If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
number_magnitudes_proportional(A, B, C, D)
given any number of magnitudes A, B, C, D, E, F be proportional, so that, as A is to B, so is C to D and E to F
prove as A is to B, so are A, C, E to B, D, F
given any number of magnitudes A, B, C, D, E, F be proportional, so that, as A is to B, so is C to D and E to F
assert I say that, as A is to B, so are A, C, E to B, D, F
construct Of A, C, E let equimultiplesG, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.
assert Then since, as A is to B, so is C to D, and E to F, and of A, C, E equimultiples G, H, K have been taken, and of B, D, F other, chance, equimultiples L, M, N,
assert if G is in excess of L, H is also in excess of M, and K of N, if =, =, and if less, less
assert So that, in addition, if G is in excess of L, then G, H, K are in excess of L, M, N, if =, equal, and if less, less
∵there_number_magnitudesassert Now G and G, H, K are equimultiples of A and A, C, E, since, if any number of magnitudes whatever are equimultiples of any magnitudes equal in multitude, whatever multiple one of the magnitudes is of one, that multiple also will all be of all
assert The same reason L and L, M, N are also equimultiples of B and B, D, F
