If a righte line passinge by the centre of a circle, doo crosse an other right line within the same circle, passinge beside the centre, if he deuide the saide line into twoo equal partes, then doo they make all their angles righte. And contrarie waies, if they make all their angles righte, then doth the longer line cutte the shorter in twoo partes. _Example._ [Illustration] The circle is A.B.C.D, the line that passeth by the centre, is A.E.C, the line that goeth beside the centre is D.B. Nowe saye I, that the line A.E.C, dothe cutte that other line D.B. into twoo iuste partes, and therefore all their four angles ar righte angles. And contrarye wayes, bicause all their angles are righte angles, therfore it muste be true, that the greater cutteth the lesser into two equal partes, accordinge as the Theoreme would. _The xlix. Theoreme._ If twoo right lines drawen in a circle doo crosse one an other, and doo not passe by the centre, euery of them dothe not deuide the other into equall partions. _Example._ [Illustration] The circle is A.B.C.D, and the centre is E, the one line A.C, and the other is B.D, which two lines crosse one an other, but yet they go not by the centre, wherefore accordinge to the woordes of the theoreme, eche of theim doth cuytte the other into equall portions. For as you may easily iudge, A.C. hath one portiõ lõger and an other shorter, and so like wise B.D. Howbeit, it is not so to be vnderstãd, but one of them may be deuided into ij. euẽ parts, but bothe to bee cutte equally in the middle, is not possible, onles both passe through the cẽtre, therfore much rather whẽ bothe go beside the centre, it can not be that eche of theym shoulde be iustely parted into ij. euen partes. _The L. Theoreme._ If two circles crosse and cut one an other, then haue not they both one centre. _Example._ [Illustration] This theoreme seemeth of it selfe so manifest, that it neadeth nother demonstration nother declaraciõ. Yet for the plaine vnderstanding of it, I haue sette forthe a figure here, where ij. circles be drawẽ, so that one of them doth crosse the other (as you see) in the pointes B. and G, and their centres appear at the firste sighte to bee diuers. For the centre of the one is F, and the centre of the other is E, which diffre as farre asondre as the edges of the circles, where they bee most distaunte in sonder. _The Li. Theoreme._ If two circles be so drawen, that one of them do touche the other, then haue they not one centre. _Example._ [Illustration] There are two circles made, as you see, the one is A.B.C, and hath his centre by G, the other is B.D.E, and his centre is by F, so that it is easy enough to perceaue that their centres doe dyffer as muche a sonder, as the halfe diameter of the greater circle is lõger then the half diameter of the lesser circle. And so must it needes be thought and said of all other circles in lyke kinde. _The .lij. theoreme._