If the Peripheries be outwardly contiguall, the matter is more easie, and by the judgement of Euclide it deserved not a demonstration, as here.
The third part is apparent out of the first: Otherwise those which cut one another should be concentrickes. For, by the [7 e], the center being found: And by the [9 e], three right lines being drawne from the center unto three points of
the sections, the three raies must be equall, as here.
The fourth part is demonstrated after the same manner: Because otherwise the Part must be greater then the whole. For let the right line aeio, be drawne by the centers a and e: And let the particular raies be eu, and au. Here two sides ue, and ea, of the triangle uea, by the [9 e vj], are greater than ua: And therefore also then ao; Take away ae, the remainder ue, shall be greater than eo. But ei is equall to eu. Wherefore ei is greater than eo, the part, than the whole.
The same will fall out, if the touch be without, as here: For, by the [9 e vj], ea and ia, are greater than ie. But eo and iu, are equall to ea, and ia. Wherefore eo, and iu, are greater than ie, the parts than the whole.
Of right lines and Peripheries joyntly the rate is but one.