14. If inscripts be equall, they be equally distant from the center: And contrariwise. 13 p iij.
The diameters in the same circle, by the [28 e iiij], are equall: And they are equally distant from the center, seeing they are by the center, or rather are no whit at all distant from it: Other inscripts are judged to be equall, greater, or lesser one than another, by the diameter, or by the diameters center.
Euclide doth demonstrate this proposition thus: Let first ae and io be equall; I say they are equidistant from the center. For let uy, and us, be perpendiculars: They shall cut the assigned ae, & io, into halfes, by the [5 e xj]: And ya and si are equall, because they are the halfes of equals. Now let the raies of the circle be ua, and ui: Their quadrates by the [9 e xij], are equall to the paire of quadrates of the shankes, which paires are therefore equall betweene themselves. Take from equalls the quadrates ya, and si, there shall remaine yu, and us, equalls: and therefore the sides are equall, by the [4 e 12].
The converse likewise is manifest: For the perpendiculars given do halfe them: And the halfes as before are equall.
15 Of unequall inscripts the diameter is the greatest: And that which is next to the diameter, is greater than that which is farther off from it: That which is farthest off from it, is the least: And that which is next to the least, is lesser than that which is farther off: And those two onely which are on each side of the diameter are equall è 15 e iij.
This proposition consisteth of five members: The first is, The diameter is the greatest inscript: The second, That which is next to the diameter is greater than that which is farther off: The third, That which is farthest off from the diameter is the least: The fourth, That next to the least is lesser, than that farther off: The fifth, That two onely on each side of the diameter are equall betweene themselves. All which are manifest, out of that same argument of equalitie, that is the center the beginning of decreasing, and the
end of increasing. For looke how much farther off you goe from the center, or how much nearer you come unto it, so much lesser or greater doe you make the inscript.
Let there be in a circle; many inscripts, of which one, to wit, ae, let it be the diameter: I say, that it is of them all the greatest or longest. But let io, be nearer to the diameter, (or as in the former Elements was said) nearer to the center, than uy. I say that io, is longer than uy. Moreover, let uy, be the farthest off from the same diameter or center; I say the same uy, is the shortest of them all. Now to this shortest uy, let io, be nearer than ae; I say therefore that io, also is lesser than ae. Let at length io, be not the diameter: I say that beyond the diameter ae, there may onely a line be inscribed equall unto it, such as is sr. And those equal betweene themselves on each side of the diametry may only be given, not three, nor more. And after the same manner also, onely one beyond the diameter, may possibly be equall to uy, to wit, that which is as farre off from the diameter as it is; and so in others.