The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 9

The angle appointed, maie be a sharpe angle, a right angle, other a blunte angle, so that the worke must be diuersely handeled according to the diuersities of the angles, but consideringe the hardenes of those seuerall woorkes, I wyll omitte them for a more meter time, and at this tyme wyll shewe you one light waye which serueth for all kindes of angles, and that is this. When the line is proposed, and the angle a.s.signed, you shall ioyne that line proposed so to the other twoo lines contayninge the angle a.s.signed, that you shall make a triangle of theym, for the easy dooinge whereof, you may enlarge or shorten as you see cause, anye of the two lynes contayninge the angle appointed. And when you haue made a triangle of those iij.

lines, then accordinge to the doctrine of the seu? and tw?ty coclusi, make a circle about that triangle. And so haue you wroughte the request of this conclusion. Whyche yet you maye woorke by the twenty and eight conclusion also, so that of your line appointed, you make one side of the trigle be equal to y^e gle a.s.signed as youre selfe mai easily gesse.

[Ill.u.s.tration]

_Example._

First for example of a sharpe gle let A. std & B.C shal be y^e lyne a.s.signed. Th? do I make a triangle, by adding B.C, as a thirde side to those other ij. which doo include the gle a.s.signed, and that trigle is D.E.F, so y^t E.F. is the line appointed, and D. is the angle a.s.signed. Then doo I drawe a portion of a circle about that triangle, from the one ende of that line a.s.signed vnto the other, that is to saie, from E.

a long by D. vnto F, whiche portion is euermore greatter then the halfe of the circle, by reason that the angle is a sharpe angle. But if the angle be right (as in the second exaumple you see it) then shall the portion of the circle that containeth that angle, euer more be the iuste halfe of a circle. And when the angle is a blunte angle, as the thirde exaumple dooeth propounde, then shall the portion of the circle euermore be lesse then the halfe circle. So in the seconde example, G. is the right angle a.s.signed, and H.K. is the lyne appointed, and L.M.N. the portion of the circle aunsweryng thereto. In the third exaumple, O. is the blunte corner a.s.signed, P.Q. is the line, and R.S.T. is the portion of the circle, that containeth that blut corner, and is drawen on R.T. the line appointed.

THE x.x.xII. CONCLVSION.

To cutte of from a circle appointed, a portion containyng an angle equall to a right lyned angle a.s.signed.

When the angle and the circle are a.s.signed, first draw a touch line vnto that circle, and then drawe an other line from the p.r.i.c.ke of the touchyng to one side of the circle, so that thereby those two lynes do make an angle equall to the angle a.s.signed. Then saie I that the portion of the circle of the contrarie side to the angle drawen, is the parte that you seke for.

_Example._

[Ill.u.s.tration]

A. is the angle appointed, and D.E.F. is the circle a.s.signed, fr which I must cut away a porti that doth contain an angle equall to this angle A. Therfore first I do draw a touche line to the circle a.s.signed, and that touch line is B.C, the very p.r.i.c.ke of the touche is D, from whiche D. I drawe a lyne D.E, so that the angle made of those two lines be equall to the angle appointed. Then say I, that the arch of the circle D.F.E, is the arche that I seke after. For if I doo deuide that arche in the middle (as here is done in F.) and so draw thence two lines, one to D, and the other to E, then will the angle F, be equall to the angle a.s.signed.

THE x.x.xIII. CONCLVSION.

To make a square quadrate in a circle a.s.signed.

Draw .ij. diameters in the circle, so that they runne a crosse, and that they make .iiij. right angles. Then drawe .iiij. lines, that may ioyne the .iiij. ends of those diameters, one to an other, and then haue you made a square quadrate in the circle appointed.

_Example._

[Ill.u.s.tration]

A.B.C.D. is the circle a.s.signed, and A.C. and B.D. are the two diameters which crosse in the centre E, and make .iiij. right corners. Then do I make fowre other lines, that is A.B, B.C, C.D, and D.A, which do ioyne together the fowre endes of the ij.

diameters. And so is the square quadrate made in the circle a.s.signed, as the conclusion willeth.

THE x.x.xIIII. CONCLVSION.

To make a square quadrate aboute annye circle a.s.signed.

Drawe two diameters in crosse waies, so that they make foure righte angles in the centre. Then with your compa.s.se take the length of the halfe diameter, and set one foote of the compas in eche end of the compas, so shall you haue viij. archelines. Then yf you marke the p.r.i.c.kes wherin those arch lines do crosse, and draw betwene those iiij. p.r.i.c.kes iiij right lines, then haue you made the square quadrate accordinge to the request of the conclusion.

_Example._

[Ill.u.s.tration]

A.B.C. is the circle a.s.signed in which first I draw two diameters, in crosse waies, making iiij. righte angles, and those ij. diameters are A.C. and B.D. Then sette I my compa.s.se (whiche is opened according to the semidiameter of the said circle) fixing one foote in the end of euery semidiameter, and drawe with the other foote twoo arche lines, one on euery side.

As firste, when I sette the one foote in A, then with the other foote I doo make twoo arche lines, one in E, and an other in F.

Then sette I the one foote of the compa.s.se in B, and drawe twoo arche lines F. and G. Like wise setting the compa.s.se foote in C, I drawe twoo other arche lines, G. and H, and on D. I make twoo other, H. and E. Then frome the crossinges of those eighte arche lines I drawe iiij. straighte lynes, that is to saye, E.F, and F.G, also G.H, and H.E, whiche iiij. straighte lynes do make the square quadrate that I should draw about the circle a.s.signed.

THE x.x.xV. CONCLVSION.

To draw a circle in any square quadrate appointed.

Fyrste deuide euery side of the quadrate into twoo equall partes, and so drawe two lynes betwene eche two contrary poinctes, and where those twoo lines doo crosse, there is the centre of the circle. Then sette the foote of the compa.s.se in that point, and stretch forth the other foot, according to the length of halfe one of those lines, and so make a compas in the square quadrate a.s.signed.

_Example._

[Ill.u.s.tration]

A.B.C.D. is the quadrate appointed, in whiche I muste make a circle. Therefore first I do deuide euery side in ij. equal partes, and draw ij. lines acrosse, betwene eche ij. ctrary p.r.i.c.kes, as you se E.G, and F.H, whiche mete in K, and therfore shal K, be the centre of the circle. Then do I set one foote of the compas in K. and op? the other as wide as K.E, and so draw a circle, which is made accordinge to the conclusion.

THE x.x.xVI. CONCLVSION.

To draw a circle about a square quadrate.

Draw ij. lines betwene the iiij. corners of the quadrate, and where they mete in crosse, ther is the centre of the circle that you seeke for. Th? set one foot of the compas in that centre, and extend the other foote vnto one corner of the quadrate, and so may you draw a circle which shall iustely inclose the quadrate proposed.

_Example._

[Ill.u.s.tration]

A.B.C.D. is the square quadrate proposed, about which I must make a circle. Therfore do I draw ij. lines crosse the square quadrate from angle to angle, as you se A.C. & B.D. And where they ij. do crosse (that is to say in E.) there set I the one foote of the compas as in the centre, and the other foote I do extend vnto one angle of the quadrate, as for exple to A, and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.

THE x.x.xVII. CONCLVSION.

To make a twileke triangle, whiche shall haue euery of the ij. angles that lye about the ground line, double to the other corner.

Fyrste make a circle, and deuide the circ.u.mference of it into fyue equall partes. And thenne drawe frome one p.r.i.c.ke (which you will) two lines to ij. other p.r.i.c.kes, that is to say to the iij.

and iiij. p.r.i.c.ke, counting that for the first, wherhence you drewe both those lines, Then drawe the thyrde lyne to make a triangle with those other twoo, and you haue doone according to the conclusion, and haue made a twelike trigle, whose ij.

corners about the grounde line, are eche of theym double to the other corner.

_Example._

[Ill.u.s.tration]

A.B.C. is the circle, whiche I haue deuided into fiue equal portions. And from one of the p.r.i.c.kes (which is A,) I haue draw?

ij. lines, A.B. and B.C, whiche are drawen to the third and iiij. p.r.i.c.kes. Then draw I the third line C.B, which is the grounde line, and maketh the triangle, that I would haue, for the gle C. is double to the angle A, and so is the angle B.

also.

THE x.x.xVIII. CONCLVSION.

To make a cinkangle of equall sides, and equall corners in any circle appointed.