If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.
magnitude_ratio_equimultiples()
prove as E is to G, so is F to H
assert Let a first magnitude A have to a second B the same ratio as a third C to a fourth D
construct And let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B, D
assert I say that, as E is to G, so is F to H
construct Let equimultiples K, L be taken of E, F, and other, chance, equimultiples M, N of G, H.
∵magnitude_multiple_equimultiplesassertE is the same multiple of A that F is of C, and equimultiples K, L of E, F have been taken → therefore K is the same multiple of A that L is of C
assert The same reason
assertM is the same multiple of B that N is of D
assert since, as A is to B, so is C to D, and of A, C equimultiples K, L have been taken, and of B, D other, chance, equimultiples M, N,
∵magnitudes_said_ratio_defassert if K is in excess of M, L also is in excess of N, if it is =, =, and if less, less
assertK, L are equimultiples of E, F, and M, N other, chance, equimultiples of G, H
