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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...

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...

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