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 ...
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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 ...
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"Proportions Of Numbers And Magnitudes." Essayworld.com. June 9, 2008. Accessed November 22, 2024. http://www.essayworld.com/essays/Proportions-Of-Numbers-And-Magnitudes/84966.
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