30. To two right lines given one may so continue or joyne the third, that the oblong of the continued and the continuation may be equall to the quadrate remaining. Vitellio 127 p j.
As in the first figure, if the first of the lines given be eo, the second ia, the third oa.
Now are we come to Circular Geometry, that is to the Geometry of Circles or Peripheries cut and touching one another: And of Right lines and Peripheries.
31. If peripheries doe either cut or touch one another, they are eccentrickes: And they doe cut one another in two points onely, and these by the touch point doe continue their diameters, 5. 6. 10, 11, 12 p iij.
All these might well have beene asked: But they have also their demonstrations, ex impossibili, not very difficult.
The first part is manifest, because the part should be equall to the whole, if the Center were the same to both, as a. For two raies are equall to the common raie ao: And therefore ae and ai, that is, the part and the whole, are equall one to another.
The second part is demonstrated as the first: For otherwise the part must be equall to the whole, as here ae and ai, the raies of the lesser periphery; And ae, and ao, the raies of the greater are equall. Wherefore ai, should be equall to ao the Part to the whole.