Kitap 9
7
Let it be measured by D; and, as many times as D measures A, so many units let there be in E.
7
Since then D measures A according to the units in E, therefore E by multiplying D has made A. [VII. Def. 15]
7
And, since A by multiplying B has made C, and A is the product of D, E, therefore the product of D, E by multiplying B has made C.
7
Therefore C is solid, and D, E, B are its sides. Q. E. D.
PROPOSITION 8.
8
If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit will be square, as will also those which successively leave out one; the fourth will be cube, as will also all those which leave out two; and the seventh will be at once cube and square, as will also those which leave out five.
8
Let there be as many numbers as we please, A, B, C, D, E, F, beginning from an unit and in continued proportion; I say that B, the third from the unit, is square, as are also all those which leave out one; C, the fourth, is cube, as are also all those which leave out two; and F, the seventh, is at once cube and square, as are also all those which leave out five.
8
For since, as the unit is to A, so is A to B, therefore the unit measures the number A the same number of times that A measures B. [VII. Def. 20]
8
But the unit measures the number A according to the units in it; therefore A also measures B according to the units in A.
8
Therefore A by multiplying itself has made B; therefore B is square.
8
And, since B, C, D are in continued proportion, and B is square, therefore D is also square. [VIII. 22]
8
For the same reason F is also square.
8
Similarly we can prove that all those which leave out one are square.
8
I say next that C, the fourth from the unit, is cube, as are also all those which leave out two.
8
For since, as the unit is to A, so is B to C, therefore the unit measures the number A the same number of times that B measures C.
8
But the unit measures the number A according to the units in A; therefore B also measures C according to the units in A.
