Book 3
16
Let it fall as AE; I say next that into the space between the straight line AE and the circumference CHA another straight line cannot be interposed.
16
For, if possible, let another straight line be so interposed, as FA, and let DG be drawn from the point D perpendicular to FA.
16
Then, since the angle AGD is right, and the angle DAG is less than a right angle, AD is greater than DG. [I. 19]
16
But DA is equal to DH; therefore DH is greater than DG, the less than the greater: which is impossible.
16
Therefore another straight line cannot be interposed into the space between the straight line and the circumference.
16
I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilineal angle, and the remaining angle contained by the circumference CHA and the straight line AE is less than any acute rectilineal angle.
16
For, if there is any rectilineal angle greater than the angle contained by the straight line BA and the circumference CHA, and any rectilineal angle less than the angle contained by the circumference CHA and the straight line AE, then into the space between the circumference and the straight line AE a straight line will be interposed such as will make an angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, and another angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE.
16
But such a straight line cannot be interposed;
16
therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE.—
