If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.

Let A, B, C, D be four numbers in proportion, so that, as A is to B, so is C to D; and let

For let A by multiplying C make G.

Since, then, A by multiplying C has made G, and by multiplying D has made E, the number A by multiplying the two numbers C, D has made G, E.

Therefore, as C is to D, so is G to E. [VII. 17]

But, as C is to D, so is A to B; therefore also, as A is to B, so is G to E.