The .lxvi. theoreme.Euerie figure of foure sides, drawen in a circle, hath his two contrarie angles equall vnto two right angles.Example.see textThe circle isA.B.C.D,and the figure of foure sides in it, is made of the sidesB.C,andC.D,andD.A,andA.B.Now if you take any two angles that be contrary, as the angle byA,and the angle byC,I saie that those .ij. be equall to .ij. right angles. Also if you take the angle byB,and the angle byD,whiche two are also contray, those two angles are like waies equall to two right angles. But if any man will take the angle byA,with the angle byB,orD,they can not be accompted contrary, no more is not the angle byC.estemed contray to the angle byB,or yet to the angle byD,for they onely be accomptedcontrary angles, whiche haue no one line common to them bothe. Suche is the angle byA,in respect of the angle byC,for there both lynes be distinct, where as the angle byA,and the angle byD,haue one common lineA.D,and therfore can not be accompted contraryangles, Sothe angle byD,and the angle byC,haueD.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.see textCantles 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 beA.E.C,on whiche I draw a cantle of a circle, whiche isA.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 isA.D.C,so if an angle were made in it, that angle would be greatter then the angle made in the cantleA.B.C,and therforecannot they be called lyke cantels, but and if any other cantle were made greater then the first, then would the angleinit 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 equalrighte lynes, are equall together.Example.What isment bylike cantles you haue heardbefore. andit 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.see textNowe for the example of the Theoreme, I haue set forthe diuers varieties of cantles of circles, amongest which the first and seconde are made vpõ 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.see textFirste I haue sette for an exaumple twoo equall circles, thatisA.B.C.D,whose centre isK,and the second circleE.F.G.H,and his centreL,and in eche of thẽ 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 isB.C.D.the one, andF.G.H.the other. Now saieth the Theoreme, that if the angleB.A.D,be equall to the angleF.E.H,then are they made in equall circles, and on equall arch lines of their circumference. Also if the angleB.K.D,be equal to the angleF.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.This Theoreme doth but conuert the sentence of the last Theoremebefore, and therfore is to be vnderstande by the same examples, for as that saith, that equall angles occupie equall archelynes, so this saith, that equal arche lines causeth equal angles, consideringe all other circumstances, as was taughte in the laste theoreme before, so that this theoreme dooeth affirming speake of the equalitie of those angles, of which the laste theoreme spake conditionally. And where the laste theoreme spake affirmatiuely of the arche lines, this theoreme speaketh conditionally of them, as thus: If the arche lineB.C.D.be equall to the other arche lineF.G.H,then is that angleB.A.D.equall to the other angleF.E.H.Or els thus may you declare it causally: Bicause the arche lineB.C.D,is equal to the other arche lineF.G.H,therefore is the angleB.K.D.equall to the angleF.L.H,consideringe that they are made on the centres of equall circles. And so of the other angles, bicause those two arche lines aforesaid ar equal, therfore the angleD.A.B,is equall to the angleF.E.H,for as muche as they are made on those equall arche lines, and also on the circumference of equall circles. And thus these theoremes doo one declare an other, and one verifie the other.The lxxi. Theoreme.In equal circles, equall right lines beinge drawen, doo cutte awaye equalle arche lines frome their circumferences, so that the greater arche line of the one is equall to the greater arche line of the other, and the lesser to the lesser.Example.see textThe circleA.B.C.D,is made equall to the circleE.F.G.H,and the right lineB.D.is equal to the righte lineF.H,wherfore it foloweth, that the ij. arche lines of the circleA.B.D,whiche are cut from his circumference by the right lineB.D,are equall to two other arche lines of the circleE.F.H,being cutte frome hiscircumference, by the right lineF.H.that is to saye, that the arche lineB.A.D,beinge the greater arch line of the firste circle, is equall to the arche lineF.E.H,beynge the greater arche line of the other circle. And so in like manner the lesser arche line of the firste circle, beyngeB.C.D,is equal to the lesser arche line of the seconde circle, that isF.G.H.The lxxij. Theoreme.In equall circles, vnder equall arche lines the right lines that bee drawen are equall togither.Example.This Theoreme is none other, but the conuersion of the laste Theoreme beefore, and therefore needeth none other example. For as that did declare the equalitie of the arche lines, by the equalitie of the righte lines, sodothethis Theoremedeclare the equalnes of the right lines to ensue of the equalnes of the arche lines, and therefore declareth that right lyneB.D,to be equal to the other right lineF.H,bicause they both are drawen vnder equall arche lines, that is to saye, the one vnderB.A.D,and thother vnderF.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 yeangle made in a semicircle. Also it is meet to noteytalangles that be made in yepart 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.see textFirstethe circle is, as you see,A.B.C.D,and his centreE,his diameter isA.D,Thenis ther a line drawẽ fromA.toB,and so forth vntoF,which is without the circle: and an other line also fromeB.toD,whiche maketh two cantles of the whole circle.The greater cantle isD.A.B,and the lesser cantle isB.C.D,Inwhiche lesser cantle also there are two lines that make an angle, the one line isB.C,and the other line isC.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 cãtleB.A.D,in which is but one angle made, and that is the angle byA,which is made of a lineA.B,and the lineA.D,Andthis 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 lineD.B,and the arche lineD.A.B,are the twoo angles of this cantle, whereof the one is byD,and the other is byB.Wher you must remẽbre, that the ãgle byD.is made of the right lineB.D,and the arche lineD.A.And this angle is diuided by an other right lineA.E.D,which in this case must be omitted as no line. Also the ãgle byB.is made of the right lineD.B,and of the arch line.B.A,& although it be deuided with ij. other right lines, of wchthe one is the right lineB.A,& thother the right lineB.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 yeright lineB.D,& of yearch lineB.C,the secõd is made of the right line.D.B,& of the arch lineD.C.Then ar ther ij. other lines, wchdeuide those ij. corners, ytis the lineB.C,& the lineC.D,wchij. lines do meetin the poynteC,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. Andinlyke 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 byA.andD,and be made of the right lineA.D,beeyng the diameter, and of the halfe circumference of the circle, but by the angle made in the semicircle is that angle byB,whiche is made of the righte lineA.B,and that other right lineB.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 byB,beyng made at the right lineA.B,and the righte lineB.D,is a iuste righte angle, because it is made in a semicircle. But the angle made byA,which is made of the right lineA.B,and of the right lineA.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 byC,beyng made of the righte lineB.C,and of the right lineC.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 byB.andD,as is before declared, are greatter eche of them then a right angle. And theangles of the lesser cantle, whiche are by the same lettersB,andD,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.see textThe circle isA.B.C.D,and the touche line isE.F.The pointe of the touchyng isD,from which point I suppose the lineD.B,to be drawen crosse the circle, and to diuide it into .ij. cantles, wherof the greater isB.A.D,and the lesser isB.C.D,and in ech of them an angle is drawen, for in the greater cantle the angle is byA,and is made of the right linesB.A,andA.D,in the lesser cantle the angle is byC,and is made of yeright linesB.C,andC.D.Now saith the Theoreme that the angleB.D.F,is equall to the angle made in the cantle on the other side of the said line, that is to saie, in the cantleB.A.D,so that the angleB.D.F,is equall to the angleB.A.D,because the angleB.D.F,is on the one side of the lineB.D,(whiche is accordingto the supposition of the Theoreme drawen crosse the circle) and the angleB.A.D,is in the cãtle on the other side. Likewaies the angleB.D.E,beyng on the one side of the lineB.D,must be equall to the angleB.C.D,(that is the ãgle byC,) whiche is made in the cãtle on the other side of the right lineB.D.The profe of all these I do reserue, as I haue often saide, to a conuenient boke, wherein they shall be all set at large.The .lxxv. Theoreme.In any circle when .ij. right lines do crosse one an other, the likeiamme that is made of the portions of the one line, shall be equall to the lykeiamme made of the partes of the other lyne.see textBecause this Theoreme doth serue to many vses, and wold be wel vnderstande, I haue set forth .ij. examples of it. In the firste, the lines by their crossyng do make their portions somewhat toward an equalitie.In the second the portiõs of the lynes be very far frõ an equalitie, and yet in bothe these and in all other yeTheoreme is true. In the first exãple the circle isA.B.C.D,in which thone lineA.C,doth crosse thother lineB.D,in yepointE.Now if you do make one likeiãme or lõgsquare ofD.E,&E.B,being ye.ij.portions of the lineD.B,that longsquare shall be equall to the other longsquare made ofA.E,andE.C,beyng the portions of the other lineA.C.Lykewaies in the second example, the circle isF.G.H.K,in whiche the lineF.H,doth crosse the other lineG.K,in the pointeL.Wherfore if you make a lykeiamme or longsquare of the two partes of the lineF.H,that is to saye, ofF.L,andL.H,that longsquare will be equall to an other longsquare made of the two partes of the lineG.K.which partes areG.L,andL.K.Those longsquares haue I set foorth vnder the circles containyng their sides, that you maie somewhat whet your own wit in practisyng this Theoreme, accordyng to the doctrine of the nineteenth conclusion.The .lxxvi. Theoreme.If a pointe be marked without a circle, and from that pointe two right lines drawen to the circle, so that the one of them doe runne crosse the circle, and the other doe touche the circle onely, the long square that is made of that whole lyne which crosseth the circle, and the portion of it, that lyeth betwene the vtter circumference of the circle and the pointe, shall be equall to the full square of the other lyne, that onely toucheth the circle.Example.The circle isD.B.C,and the pointe without the circle isA,from whiche pointe there is drawen one line crosse the circle, and that isA.D.C,and an other lyne is drawn from the saidpricke to the marge or edge of the circumference of the circle, and doeth only touche it, that is the lineA.B.And of that first lineA.D.C,you maie perceiue one part of it, whiche isA.D,to lie without the circle, betweene the vtter circumference of it, and the pointe assigned, whiche wasA.Noweconcernyng the meanyng of the Theoreme, if you make a longsquare of the whole lineA.C,and of that parte of it that lyeth betwene the circumference and the point, (whiche isA.D,) that longesquare shall be equall to the full square of the touche lineA.B,accordyng not onely as this figure sheweth, but also the saied nyneteenth conclusion dooeth proue, if you lyste to examyne the one by the other.see textsee textsee textThe lxxvii. Theoreme.If a pointe be assigned without a circle, and from that pointe .ij. right lynes be drawen to the circle, so that the one doe crosse the circle, and the other dooe ende at the circumference, and that the longsquare of the line which crosseththe circle made with the portiõ of the same line beyng without the circle betweene the vtter circumference and the pointe assigned, doe equally agree with the iuste square of that line that endeth at the circumference, then is that lyne so endyng on the circumference a touche line vnto that circle.Example.In as muche as this Theoreme is nothyng els but the sentence of the last Theoreme before conuerted, therfore it shall not be nedefull to vse any other example then the same, for as in that other Theoreme because the one line is a touche lyne, therfore it maketh a square iust equal with the longsquare made of that whole line, whiche crosseth the circle, and his portion liyng without the same circle. So saith this Theoreme: that if the iust square of the line that endeth on the circumference, be equall to that longsquare whiche is made as for his longer sides of the whole line, which commeth from the pointt assigned, and crosseth the circle, and for his other shorter sides is made of the portion of the same line, liyng betwene the circumference of the circle and the pointe assigned, then is that line whiche endeth on the circumference a right touche line, that is to saie, yf the full square of the right lineA.B,be equall to the longsquare made of the whole lineA.C,as one of his lines, and of his portionA.D,as his other line, then must it nedes be, that the lyneA.B,is a right touche lyne vnto the circleD.B.C.And thus for this tyme I make an ende of the Theoremes.FINIS,
The .lxvi. theoreme.Euerie figure of foure sides, drawen in a circle, hath his two contrarie angles equall vnto two right angles.Example.see textThe circle isA.B.C.D,and the figure of foure sides in it, is made of the sidesB.C,andC.D,andD.A,andA.B.Now if you take any two angles that be contrary, as the angle byA,and the angle byC,I saie that those .ij. be equall to .ij. right angles. Also if you take the angle byB,and the angle byD,whiche two are also contray, those two angles are like waies equall to two right angles. But if any man will take the angle byA,with the angle byB,orD,they can not be accompted contrary, no more is not the angle byC.estemed contray to the angle byB,or yet to the angle byD,for they onely be accomptedcontrary angles, whiche haue no one line common to them bothe. Suche is the angle byA,in respect of the angle byC,for there both lynes be distinct, where as the angle byA,and the angle byD,haue one common lineA.D,and therfore can not be accompted contraryangles, Sothe angle byD,and the angle byC,haueD.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.see textCantles 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 beA.E.C,on whiche I draw a cantle of a circle, whiche isA.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 isA.D.C,so if an angle were made in it, that angle would be greatter then the angle made in the cantleA.B.C,and therforecannot they be called lyke cantels, but and if any other cantle were made greater then the first, then would the angleinit 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 equalrighte lynes, are equall together.Example.What isment bylike cantles you haue heardbefore. andit 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.see textNowe for the example of the Theoreme, I haue set forthe diuers varieties of cantles of circles, amongest which the first and seconde are made vpõ 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.see textFirste I haue sette for an exaumple twoo equall circles, thatisA.B.C.D,whose centre isK,and the second circleE.F.G.H,and his centreL,and in eche of thẽ 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 isB.C.D.the one, andF.G.H.the other. Now saieth the Theoreme, that if the angleB.A.D,be equall to the angleF.E.H,then are they made in equall circles, and on equall arch lines of their circumference. Also if the angleB.K.D,be equal to the angleF.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.This Theoreme doth but conuert the sentence of the last Theoremebefore, and therfore is to be vnderstande by the same examples, for as that saith, that equall angles occupie equall archelynes, so this saith, that equal arche lines causeth equal angles, consideringe all other circumstances, as was taughte in the laste theoreme before, so that this theoreme dooeth affirming speake of the equalitie of those angles, of which the laste theoreme spake conditionally. And where the laste theoreme spake affirmatiuely of the arche lines, this theoreme speaketh conditionally of them, as thus: If the arche lineB.C.D.be equall to the other arche lineF.G.H,then is that angleB.A.D.equall to the other angleF.E.H.Or els thus may you declare it causally: Bicause the arche lineB.C.D,is equal to the other arche lineF.G.H,therefore is the angleB.K.D.equall to the angleF.L.H,consideringe that they are made on the centres of equall circles. And so of the other angles, bicause those two arche lines aforesaid ar equal, therfore the angleD.A.B,is equall to the angleF.E.H,for as muche as they are made on those equall arche lines, and also on the circumference of equall circles. And thus these theoremes doo one declare an other, and one verifie the other.The lxxi. Theoreme.In equal circles, equall right lines beinge drawen, doo cutte awaye equalle arche lines frome their circumferences, so that the greater arche line of the one is equall to the greater arche line of the other, and the lesser to the lesser.Example.see textThe circleA.B.C.D,is made equall to the circleE.F.G.H,and the right lineB.D.is equal to the righte lineF.H,wherfore it foloweth, that the ij. arche lines of the circleA.B.D,whiche are cut from his circumference by the right lineB.D,are equall to two other arche lines of the circleE.F.H,being cutte frome hiscircumference, by the right lineF.H.that is to saye, that the arche lineB.A.D,beinge the greater arch line of the firste circle, is equall to the arche lineF.E.H,beynge the greater arche line of the other circle. And so in like manner the lesser arche line of the firste circle, beyngeB.C.D,is equal to the lesser arche line of the seconde circle, that isF.G.H.The lxxij. Theoreme.In equall circles, vnder equall arche lines the right lines that bee drawen are equall togither.Example.This Theoreme is none other, but the conuersion of the laste Theoreme beefore, and therefore needeth none other example. For as that did declare the equalitie of the arche lines, by the equalitie of the righte lines, sodothethis Theoremedeclare the equalnes of the right lines to ensue of the equalnes of the arche lines, and therefore declareth that right lyneB.D,to be equal to the other right lineF.H,bicause they both are drawen vnder equall arche lines, that is to saye, the one vnderB.A.D,and thother vnderF.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 yeangle made in a semicircle. Also it is meet to noteytalangles that be made in yepart 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.see textFirstethe circle is, as you see,A.B.C.D,and his centreE,his diameter isA.D,Thenis ther a line drawẽ fromA.toB,and so forth vntoF,which is without the circle: and an other line also fromeB.toD,whiche maketh two cantles of the whole circle.The greater cantle isD.A.B,and the lesser cantle isB.C.D,Inwhiche lesser cantle also there are two lines that make an angle, the one line isB.C,and the other line isC.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 cãtleB.A.D,in which is but one angle made, and that is the angle byA,which is made of a lineA.B,and the lineA.D,Andthis 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 lineD.B,and the arche lineD.A.B,are the twoo angles of this cantle, whereof the one is byD,and the other is byB.Wher you must remẽbre, that the ãgle byD.is made of the right lineB.D,and the arche lineD.A.And this angle is diuided by an other right lineA.E.D,which in this case must be omitted as no line. Also the ãgle byB.is made of the right lineD.B,and of the arch line.B.A,& although it be deuided with ij. other right lines, of wchthe one is the right lineB.A,& thother the right lineB.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 yeright lineB.D,& of yearch lineB.C,the secõd is made of the right line.D.B,& of the arch lineD.C.Then ar ther ij. other lines, wchdeuide those ij. corners, ytis the lineB.C,& the lineC.D,wchij. lines do meetin the poynteC,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. Andinlyke 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 byA.andD,and be made of the right lineA.D,beeyng the diameter, and of the halfe circumference of the circle, but by the angle made in the semicircle is that angle byB,whiche is made of the righte lineA.B,and that other right lineB.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 byB,beyng made at the right lineA.B,and the righte lineB.D,is a iuste righte angle, because it is made in a semicircle. But the angle made byA,which is made of the right lineA.B,and of the right lineA.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 byC,beyng made of the righte lineB.C,and of the right lineC.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 byB.andD,as is before declared, are greatter eche of them then a right angle. And theangles of the lesser cantle, whiche are by the same lettersB,andD,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.see textThe circle isA.B.C.D,and the touche line isE.F.The pointe of the touchyng isD,from which point I suppose the lineD.B,to be drawen crosse the circle, and to diuide it into .ij. cantles, wherof the greater isB.A.D,and the lesser isB.C.D,and in ech of them an angle is drawen, for in the greater cantle the angle is byA,and is made of the right linesB.A,andA.D,in the lesser cantle the angle is byC,and is made of yeright linesB.C,andC.D.Now saith the Theoreme that the angleB.D.F,is equall to the angle made in the cantle on the other side of the said line, that is to saie, in the cantleB.A.D,so that the angleB.D.F,is equall to the angleB.A.D,because the angleB.D.F,is on the one side of the lineB.D,(whiche is accordingto the supposition of the Theoreme drawen crosse the circle) and the angleB.A.D,is in the cãtle on the other side. Likewaies the angleB.D.E,beyng on the one side of the lineB.D,must be equall to the angleB.C.D,(that is the ãgle byC,) whiche is made in the cãtle on the other side of the right lineB.D.The profe of all these I do reserue, as I haue often saide, to a conuenient boke, wherein they shall be all set at large.The .lxxv. Theoreme.In any circle when .ij. right lines do crosse one an other, the likeiamme that is made of the portions of the one line, shall be equall to the lykeiamme made of the partes of the other lyne.see textBecause this Theoreme doth serue to many vses, and wold be wel vnderstande, I haue set forth .ij. examples of it. In the firste, the lines by their crossyng do make their portions somewhat toward an equalitie.In the second the portiõs of the lynes be very far frõ an equalitie, and yet in bothe these and in all other yeTheoreme is true. In the first exãple the circle isA.B.C.D,in which thone lineA.C,doth crosse thother lineB.D,in yepointE.Now if you do make one likeiãme or lõgsquare ofD.E,&E.B,being ye.ij.portions of the lineD.B,that longsquare shall be equall to the other longsquare made ofA.E,andE.C,beyng the portions of the other lineA.C.Lykewaies in the second example, the circle isF.G.H.K,in whiche the lineF.H,doth crosse the other lineG.K,in the pointeL.Wherfore if you make a lykeiamme or longsquare of the two partes of the lineF.H,that is to saye, ofF.L,andL.H,that longsquare will be equall to an other longsquare made of the two partes of the lineG.K.which partes areG.L,andL.K.Those longsquares haue I set foorth vnder the circles containyng their sides, that you maie somewhat whet your own wit in practisyng this Theoreme, accordyng to the doctrine of the nineteenth conclusion.The .lxxvi. Theoreme.If a pointe be marked without a circle, and from that pointe two right lines drawen to the circle, so that the one of them doe runne crosse the circle, and the other doe touche the circle onely, the long square that is made of that whole lyne which crosseth the circle, and the portion of it, that lyeth betwene the vtter circumference of the circle and the pointe, shall be equall to the full square of the other lyne, that onely toucheth the circle.Example.The circle isD.B.C,and the pointe without the circle isA,from whiche pointe there is drawen one line crosse the circle, and that isA.D.C,and an other lyne is drawn from the saidpricke to the marge or edge of the circumference of the circle, and doeth only touche it, that is the lineA.B.And of that first lineA.D.C,you maie perceiue one part of it, whiche isA.D,to lie without the circle, betweene the vtter circumference of it, and the pointe assigned, whiche wasA.Noweconcernyng the meanyng of the Theoreme, if you make a longsquare of the whole lineA.C,and of that parte of it that lyeth betwene the circumference and the point, (whiche isA.D,) that longesquare shall be equall to the full square of the touche lineA.B,accordyng not onely as this figure sheweth, but also the saied nyneteenth conclusion dooeth proue, if you lyste to examyne the one by the other.see textsee textsee textThe lxxvii. Theoreme.If a pointe be assigned without a circle, and from that pointe .ij. right lynes be drawen to the circle, so that the one doe crosse the circle, and the other dooe ende at the circumference, and that the longsquare of the line which crosseththe circle made with the portiõ of the same line beyng without the circle betweene the vtter circumference and the pointe assigned, doe equally agree with the iuste square of that line that endeth at the circumference, then is that lyne so endyng on the circumference a touche line vnto that circle.Example.In as muche as this Theoreme is nothyng els but the sentence of the last Theoreme before conuerted, therfore it shall not be nedefull to vse any other example then the same, for as in that other Theoreme because the one line is a touche lyne, therfore it maketh a square iust equal with the longsquare made of that whole line, whiche crosseth the circle, and his portion liyng without the same circle. So saith this Theoreme: that if the iust square of the line that endeth on the circumference, be equall to that longsquare whiche is made as for his longer sides of the whole line, which commeth from the pointt assigned, and crosseth the circle, and for his other shorter sides is made of the portion of the same line, liyng betwene the circumference of the circle and the pointe assigned, then is that line whiche endeth on the circumference a right touche line, that is to saie, yf the full square of the right lineA.B,be equall to the longsquare made of the whole lineA.C,as one of his lines, and of his portionA.D,as his other line, then must it nedes be, that the lyneA.B,is a right touche lyne vnto the circleD.B.C.And thus for this tyme I make an ende of the Theoremes.FINIS,
The .lxvi. theoreme.Euerie figure of foure sides, drawen in a circle, hath his two contrarie angles equall vnto two right angles.Example.see textThe circle isA.B.C.D,and the figure of foure sides in it, is made of the sidesB.C,andC.D,andD.A,andA.B.Now if you take any two angles that be contrary, as the angle byA,and the angle byC,I saie that those .ij. be equall to .ij. right angles. Also if you take the angle byB,and the angle byD,whiche two are also contray, those two angles are like waies equall to two right angles. But if any man will take the angle byA,with the angle byB,orD,they can not be accompted contrary, no more is not the angle byC.estemed contray to the angle byB,or yet to the angle byD,for they onely be accomptedcontrary angles, whiche haue no one line common to them bothe. Suche is the angle byA,in respect of the angle byC,for there both lynes be distinct, where as the angle byA,and the angle byD,haue one common lineA.D,and therfore can not be accompted contraryangles, Sothe angle byD,and the angle byC,haueD.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.see textCantles 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 beA.E.C,on whiche I draw a cantle of a circle, whiche isA.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 isA.D.C,so if an angle were made in it, that angle would be greatter then the angle made in the cantleA.B.C,and therforecannot they be called lyke cantels, but and if any other cantle were made greater then the first, then would the angleinit 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 equalrighte lynes, are equall together.Example.What isment bylike cantles you haue heardbefore. andit 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.see textNowe for the example of the Theoreme, I haue set forthe diuers varieties of cantles of circles, amongest which the first and seconde are made vpõ 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.see textFirste I haue sette for an exaumple twoo equall circles, thatisA.B.C.D,whose centre isK,and the second circleE.F.G.H,and his centreL,and in eche of thẽ 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 isB.C.D.the one, andF.G.H.the other. Now saieth the Theoreme, that if the angleB.A.D,be equall to the angleF.E.H,then are they made in equall circles, and on equall arch lines of their circumference. Also if the angleB.K.D,be equal to the angleF.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.This Theoreme doth but conuert the sentence of the last Theoremebefore, and therfore is to be vnderstande by the same examples, for as that saith, that equall angles occupie equall archelynes, so this saith, that equal arche lines causeth equal angles, consideringe all other circumstances, as was taughte in the laste theoreme before, so that this theoreme dooeth affirming speake of the equalitie of those angles, of which the laste theoreme spake conditionally. And where the laste theoreme spake affirmatiuely of the arche lines, this theoreme speaketh conditionally of them, as thus: If the arche lineB.C.D.be equall to the other arche lineF.G.H,then is that angleB.A.D.equall to the other angleF.E.H.Or els thus may you declare it causally: Bicause the arche lineB.C.D,is equal to the other arche lineF.G.H,therefore is the angleB.K.D.equall to the angleF.L.H,consideringe that they are made on the centres of equall circles. And so of the other angles, bicause those two arche lines aforesaid ar equal, therfore the angleD.A.B,is equall to the angleF.E.H,for as muche as they are made on those equall arche lines, and also on the circumference of equall circles. And thus these theoremes doo one declare an other, and one verifie the other.The lxxi. Theoreme.In equal circles, equall right lines beinge drawen, doo cutte awaye equalle arche lines frome their circumferences, so that the greater arche line of the one is equall to the greater arche line of the other, and the lesser to the lesser.Example.see textThe circleA.B.C.D,is made equall to the circleE.F.G.H,and the right lineB.D.is equal to the righte lineF.H,wherfore it foloweth, that the ij. arche lines of the circleA.B.D,whiche are cut from his circumference by the right lineB.D,are equall to two other arche lines of the circleE.F.H,being cutte frome hiscircumference, by the right lineF.H.that is to saye, that the arche lineB.A.D,beinge the greater arch line of the firste circle, is equall to the arche lineF.E.H,beynge the greater arche line of the other circle. And so in like manner the lesser arche line of the firste circle, beyngeB.C.D,is equal to the lesser arche line of the seconde circle, that isF.G.H.The lxxij. Theoreme.In equall circles, vnder equall arche lines the right lines that bee drawen are equall togither.Example.This Theoreme is none other, but the conuersion of the laste Theoreme beefore, and therefore needeth none other example. For as that did declare the equalitie of the arche lines, by the equalitie of the righte lines, sodothethis Theoremedeclare the equalnes of the right lines to ensue of the equalnes of the arche lines, and therefore declareth that right lyneB.D,to be equal to the other right lineF.H,bicause they both are drawen vnder equall arche lines, that is to saye, the one vnderB.A.D,and thother vnderF.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 yeangle made in a semicircle. Also it is meet to noteytalangles that be made in yepart 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.see textFirstethe circle is, as you see,A.B.C.D,and his centreE,his diameter isA.D,Thenis ther a line drawẽ fromA.toB,and so forth vntoF,which is without the circle: and an other line also fromeB.toD,whiche maketh two cantles of the whole circle.The greater cantle isD.A.B,and the lesser cantle isB.C.D,Inwhiche lesser cantle also there are two lines that make an angle, the one line isB.C,and the other line isC.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 cãtleB.A.D,in which is but one angle made, and that is the angle byA,which is made of a lineA.B,and the lineA.D,Andthis 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 lineD.B,and the arche lineD.A.B,are the twoo angles of this cantle, whereof the one is byD,and the other is byB.Wher you must remẽbre, that the ãgle byD.is made of the right lineB.D,and the arche lineD.A.And this angle is diuided by an other right lineA.E.D,which in this case must be omitted as no line. Also the ãgle byB.is made of the right lineD.B,and of the arch line.B.A,& although it be deuided with ij. other right lines, of wchthe one is the right lineB.A,& thother the right lineB.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 yeright lineB.D,& of yearch lineB.C,the secõd is made of the right line.D.B,& of the arch lineD.C.Then ar ther ij. other lines, wchdeuide those ij. corners, ytis the lineB.C,& the lineC.D,wchij. lines do meetin the poynteC,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. Andinlyke 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 byA.andD,and be made of the right lineA.D,beeyng the diameter, and of the halfe circumference of the circle, but by the angle made in the semicircle is that angle byB,whiche is made of the righte lineA.B,and that other right lineB.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 byB,beyng made at the right lineA.B,and the righte lineB.D,is a iuste righte angle, because it is made in a semicircle. But the angle made byA,which is made of the right lineA.B,and of the right lineA.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 byC,beyng made of the righte lineB.C,and of the right lineC.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 byB.andD,as is before declared, are greatter eche of them then a right angle. And theangles of the lesser cantle, whiche are by the same lettersB,andD,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.see textThe circle isA.B.C.D,and the touche line isE.F.The pointe of the touchyng isD,from which point I suppose the lineD.B,to be drawen crosse the circle, and to diuide it into .ij. cantles, wherof the greater isB.A.D,and the lesser isB.C.D,and in ech of them an angle is drawen, for in the greater cantle the angle is byA,and is made of the right linesB.A,andA.D,in the lesser cantle the angle is byC,and is made of yeright linesB.C,andC.D.Now saith the Theoreme that the angleB.D.F,is equall to the angle made in the cantle on the other side of the said line, that is to saie, in the cantleB.A.D,so that the angleB.D.F,is equall to the angleB.A.D,because the angleB.D.F,is on the one side of the lineB.D,(whiche is accordingto the supposition of the Theoreme drawen crosse the circle) and the angleB.A.D,is in the cãtle on the other side. Likewaies the angleB.D.E,beyng on the one side of the lineB.D,must be equall to the angleB.C.D,(that is the ãgle byC,) whiche is made in the cãtle on the other side of the right lineB.D.The profe of all these I do reserue, as I haue often saide, to a conuenient boke, wherein they shall be all set at large.The .lxxv. Theoreme.In any circle when .ij. right lines do crosse one an other, the likeiamme that is made of the portions of the one line, shall be equall to the lykeiamme made of the partes of the other lyne.see textBecause this Theoreme doth serue to many vses, and wold be wel vnderstande, I haue set forth .ij. examples of it. In the firste, the lines by their crossyng do make their portions somewhat toward an equalitie.In the second the portiõs of the lynes be very far frõ an equalitie, and yet in bothe these and in all other yeTheoreme is true. In the first exãple the circle isA.B.C.D,in which thone lineA.C,doth crosse thother lineB.D,in yepointE.Now if you do make one likeiãme or lõgsquare ofD.E,&E.B,being ye.ij.portions of the lineD.B,that longsquare shall be equall to the other longsquare made ofA.E,andE.C,beyng the portions of the other lineA.C.Lykewaies in the second example, the circle isF.G.H.K,in whiche the lineF.H,doth crosse the other lineG.K,in the pointeL.Wherfore if you make a lykeiamme or longsquare of the two partes of the lineF.H,that is to saye, ofF.L,andL.H,that longsquare will be equall to an other longsquare made of the two partes of the lineG.K.which partes areG.L,andL.K.Those longsquares haue I set foorth vnder the circles containyng their sides, that you maie somewhat whet your own wit in practisyng this Theoreme, accordyng to the doctrine of the nineteenth conclusion.The .lxxvi. Theoreme.If a pointe be marked without a circle, and from that pointe two right lines drawen to the circle, so that the one of them doe runne crosse the circle, and the other doe touche the circle onely, the long square that is made of that whole lyne which crosseth the circle, and the portion of it, that lyeth betwene the vtter circumference of the circle and the pointe, shall be equall to the full square of the other lyne, that onely toucheth the circle.Example.The circle isD.B.C,and the pointe without the circle isA,from whiche pointe there is drawen one line crosse the circle, and that isA.D.C,and an other lyne is drawn from the saidpricke to the marge or edge of the circumference of the circle, and doeth only touche it, that is the lineA.B.And of that first lineA.D.C,you maie perceiue one part of it, whiche isA.D,to lie without the circle, betweene the vtter circumference of it, and the pointe assigned, whiche wasA.Noweconcernyng the meanyng of the Theoreme, if you make a longsquare of the whole lineA.C,and of that parte of it that lyeth betwene the circumference and the point, (whiche isA.D,) that longesquare shall be equall to the full square of the touche lineA.B,accordyng not onely as this figure sheweth, but also the saied nyneteenth conclusion dooeth proue, if you lyste to examyne the one by the other.see textsee textsee textThe lxxvii. Theoreme.If a pointe be assigned without a circle, and from that pointe .ij. right lynes be drawen to the circle, so that the one doe crosse the circle, and the other dooe ende at the circumference, and that the longsquare of the line which crosseththe circle made with the portiõ of the same line beyng without the circle betweene the vtter circumference and the pointe assigned, doe equally agree with the iuste square of that line that endeth at the circumference, then is that lyne so endyng on the circumference a touche line vnto that circle.Example.In as muche as this Theoreme is nothyng els but the sentence of the last Theoreme before conuerted, therfore it shall not be nedefull to vse any other example then the same, for as in that other Theoreme because the one line is a touche lyne, therfore it maketh a square iust equal with the longsquare made of that whole line, whiche crosseth the circle, and his portion liyng without the same circle. So saith this Theoreme: that if the iust square of the line that endeth on the circumference, be equall to that longsquare whiche is made as for his longer sides of the whole line, which commeth from the pointt assigned, and crosseth the circle, and for his other shorter sides is made of the portion of the same line, liyng betwene the circumference of the circle and the pointe assigned, then is that line whiche endeth on the circumference a right touche line, that is to saie, yf the full square of the right lineA.B,be equall to the longsquare made of the whole lineA.C,as one of his lines, and of his portionA.D,as his other line, then must it nedes be, that the lyneA.B,is a right touche lyne vnto the circleD.B.C.And thus for this tyme I make an ende of the Theoremes.FINIS,
see text
The circle isA.B.C.D,and the figure of foure sides in it, is made of the sidesB.C,andC.D,andD.A,andA.B.Now if you take any two angles that be contrary, as the angle byA,and the angle byC,I saie that those .ij. be equall to .ij. right angles. Also if you take the angle byB,and the angle byD,whiche two are also contray, those two angles are like waies equall to two right angles. But if any man will take the angle byA,with the angle byB,orD,they can not be accompted contrary, no more is not the angle byC.estemed contray to the angle byB,or yet to the angle byD,for they onely be accomptedcontrary angles, whiche haue no one line common to them bothe. Suche is the angle byA,in respect of the angle byC,for there both lynes be distinct, where as the angle byA,and the angle byD,haue one common lineA.D,and therfore can not be accompted contraryangles, Sothe angle byD,and the angle byC,haueD.C,as a common line, and therefore be not contrary angles. And this maie you iudge of the residewe, by like reason.
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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 beA.E.C,on whiche I draw a cantle of a circle, whiche isA.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 isA.D.C,so if an angle were made in it, that angle would be greatter then the angle made in the cantleA.B.C,and therforecannot they be called lyke cantels, but and if any other cantle were made greater then the first, then would the angleinit 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.
What isment bylike cantles you haue heardbefore. andit 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.
see text
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 vpõ 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.
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Firste I haue sette for an exaumple twoo equall circles, thatisA.B.C.D,whose centre isK,and the second circleE.F.G.H,and his centreL,and in eche of thẽ 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 isB.C.D.the one, andF.G.H.the other. Now saieth the Theoreme, that if the angleB.A.D,be equall to the angleF.E.H,then are they made in equall circles, and on equall arch lines of their circumference. Also if the angleB.K.D,be equal to the angleF.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.
This Theoreme doth but conuert the sentence of the last Theoremebefore, and therfore is to be vnderstande by the same examples, for as that saith, that equall angles occupie equall archelynes, so this saith, that equal arche lines causeth equal angles, consideringe all other circumstances, as was taughte in the laste theoreme before, so that this theoreme dooeth affirming speake of the equalitie of those angles, of which the laste theoreme spake conditionally. And where the laste theoreme spake affirmatiuely of the arche lines, this theoreme speaketh conditionally of them, as thus: If the arche lineB.C.D.be equall to the other arche lineF.G.H,then is that angleB.A.D.equall to the other angleF.E.H.Or els thus may you declare it causally: Bicause the arche lineB.C.D,is equal to the other arche lineF.G.H,therefore is the angleB.K.D.equall to the angleF.L.H,consideringe that they are made on the centres of equall circles. And so of the other angles, bicause those two arche lines aforesaid ar equal, therfore the angleD.A.B,is equall to the angleF.E.H,for as muche as they are made on those equall arche lines, and also on the circumference of equall circles. And thus these theoremes doo one declare an other, and one verifie the other.
see text
The circleA.B.C.D,is made equall to the circleE.F.G.H,and the right lineB.D.is equal to the righte lineF.H,wherfore it foloweth, that the ij. arche lines of the circleA.B.D,whiche are cut from his circumference by the right lineB.D,are equall to two other arche lines of the circleE.F.H,being cutte frome hiscircumference, by the right lineF.H.that is to saye, that the arche lineB.A.D,beinge the greater arch line of the firste circle, is equall to the arche lineF.E.H,beynge the greater arche line of the other circle. And so in like manner the lesser arche line of the firste circle, beyngeB.C.D,is equal to the lesser arche line of the seconde circle, that isF.G.H.
This Theoreme is none other, but the conuersion of the laste Theoreme beefore, and therefore needeth none other example. For as that did declare the equalitie of the arche lines, by the equalitie of the righte lines, sodothethis Theoremedeclare the equalnes of the right lines to ensue of the equalnes of the arche lines, and therefore declareth that right lyneB.D,to be equal to the other right lineF.H,bicause they both are drawen vnder equall arche lines, that is to saye, the one vnderB.A.D,and thother vnderF.E.H,and those two arch lines are estimed equall by the theoreme laste before, and shal be proued in the booke of proofes.
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 yeangle made in a semicircle. Also it is meet to noteytalangles that be made in yepart 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.
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Firstethe circle is, as you see,A.B.C.D,and his centreE,his diameter isA.D,Thenis ther a line drawẽ fromA.toB,and so forth vntoF,which is without the circle: and an other line also fromeB.toD,whiche maketh two cantles of the whole circle.The greater cantle isD.A.B,and the lesser cantle isB.C.D,Inwhiche lesser cantle also there are two lines that make an angle, the one line isB.C,and the other line isC.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 cãtleB.A.D,in which is but one angle made, and that is the angle byA,which is made of a lineA.B,and the lineA.D,Andthis 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 lineD.B,and the arche lineD.A.B,are the twoo angles of this cantle, whereof the one is byD,and the other is byB.Wher you must remẽbre, that the ãgle byD.is made of the right lineB.D,and the arche lineD.A.And this angle is diuided by an other right lineA.E.D,which in this case must be omitted as no line. Also the ãgle byB.is made of the right lineD.B,and of the arch line.B.A,& although it be deuided with ij. other right lines, of wchthe one is the right lineB.A,& thother the right lineB.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 yeright lineB.D,& of yearch lineB.C,the secõd is made of the right line.D.B,& of the arch lineD.C.Then ar ther ij. other lines, wchdeuide those ij. corners, ytis the lineB.C,& the lineC.D,wchij. lines do meetin the poynteC,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. Andinlyke 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 byA.andD,and be made of the right lineA.D,beeyng the diameter, and of the halfe circumference of the circle, but by the angle made in the semicircle is that angle byB,whiche is made of the righte lineA.B,and that other right lineB.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 byB,beyng made at the right lineA.B,and the righte lineB.D,is a iuste righte angle, because it is made in a semicircle. But the angle made byA,which is made of the right lineA.B,and of the right lineA.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 byC,beyng made of the righte lineB.C,and of the right lineC.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 byB.andD,as is before declared, are greatter eche of them then a right angle. And theangles of the lesser cantle, whiche are by the same lettersB,andD,but be on the other side of the corde, are lesser eche of them then a right angle, and be therfore sharpe corners.
see text
The circle isA.B.C.D,and the touche line isE.F.The pointe of the touchyng isD,from which point I suppose the lineD.B,to be drawen crosse the circle, and to diuide it into .ij. cantles, wherof the greater isB.A.D,and the lesser isB.C.D,and in ech of them an angle is drawen, for in the greater cantle the angle is byA,and is made of the right linesB.A,andA.D,in the lesser cantle the angle is byC,and is made of yeright linesB.C,andC.D.Now saith the Theoreme that the angleB.D.F,is equall to the angle made in the cantle on the other side of the said line, that is to saie, in the cantleB.A.D,so that the angleB.D.F,is equall to the angleB.A.D,because the angleB.D.F,is on the one side of the lineB.D,(whiche is accordingto the supposition of the Theoreme drawen crosse the circle) and the angleB.A.D,is in the cãtle on the other side. Likewaies the angleB.D.E,beyng on the one side of the lineB.D,must be equall to the angleB.C.D,(that is the ãgle byC,) whiche is made in the cãtle on the other side of the right lineB.D.The profe of all these I do reserue, as I haue often saide, to a conuenient boke, wherein they shall be all set at large.
see text
Because this Theoreme doth serue to many vses, and wold be wel vnderstande, I haue set forth .ij. examples of it. In the firste, the lines by their crossyng do make their portions somewhat toward an equalitie.In the second the portiõs of the lynes be very far frõ an equalitie, and yet in bothe these and in all other yeTheoreme is true. In the first exãple the circle isA.B.C.D,in which thone lineA.C,doth crosse thother lineB.D,in yepointE.Now if you do make one likeiãme or lõgsquare ofD.E,&E.B,being ye.ij.portions of the lineD.B,that longsquare shall be equall to the other longsquare made ofA.E,andE.C,beyng the portions of the other lineA.C.Lykewaies in the second example, the circle isF.G.H.K,in whiche the lineF.H,doth crosse the other lineG.K,in the pointeL.Wherfore if you make a lykeiamme or longsquare of the two partes of the lineF.H,that is to saye, ofF.L,andL.H,that longsquare will be equall to an other longsquare made of the two partes of the lineG.K.which partes areG.L,andL.K.Those longsquares haue I set foorth vnder the circles containyng their sides, that you maie somewhat whet your own wit in practisyng this Theoreme, accordyng to the doctrine of the nineteenth conclusion.
The circle isD.B.C,and the pointe without the circle isA,from whiche pointe there is drawen one line crosse the circle, and that isA.D.C,and an other lyne is drawn from the saidpricke to the marge or edge of the circumference of the circle, and doeth only touche it, that is the lineA.B.And of that first lineA.D.C,you maie perceiue one part of it, whiche isA.D,to lie without the circle, betweene the vtter circumference of it, and the pointe assigned, whiche wasA.Noweconcernyng the meanyng of the Theoreme, if you make a longsquare of the whole lineA.C,and of that parte of it that lyeth betwene the circumference and the point, (whiche isA.D,) that longesquare shall be equall to the full square of the touche lineA.B,accordyng not onely as this figure sheweth, but also the saied nyneteenth conclusion dooeth proue, if you lyste to examyne the one by the other.
In as muche as this Theoreme is nothyng els but the sentence of the last Theoreme before conuerted, therfore it shall not be nedefull to vse any other example then the same, for as in that other Theoreme because the one line is a touche lyne, therfore it maketh a square iust equal with the longsquare made of that whole line, whiche crosseth the circle, and his portion liyng without the same circle. So saith this Theoreme: that if the iust square of the line that endeth on the circumference, be equall to that longsquare whiche is made as for his longer sides of the whole line, which commeth from the pointt assigned, and crosseth the circle, and for his other shorter sides is made of the portion of the same line, liyng betwene the circumference of the circle and the pointe assigned, then is that line whiche endeth on the circumference a right touche line, that is to saie, yf the full square of the right lineA.B,be equall to the longsquare made of the whole lineA.C,as one of his lines, and of his portionA.D,as his other line, then must it nedes be, that the lyneA.B,is a right touche lyne vnto the circleD.B.C.And thus for this tyme I make an ende of the Theoremes.