Section: Book 05: Proportion
The theory of proportion set out in this book is
generally attributed to Eudoxus of Cnidus (408 BC  355 BC). The novel feature of this theory is its ability to deal with irrational magnitudes, which had hitherto been a major stumbling block for Greek mathematicians.
This book contains $25$ propositions and one corollary, and unlike other books of Euclid's "Elements", no compass and ruler constructions. Throughout the book, $\alpha$, $\beta$, $\gamma$, etc., denote general positive real numbers (possibly irrational magnitudes), whereas \(m\), \(n\), \(l\), etc., denote positive integers. Euclid limits himself to positive real numbers in order to preserve the visual aid of geometry  his "magnitudes" always mean "lengths" or "volumes". Nevertheless, most of his results hold also for the general case, when the magnitudes become negative.
The following table lists results from this book which are also known in modern mathematics, but which were proven by Euclid purely geometrically about 2500 years ago:
Euclid's Elements 
Corresponding Contemporary Results 
Def. 5.08, Def. 5.09, and Def. 5.10 
geometric progression. 
Prop. 5.01, Prop. 5.02, and Prop. 5.06 
distributivity law for real numbers. 
Prop. 5.03 
multiplication of real numbers is associative. 
Prop. 5.04 
multiplication of real numbers is cancellative. 
Prop. 5.07 and Cor. 5.07, Prop. 5.09 
existence and uniqueness of inverse real numbers with respect to multiplication 
Prop. 5.08, Prop. 5.10, Prop. 5.13, Prop. 5.14, Prop. 5.20, Prop. 5.21, Prop. 5.22, Prop. 5.23, Prop. 5.24, and Prop. 5.25 
some rules of calculation with inequalities (i.e. those related to inequalities involving fractions) and following from the properties of real numbers and onwards 
Prop. 5.11, socalled Common Notion 1.1 
special cases of the transitivity of any equivalence relation. 
Prop. 5.12, Prop. 5.17, Prop. 5.18, Prop. 5.19 
proportions between rations involving sums 
Prop. 5.15 
application of the existence and uniqueness of the real number $1$ 
Prop. 5.16 
special case of multiplication of real numbers is cancellative. 
Table of Contents
 Subsection: Definitions from Book 5
 Subsection: Propositions from Book 5
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"