Part 7 (2/2)
THE XXIII. CONCLVSION.
To find the commen centre belongyng to anye three p.r.i.c.kes 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 circ.u.mference of al circles is compa.s.sed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. p.r.i.c.kes in an exact right line can not bee brought into the circ.u.mference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their cmon centre. Op? your compas so wide, that it be somewhat more then the halfe distance of two of those p.r.i.c.kes. Then sette the one foote of the compas in the one p.r.i.c.ke, and with the other foot draw an arche lyne toward the other p.r.i.c.ke, Then againe putte the foot of your compas in the second p.r.i.c.ke, 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 p.r.i.c.ks, so do with the middle p.r.i.c.ke, 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_
[Ill.u.s.tration]
The iij. p.r.i.c.kes I haue set to be A.B, and C, whiche I wold bring into the edg of one common circle, by finding a centre cmen to them all, fyrst therefore I open my cpas, so that thei occupye more then y^e halfe distance betwene ij. p.r.i.c.ks (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 settig 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 cpa.s.se 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. p.r.i.c.kes. Now therfore if you set one foot of your compas in H, and extend the other to any of the iij. p.r.i.c.ks, you may draw a circle w^{ch} shal enclose those iij. p.r.i.c.ks in the edg of his circufer?ce & thus haue you attained y^e vse of this cclusi.
THE XXIIII. CONCLVSION.
To drawe a touche line onto a circle, from any poincte a.s.signed.
Here must you vnderstand that the p.r.i.c.ke must be without the circle, els the conclusion is not possible. But the p.r.i.c.ke 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 p.r.i.c.ke 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 fr the p.r.i.c.ke 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 p.r.i.c.ke vnto the edge of the greatter circle, that line I say is a touche line, drawen from the point a.s.signed, according to the meaning of this conclusion.
[Ill.u.s.tration]
_Example._
Let the circle be called B.C.D, and his c?tre E, and y^e p.r.i.c.k a.s.signed A, op? your cpas now of such widenes, y^t the one foote may be set in E, w^{ch} is y^e c?tre of y^e circle, & y^e other in A, w^{ch} is y^e pointe a.s.signed, & so make an other greter circle (as here is A.F.G) th? draw a line from A. vnto E, and wher that line doth cross y^e inner circle (w^{ch} heere is in the p.r.i.c.k B.) there erect a plub 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 p.r.i.c.ke, and so drawing a right line, as this example sheweth, where the circle is E, the p.r.i.c.ke a.s.signed 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.
[Ill.u.s.tration]
THE XXV. CONCLVSION.
When you haue any peece of the circ.u.mference of a circle a.s.signed, 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 circ.u.mference, 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 cpa.s.se 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 circuference by the self patr, vsing it as it wer a croked ruler. Then make .iij. p.r.i.c.kes 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 c?tre 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 compa.s.se, you haue youre whole intent.
[Ill.u.s.tration]
THE XXVI. CONCLVSION.
To finde the centre to any arche of a circle.
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