Book:Euclid/The Elements/Book V

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Euclid: The Elements: Book V

Published $c. 300 B.C.E$.


Contents

Book $\text{V}$: Theory of Proportions

Definitions
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
Porism to 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
Porism to 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 done 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