Proportions of Numbers and Magnitudes

Proportions of Numbers and Magnitudes In the Elements, Euclid devotes a book to magnitudes (Five), and he devotes a book to numbers (Seven). Both magnitudes and numbers represent quantity, however; magnitude is continuous while number is discrete. That is, numbers are composed of units which can be used to divide the whole, while magnitudes can not be distinguished as parts from a whole, therefore; numbers can be more accurately compared because there is a standard …

…Thus, the phrase '..... alike exceeding, alike equal to, or alike falls short of...' is replaced with '......same multiple, same part, or same parts....' Numbers are representations of magnitude. They are more easily compared, but the proportion of numbers is fundamentally the same as that of magnitudes, since a proportion is generally a similarity between ratios. A proportion of numbers is therefore included in the proportion of magnitudes as a specific case.