The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 10

Deuide the circle appointed into fiue equall partes, as you didde in the laste conclusion, and drawe ij. lines from euery p.r.i.c.ke to the other ij. that are nexte vnto it. And so shall you make a cinkangle after the meanynge of the conclusion.

_Example._

Yow se here this circle A.B.C.D.E. deuided into fiue equall portions. And from eche p.r.i.c.ke ij. lines drawen to the other ij.

nexte p.r.i.c.kes, so from A. are drawen ij. lines, one to B, and the other to E, and so from C. one to B. and an other to D, and likewise of the reste. So that you haue not only learned hereby how to make a sinkangle in anye circle, but also how you shal make a like figure spedely, whanne and where you will, onlye drawinge the circle for the intente, readylye to make the other figure (I meane the cinkangle) thereby.

[Ill.u.s.tration]

THE x.x.xIX. CONCLVSION.

How to make a cinkangle of equall sides and equall angles about any circle appointed.

Deuide firste the circle as you did in the last conclusion into fiue equall portions, and draw fiue semidiameters in the circle.

Then make fiue touche lines, in suche sorte that euery touche line make two right angles with one of the semidiameters. And those fiue touche lines will make a cinkangle of equall sides and equall angles.

[Ill.u.s.tration]

_Example._

A.B.C.D.E. is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is draw? a semidiameter, as you see. Then doo I make a touche line in the p.r.i.c.ke B, whiche is F.G, making ij. right angles with the semidiameter B, and lyke waies on C. is made G.H, on D. standeth H.K, and on E, is set K.L, so that of those .v. touche lynes are made the .v.

sides of a cinkeangle, accordyng to the conclusion.

An other waie.

Another waie also maie you drawe a cinkeangle aboute a circle, drawyng first a cinkeangle in the circle (whiche is an easie thyng to doe, by the doctrine of the .x.x.xvij. conclusion) and then drawing .v. touche lines whiche shall be iuste paralleles to the .v. sides of the cinkeangle in the circle, forseeyng that one of them do not crosse ouerthwarte an other and then haue you done. The exaumple of this (because it is easie) I leaue to your owne exercise.

THE XL. CONCLVSION.

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

Drawe a plumbe line from any one corner of the cinkeangle, vnto the middle of the side that lieth iuste against that angle. And do likewaies in drawyng an other line from some other corner, to the middle of the side that lieth against that corner also. And those two lines wyll meete in crosse in the p.r.i.c.ke of their crossyng, shall you iudge the centre of the circle to be.

Therfore set one foote of the compas in that p.r.i.c.ke, and extend the other to the end of the line that toucheth the middle of one side, whiche you liste, and so drawe a circle. And it shall be iustly made in the cinkeangle, according to the conclusion.

_Example._

The cinkeangle a.s.signed is A.B.C.D.E, in whiche I muste make a circle, wherefore I draw a right line from the one angle (as from B,) to the middle of the contrary side (whiche is E. D,) and that middle p.r.i.c.ke is F. Then lykewaies from an other corner (as from E) I drawe a right line to the middle of the side that lieth against it (whiche is B.C.) and that p.r.i.c.ke is G. Nowe because that these two lines do crosse in H, I saie that H. is the centre of the circle, whiche I would make. Therfore I set one foote of the compa.s.se in H, and extend the other foote vnto G, or F. (whiche are the endes of the lynes that lighte in the middle of the side of that cinkeangle) and so make I the circle in the cinkangle, right as the cclusion meaneth.

[Ill.u.s.tration]

THE XLI. CONCLVSION

To make a circle about any a.s.signed cinkeangle of equall sides, and equall corners.

Drawe .ij. lines within the cinkeangle, from .ij. corners to the middle on tbe .ij. contrary sides (as the last conclusion teacheth) and the pointe of their crossyng shall be the centre of the circle that I seke for. Then sette I one foote of the compas in that centre, and the other foote I extend to one of the angles of the cinkangle, and so draw I a circle about the cinkangle a.s.signed.

_Example._

A.B.C.D.E, is the cinkangle a.s.signed, about which I would make a circle. Therfore I drawe firste of all two lynes (as you see) one fr E. to G, and the other fr C. to F, and because thei do meete in H, I saye that H. is the centre of the circle that I woulde haue, wherfore I sette one foote of the compa.s.se in H.

and extende the other to one corner (whiche happeneth fyrste, for all are like distaunte from H.) and so make I a circle aboute the cinkeangle a.s.signed.

[Ill.u.s.tration]

An other waye also.

Another waye maye I do it, thus presupposing any three corners of the cinkangle to be three p.r.i.c.kes appointed, vnto whiche I shoulde finde the centre, and then drawinge a circle touchinge them all thre, accordinge to the doctrine of the seuentene, one and twenty, and two and twenty conclusions. And when I haue founde the centre, then doo I drawe the circle as the same conclusions do teache, and this forty conclusion also.

THE XLII. CONCLVSION.

To make a siseangle of equall sides, and equall angles, in any circle a.s.signed.

Yf the centre of the circle be not knowen, then seeke oute the centre according to the doctrine of the sixtenth conclusion. And with your compas take the quant.i.tee of the semidiameter iustly.

And then sette one foote in one p.r.i.c.ke of the circuference of the circle, and with the other make a marke in the circ.u.mference also towarde both sides. Then sette one foote of the compas stedily in eche of those new p.r.i.c.kes, and point out two other p.r.i.c.kes. And if you haue done well, you shal perceaue that there will be but euen sixe such diuisions in the circ.u.mference.

Whereby it dothe well appeare, that the side of anye sisangle made in a circle, is equalle to the semidiameter of the same circle.

_Example._

[Ill.u.s.tration]

The circle is B.C.D.E.F.G, whose centre I finde to bee A.

Therefore I sette one foote of the compas in A, and do ext?d the other foote to B, thereby takinge the semidiameter. Then sette I one foote of the compas vnremoued in B, and marke with the other foote on eche side C. and G. Then from C. I marke D, and fr D, E: from E. marke I F. And then haue I but one s.p.a.ce iuste vnto G. and so haue I made a iuste siseangle of equall sides and equall angles, in a circle appointed.

THE XLIII. CONCLVSION.

To make a siseangle of equall sides, and equall angles about any circle a.s.signed.

THE XLIIII. CONCLVSION.

To make a circle in any siseangle appointed, of equall sides and equal angles.

THE XLV. CONCLVSION.

To make a circle about any sise angle limited of equall sides and equall angles.

Bicause you maye easily coniecture the makinge of these figures by that that is saide before of cinkangles, only consideringe that there is a difference in the numbre of sides, I thought beste to leue these vnto your owne deuice, that you should study in some thinges to exercise your witte withall and that you mighte haue the better occasion to perceaue what difference there is betwene eche twoo of those conclusions. For thoughe it seeme one thing to make a siseangle in a circle, and to make a circle about a siseangle, yet shall you perceaue, that is not one thinge, nother are those twoo conclusions wrought one way.

Likewaise shall you thinke of those other two conclusions. To make a siseangle about a circle, and to make a circle in a siseangle, thoughe the figures be one in fashion, when they are made, yet are they not one in working, as you may well perceaue by the x.x.xvij. x.x.xviij. x.x.xix. and xl. conclusions, in whiche the same workes are taught, touching a circle and a cinkangle, yet this muche wyll I saye, for your helpe in working, that when you shall seeke the centre in a siseangle (whether it be to make a circle in it other about it) you shall drawe the two crosselines, from one angle to the other angle that lieth againste it, and not to the middle of any side, as you did in the cinkangle.

THE XLVI. CONCLVSION.

To make a figure of fifteene equall sides and angles in any circle appointed.