Firste I haue sette for an exaumple twoo equall circles, that
is A.B.C.D, whose centre is K, and the second circle E.F.G.H, and his centre L, and in eche of thẽ is there made two angles, one on the circumference, and the other on the centre of eche circle, and they be all made on two equall arche lines, that is B.C.D. the one, and F.G.H. the other. Now saieth the Theoreme, that if the angle B.A.D, be equall to the angle F.E.H, then are they made in equall circles, and on equall arch lines of their circumference. Also if the angle B.K.D, be equal to the angle F.L.H, then be they made on the centres of equall circles, and on equall arche lines, so that you muste compare those angles together, whiche are made both on the centres, or both on the circumference, and maie not conferre those angles, wherof one is drawen on the circumference, and the other on the centre. For euermore the angle on the centre in suche sorte shall be double to the angle on the circumference, as is declared in the three score and foure Theoreme.
[ The .lxx. Theoreme.]
In equall circles, those angles whiche bee made on equall arche lynes, are euer equall together, whether they be made on the centre, or on the circumference.
Example.
This Theoreme doth but conuert the sentence of the last Theoreme
before, and therfore is to be vnderstande by the same examples, for as that saith, that equall angles occupie equall archelynes, so this saith, that equal arche lines causeth equal angles, consideringe all other circumstances, as was taughte in the laste theoreme before, so that this theoreme dooeth affirming speake of the equalitie of those angles, of which the laste theoreme spake conditionally. And where the laste theoreme spake affirmatiuely of the arche lines, this theoreme speaketh conditionally of them, as thus: If the arche line B.C.D. be equall to the other arche line F.G.H, then is that angle B.A.D. equall to the other angle F.E.H. Or els thus may you declare it causally: Bicause the arche line B.C.D, is equal to the other arche line F.G.H, therefore is the angle B.K.D. equall to the angle F.L.H, consideringe that they are made on the centres of equall circles. And so of the other angles, bicause those two arche lines aforesaid ar equal, therfore the angle D.A.B, is equall to the angle F.E.H, for as muche as they are made on those equall arche lines, and also on the circumference of equall circles. And thus these theoremes doo one declare an other, and one verifie the other.