Part 25

The lxiiij. Theoreme.

If an angle be made on the centre of a circle, and an other angle made on the circumference of the same circle, and their grounde line be one common portion of the circumference, then is the angle on the centre twise so great as the other angle on the circumference.

Example.

The circle is A.B.C.D, and his centre is E: the angle on the centre is C.E.D, and the angle on the circumference is C.A.D t their commen ground line, is C.F.D. Now say I that the angle C.E.D, whiche is on the centre, is twise so greate as the angle C.A.D, which is on the circumference.

The lxv. Theoreme.

Those angles whiche be made in one cantle of a circle, must needes be equal togither.

Example.

Before I declare this theoreme by example, it shall bee needefull to declare, what is it to be vnderstande by the wordes in this theoreme. For the sentence canne not be knowen, onles the uery meaning of the wordes be firste vnderstand. Therefore when it speaketh of angles made in one cantle of a circle, it is this to be vnderstand, that the angle muste touch the circumference: and the lines that doo inclose that angle, muste be drawen to the extremities of that line, which maketh the cantle of the circle. So that if any angle do not touch the circumference, or if the lines that inclose that angle, doo not ende in the extremities of the corde line, but ende other in some other part of the said corde, or in the circumference, or that any one of them do so eande, then is not that angle accompted to be drawen in the said cantle of the circle. And this promised, nowe will I cumme to the meaninge of the theoreme. I sette forthe a circle whiche is A.B.C.D, and his centre E, in this circle I drawe a line D.C, whereby there ar made two cantels, a more and a lesser. The lesser is D.E.C, and the geater is D.A.B.C. In this greater cantle I drawe two angles, the firste is D.A.C, and the second is D.B.C which two angles by reason they are made bothe in one cantle of a circle (that is the cantle D.A.B.C) therefore are they both equall. Now doth there appere an other triangle, whose angle lighteth on the centre of the circle, and that triangle is D.E.C, whose angle is double to the other angles, as is declared in the lxiiij. Theoreme, whiche maie stande well enough with this Theoreme, for it is not made in this cantle of the circle, as the other are, by reason that his angle doth not light in the circumference of the circle, but on the centre of it.

The .lxvi. theoreme.

Euerie figure of foure sides, drawen in a circle, hath his two contrarie angles equall vnto two right angles.

Example.

The circle is A.B.C.D, and the figure of foure sides in it, is made of the sides B.C, and C.D, and D.A, and A.B. Now if you take any two angles that be contrary, as the angle by A, and the angle by C, I saie that those .ij. be equall to .ij. right angles. Also if you take the angle by B, and the angle by D, whiche two are also contray, those two angles are like waies equall to two right angles. But if any man will take the angle by A, with the angle by B, or D, they can not be accompted contrary, no more is not the angle by C. estemed contray to the angle by B, or yet to the angle by D, for they onely be accompted contrary angles, whiche haue no one line common to them bothe. Suche is the angle by A, in respect of the angle by C, for there both lynes be distinct, where as the angle by A, and the angle by D, haue one common line A.D, and therfore can not be accompted contrary angles, So the angle by D, and the angle by C, haue D.C, as a common line, and therefore be not contrary angles. And this maie you iudge of the residewe, by like reason.

The lxvij. Theoreme.

Vpon one right lyne there can not be made two cantles of circles, like and vnequall, and drawen towarde one parte.

Example.

Cantles of circles be then called like, when the angles that are made in them be equall. But now for the Theoreme, let the right line be A.E.C, on whiche I draw a cantle of a circle, whiche is A.B.C. Now saieth the Theoreme, that it is not possible to draw an other cantle of a circle, whiche shall be vnequall vnto this first cantle, that is to say, other greatter or lesser then it, and yet be lyke it also, that is to say, that the angle in the one shall be equall to the angle in the other. For as in this example you see a lesser cantle drawen also, that is A.D.C, so if an angle were made in it, that angle would be greatter then the angle made in the cantle A.B.C, and therfore can not they be called lyke cantels, but and if any other cantle were made greater then the first, then would the angle in it be lesser then that in the firste, and so nother a lesser nother a greater cantle can be made vpon one line with an other, but it will be vnlike to it also.

The .lxviij. Theoreme.

Lyke cantelles of circles made on equal righte lynes, are equall together.

Example.

What is ment by like cantles you haue heard before. and it is easie to vnderstand, that suche figures a called equall, that be of one bygnesse, so that the one is nother greater nother lesser then the other. And in this kinde of comparison, they must so agree, that if the one be layed on the other, they shall exactly agree in all their boundes, so that nother shall excede other.

Nowe for the example of the Theoreme, I haue set forthe diuers varieties of cantles of circles, amongest which the first and seconde are made vpon equall lines, and ar also both equall and like. The third couple ar ioyned in one, and be nother equall, nother like, but expressyng an absurde deformitee, whiche would folowe if this Theoreme wer not true. And so in the fourth couple you maie see, that because they are not equall cantles, therfore can not they be like cantles, for necessarily it goeth together, that all cantles of circles made vpon equall right lines, if they be like they must be equall also.

The lxix. Theoreme.

In equall circles, suche angles as be equall are made vpon equall arch lines of the circumference, whether the angle light on the circumference, or on the centre.

Example.

Firste I haue sette for an exaumple twoo equall circles, that is A.B.C.D, whose centre is K, and the second circle E.F.G.H, and his centre L, and in eche of them is there made two angles, one on the circumference, and the other on the centre of eche circle, and they be all made on two equall arche lines, that is B.C.D. the one, and F.G.H. the other. Now saieth the Theoreme, that if the angle B.A.D, be equall to the angle F.E.H, then are they made in equall circles, and on equall arch lines of their circumference. Also if the angle B.K.D, be equal to the angle F.L.H, then be they made on the centres of equall circles, and on equall arche lines, so that you muste compare those angles together, whiche are made both on the centres, or both on the circumference, and maie not conferre those angles, wherof one is drawen on the circumference, and the other on the centre. For euermore the angle on the centre in suche sorte shall be double to the angle on the circumference, as is declared in the three score and foure Theoreme.

The .lxx. Theoreme.

In equall circles, those angles whiche bee made on equall arche lynes, are euer equall together, whether they be made on the centre, or on the circumference.

Example.