D.B, and the
arche line D.A.B, are the twoo angles of this cantle, whereof
the one is by D, and the other is by B. Wher you must remembre,
that the angle by D. is made of the right line B.D, and the arche
line D.A. And this angle is diuided by an other right line
A.E.D, which in this case must be omitted as no line. Also the
angle by B. is made of the right line D.B, and of the arch line
.B.A, & although it be deuided with ij. other right lines, of
w^{ch} the one is the right line B.A, & thother the right line
B.E, yet in this case they ar not to be considered. And by this
may you perceaue also which be the angles of the lesser cantle,
the first of them is made of y^e right line B.D, & of y^e arch
line B.C, the second is made of the right line .D.B, & of the
arch line D.C. Then ar ther ij. other lines, w^{ch} deuide those
ij. corners, y^t is the line B.C, & the line C.D, w^{ch} ij.
lines do meet in the poynte C, and there make an angle, whiche
is called an angle made in that lesser cantle, but yet is not
any angle of that cantle. And so haue you heard the difference
betweene an angle in a cantle, and an angle of a cantle. And in
lyke sorte shall you iudg of the angle made in a semicircle,
whiche is distinct from the angles of the semicircle. For in this
figure, the angles of the semicircle are those angles which be
by A. and D, and be made of the right line A.D, beeyng the
diameter, and of the halfe circumference of the circle, but by
the angle made in the semicircle is that angle by B, whiche is
made of the righte line A.B, and that other right line B.D,
whiche as they mete in the circumference, and make an angle, so
they ende with their other extremities at the endes of the
diameter. These thynges premised, now saie I touchyng the
Theoreme, that euerye angle that is made in a semicircle, is a
right angle, and if it be made in any cantle of a circle, then
must it neds be other a blunt angle, or els a sharpe angle, and in
no wise a righte angle. For if the cantle wherein the angle is
made, be greater then the halfe circle, then is that angle a
sharpe angle. And generally the greater the cantle is, the lesser
is the angle comprised in that cantle: and contrary waies, the
lesser any cantle is, the greater is the angle that is made in
it. Wherfore it must nedes folowe, that the angle made in a
cantle lesse then a semicircle, must nedes be greater then a
right angle. So the angle by B, beyng made at the right line
A.B, a