Chapter III: Part 3

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._

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.

And where all those lines do mete (that is to saie M. G,) I set the one foote of my compasse, 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 from eche angle to their middle pricke. 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 length of the perpendicular, and so drawe your circle.

_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 perpendicular 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 triangle according to the meaning of the conclusion.

THE XXVIII. CONCLVSION.

To drawe a circle about any triangle assigned.

Fyrste deuide two sides of the triangle equally in half and from those ij. prickes 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 compasse 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._

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. prickes, and then (as you learned there) yf you seeke out the centre to those three prickes, and so make it a circle to include those thre prickes in his circumference, you shall perceaue that the same circle shall iustelye include the triangle proposed.

_Example._

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 prickes. that centre as you se is D, for in D. doth the right lines, that passe 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 triangle, 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 assigned.

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 assigned, 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 pricke 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 circumference (beside the touche line) there set I two prickes, 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 assigned.

_Example._

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 assigned, 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 common sentence saith. Then 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 XXX. CONCLVSION.

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

First draw forth in length the one side of the triangle assigned 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 circumference, whiche shal enclose those ij. angles, then 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 assigned, and that triangle is drawen about a circle apointed, as the conclusion did wil.

_Example._

A.B.C, is the triangle assigned, 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.

Then drawe I in the circle appointed a semidiameter, which is here H.F, for F. is the centre 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 centre 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 assigned, 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 XXXI. CONCLVSION.

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

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 assigned, you shall ioyne that line proposed so to the other twoo lines contayninge the angle assigned, 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 seuen and twenty coclusion, 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 triangle be equal to y^e angle assigned as youre selfe mai easily gesse.

_Example._

First for example of a sharpe angle let A. stand & B.C shal be y^e lyne assigned. Then do I make a triangle, by adding B.C, as a thirde side to those other ij. which doo include the angle assigned, and that triangle is D.E.F, so y^t E.F. is the line appointed, and D. is the angle assigned. Then doo I drawe a portion of a circle about that triangle, from the one ende of that line assigned 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 assigned, 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 assigned, P.Q. is the line, and R.S.T. is the portion of the circle, that containeth that blunt corner, and is drawen on R.T. the line appointed.

THE XXXII. CONCLVSION.

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

When the angle and the circle are assigned, first draw a touch line vnto that circle, and then drawe an other line from the pricke of the touchyng to one side of the circle, so that thereby those two lynes do make an angle equall to the angle assigned. 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._

A. is the angle appointed, and D.E.F. is the circle assigned, from which I must cut away a portion that doth contain an angle equall to this angle A. Therfore first I do draw a touche line to the circle assigned, and that touch line is B.C, the very pricke 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 assigned.

THE XXXIII. CONCLVSION.

To make a square quadrate in a circle assigned.

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._

A.B.C.D. is the circle assigned, 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 assigned, as the conclusion willeth.

THE XXXIIII. CONCLVSION.

To make a square quadrate aboute annye circle assigned.

Drawe two diameters in crosse waies, so that they make foure righte angles in the centre. Then with your compasse 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 prickes wherin those arch lines do crosse, and draw betwene those iiij. prickes iiij right lines, then haue you made the square quadrate accordinge to the request of the conclusion.

_Example._

A.B.C. is the circle assigned 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 compasse (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 compasse in B, and drawe twoo arche lines F. and G. Like wise setting the compasse 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 assigned.

THE XXXV. 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 compasse 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 assigned.

_Example._

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. contrary prickes, 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 open the other as wide as K.E, and so draw a circle, which is made accordinge to the conclusion.

THE XXXVI. 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. Then 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._

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 example to A, and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.

THE XXXVII. 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 circumference of it into fyue equall partes. And thenne drawe frome one pricke (which you will) two lines to ij. other prickes, that is to say to the iij. and iiij. pricke, 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 triangle, whose ij. corners about the grounde line, are eche of theym double to the other corner.

_Example._

A.B.C. is the circle, whiche I haue deuided into fiue equal portions. And from one of the prickes (which is A,) I haue drawen ij. lines, A.B. and B.C, whiche are drawen to the third and iiij. prickes. Then draw I the third line C.B, which is the grounde line, and maketh the triangle, that I would haue, for the angle C. is double to the angle A, and so is the angle B. also.

THE XXXVIII. CONCLVSION.

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

Deuide the circle appointed into fiue equall partes, as you didde in the laste conclusion, and drawe ij. lines from euery pricke 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 pricke ij. lines drawen to the other ij. nexte prickes, 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.

THE XXXIX. 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.

_Example._

A.B.C.D.E. is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is drawen a semidiameter, as you see. Then doo I make a touche line in the pricke 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 .xxxvij. 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 pricke of their crossyng, shall you iudge the centre of the circle to be. Therfore set one foote of the compas in that pricke, 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 assigned 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 pricke 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 pricke 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 compasse 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 conclusion meaneth.

THE XLI. CONCLVSION

To make a circle about any assigned 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 assigned.

_Example._

A.B.C.D.E, is the cinkangle assigned, about which I would make a circle. Therfore I drawe firste of all two lynes (as you see) one from E. to G, and the other from 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 compasse 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 assigned.

An other waye also.

Another waye maye I do it, thus presupposing any three corners of the cinkangle to be three prickes 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 assigned.

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 quantitee of the semidiameter iustly. And then sette one foote in one pricke of the circumference of the circle, and with the other make a marke in the circumference also towarde both sides. Then sette one foote of the compas stedily in eche of those new prickes, and point out two other prickes. And if you haue done well, you shal perceaue that there will be but euen sixe such diuisions in the circumference. 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._

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 extend 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 from D, E: from E. marke I F. And then haue I but one space 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 assigned.

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 xxxvij. xxxviij. xxxix. 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.

This rule is generall, that how many sides the figure shall
haue, that shall be drawen in any circle, into so many partes
iustely muste the circles bee deuided. And therefore it is the
more easier woorke commonly, to drawe a figure in a circle, then
to make a circle in an other figure. Now therefore to end this
conclusion, deuide the circle firste into fiue partes, and
then eche of them into three partes againe: Or els
first deuide it into three partes, and then ech
of them into fiue other partes, as you
list, and canne most readilye.
Then draw lines betwene
euery two prickes
that be nighest
togither, and
ther wil appear rightly drawen the figure, of fiftene sides,
and angles equall. And so do with any other figure
of what numbre of sides so euer it bee.

+FINIS.+

THE SECOND BOOKE
+OF THE PRINCIPLES+
_of Geometry, containing certaine_
_Theoremes, whiche may be cal-_
led Approued truthes. And be as
it were the moste certaine
groundes, wheron the
practike conclusions
of Geometry ar
founded.

[Leaf]

Whervnto are annexed certaine declarations by
examples, for the right vnderstanding of the
same, to the ende that the simple reader
might not iustly complain of hardnes
or obscuritee, and for the same
cause ar the demonstra-
tions and iust profes
omitted, vntill a
more conueni-
ent time.

1551.

If truthe maie trie it selfe,
By Reasons prudent skyll,
If reason maie preuayle by right,
And rule the rage of will,
I dare the triall byde,
For truthe that I pretende.
And though some lyst at me repine,
Iuste truthe shall me defende.

THE PREFACE VNTO
the Theoremes.

I Doubt not gentle reader, but as my argument is straunge and vnacquainted with the vulgare toungue, so shall I of many men be straungly talked of, and as straungly iudged. Some men will saye peraduenture, I mighte haue better imployed my tyme in some pleasaunte historye, comprisinge matter of chiualrye. Some other wolde more haue preised my trauaile, if I hadde spente the like time in some morall matter, other in deciding some controuersy of religion. And yet some men (as I iudg) will not mislike this kind of mater, but then will they wishe that I had vsed a more certaine order in placinge bothe the Propositions and Theoremes, and also a more exacter proofe of eche of theim bothe, by demonstrations mathematicall. Some also will mislike my shortenes and simple plainesse, as other of other affections diuersely shall espye somwhat that they shall thinke blame worthy, and shal misse somewhat, that thei wold with to haue bene here vsed, so that euerie manne shall giue his verdicte of me according to his phantasie, vnto whome ioinctly, I make this my firste answere: that as they ar many and in opinions verie diuers, so were it scarse possible to please them all with anie one argumente, of what kinde so euer it were. And for my seconde aunswere, I saye thus. That if annye one argumente mighte please them all, then should thei be thankfull vnto me for this kind of matter. For nother is there anie matter more straunge in the englishe tungue, then this whereof neuer booke was written before now, in that tungue, and therefore oughte to delite all them, that desire to vnderstand strange matters, as most men commonlie doo. And againe the practise is so pleasaunt in vsinge, and so profitable in appliynge, that who so euer dothe delite in anie of bothe, ought not of right to mislike this arte. And if any manne shall like the arte welle for it selfe, but shall mislyke the fourme that I haue vsed in teachyng of it, to hym I shall saie, Firste, that I dooe wishe with hym that some other man, whiche coulde better haue doone it, hadde shewed his good will, and vsed his diligence in suche sorte, that I myght haue bene therby occasioned iustely to haue left of my laboure, or after my trauaile to haue suppressed my bookes. But sithe no manne hath yet attempted the like, as far as I canne learne, I truste all suche as bee not exercised in the studie of Geometrye, shall finde greate ease and furtheraunce by this simple, plaine, and easie forme of writinge. And shall perceaue the exacte woorkes of Theon, and others that write on Euclide, a great deale the soner, by this blunte delineacion afore hande to them taughte. For I dare presuppose of them, that thing which I haue sette in my selfe, and haue marked in others, that is to saye, that it is not easie for a man that shall trauaile in a straunge arte, to vnderstand at the beginninge bothe the thing that is taught and also the iuste reason whie it is so. And by experience of teachinge I haue tried it to bee true, for whenne I haue taughte the proposition, as it is imported in meaninge, and annexed the demonstration with all, I didde perceaue that it was a greate trouble and a painefull vexacion of mynde to the learner, to comprehend bothe those thinges at ones. And therfore did I proue firste to make them to vnderstande the sence of the propositions, and then afterward did they conceaue the demonstrations muche soner, when they hadde the sentence of the propositions first ingrafted in their mindes. This thinge caused me in bothe these bookes to omitte the demonstrations, and to vse onlye a plaine forme of declaration, which might best serue for the firste introduction. Whiche example hath beene vsed by other learned menne before nowe, for not only Georgius Ioachimus Rheticus, but also Boetius that wittye clarke did set forth some whole books of Euclide, without any demonstration or any other declaration at al. But & if I shal hereafter perceaue that it maie be a thankefull trauaile to sette foorth the propositions of geometrie with demonstrations, I will not refuse to dooe it, and that with sundry varietees of demonstrations, bothe pleasaunt and profitable also. And then will I in like maner prepare to sette foorth the other bookes, whiche now are lefte vnprinted, by occasion not so muche of the charges in cuttyng of the figures, as for other iuste hynderances, whiche I truste hereafter shall bee remedied. In the meane season if any man muse why I haue sette the Conclusions beefore the Teoremes, seynge many of the Theoremes seeme to include the cause of some of the conclusions, and therfore oughte to haue gone before them, as the cause goeth before the effecte. Here vnto I saie, that although the cause doo go beefore the effect in order of nature, yet in order of teachyng the effect must be fyrst declared, and than the cause therof shewed, for so that men best vnderstand things First to lerne that such thinges ar to be wrought, and secondarily what thei ar, and what thei do import, and than thirdly what is the cause therof. An other cause why y^t the theoremes be put after the conclusions is this, whan I wrote these first connclusions (which was .iiiij. yeres passed) I thought not then to haue added any theoremes, but next vnto y^e conclusions to haue taught the order how to haue applied them to work, for drawing of plottes & such like vses. But afterward considering the great commoditie y^t thei serue for, and the light that thei do geue to all sortes of practise geometricall, besyde other more notable benefites, whiche shall be declared more specially in a place conuenient, I thoughte beste to geue you some taste of theym, and the pleasaunt contemplation of suche geometrical propositions, which might serue diuerselye in other bookes for the demonstrations and proofes of all Geometricall woorkes. And in theim, as well as in the propositions, I haue drawen in the Linearie examples many tymes more lynes, than be spoken of in the explication of them, whiche is doone to this intent, that yf any manne lyst to learne the demonstrations by harte, (as somme learned men haue iudged beste to doo) those same men should find the Linearye exaumples to serue for this purpose, and to wante no thyng needefull to the iuste proofe, whereby this booke may bee wel approued to be more complete then many men wolde suppose it.

And thus for this tyme I wyll make an ende without any larger declaration of the commoditiees of this arte, or any farther answeryng to that may bee obiected agaynst my handelyng of it, wyllyng them that myslike it, not to medle with it: and vnto those that will not disdaine the studie of it, I promise all suche aide as I shall be able to shewe for their farther procedyng both in the same, and in all other commoditees that thereof maie ensue. And for their incouragement I haue here annexed the names and brefe argumentes of suche bookes, as I intende (God willyng) shortly to sette forth, if I shall perceaue that my paynes maie profyte other, as my desyre is.

+The brefe argumentes of suche bokes as ar appoynted shortly
to be set forth by the author herof.+

THE seconde part of Arithmetike, teachyng the workyng by fractions, with extraction of rootes both square and cubike: And declaryng the rule of allegation, with sundrye plesaunt exaumples in metalles and other thynges. Also the rule of false position, with dyuers examples not onely vulgar, but some appertaynyng to the rule of Algeber, applied vnto quantitees partly rationall, and partly surde.

THE arte of Measuryng by the quadrate geometricall, and the disorders committed by vsyng the same, not only reueled but reformed also (as muche as to the instrument pertayneth) by the deuise of a new quadrate newely inuented by the author hereof.

THE arte of measuryng by the astronomers staffe, and by the astronomers ryng, and the form of makyng them both.

THE arte of makyng of Dials, bothe for the daie and the nyght, with certayn new formes of fixed dialles for the moon and other for the sterres, whiche may bee sette in glasse windowes to serue by daie and by night. And howe you may by those dialles knowe in what degree of the Zodiake not only the sonne, but also the moone is. And how many howrs old she is. And also by the same dial to know whether any eclipse shall be that moneth, of the sonne or of the moone.

The makyng and vse of an instrument, wherby you maye not onely measure the distance at ones of all places that you can see togyther, howe muche eche one is from you, and euery one from other, but also therby to drawe the plotte of any countreie that you shall come in, as iustely as maie be, by mannes diligence and labour.

THE vse bothe of the Globe and the Sphere, and therin also of the arte of Nauigation, and what instrumentes serue beste thervnto, and of the trew latitude and longitude of regions and townes.

Euclides woorkes in foore partes, with diuers demonstrations Arithmeticall and Geometricall or Linearie. The fyrst parte of platte formes. The second of numbres and quantitees surde or irrationall. The third of bodies and solide formes. The fourthe of perspectiue, and other thynges thereto annexed.

BESIDE these I haue other sundrye woorkes partely ended, and partely to bee ended, Of the peregrination of man, and the originall of al nations, The state of tymes, and mutations of realmes, The image of a perfect common welth, with diuers other woorkes in naturall sciences, Of the wonderfull workes and effectes in beastes, plantes, and minerals, of whiche at this tyme, I will omitte the argumentes, beecause thei doo appertaine littel to this arte, and handle other matters in an other sorte.

To haue, or leaue,
Nowe maie you chuse,
No paine to please,
Will I refuse.

The Theoremes of Geometry, before
_WHICHE ARE SET FORTHE_
_certaine grauntable requestes_
_which serue for demonstrations_
Mathematicall.

[Sidenote: I.]

That from any pricke to one other, there may be drawen a
right line.

As for example A--------B. A. being the one pricke, and B. the other, you maye drawe betwene them from the one to the other, that is to say, frome A. vnto B, and from B. to A.

[Sidenote: II.]

That any right line of measurable length may be drawen forth
longer, and straight.

Example of A.B, which as it is a line of measurable lengthe, so may it be drawen forth farther, as for example vnto C, and that in true streightenes without crokinge.

[Sidenote: III.]

That vpon any centre, there may be made a circle of anye
quantitee that a man wyll.

Let the centre be set to be A, what shal hinder a man to drawe a circle aboute it, of what quantitee that he lusteth, as you se the forme here: other bygger or lesse, as it shall lyke him to doo:

That all right angles be equall eche to other.

Set for an example A. and B, of which two though A. seme the greatter angle to some men of small experience, it happeneth only bicause that the lines aboute A, are longer then the lines about B, as you may proue by drawing them longer, for so that B. seme the greater angle yf you make his lines longer then the lines that make the angle A. And to proue it by demonstration, I say thus. If any ij. right corners be not equal, then one right corner is greater then an other, but that corner which is greatter then a right angle, is a blunt corner (by his definition) so must one corner be both a right corner and a blunt corner also, which is not possible: And againe: the lesser right corner must be a sharpe corner, by his definition, bicause it is lesse then a right angle. which thing is impossible. Therefore I conclude that all right angles be equall.

Yf one right line do crosse two other right lines, and make
ij. inner corners of one side lesser then ij. righte corners,
it is certaine, that if those two lines be drawen forth
right on that side that the sharpe inner corners be, they
wil at length mete togither, and crosse on an other.

The ij. lines beinge as A.B. and C.D, and the third line crossing them as dooth heere E.F, making ij inner cornes (as ar G.H.) lesser then two right corners, sith ech of them is lesse then a right corner, as your eyes maye iudge, then say I, if those ij. lines A.B. and C.D. be drawen in lengthe on that side that G. and H. are, the will at length meet and crosse one an other.

Two right lines make no platte forme.

A platte forme, as you harde before, hath bothe length and bredthe, and is inclosed with lines as with his boundes, but ij. right lines cannot inclose al the bondes of any platte forme. Take for an example firste these two right lines A.B. and A.C. whiche meete togither in A, but yet cannot be called a platte forme, bicause there is no bond from B. to C, but if you will drawe a line betwene them twoo, that is frome B. to C, then will it be a platte forme, that is to say, a triangle, but then are there iij. lines, and not only ij. Likewise may you say of D.E. and F.G, whiche doo make a platte forme, nother yet can they make any without helpe of two lines more, whereof the one must be drawen from D. to F, and the other frome E. to G, and then will it be a longe rquare. So then of two right lines can bee made no platte forme. But of ij. croked lines be made a platte forme, as you se in the eye form. And also of one right line, & one croked line, maye a platte fourme bee made, as the semicircle F. doothe sette forth.

Certayn common sentences manifest to
sence, and acknowledged of all men.

_The firste common sentence._

What so euer things be equal to one other thinge, those same
bee equall betwene them selues.

Examples therof you may take both in greatnes and also in numbre. First (though it pertaine not proprely to geometry, but to helpe the vnderstandinge of the rules, whiche may bee wrought by bothe artes) thus may you perceaue. If the summe of monnye in my purse, and the mony in your purse be equall eche of them to the mony that any other man hathe, then must needes your mony and mine be equall togyther. Likewise, if anye ij. quantities, as A. and B, be equal to an other, as vnto C, then muste nedes A. and B. be equall eche to other, as A. equall to B, and B. equall to A, whiche thinge the better to perceaue, tourne these quantities into numbre, so shall A. and B. make sixteene, and C. as many. As you may perceaue by multipliyng the numbre of their sides togither.

_The seconde common sentence._

And if you adde equall portions to thinges that be equall,
what so amounteth of them shall be equall.

Example, Yf you and I haue like summes of mony, and then receaue eche of vs like summes more, then our summes wil be like styll. Also if A. and B. (as in the former example) bee equall, then by adding an equal portion to them both, as to ech of them, the quarter of A. (that is foure) they will be equall still.

_The thirde common sentence._

And if you abate euen portions from things that are equal,
those partes that remain shall be equall also.

This you may perceaue by the last example. For that that was added there, is subtracted heere. and so the one doothe approue the other.

_The fourth common sentence._

If you abate equalle partes from vnequal thinges, the
remainers shall be vnequall.