Part 23

Example.

This theoreme seemeth of it selfe so manifest, that it neadeth nother demonstration nother declaracion. Yet for the plaine vnderstanding of it, I haue sette forthe a figure here, where ij. circles be drawen, so that one of them doth crosse the other (as you see) in the pointes B. and G, and their centres appear at the firste sighte to bee diuers. For the centre of the one is F, and the centre of the other is E, which diffre as farre asondre as the edges of the circles, where they bee most distaunte in sonder.

The Li. Theoreme.

If two circles be so drawen, that one of them do touche the other, then haue they not one centre.

Example.

There are two circles made, as you see, the one is A.B.C, and hath his centre by G, the other is B.D.E, and his centre is by F, so that it is easy enough to perceaue that their centres doe dyffer as muche a sonder, as the halfe diameter of the greater circle is longer then the half diameter of the lesser circle. And so must it needes be thought and said of all other circles in lyke kinde.

The .lij. theoreme.

If a certaine pointe be assigned in the diameter of a circle, distant from the centre of the said circle, and from that pointe diuerse lynes drawen to the edge and circumference of the same circle, the longest line is that whiche passeth by the centre, and the shortest is the residew of the same line. And of al the other lines that is euer the greatest, that is nighest to the line, which passeth by the centre. And contrary waies, that is the shortest, that is farthest from it. And amongest them all there can be but onely .ij. equall together, and they must nedes be so placed, that the shortest line shall be in the iust middle betwixte them.

Example.

The circle is A.B.C.D.E.H, and his centre is F, the diameter is A.E, in whiche diameter I haue taken a certain point distaunt from the centre, and that pointe is G, from which I haue drawen .iiij. lines to the circumference, beside the two partes of the diameter, whiche maketh vp vi. lynes in all. Nowe for the diuersitee in quantitie of these lynes, I saie accordyng to the Theoreme, that the line whiche goeth by the centre is the longest line, that is to saie, A.G, and the residewe of the same diameter beeyng G.E, is the shortest lyne. And of all the other that lyne is longest, that is neerest vnto that parte of the diameter whiche gooeth by the centre, and that is shortest, that is farthest distant from it, wherefore I saie, that G.B, is longer then G.C, and therfore muche more longer then G.D, sith G.C, also is longer then G.D, and by this maie you soone perceiue, that it is not possible to drawe .ij. lynes on any one side of the diameter, whiche might be equall in lengthe together, but on the one side of the diameter maie you easylie make one lyne equall to an other, on the other side of the same diameter, as you see in this example G.H, to bee equall to G.D, betweene whiche the lyne G.E, (as the shortest in all the circle) doothe stande euen distaunte from eche of them, and it is the precise knoweledge of their equalitee, if they be equally distaunt from one halfe of the diameter. Where as contrary waies if the one be neerer to any one halfe of the diameter then the other is, it is not possible that they two may be equall in lengthe, namely if they dooe ende bothe in the circumference of the circle, and be bothe drawen from one poynte in the diameter, so that the saide poynte be (as the Theoreme doeth suppose) somewhat distaunt from the centre of the said circle. For if they be drawen from the centre, then must they of necessitee be all equall, howe many so euer they bee, as the definition of a circle dooeth importe, withoute any regarde how neere so euer they be to the diameter, or how distante from it. And here is to be noted, that in this Theoreme, by neerenesse and distaunce is vnderstand the nereness and distaunce of the extreeme partes of those lynes where they touche the circumference. For at the other end they do all meete and touche.

The .liij. Theoreme.

If a pointe bee marked without a circle, and from it diuerse lines drawen crosse the circle, to the circumference on the other side, so that one of them passe by the centre, then that line whiche passeth by the centre shall be the loongest of them all that crosse the circle. And of the other lines those are longest, that be nexte vnto it that passeth by the centre. And those ar shortest, that be farthest distant from it. But among those partes of those lines, whiche ende in the outewarde circumference, that is most shortest, whiche is parte of the line that passeth by the centre, and amongeste the othere eche, of them, the nerer they are vnto it, the shorter they are, and the farther from it, the longer they be. And amongest them all there can not be more then .ij. of any one length, and they two muste be on the two contrarie sides of the shortest line.

Example.

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.