Let the three straight lines A and B and C be proportional, so that A is to B as B is to C.
I say that the rectangle A by C equals the square on B.
Make D equal to B.
Then, since A is to B as B is to C, and B equals D, therefore A is to B as D is to C.
But, if four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means.
Therefore the rectangle A by C equals the rectangle B by D. But the rectangle B by D is the square on B, for B equals D, therefore the rectangle A by C equals the square on B.
Next, let the rectangle A by C equal the square on B.
I say that A is to B as B is to C.
With the same construction, since the rectangle A by C equals the square on B, while the square on B is the rectangle B by D, for B equals D, therefore the rectangle A by C equals the rectangle B by D.
But, if the rectangle contained by the extremes equals that contained by the means, then the four straight lines are proportional.
Therefore A is to B as D is to C.
But B equals D, therefore A is to B as B is to C.
Therefore, if three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional.