Beginning of Essay
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 ....

Middle of Essay
.... 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 ....