The next problem—
Prop. 30. To cut a given straight line in extreme and mean ratio, leads to the equation
ax + x² = a².
This is, therefore, only a special case of the last, and is, besides, an old acquaintance, being essentially the same problem as that proposed in II. 11.
Prop. 30 may therefore be solved in two ways, either by aid of Prop. 29 or by aid of II. 11. Euclid gives both solutions.
§ 71. Prop. 31 (Theorem). In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly-described figures on the sides containing the right angle,—is a pretty generalization of the theorem of Pythagoras (I. 47).
Leaving out the next proposition, which is of little interest, we come to the last in this book.
Prop. 33. In equal circles angles, whether at the centres or the circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.
Of this, the part relating to angles at the centre is of special importance; it enables us to measure angles by arcs.