# Book:Euclid/The Elements/Book V

## Euclid: *The Elements: Book V*

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

### Contents

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

## Historical Note

Book $\text{V}$ of Euclid's *The Elements* was supposedly created by Eudoxus of Cnidus.

It is an attempt to give a rigorous treatment of the concept of certain irrational quantities such as the geometric mean of two integers.

It has been suggested that this work was one of the main precursors and influences behind the development of the mathematical discipline of set theory.

## Critical View

With the development of algebra and the definition of a ratio as a fraction, and a proportion as an equality of ratios, the importance of this book has been considerably reduced.

As Augustus De Morgan put it:

*...simple propositions of concrete arithmetic, covered in language which makes them unintelligible to modern ears.*

The modern student of mathematics, having been raised on the definition of rational numbers in the context of field theory, is perhaps excused for considering this book as little more than pointless piffle.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: The golden mean