|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.|
When b is taken to have unit length, this construction gives the construction for the square root of a.
Next proposition: VI.14