The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 8

If so be it that you desire to find the centre by any other way then by those .iij. p.r.i.c.kes, consideryng that sometimes you can not haue so much s.p.a.ce 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._

[Ill.u.s.tration]

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 circ.u.mference of the circle. And that circle is contained in a triangle, whose circ.u.mference 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 circ.u.mference doeth touche euery side of the quadrate, and lykewaies of other figures.

_Examples are these. A.B.C.D.E.F._

[Ill.u.s.tration: A. is a circle in a triangle.

B. a triangle in a circle.

C. a quadrate in a circle.

D. a circle in a quadrate.]

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 couted to be drawen in the circle, because one of his corners doth not touche the circ.u.mference 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 p.r.i.c.ke of any of the sides, and so drawe a compas, whiche shall touche euery side of the triangle, but shall not pa.s.se with out any of them.

_Example._

The triangle is A.B.C, whose sides I do part into .ij. equall partes, eche by it selfe in these pointes D.E.F, puttyng F.

betwene A.B, and D. betwene B.C, and E. betwene A.C. Then draw I a line from C. to F, and an other from A. to D, and the third from B. to E.

[Ill.u.s.tration]

And where all those lines do mete (that is to saie M. G,) I set the one foote of my compa.s.se, because it is the common centre, and so drawe a circle accordyng to the distaunce of any of the sides of the triangle. And then find I that circle to agree iustely to all the sides of the triangle, so that the circle is iustely made in the triangle, as the conclusion did purporte.

And this is euer true, when the triangle hath all thre sides equall, other at the least .ij. sides lyke long. But in the other kindes of triangles you must deuide euery angle in the middle, as the third conclusion teaches you. And so drawe lines fr eche angle to their middle p.r.i.c.ke. And where those lines do crosse, there is the common centre, from which you shall draw a perpendicular to one of the sides. Then sette one foote of the compas in that centre, and stretche the other foote accordyng to the l?gth of the perpendicular, and so drawe your circle.

[Ill.u.s.tration]

_Example._

The triangle is A.B.C, whose corners I haue diuided in the middle with D.E.F, and haue drawen the lines of diuision A.D. B.E, and C.F, which crosse in G, therfore shall G. be the common centre. Then make I one perp?dicular from G. vnto the side B.C, and that is G.H. Now sette I one fote of the compas in G, and extend the other foote vnto H. and so drawe a compas, whiche wyll iustly answere to that trigle according to the meaning of the conclusion.

THE XXVIII. CONCLVSION.

To drawe a circle about any trigle a.s.signed.

Fyrste deuide two sides of the triangle equally in half and from those ij. p.r.i.c.kes erect two perpendiculars, which muste needes meet in crosse, and that point of their meting is the centre of the circle that must be drawen, therefore sette one foote of the compa.s.se in that pointe, and extend the other foote to one corner of the triangle, and so make a circle, and it shall touche all iij. corners of the triangle.

_Example._

[Ill.u.s.tration]

A.B.C. is the triangle, whose two sides A.C. and B.C. are diuided into two equall partes in D. and E, settyng D. betwene B. and C, and E. betwene A. and C. And from eche of those two pointes is ther erected a perpendicular (as you se D.F, and E.F.) which mete, and crosse in F, and stretche forth the other foot of any corner of the triangle, and so make a circle, that circle shal touch euery corner of the triangle, and shal enclose the whole triangle, accordinge, as the conclusion willeth.

An other way to do the same.

And yet an other waye may you doo it, accordinge as you learned in the seuententh conclusion, for if you call the three corners of the triangle iij. p.r.i.c.kes, and then (as you learned there) yf you seeke out the centre to those three p.r.i.c.kes, and so make it a circle to include those thre p.r.i.c.kes in his circ.u.mference, you shall perceaue that the same circle shall iustelye include the triangle proposed.

_Example._

[Ill.u.s.tration]

A.B.C. is the triangle, whose iij. corners I count to be iij.

pointes. Then (as the seuentene conclusion doth teache) I seeke a common centre, on which I may make a circle, that shall enclose those iij p.r.i.c.kes. that centre as you se is D, for in D.

doth the right lines, that pa.s.se by the angles of the arche lines, meete and crosse. And on that centre as you se, haue I made a circle, which doth inclose the iij. angles of the trigle, and consequentlye the triangle itselfe, as the conclusion dydde intende.

THE XXIX. CONCLVSION.

To make a triangle in a circle appoynted whose corners shal be equall to the corners of any triangle a.s.signed.

When I will draw a triangle in a circle appointed, so that the corners of that triangle shall be equall to the corners of any triangle a.s.signed, then must I first draw a tuche lyne vnto that circle, as the twenty conclusion doth teach, and in the very poynte of the touche muste I make an angle, equall to one angle of the triangle, and that inwarde toward the circle: likewise in the same p.r.i.c.ke must I make an other angle w^t the other halfe of the touche line, equall to an other corner of the triangle appointed, and then betwen those two corners will there resulte a third angle, equall to the third corner of that triangle. Nowe where those two lines that entre into the circle, doo touche the circ.u.mference (beside the touche line) there set I two p.r.i.c.kes, and betwene them I drawe a thyrde line. And so haue I made a triangle in a circle appointed, whose corners bee equall to the corners of the triangle a.s.signed.

_Example._

[Ill.u.s.tration]

A.B.C, is the triangle appointed, and F.G.H. is the circle, in which I muste make an other triangle, with lyke angles to the angles of A.B.C. the triangle appointed. Therefore fyrst I make the touch lyne D.F.E. And then make I an angle in F, equall to A, whiche is one of the angles of the triangle. And the lyne that maketh that angle with the touche line, is F.H, whiche I drawe in lengthe vntill it touche the edge of the circle. Then againe in the same point F, I make an other corner equall to the angle C. and the line that maketh that corner with the touche line, is F.G. whiche also I drawe foorthe vntill it touche the edge of the circle. And then haue I made three angles vpon that one touch line, and in y^t one point F, and those iij. angles be equall to the iij. angles of the triangle a.s.signed, whiche thinge doth plainely appeare, in so muche as they bee equall to ij. right angles, as you may gesse by the fixt theoreme. And the thre angles of euerye triangle are equill also to ij. righte angles, as the two and twenty theoreme dothe show, so that bicause they be equall to one thirde thinge, they must needes be equal togither, as the cmon sentence saith. Th? do I draw a line frome G. to H, and that line maketh a triangle F.G.H, whole angles be equall to the angles of the triangle appointed. And this triangle is drawn in a circle, as the conclusion didde wyll. The proofe of this conclusion doth appeare in the seuenty and iiij. Theoreme.

THE x.x.x. CONCLVSION.

To make a triangle about a circle a.s.signed which shall haue corners, equall to the corners of any triangle appointed.

First draw forth in length the one side of the triangle a.s.signed so that therby you may haue ij. vtter angles, vnto which two vtter angles you shall make ij. other equall on the centre of the circle proposed, drawing thre halfe diameters frome the circ.u.mference, whiche shal enclose those ij. angles, th? draw iij. touche lines which shall make ij. right angles, eche of them with one of those semidiameters. Those iij. lines will make a triangle equally cornered to the triangle a.s.signed, and that triangle is draw? about a circle apointed, as the cclusi did wil.

_Example._

A.B.C, is the triangle a.s.signed, and G.H.K, is the circle appointed, about which I muste make a triangle hauing equall angles to the angles of that triangle A.B.C. Fyrst therefore I draw A.C. (which is one of the sides of the triangle) in length that there may appeare two vtter angles in that triangle, as you se B.A.D, and B.C.E.

[Ill.u.s.tration]

Then drawe I in the circle appointed a semidiameter, which is here H.F, for F. is the c?tre of the circle G.H.K. Then make I on that centre an angle equall to the vtter angle B.A.D, and that angle is H.F.K. Like waies on the same c?tre by drawyng an other semidiameter, I make an other angle H.F.G, equall to the second vtter angle of the triangle, whiche is B.C.E. And thus haue I made .iij. semidiameters in the circle appointed. Then at the ende of eche semidiameter, I draw a touche line, whiche shall make righte angles with the semidiameter. And those .iij.

touch lines mete, as you see, and make the trianagle L.M.N, whiche is the triangle that I should make, for it is drawen about a circle a.s.signed, and hath corners equall to the corners of the triangle appointed, for the corner M. is equall to C.

Likewaies L. to A, and N. to B, whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.

THE x.x.xI. CONCLVSION.

To make a portion of a circle on any right line a.s.signed, whiche shall conteine an angle equall to a right lined angle appointed.