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 are in same ratio with c
and d, that is, a : b :: c : d, only if:
(m)a > (n)b and (m)c > (n)d, or
(m)a = (n)b and (m)c = (n)d, or
(m)a < (n)b and (m)c < (n)d.

Though, because magnitudes are continuous quantities, and an exact measurement
of magnitudes is impossible, it is not possible to say by how much one exceeds
the other, nor is it possible to determine if a > b by the same amount that c >
d.

Now, it is important to realize that taking equimultiples is not a test to see
if magnitudes are in the same ratio, but rather it is a condition that defines
it. And because of the phrase "any equimultiples whatever," it would be correct
to say that if a and b are in same ratio with c and d, then any one of the three
instances above, m and n being "any equimultiples whatever," are true. Likewise,
as stated in proposition 4, the corresponding equimultiples are also in
proportion. It would be incorrect, however; to say that equimultiples are taken
of the original magnitudes to show that they are in same ratio. The two
instances coexist. Furthermore, if there is any one possibility of taking "any
equimultiple whatever," and not having any one of the above three instances come
true, then the instance is not that of same ratio, but rather that of greater or
lesser ratio as is stated in definition 7, Book 5.

In Book Seven, number replaces magnitude as the substance of ratios and
proportions. A number is a multitude composed of units. Definition 20 states
that "Numbers are proportional when the first is the same multiple, or the same
part, or the same parts, of the second that the third is of the fourth." Thus,
there are three instances of numerical proportions:
same multiple- 18 : 6 :: 6 : 2
same part- 2 : 4 :: 4 : 8
same parts- 5 : 6 :: 15 : 18.

Compared to the definition of proportion in Book 5, this one is much less
complex and more easily comprehended because using numbers is more exact and
concrete. First of all, there is no taking of equimultiples of the antecedents
and consequents of two ratios. This is because the taking of equimultiples is a
necessary condition when it is only possible to say that one magnitude is
greater, lesser, or equal to another. With numbers, however; there is a more
specific relationship. Two is less than five by three units. It is necessary
to state by how many, which then limits the comparison. For instance, in the
example above of "same multiples," one can see that eighteen is three multiples
of six and that six is three multiples