be more
then .ij. of any one length, and they two muste be on the two
contrarie sides of the shortest line.

_Example._

[Illustration]

Take the circle to be A.B.C, and the point assigned without it
to be D. Now say I, that if there be drawen sundrie lines from
D, and crosse the circle, endyng in the circumference on the
contrary side, as here you see, D.A, D.E, D.F, and D.B, then of
all these lines the longest must needes be D.A, which goeth by
the centre of the circle, and the nexte vnto it, that is D.E, is
the longest amongest the rest. And contrarie waies, D.B, is the
shorteste, because it is farthest distaunt from D.A. And so maie
you iudge of D.F, because it is nerer vnto D.A, then is D.B,
therefore is it longer then D.B. And likewaies because it is
farther of from D.A, then is D.E, therfore is it shorter then
D.E. Now for those partes of the lines whiche bee withoute the
circle (as you see) D.C, is the shortest. because it is the
parte of that line which passeth by the centre, And D.K, is next
to it in distance, and therefore also in shortnes, so D.G, is
farthest from it in distance, and therfore is the longest of
them. Now D.H, beyng nerer then D.G, is also shorter then it,
and beynge farther of, then D.K, is longer then it. So that for
this parte of the theoreme (as I think) you do plainly perceaue
the truthe thereof, so the residue hathe no difficulte. For
seing that the nearer any line is to D.C, (which ioyneth with
the diameter) the shorter it is and the farther of from it, the
longer it is. And seyng two lynes can not be of like distaunce
beinge bothe on one side, therefore if they shal be of one
lengthe, and consequently of one distaunce, they must needes bee
on contrary sides of the saide line D.C. And so appeareth the
meaning of the whole Theoreme.

And of this Theoreme dothe there folowe an other lyke. whiche
you maye calle other a theoreme by it selfe, or else a Corollary
vnto this laste theoreme, I passe not so muche for the name. But
his sentence is this: _when so euer any lynes be drawen frome
any pointe, withoute a circle, whether they crosse the circle,
or eande in the utter edge of his circumference, those two lines
that bee equally distaunt from the least line are equal
togither, and contrary waies, if they be equall togither, they
ar also equally distant from that least line._

For the declaracion of this proposition, it shall not need to
vse any other example, then that whi