Book 7
24
Let a number measure them, and let it be E.
24
Now, since C, A are prime to one another, and a certain number E measures C, therefore A, E are prime to one another. [VII. 23]
24
As many times, then, as E measures D, so many units let there be in F; therefore F also measures D according to the units in E. [VII. 16]
24
Therefore E by multiplying F has made D. [VII. Def. 15]
24
But, further, A by multiplying B has also made D; therefore the product of E, F is equal to the product of A, B.
24
But, if the product of the extremes be equal to that of the means, the four numbers are proportional; [VII. 19] therefore, as E is to A, so is B to F.
24
But A, E are prime to one another, numbers which are prime to one another are also the least of those which have the same ratio, [VII. 21] and the least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures B.
24
But it also measures C; therefore E measures B, C which are prime to one another: which is impossible. [VII. Def. 12]
24
Therefore no number will measure the numbers C, D.
24
Therefore C, D are prime to one another. Q. E. D. ὁ ἐξ αὐτῶν γενόμενος, literally the (number) produced from them, will henceforth be translated as their product.
PROPOSITION 25.
25
If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.
25
Let A, B be two numbers prime to one another, and let A by multiplying itself make C: I say that B, C are prime to one another.
25
For let D be made equal to A.
25
Since A, B are prime to one another, and A is equal to D, therefore D, B are also prime to one another.
25
Therefore each of the two numbers D, A is prime to B; therefore the product of D, A will also be prime to B. [VII. 24]
25
But the number which is the product of D, A is C.
