example. For
as that did declare the equalitie of the arche lines, by the
equalitie of the righte lines, so dothe this Theoreme declare
the equalnes of the right lines to ensue of the equalnes of the
arche lines, and therefore declareth that right lyne B.D, to be
equal to the other right line F.H, bicause they both are drawen
vnder equall arche lines, that is to saye, the one vnder B.A.D,
and thother vnder F.E.H, and those two arch lines are estimed
equall by the theoreme laste before, and shal be proued in the
booke of proofes.

_The lxxiij. Theoreme._

In euery circle, the angle that is made in the halfe circle,
is a iuste righte angle, and the angle that is made in a
cantle greater then the halfe circle, is lesser thanne a
righte angle, but that angle that is made in a cantle,
lesser then the halfe circle, is greatter then a right
angle. And moreouer the angle of the greater cantle is
greater then a righte angle and the angle of the lesser
cantle is lesser then a right angle.

_Example._

In this proposition, it shal be meete to note, that there is a
greate diuersite betwene an angle of a cantle, and an angle made
in a cantle, and also betwene the angle of a semicircle, and y^e
angle made in a semicircle. Also it is meet to note y^t al
angles that be made in y^e part of a circle, ar made other in a
semicircle, (which is the iuste half circle) or els in a cantle
of the circle, which cantle is other greater or lesser then the
semicircle is, as in this figure annexed you maye perceaue
euerye one of the thinges seuerallye.

[Illustration]

Firste the circle is, as you see, A.B.C.D, and his centre E, his
diameter is A.D, Then is ther a line drawen from A. to B, and so
forth vnto F, which is without the circle: and an other line
also frome B. to D, whiche maketh two cantles of the whole
circle. The greater cantle is D.A.B, and the lesser cantle is
B.C.D, In whiche lesser cantle also there are two lines that
make an angle, the one line is B.C, and the other line is C.D.
Now to showe the difference of an angle in a cantle, and an
angle of a cantle, first for an example I take the greter cantle
B.A.D, in which is but one angle made, and that is the angle by
A, which is made of a line A.B, and the line A.D, And this angle
is therfore called an angle in a cantle. But now the same cantle
hathe two other angles, which be called the angles of that
cantle, so the twoo angles made of the righte line