# Book:Euclid/The Elements/Book X

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## Euclid:

## Euclid: *The Elements: Book X*

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

### Contents

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

## Historical Note

Most, if not all, of the content of Book $\text X$ of Euclid's *The Elements* appears to have originated with Theaetetus of Athens.

## Notes

The term "rational" (ῥητός) is often used in this book. However, its meaning differs from the modern-day usage of rationality. Therefore, the following nomenclature is used instead in $\mathsf{Pr} \infty \mathsf{fWiki}$ (in the following, $x$ is strictly positive):

- If $x \in \Q$, $x$ is called
**rational**(ῥητός). - If $x^2 \in \Q$, $x$ is called
**rationally expressible**(ῥητός). - If $x \notin \Q$ but $x^2 \in \Q$, $x$ is called
**linearly irrational**(ῥητός). - If $x \notin \Q$ and $x^2 \notin \Q$, $x$ is called
**squarely irrational**(ἄλογος).