Part 18 (2/2)
And contrarie waies, if they make all their angles righte, then doth the longer line cutte the shorter in twoo partes.
_Example._
[Ill.u.s.tration]
The circle is A.B.C.D, the line that pa.s.seth 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 pa.s.se by the centre, euery of them dothe not deuide the other into equall partions.
_Example._
[Ill.u.s.tration]
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 porti lger and an other shorter, and so like wise B.D.
Howbeit, it is not so to be vnderstd, but one of them may be deuided into ij. eu? parts, but bothe to bee cutte equally in the middle, is not possible, onles both pa.s.se through the c?tre, therfore much rather wh? 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._
[Ill.u.s.tration]
This theoreme seemeth of it selfe so manifest, that it neadeth nother demonstration nother declaraci. Yet for the plaine vnderstanding of it, I haue sette forthe a figure here, where ij. circles be draw?, 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._
[Ill.u.s.tration]
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 lger 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._
If a certaine pointe be a.s.signed in the diameter of a circle, distant from the centre of the said circle, and from that pointe diuerse lynes drawen to the edge and circ.u.mference of the same circle, the longest line is that whiche pa.s.seth by the centre, and the shortest is the residew of the same line. And of al the other lines that is euer the greatest, that is nighest to the line, which pa.s.seth by the centre. And ctrary waies, that is the shortest, that is farthest from it. And amongest th? all there can be but onely .ij. equall together, and they must nedes be so placed, that the shortest line shall be in the iust middle betwixte them.
_Example._
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