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Beginning of Paper
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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
accu ....
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Middle of Paper
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.... 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 ma ....
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ADDITIONAL FEATURED ESSAYS
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