Book 7
16
Therefore C is equal to D. Q. E. D. The Greek has οἰ γενόμενοι ἐξ αὐτῶν, the (numbers) produced from them. By from them Euclid means from the original numbers, though this is not very clear even in the Greek. I think ambiguity is best avoided by leaving out the words.
PROPOSITION 17.
17
If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.
17
For let the number A by multiplying the two numbers B, C make D, E; I say that, as B is to C, so is D to E.
17
For, since A by multiplying B has made D, therefore B measures D according to the units in A.
17
But the unit F also measures the number A according to the units in it; therefore the unit F measures the number A the same number of times that B measures D.
17
Therefore, as the unit F is to the number A, so is B to D. [VII. Def. 20]
17
For the same reason, as the unit F is to the number A, so also is C to E; therefore also, as B is to D, so is C to E.
17
Therefore, alternately, as B is to C, so is D to E. [VII. 13] Q. E. D.
PROPOSITION 18.
18
If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.
18
For let two numbers A, B by multiplying any number C make D, E; I say that, as A is to B, so is D to E.
18
For, since A by multiplying C has made D, therefore also C by multiplying A has made D. [VII. 16] For the same reason also C by multiplying B has made E.
18
Therefore the number C by multiplying the two numbers A, B has made D, E.
18
Therefore, as A is to B, so is D to E. [VII. 17]
PROPOSITION 19.
19
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.
