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# An Analysis of Magnitudes and Numbers in Euclid's Elements

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 Euclids 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 55 ma nb mc nd 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 ma nb and mc

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