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.*

Since the angle *ADC* is an angle in a semicircle, it is right.

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.*

Therefore a mean proportional *BD* has been found to the two given straight lines *AB* and *BC.*

Q.E.F.

When *b* is taken to have unit length, this construction gives the construction for the square root of *a.*