Part 10

Take the circle to be A.B.C, and the point assigned without it to be D. Now say I, that if there be drawen sundrie lines from D, and crosse the circle, endyng in the circumference on the contrary side, as here you see, D.A, D.E, D.F, and D.B, then of all these lines the longest must needes be D.A, which goeth by the centre of the circle, and the nexte vnto it, that is D.E, is the longest amongest the rest. And contrarie waies, D.B, is the shorteste, because it is farthest distaunt from D.A. And so maie you iudge of D.F, because it is nerer vnto D.A, then is D.B, therefore is it longer then D.B. And likewaies because it is farther of from D.A, then is D.E, therfore is it shorter then D.E. Now for those partes of the lines whiche bee withoute the circle (as you see) D.C, is the shortest. because it is the parte of that line which passeth by the centre, And D.K, is next to it in distance, and therefore also in shortnes, so D.G, is farthest from it in distance, and therfore is the longest of them. Now D.H, beyng nerer then D.G, is also shorter then it, and beynge farther of, then D.K, is longer then it. So that for this parte of the theoreme (as I think) you do plainly perceaue the truthe thereof, so the residue hathe no difficulte. For seing that the nearer any line is to D.C, (which ioyneth with the diameter) the shorter it is and the farther of from it, the longer it is. And seyng two lynes can not be of like distaunce beinge bothe on one side, therefore if they shal be of one lengthe, and consequently of one distaunce, they must needes bee on contrary sides of the saide line D.C. And so appeareth the meaning of the whole Theoreme.

And of this Theoreme dothe there folowe an other lyke. whiche you maye calle other a theoreme by it selfe, or else a Corollary vnto this laste theoreme, I passe not so muche for the name. But his sentence is this: _when so euer any lynes be drawen frome any pointe, withoute a circle, whether they crosse the circle, or eande in the utter edge of his circumference, those two lines that bee equally distaunt from the least line are equal togither, and contrary waies, if they be equall togither, they ar also equally distant from that least line._

For the declaracion of this proposition, it shall not need to vse any other example, then that which is brought for the explication of this laste theoreme, by whiche you may without any teachinge easyly perceaue both the meanyng and also the truth of this proposition.

_The Liiij. Theoreme._

If a point be set forthe in a circle, and from that pointe vnto the circumference many lines drawen, of which more then two are equal togither, then is that point the centre of that circle.

_Example._

The circle is A.B.C, and within it I haue sette fourth for an example three prickes, which are D.E. and F, from euery one of them I haue drawen (at the leaste) iiij. lines vnto the circumference of the circle but frome D, I haue drawen more, yet maye it appear readily vnto your eye, that of all the lines whiche be drawen from E. and F, vnto the circumference, there are but twoo equall, and more can not bee, for G.E. nor E.H. hath none other equal to theim, nor canne not haue any beinge drawen from the same point E. No more can L.F, or F.K, haue anye line equall to either of theim, beinge drawen from the same pointe F. And yet from either of those two poinctes are there drawen twoo lines equall togither, as A.E, is equall to E.B, and B.F, is equall to F.C, but there can no third line be drawen equall to either of these two couples, and that is by reason that they be drawen from a pointe distaunte from the centre of the circle. But from D, althoughe there be seuen lines drawen, to the circumference, yet all bee equall, bicause it is the centre of the circle. And therefore if you drawe neuer so mannye more from it vnto the circumference, all shall be equal, so that this is the priuilege (as it were of the centre) and therfore no other point can haue aboue two equal lines drawen from it vnto the circumference. And from all pointes you maye drawe ij. equall lines to the circumference of the circle, whether that pointe be within the circle or without it.

_The lv. Theoreme._

No circle canne cut an other circle in more pointes then two.

_Example._

The first circle is A.B.F.E, the second circle is B.C.D.E, and they crosse one an other in B. and in E, and in no more pointes. Nother is it possible that they should, but other figures ther be, which maye cutte a circle in foure partes, as you se in this example. Where I haue set forthe one tunne forme, and one eye forme, and eche of them cutteth euery of their two circles into foure partes. But as they be irregulare formes, that is to saye, suche formes as haue no precise measure nother proportion in their draughte, so can there scarcely be made any certaine theorem of them. But circles are regulare formes, that is to say, such formes as haue in their protracture a iuste and certaine proportion, so that certain and determinate truths may be affirmed of them, sith they ar vniforme and vnchaungable.

_The lvi. Theoreme._

If two circles be so drawen, that the one be within the other, and that they touche one an other: If a line bee drawen by bothe their centres, and so forthe in lengthe, that line shall runne to that pointe, where the circles do touche.

_Example._

The one circle, which is the greattest and vttermost is A.B.C, the other circle that is y^e lesser, and is drawen within the firste, is A.D.E. The centre of the greater circle is F, and the centre of the lesser circle is G, the pointe where they touche is A. And now you may see the truthe of the theoreme so plainely, that it needeth no farther declaracion. For you maye see, that drawinge a line from F. to G, and so forth in lengthe, vntill it come to the circumference, it wyll lighte in the very poincte A, where the circles touche one an other.

_The Lvij. Theoreme._

If two circles bee drawen so one withoute an other, that their edges doo touche and a right line bee drawnenne frome the centre of the one to the centre of the other, that line shall passe by the place of their touching.

_Example._

The firste circle is A.B.E, and his centre is K, The second circle is D.B.C, and his centre is H, the point wher they do touch is B. Nowe doo you se that the line K.H, whiche is drawen from K, that is centre of the firste circle, vnto H, beyng centre of the second circle, doth passe (as it must nedes by the pointe B,) whiche is the verye poynte wher they do to touche together.

_The .lviij. theoreme._

One circle can not touche an other in more pointes then one, whether they touche within or without.

_Example._

For the declaration of this Theoreme, I haue drawen iiij. circles, the first is A.B.C, and his centre H. the second is A.D.G, and his centre F. the third is L.M, and his centre K. the .iiij. is D.G.L.M, and his centre E. Nowe as you perceiue the second circle A.D.G, toucheth the first in the inner side, in so much as it is drawen within the other, and yet it toucheth him but in one point, that is to say in A, so lykewaies the third circle L.M, is drawen without the firste circle and toucheth hym, as you maie see, but in one place. And now as for the .iiij. circle, it is drawen to declare the diuersitie betwene touchyng and cuttyng, or crossyng. For one circle maie crosse and cutte a great many other circles, yet can be not cutte any one in more places then two, as the fiue and fiftie Theoreme affirmeth.

_The .lix. Theoreme._

In euerie circle those lines are to be counted equall, whiche are in lyke distaunce from the centre, And contrarie waies they are in lyke distance from the centre, whiche be equall.

_Example._

In this figure you see firste the circle drawen, whiche is A.B.C.D, and his centre is E. In this circle also there are drawen two lines equally distaunt from the centre, for the line A.B, and the line D.C, are iuste of one distaunce from the centre, whiche is E, and therfore are they of one length. Again thei are of one lengthe (as shall be proued in the boke of profes) and therefore their distaunce from the centre is all one.

_The lx. Theoreme._

In euerie circle the longest line is the diameter, and of all the other lines, thei are still longest that be nexte vnto the centre, and they be the shortest, that be farthest distaunt from it.

_Example._

In this circle A.B.C.D, I haue drawen first the diameter, whiche is A.D, whiche passeth (as it must) by the centre E, Then haue I drawen ij. other lines as M.N, whiche is neerer the centre, and F.G, that is farther from the centre. The fourth line also on the other side of the diameter, that is B.C, is neerer to the centre then the line F.G, for it is of lyke distance as is the lyne M.N. Nowe saie I, that A.D, beyng the diameter, is the longest of all those lynes, and also of any other that maie be drawen within that circle, And the other line M.N, is longer then F.G. Also the line F.G, is shorter then the line B.C, for because it is farther from the centre then is the lyne B.C. And thus maie you iudge of al lines drawen in any circle, how to know the proportion of their length, by the proportion of their distance, and contrary waies, howe to discerne the proportion of their distance by their lengthes, if you knowe the proportion of their length. And to speake of it by the waie, it is a maruaylouse thyng to consider, that a man maie knowe an exacte proportion betwene two thynges, and yet can not name nor attayne the precise quantitee of those two thynges, As for exaumple, If two squares be sette foorthe, whereof the one containeth in it fiue square feete, and the other contayneth fiue and fortie foote, of like square feete, I am not able to tell, no nor yet anye manne liuyng, what is the precyse measure of the sides of any of those .ij. squares, and yet I can proue by vnfallible reason, that their sides be in a triple proportion, that is to saie, that the side of the greater square (whiche containeth .xlv. foote) is three tymes so long iuste as the side of the lesser square, that includeth but fiue foote. But this seemeth to be spoken out of ceason in this place, therfore I will omitte it now, reseruyng the exacter declaration therof to a more conuenient place and time, and will procede with the residew of the Theoremes appointed for this boke.

_The .lxi. Theoreme._

If a right line be drawen at any end of a diameter in perpendicular forme, and do make a right angle with the diameter, that right line shall light without the circle, and yet so iointly knitte to it, that it is not possible to draw any other right line betwene that saide line and the circumference of the circle. And the angle that is made in the semicircle is greater then any sharpe angle that may be made of right lines, but the other angle without, is lesser then any that can be made of right lines.

_Example._

In this circle A.B.C, the diameter is A.C, the perpendicular line, which maketh a right angle with the diameter, is C.A, whiche line falleth without the circle, and yet ioyneth so exactly vnto it, that it is not possible to draw an other right line betwene the circumference of the circle and it, whiche thyng is so plainly seene of the eye, that it needeth no farther declaracion. For euery man wil easily consent, that betwene the croked line A.F, (whiche is a parte of the circumference of the circle) and A.E (which is the said perpendicular line) there can none other line bee drawen in that place where they make the angle. Nowe for the residue of the theoreme. The angle D.A.B, which is made in the semicircle, is greater then anye sharpe angle that may bee made of ryghte lines. and yet is it a sharpe angle also, in as much as it is lesser then a right angle, which is the angle E.A.D, and the residue of that right angle, which lieth without the circle, that is to saye, E.A.B, is lesser then any sharpe angle that can be made of right lines also. For as it was before rehersed, there canne no right line be drawen to the angle, betwene the circumference and the right line E.A. Then must it needes folow, that there can be made no lesser angle of righte lines. And againe, if ther canne be no lesser then the one, then doth it sone appear, that there canne be no greater then the other, for they twoo doo make the whole right angle, so that if anye corner coulde be made greater then the one parte, then shoulde the residue bee lesser then the other parte, so that other bothe partes muste be false, or els bothe graunted to be true.

_The lxij. Theoreme._

If a right line doo touche a circle, and an other right line drawen frome the centre of the circle to the pointe where they touche, that line whiche is drawenne frome the centre, shall be a perpendicular line to the touch line.

_Example._

The circle is A.B.C, and his centre is F. The touche line is D.E, and the point wher they touch is C. Now by reason that a right line is drawen frome the centre F. vnto C, which is the point of the touche, therefore saith the theoreme, that the sayde line F.C, muste needes bee a perpendicular line vnto the touche line D.E.

_The lxiij. Theoreme._

If a righte line doo touche a circle, and an other right line be drawen from the pointe of their touchinge, so that it doo make righte corners with the touche line, then shal the centre of the circle bee in that same line, so drawen.

_Example._

The circle is A.B.C, and the centre of it is G. The touche line is D.C.E, and the pointe where it toucheth, is C. Nowe it appeareth manifest, that if a righte line be drawen from the pointe where the touch line doth ioine with the circle, and that the said lyne doo make righte corners with the touche line, then muste it needes go by the centre of the circle, and then consequently it must haue the sayde centre in him. For if the saide line shoulde go beside the centre, as F.C. doth, then dothe it not make righte angles with the touche line, which in the theoreme is supposed.

_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._

This Theoreme doth but conuert the sentence of the last Theoreme before, 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 line B.C.D. be equall to the other arche line F.G.H, then is that angle B.A.D. equall to the other angle F.E.H. Or els thus may you declare it causally: Bicause the arche line B.C.D, is equal to the other arche line F.G.H, therefore is the angle B.K.D. equall to the angle F.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 angle D.A.B, is equall to the angle F.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._

The circle A.B.C.D, is made equall to the circle E.F.G.H, and the right line B.D. is equal to the righte line F.H, wherfore it foloweth, that the ij. arche lines of the circle A.B.D, whiche are cut from his circumference by the right line B.D, are equall to two other arche lines of the circle E.F.H, being cutte frome his circumference, by the right line F.H. that is to saye, that the arche line B.A.D, beinge the greater arch line of the firste circle, is equall to the arche line F.E.H, beynge the greater arche line of the other circle. And so in like manner the lesser arche line of the firste circle, beynge B.C.D, is equal to the lesser arche line of the seconde circle, that is F.G.H.

_The lxxij. Theoreme._