Kitap 9
12
But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures C.
12
Let it measure it according to G; therefore E by multiplying G has made C.
12
But, further, by the theorem before this, A has also by multiplying B made C. [IX. 11 and Por.]
12
Therefore the product of A, B is equal to the product of E, G.
12
Therefore, as A is to E, so is G to B. [VII. 19]
12
But A, E are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent: [VII. 20] therefore E measures B.
12
Let it measure it according to H; therefore E by multiplying H has made B.
12
But further A has also by multiplying itself made B; [IX. 8] therefore the product of E, H is equal to the square on A.
12
Therefore, as E is to A, so is A to H. [VII. 19]
12
But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A, as antecedent antecedent.
12
But, again, it also does not measure it: which is impossible.
12
Therefore E, A are not prime to one another.
12
Therefore they are composite to one another.
12
But numbers composite to one another are measured by some number. [VII. Def. 14]
