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, so-called 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.|