# Book:Euclid/The Elements/Book VIII

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## Euclid:

## Euclid: *The Elements: Book VIII*

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

### Contents

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

## Critical View

Once the more-or-less profound results about the structure of a geometric sequence of integers has been comprehended, the rest of the book consists of simple examples.

Hence, once the initial few results have been established in contemporary language (and to as high a power as is desired), the Euclidean proofs of these results, and indeed the results themselves, become all but irrelevant.