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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