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 unit representing one of something. Numbers allow for measurement and degrees of ordinal position through which one can better compare quantity. In short, magnitudes tell you how much there is, and numbers tell you how many there are. This is cause for differences in comparison among them.
Euclid's definition five in Book Five of the Elements states that ' Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.'
From this it follows that magnitudes in the same ratio are proportional. Thus, we can use the following algebraic proportion to represent definition 5.5:
(m)a : (n)b :: (m)c : (n)d.
However, it is necessary to be more specific because of the way in which the definition was worded with the phrase 'the former equimultiples alike exceed, are alike equal to, or alike fall short of....'. Thus, if we take any four magnitudes a, b, c, d, it is defined that if equimultiple m is taken of a and c, and equimultiple n is taken of c and d, then a and b...