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Proposition 17

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.
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. | I.3
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. | V.7 V.11
But, if four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means. | VI.16
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. | VI.16
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.
Q.E.D.

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This is obviously a special case of the previous proposition. It is used very frequently in Books X and XIII.

Book VI Introduction - Proposition VI.16 - Proposition VI.18.

© 1996
D.E.Joyce
Clark University