Book:Euclid/The Elements/Book V

Contents

 * Book V: Theory of Proportions
 * Definitions
 * Proposition 1: Multiplication of Numbers Distributes over Addition (1)
 * Proposition 2: Multiplication of Numbers Distributes over Addition (2)
 * Proposition 3: Multiplication of Numbers is Associative
 * Proposition 4: Multiples of Terms in Equal Ratios
 * Propositions 5 and 6: Multiplication of Numbers Distributes 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

Comment
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.