25 If of two right lines, the greater be made the diameter of a circle, and the lesser jointly bounded with the greater and inscribed, be knit together, the power of the greater shall be more than the power of the lesser by the quadrate of that which knitteth them both together. ad 13 p. x.
As in this example; The power of the diameter ae, is greater than the power of ei, by the quadrate of ai. For the triangle aei, shall be a rectangle; And by the [9 e xij.] ae, the greater shall be of
power equall to the shankes. Out of an angle in a semicircle Euclide raiseth two notable fabrickes; to wit, the invention of a meane proportionall betweene two lines given: And the Reason or rate in opposite sections. The genesis or invention of the meane proportionall, of which we heard at the [9 e viij]. is thus:
26 If a right line continued or continually made of two right lines given, be made the diameter of a circle, the perpendicular from the point of their continuation unto the periphery, shall be the meane proportionall betweene the two lines given. 13 p vj.
As for example, let the assigned right lines be ae, and ei, of the which aei, is continued. And let eo, be perpendicular from the periphery aoi, unto e, the point of continuation or joyning together of the lines given. This eo, say I, shall be the meane proportionall: Because drawing the right lines ao, and io, you shall make a rectangled triangle, seeing that aoi, is an angle in a semicircle: And, by the [9 e viij]. oe, shall be proportionall betweene ae, and ei.
So if the side of a quadrate of 10. foote content, were sought; let the sides 1. foote and 10. foote an oblong equall to that same quadrate, be continued; the meane proportionall shall be the side of the quadrate, that is, the power of it shall be 10. foote. The reason of the angles in opposite sections doth follow.
27 The angles in opposite sections are equall in the alterne angles made of the secant and touch line. 32. p iij.
If the sections be equall or alike, then are they the sections of a semicircle, and the matter is plaine by the [21 e]. But if they be unequall or unlike the argument of demonstration