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Proposition 13

To find a mean proportional to two given straight lines.
Let AB and BC be the two given straight lines. It is required to find a mean proportional to AB and BC.
| Place them in a straight line, and describe the semicircle ADC on AC. Draw BD from the point B at right angles to the straight line AC, and join AD and DC. | I.11
Since the angle ADC is an angle in a semicircle, it is right. | III.31
And, since, in the right-angled triangle ADC, BD has been drawn from the right angle perpendicular to the base, therefore BD is a mean proportional between the segments of the base, AB and BC. | VI.8,Cor
Therefore a mean proportional BD has been found to the two given straight lines AB and BC.
Q.E.F.

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This construction of the mean proportional was used before in II.4 to find a square equal to a given rectangle. By proposition VI.17 coming up, the two constructions are equivalent. That is the mean proportional between two lines is the side of a square equal to the rectangle contained by the two lines. Algebraically, a : x = x : b if and only if ab = x2.

This construction is used in the proofs of propositions VI.25, X.27, and X.28.

Book VI Introduction - Proposition VI.12 - Proposition VI.14.

© 1996
D.E.Joyce
Clark University