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 ye angle made in a semicircle. Also it is meet to note yt al angles that be made in ye 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.
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 drawẽ 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 cãtle 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 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 remẽbre, that the ãgle 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 ãgle 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 wch the one is the right line B.A, & thother the right line B.E, yet in this case they ar not to be cõsidered. And by this may you perceaue also which be the angles of the lesser cantle, the first of thẽ is made of ye right line B.D, & of ye arch line B.C, the secõd is made of the right line .D.B, & of the arch line D.C. Then ar ther ij. other lines, wch deuide those ij. corners, yt is the line B.C, & the line C.D, wch 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 ãgle made in a semicircle, whiche is distinct frõ 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 cãtle of a circle, thẽ must it neds be other a blũt ãgle, 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 cãtle 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, and the righte line B.D, is a iuste righte angle, because it is made in a semicircle. But the angle made by A, which is made of the right line A.B, and of the right line A.D, is lesser then a righte angle, and is named a sharpe angle, for as muche as it is made in a cantle of a circle, greater then a semicircle. And contrary waies, the angle by C, beyng made of the righte line B.C, and of the right line C.D, is greater then a right angle, and is named a blunte angle, because it is made in a cantle of a circle, lesser then a semicircle. But now touchyng the other angles of the cantles, I saie accordyng to the Theoreme, that the .ij. angles of the greater cantle, which are by B. and D, as is before declared, are greatter eche of them then a right angle. And the
angles of the lesser cantle, whiche are by the same letters B, and D, but be on the other side of the corde, are lesser eche of them then a right angle, and be therfore sharpe corners.
[ The lxxiiij. Theoreme.]
If a right line do touche a circle, and from the pointe where they touche, a righte lyne be drawen crosse the circle, and deuide it, the angles that the saied lyne dooeth make with the touche line, are equall to the angles whiche are made in the cantles of the same circle, on the contrarie sides of the lyne aforesaid.
Example.
