Part 9

Example.

Let the circle be A.B.C.D, whose centre I shall seke. First therfore I draw a corde crosse the circle, that is A.C. Then do I deuide that corde in the middle, in E, and likewaies also do I deuide his arche line A.B.C, in the middle, in the pointe B. Afterward I drawe a line from B. to E, and so crosse the circle, whiche line is B.D, in which line is the centre that I seeke for. Therefore if I parte that line B.D, in the middle in to two equall portions, that middle pricke (which here is F) is the verye centre of the sayde circle that I seke. This conclusion may other waies be wrought, as the moste part of conclusions haue sondry formes of practise, and that is, by makinge thre prickes in the circumference of the circle, at liberty where you wyll, and then findinge the centre to those thre pricks, Which worke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.

THE XXIII. CONCLVSION.

To find the commen centre belongyng to anye three prickes appointed, if they be not in an exacte right line.

It is to be noted, that though euery small arche of a greate circle do seeme to be a right lyne, yet in very dede it is not so, for euery part of the circumference of al circles is compassed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. prickes in an exact right line can not bee brought into the circumference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their common centre. Open your compas so wide, that it be somewhat more then the halfe distance of two of those prickes. Then sette the one foote of the compas in the one pricke, and with the other foot draw an arche lyne toward the other pricke, Then againe putte the foot of your compas in the second pricke, and with the other foot make an arche line, that may crosse the firste arch line in ij. places. Now as you haue done with those two pricks, so do with the middle pricke, and the thirde that remayneth. Then draw ij. lines by the poyntes where those arche lines do crosse, and where those two lines do meete, there is the centre that you seeke for.

Example

The iij. prickes I haue set to be A.B, and C, whiche I wold bring into the edg of one common circle, by finding a centre commen to them all, fyrst therefore I open my compas, so that thei occupye more then y^e halfe distance betwene ij. pricks (as are A.B.) and so settinge one foote in A. and extendinge the other toward B, I make the arche line D.E. Likewise setting one foot in B, and turninge the other toward A, I draw an other arche line that crosseth the first in D. and E. Then from D. to E, I draw a right lyne D.H. After this I open my compasse to a new distance, and make ij. arche lines betwene B. and C, whiche crosse one the other in F. and G, by whiche two pointes I draw an other line, that is F.H. And bycause that the lyne D.H. and the lyne F.H. doo meete in H, I saye that H. is the centre that serueth to those iij. prickes. Now therfore if you set one foot of your compas in H, and extend the other to any of the iij. pricks, you may draw a circle w^{ch} shal enclose those iij. pricks in the edg of his circumference & thus haue you attained y^e vse of this conclusion.

THE XXIIII. CONCLVSION.

To drawe a touche line onto a circle, from any poincte assigned.

Here must you vnderstand that the pricke must be without the circle, els the conclusion is not possible. But the pricke or poinct beyng without the circle, thus shall you procede: Open your compas, so that the one foote of it maie be set in the centre of the circle, and the other foote on the pricke appoincted, and so draw an other circle of that largenesse about the same centre: and it shall gouerne you certainly in makyng the said touche line. For if you draw a line from the pricke appointed vnto the centre of the circle, and marke the place where it doeth crosse the lesser circle, and from that poincte erect a plumbe line that shall touche the edge of the vtter circle, and marke also the place where that plumbe line crosseth that vtter circle, and from that place drawe an other line to the centre, takyng heede where it crosseth the lesser circle, if you drawe a plumbe line from that pricke vnto the edge of the greatter circle, that line I say is a touche line, drawen from the point assigned, according to the meaning of this conclusion.

Example.

Let the circle be called B.C.D, and his centre E, and y^e prick assigned A, open your compas now of such widenes, y^t the one foote may be set in E, w^{ch} is y^e centre of y^e circle, & y^e other in A, w^{ch} is y^e pointe assigned, & so make an other greter circle (as here is A.F.G) then draw a line from A. vnto E, and wher that line doth cross y^e inner circle (w^{ch} heere is in the prick B.) there erect a plumb line vnto the line. A.E. and let that plumb line touch the vtter circle, as it doth here in the point F, so shall B.F. bee that plumbe lyne. Then from F. vnto E. drawe an other line whiche shal be F.E, and it will cutte the inner circle, as it doth here in the point C, from which pointe C. if you erect a plumb line vnto A, then is that line A.C, the touche line, whiche you shoulde finde. Not withstandinge that this is a certaine waye to fynde any touche line, and a demonstrable forme, yet more easyly by many folde may you fynde and make any suche line with a true ruler, layinge the edge of the ruler to the edge of the circle and to the pricke, and so drawing a right line, as this example sheweth, where the circle is E, the pricke assigned is A. and the ruler C.D. by which the touch line is drawen, and that is A.B, and as this way is light to doo, so is it certaine inoughe for any kinde of workinge.

THE XXV. CONCLVSION.

When you haue any peece of the circumference of a circle assigned, howe you may make oute the whole circle agreynge therevnto.

First seeke out of the centre of that arche, according to the doctrine of the seuententh conclusion, and then setting one foote of your compas in the centre, and extending the other foot vnto the edge of the arche or peece of the circumference, it is easy to drawe the whole circle.

Example.

A peece of an olde pillar was found, like in forme to thys figure A.D.B. Now to knowe howe muche the compasse of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make in a table the like draught of y^t circumference by the self patron, vsing it as it wer a croked ruler. Then make .iij. prickes in that arche line, as I haue made, C. D. and E. And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that centre is here F, nowe settyng one foote of your compas in F, and the other in C. D, other in E, and so makyng a compasse, you haue youre whole intent.

THE XXVI. CONCLVSION.

To finde the centre to any arche of a circle.

If so be it that you desire to find the centre by any other way then by those .iij. prickes, consideryng that sometimes you can not haue so much space in the thyng where the arche is drawen, as should serue to make those .iiij. bowe lines, then shall you do thus: Parte that arche line into two partes, equall other vnequall, it maketh no force, and vnto ech portion draw a corde, other a stringline. And then accordyng as you dyd in one arche in the .xvi. conclusion, so doe in bothe those arches here, that is to saie, deuide the arche in the middle, and also the corde, and drawe then a line by those two deuisions, so then are you sure that that line goeth by the centre. Afterward do lykewaies with the other arche and his corde, and where those .ij. lines do crosse, there is the centre, that you seke for.

Example.

The arche of the circle is A.B.C, vnto whiche I must seke a centre, therfore firste I do deuide it into .ij. partes, the one of them is A.B, and the other is B.C. Then doe I cut euery arche in the middle, so is E. the middle of A.B, and G. is the middle of B.C. Likewaies, I take the middle of their cordes, whiche I mark with F. and H, settyng F. by E, and H. by G. Then drawe I a line from E. to F, and from G. to H, and they do crosse in D, wherefore saie I, that D. is the centre, that I seke for.

THE XXVII. CONCLVSION.

To drawe a circle within a triangle appoincted.

For this conclusion and all other lyke, you muste vnderstande, that when one figure is named to be within an other, that it is not other waies to be vnderstande, but that eyther euery syde of the inner figure dooeth touche euerie corner of the other, other els euery corner of the one dooeth touche euerie side of the other. So I call that triangle drawen in a circle, whose corners do touche the circumference of the circle. And that circle is contained in a triangle, whose circumference doeth touche iustely euery side of the triangle, and yet dooeth not crosse ouer any side of it. And so that quadrate is called properly to be drawen in a circle, when all his fower angles doeth touche the edge of the circle, And that circle is drawen in a quadrate, whose circumference doeth touche euery side of the quadrate, and lykewaies of other figures.

Examples are these. A.B.C.D.E.F.

In these .ij. last figures E. and F, the circle is not named to be drawen in a triangle, because it doth not touche the sides of the triangle, neither is the triangle counted to be drawen in the circle, because one of his corners doth not touche the circumference of the circle, yet (as you see) the circle is within the triangle, and the triangle within the circle, but nother of them is properly named to be in the other. Now to come to the conclusion. If the triangle haue all .iij. sides lyke, then shall you take the middle of euery side, and from the contrary corner drawe a right line vnto that poynte, and where those lines do crosse one an other, there is the centre. Then set one foote of the compas in the centre and stretche out the other to the middle pricke of any of the sides, and so drawe a compas, whiche shall touche euery side of the triangle, but shall not passe with out any of them.

Example.