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 of two. 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.
|