# Book:Euclid/The Elements/Book IX

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## Euclid:

## Euclid: *The Elements: Book IX*

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

### Contents

- 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)