ProofWiki:Books/Euclid/The Elements/Book X

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Euclid: The Elements: Book X

Published c. 300 B.C.E.


Contents

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

Definitions
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 Rational Numbers is Rational
Proposition $20$: Quotient of Rational 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


Definitions $\text{II}$
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


Definitions $\text{III}$
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