatest books ever written.

*It is also one of the dullest, and by any educational standards whatever, it ought to have been a student's and teacher's nightmare throughout the twenty-three centuries of its existence.*

## Sources

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 1993: Richard J. Trudeau:
*Introduction to Graph Theory*... (previous) ... (next): $1$. Pure Mathematics: Euclidean geometry as pure mathematics - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Euclid

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