part x, and the altitude y, we have to determine x and y in the first

case from the equations

(a - x)y = k^2,

x p -- = --, y q

k^2 being the given size of the first, and p and q the base and altitude of the parallelogram which determine the shape of the second of the required parallelograms.

If we substitute the value of y, we get

pk^2 (a - x)x = ----, q

or,

ax - x^2 = b^2,

where a and b are known quantities, taking b^2 = pk^2/q.

The second case (Prop. 29) gives rise, in the same manner, to the quadratic

ax + x^2 = b^2.

The next problem--

Prop. 30. _To cut a given straight line in extreme and mean ratio_, leads to the equation

ax + x^2 = a^2.

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.

S 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.

With this closes that part of the _Elements_ which is devoted to the study of figures in a plane.