THE PRACTIKE WORKINGE OFsondry conclusions geometrical.

THE PRACTIKE WORKINGE OFsondry conclusions geometrical.THE FYRST CONCLVSION.To make a threlike triangleonany lyne measurable.Take the iuste lẽgth of the lyne with your cõpasse, and stay the one foot of the compas in one of the endes of that line, turning the other vp or doun at your will, drawyng the arche of a circle against the midle of the line, and doo like wise with the same cõpasse vnaltered, at the other end of the line, and wher these ij. croked lynes doth crosse, frome thence drawe a lyne to ech end of your first line, and there shall appear a threlike triangle drawen on that line.Example.A.B.is the first line, on which I wold make the threlike triangle, therfore I open the compasse as wyde as that line is long, and draw two arch lines that mete inC,then fromC,I draw ij other lines one toA,another toB,and than I haue my purpose.see textsee textsee textTHE.II. CONCLVSIONIf you wil make a twileke or a nouelike triangle on ani certaine line.Consider fyrst the length that yow will haue the other sides to containe, and to that length open your compasse, andthen worke as you did in the threleke triangle, remembryng this, that in a nouelike triangle you must take ij. lengthes besyde the fyrste lyne, and draw an arche lyne with one of thẽ at the one ende, and with the other at the other end, the exãple is as in the other before.THEIII. CONCL.To diuide an angle of right lines into ij. equal partes.see textFirst open your compasse as largely as you can, so that it do not excede the length of the shortest line ytincloseth the angle. Then set one foote of the compasse in the verye point of the angle, and with the other fote draw a compassed arch frõ the one lyne of the angle to the other, that arch shall you deuide in halfe, and thẽ draw a line frõ the ãgle to yemiddle of yearch, and so yeangle is diuided into ij. equall partes.Example.Let the triãgle beA.B.C,thẽ set I one foot of yecõpasse inB,and with the other I draw yearchD.E,which I part into ij. equall parts inF,and thẽ draw a line frõB,toF,& so I haue mine intẽt.THEIIII. CONCL.To deuide any measurable line into ij. equall partes.see textOpen your compasse to the iust lẽgth of yeline. And thẽ set one foote steddely at the one ende of the line, & wtthe other fote draw an arch of a circle against yemidle of the line, both ouer it, and also vnder it, then doo lykewaiseat the other ende of the line. And marke where those arche lines do meet crosse waies, and betwene those ij. pricks draw a line, and it shallcutthe first line in two equall portions.Example.The lyne isA.B.accordyng to which I open the compasse and make .iiij. arche lines, whiche meete inC.andD,then drawe I a lyne fromC,so haue I my purpose.This conlusion serueth for makyng of quadrates and squires, beside many other commodities, howebeit it maye bee don more readylye by this conclusion that foloweth nexte.THE FIFT CONCLVSION.To make a plumme line or any pricke that you will in any right lyne appointed.Open youre compas so that it be not wyder then from the pricke appoynted in the line to the shortest ende of the line, but rather shorter. Then sette the one foote of the compasse in the first pricke appointed, and with the other fote marke ij. other prickes, one of eche syde of that fyrste, afterwarde open your compasse to the wydenes of those ij. new prickes, and draw from them ij. arch lynes, as you did in the fyrst conclusion, for making of a threlyketriãgle. thenif you do mark their crossing, and from it drawe a line to your fyrste pricke, it shall bee a iust plum lyne on that place.see textExample.The lyne isA.B.the prick on whiche I shoulde make the plumme lyne, isC.then open I the compasse as wyde asA.C,and sette one foot inC.and with the other doo I marke outC.A.andC.B,then open I the compasse as wide asA.B,and make ij. arch lines which do crosse inD,and so haue I doone.Howe bee it, it happeneth so sommetymes, that thepricke on whiche you would make the perpendicular or plum line, is so nere the eand of your line, that you can not extende any notable length from it to thone end of the line, and if so be it then that you maie not drawe your line lenger frõ that end, then doth this conclusion require a newe ayde, for the last deuise will not serue. In suche case therfore shall you dooe thus: If your line be of any notable length, deuide it into fiue partes. And if it be not so long that it maie yelde fiue notable partes, then make an other line at will, and parte it into fiue equall portiõs: so that thre of those partes maie be found in your line. Then open your compas as wide as thre of these fiue measures be, and sette the one foote of the compas in the pricke, where you would haue the plumme line to lighte (whiche I call the first pricke,) and with the other foote drawe an arche line righte ouer the pricke, as you can ayme it: then open youre compas as wide as all fiue measures be, and set the one foote in the fourth pricke, and with the other foote draw an other arch line crosse the first, and where thei two do crosse, thense draw a line to the poinct where you woulde haue the perpendicular line to light, and you haue doone.Example.see textsee textThe line isA.B.andA.is the prick, on whiche the perpendicular line must light. Therfore I deuideA.B.into fiue partes equall, then do I open the compas to the widenesse of three partes (that isA.D.) and let one foote staie inA.and with the other I make an arche line inC.Afterwarde I open the compas as wide asA.B.(that is as wide as all fiue partes) and set one foote in the .iiij. pricke, which isE,drawyng an arch line with the other foote inC.also. Then do I draw thence a line vntoA,and so haue I doone. But and if the line be to shorte to be parted into fiue partes, I shall deuide it into iij. partes only, as you see the liueF.G,and then makeD.an other line (as isK.L.) whiche I deuide into .v. suche diuisions, asF.G.containeth .iij, then open I thecompassas wide as .iiij. partes (whiche isK.M.) and so set I one foote of the compas inF,and with the other I drawe an arch lyne towardH,then open I the cõpas as wide asK.L.(that is all .v. partes) and set one foote inG,(that is the iij. pricke) and with the other I draw an arch line towardH.also: and where those .ij. arch lines do crosse (whiche is byH.) thence draw I a line vntoF,and that maketh a very plumbe line toF.G,as my desire was. The maner of workyng of this conclusion, is like to the second conlusion, but the reason of it doth depẽd of the .xlvi. proposiciõ of yefirst boke of Euclide. An other waieyet. setone foote of the compas in the prick, on whiche you would haue the plumbe line to light, and stretche forth thother foote toward the longest end of the line, as wide as you can for the length of the line, and so draw a quarter of a compas or more, then without stirryng of the compas, set one foote of it in the same line, where as the circular line did begin, and extend thother in the circular line, settyng a marke where it doth light, then take half that quantitie more there vnto, and by that prick that endeth the last part, draw a line to the pricke assigned, and it shall be a perpendicular.see textExample.A.B.is the line appointed, to whiche I must make a perpendicular line to light in the pricke assigned, which isA.Therfore doo I set one foote of the compas inA,and extend the other vntoD.makyng a part of a circle,more then a quarter, that isD.E.Then do I set one foote of the compas vnaltered inD,and stretch the other in the circular line, and it doth light inF,this space betweneD.andF.I deuide into halfe in the prickeG,whiche halfe I take with the compas, and set it beyondF.vntoH,and thefore isH.the point, by whiche the perpendicular line must be drawn, so say I that the lineH.A,is a plumbe line toA.B,as the conclusion would.THE.VI. CONCLVSION.To drawe a streight line from any pricke that is not in a line, and to make it perpendicular to an other line.see textOpen your compas as so wide that it may extend somewhat farther, thẽ from the prick to the line, then sette the one foote of the compas in the pricke, and with the other shall you draw a cõpassed line, that shall crosse that other first line in .ij. places.Now if you deuide that arch line into .ij. equall partes, and from the middell pricke therof vnto the prick without the line you drawe a streight line, it shalbe a plumbe line to that firste lyne, accordyng to the conclusion.Example.C.is the appointed pricke, from whiche vnto the lineA.B.I must draw a perpẽdicular. Thefore I open the cõpas so wide, that it may haue one foote inC,and thother to reach ouer the line, and with ytfoote I draw an arch line as you see, betweneA.andB,which arch line I deuide in the middell in the pointD.Then drawe I a line fromC.toD,and it is perpendicular to the lineA.B,accordyng as my desire was.THE.VII. CONCLVSION.To make a plumbe lyneonany porcion of a circle, and that on the vtter or inner bughte.Mark first the prick where yeplũbe line shal lyght: and prick out of ech side of it .ij. other poinctes equally distant from that first pricke. Then set the one foote of the cõpas in one of those side prickes, and the other foote in the other side pricke, and first moue one of the feete and drawe an arche line ouer the middell pricke, then set the compas steddie with the one foote in the other side pricke, and with the other foote drawe an other arche line, that shall cut that first arche, and from the very poincte of their meetyng, drawe a right line vnto the firste pricke, where you do minde that the plumbe line shall lyghte. And so haue you performed thintent of this conclusion.see textExample.The arche of the circle on whiche I would erect a plumbe line, isA.B.C.andB.is the pricke where I would haue the plumbe line to light. Therfore I meate out two equall distaunces on eche side of that prickeB.and they areA.C.Then open I the compas as wide asA.C.and settyng one of the feete inA.with the other I drawe an arche line which goeth byG.Like waies I set one foote of the compas steddily inC.and with the other I drawe an arche line, goyng byG.also. Now consideryng thatG.is the pricke of their meetyng, it shall be also the poinct fro whiche I must drawe the plũbe line. Then draw I a right line fromG.toB.and so haue mine intent. Now asA.B.C.hath a plumbe line erected on hisvtter bought, so may I erect a plumbe line on the inner bught ofD.E.F,doynge with it as I did with the other, that is to saye, fyrste settyng forthe the pricke where the plumbe line shall light, which isE,and then markyng one other on eche syde, as areD.andF.And then proceding as I dyd in the example before.THEVIII. CONCLVSYON.How to deuide the arche of a circle into two equall partes, without measuring the arche.Deuide the corde of that line info ij. equall portions, and then from the middle prycke erecte a plumbe line, and it shal parte that arche in the middle.see textExample.The arch to be diuided ysA.D.C,the corde isA.B.C,this corde is diuided in the middle withB,from which prick if I erect a plum line asB.D,thẽ will it diuide the arch in the middle, that is to say, inD.THEIX. CONCLVSION.To do the same thynge other wise. And for shortenes of worke, if you wyl make a plumbe line without much labour, you may do it with your squyre, so that it be iustly made, for yf you applye the edge of the squyre to the line in which the prick is, and foresee the very corner of the squyre doo touche the pricke. And than frome that corner if you drawe a lyne by the other edge of the squyre, yt will be perpendicular to the former line.see textExample.A.B.is the line, on which I wold make the plumme line, or perpendicular. And therefore I marke the prick, from which the plumbe lyne muste rise, which here isC.Then do I sette one edg of my squyre (that isB.C.) to the lineA.B,so at the corner of the squyre do toucheC.iustly. And fromC.I drawe a line by the other edge of the squire, (which isC.D.) And so haue I made the plumme lineD.C,which I sought for.THEX. CONCLVSION.How to do the same thinge an other way yetsee textIf so be it that you haue an arche of suche greatnes, that your squyre wyll not suffice therto, as the arche of a brydge or of a house or window, then may you do this. Mete vnderneth the arch where yemidle of his cord wyl be, and ther set a mark.Then take a long line with a plummet, and holde the line in suche a place of the arch, that the plummet do hang iustely ouer the middle of the corde, that you didde diuide before, and then the line doth shewe you the middle of the arche.Example.The arch isA.D.B,of which I trye the midle thus. I draw a corde from one syde to the other (as here isA.B,) which I diuide in the middle inC.Thẽ take I a line with a plummet (that isD.E,) and so hold I the line that the plummetE,dooth hange ouerC,Andthen I say thatD.is the middle of the arche. And to thentent that my plummet shall point the more iustely, I doo make it sharpe at the nether ende, and so may I trust this woorke for certaine.THEXI. CONCLVSION.When any line is appointed and without it a pricke, whereby a parallel must be drawen howe you shall doo it.Take the iuste measure beetwene the line and the pricke, accordinge to which you shal open your compasse. Thẽ pitch one foote of your compasse at the one ende of the line, and with the other foote draw a bowe line right ouer the pytche of the compasse, lyke-wise doo at the other ende of the lyne, then draw a line that shall touche the vttermoste edge of bothe those bowe lines, and it will bee a true parallele to the fyrste lyne appointed.Example.see textA.B,is the line vnto which I must draw an other gemow line, which muste passe by the prickC,first I meate with my compasse the smallest distance that is fromC.to the line, and that isC.F,wherfore staying the compasse at that distaunce, I seete the one foote inA,and with the other foot I make a bowe lyne, which isD,thẽ like wise set I the one foote of the compasse inB,and with the other I make the second bow line, which isE.And then draw I a line, so that it toucheth the vttermost edge of bothe these bowe lines, and that lyne passeth by the prickeC,end is a gemowe line toA.B,as my sekyng was.THE.XII. CONCLVSION.To make a triangle of any .iij. lines, so that the lines be suche, that any .ij. of them be longer then the thirde. For this rule is generall, that any two sides of euerie triangle taken together, are longer then the other side that remaineth.If you do remember the first and seconde conclusions, then is there no difficultie in this, for it is in maner the same woorke. First cõsider the .iij. lines that you must take, and set one of thẽ for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.Example.see textI haue .iij.A.B.andC.D.andE.F.of whiche I put.C.D.for my ground line, then with my compas I take the length of.A.B.and set the one foote of my compas inC,and draw an arch line with the other foote. Likewaies I take the lẽgth ofE.F,and set one foote inD,and with the other foote I make an arch line crosse the other arche, and the pricke of their metyng (whiche isG.) shall be the thirde corner of the triangle, for in all suche kyndes of woorkynge to make a tryangle, if you haue one line drawen, there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee, for two of them must needes be at the two eandes of the lyne that is drawen.THEXIII. CONCLVSION.If you haue a line appointed, and a pointe in it limited, howe you maye make on it a righte lined angle, equall to an other right lined angle, all ready assigned.Fyrste draw a line against the corner assigned, and so is it a triangle, then take heede to the line and the pointe in it assigned, and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned, and if it bee longe enoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accordinge to the laste conlusion, makynge a triangle of thre like lynes to that assigned triangle. If it bee not longe inoughe, thenn lengthen it fyrste, and afterwarde doo as I haue sayde beefore.see textExample.Lette the angle appoynted beeA.B.C,and the corner assigned,B.Farthermore let the lymited line beeD.G,and the pricke assignedD.Fyrste therefore by drawinge the lineA.C,I make the triangleA.B.C.see textThen consideringe thatD.G,is longer thanneA.B,you shall cut out a line frõD.towardG,equal toA.B,as for exãpleD.F.Thẽ measure oute the other ij. lines and worke with thẽ according as the conclusion with the fyrste also and the second teacheth yow, and then haue you done.THEXIIII. CONCLVSION.To make a square quadrate of any righte lyne appoincted.First make a plumbe line vnto your line appointed, whiche shall light at one of the endes of it,accordyng tothe fifth conclusion, and let it be of like length as your first line is, then opẽ your compasse to the iuste length of one of them, and sette one foote of the compasse in the ende of the one line, and with the other foote draw an arche line, there as you thinke that the fowerth corner shall be, after that set the one foote of the same compasse vnsturred, in the eande of the other line, and drawe an other arche line crosse the first archeline, and the poincte that they do crosse in, is the pricke of the fourth corner of the square quadrate which you seke for, therfore draw a line from that pricke to the eande of eche line, and you shall therby haue made a square quadrate.see textExample.A.B.is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plũbe line vnto it, whiche shall lighte inA,and that plũb line isA.C,then open I my compasse as wide as the length ofA.B,orA.C,(for they must be bothe equall) and I set the one foote of thend inC,and with the other I make an arche line nigh vntoD,afterward I set the compas again with one foote inB,and with the other foote I make an arche line crosse the first arche line inD,and from the prick of their crossyng I draw .ij. lines, one toB,and an other toC,and so haue I made the square quadrate that I entended.THE.XV. CONCLVSION.To make a likeiãme equall to a triangle appointed, and that in a right lined ãgle limited.First from one of the angles of the triangle, you shall drawe a gemowe line, whiche shall be a parallele to that syde of the triangle, on whiche you will make that likeiamme. Then on one end of the side of the triangle, whiche lieth against the gemowe lyne, you shall draw forth a line vnto the gemow line, so that one angle that commeth of those .ij. lines be like to the angle which is limited vnto you. Then shall you deuide into ij. equall partes that side of the triangle whiche beareth that line, and from the pricke of that deuision, you shall raise an other line parallele to that former line, and continewe it vnto the first gemowe line, and thẽ of those .ij. last gemowe lynes, and the first gemowe line, with the halfe side of the triangle, is made a lykeiamme equall to the triangle appointed, and hath an angle lyke to an angle limited, accordyng to the conclusion.see textExample.B.C.G,is the triangle appoincted vnto, whiche I muste make an equall likeiamme. AndD,is the angle that the likeiamme must haue. Therfore first entendyng to erecte the likeiãme on the one side, that the ground line of the triangle (whiche isB.G.) I do draw a gemow line byC,and make it parallele to the ground lineB.G,and that new gemow line isA.H.Then do I raise a line fromB.vnto the gemowe line, (whiche line isA.B) and make an angle equall toD,that is the appointed angle (accordyng as the .viij. cõclusion teacheth)and that angle isB.A.E.Then to procede, I doo parte in yemiddle the said groũd lineB.G,in the prickF,frõ which prick I drawto the first gemowe line (A.H.) an other line that is parallele toA.B,and that line isE.F.Now saie I that the likeiãmeB.A.E.F,is equall to the triangleB.C.G.And also that it hath one angle (that isB.A.E.)like toD.the angle that was limitted. And so haue I mine intent. The profe of the equalnes of those two figures doeth depend of the .xli. proposition of Euclides first boke, and is the .xxxi. proposition of this second boke of Theoremis, whiche saieth, that whan a tryangle and a likeiamme be made betwene .ij. selfe same gemow lines, and haue their ground line of one length, then is the likeiamme double to the triangle, wherof it foloweth, that if .ij. suche figures so drawen differ in their ground line onely, so that the ground line of the likeiamme be but halfe the ground line of the triangle, then be those .ij. figures equall, as you shall more at large perceiue by the boke of Theoremis, in ye.xxxi. theoreme.THE.XVI. CONCLVSION.To make a likeiamme equall to a triangle appoincted, accordyng to an angle limitted, and on a line also assigned.In the last conclusion the sides of your likeiamme wer left to your libertie, though you had an angle appoincted. Nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted, which must be the side of the likeiãme. Therfore thus shall you procede. Firste accordyng to the laste conclusion, make a likeiamme in the angle appoincted, equall to the triangle that is assigned. Then with your compasse take the length of your line appointed, and set out two lines of the same length in the second gemowe lines, beginnyng at the one side of the likeiamme, and by those two prickes shall you draw an other gemowe line, whiche shall be parallele to two sides of the likeiamme. Afterward shall you draw .ij. lines more for the accomplishement of your worke, which better shall beperceaued by a shorte exaumple, then by a greate numbre of wordes, only without example,thereforeI wyl by example sette forth the whole worke.Example.see textFyrst, according to the last conclusion, I make the likeiammeE.F.C.G,equal to the triangleD,in the appoynted angle whiche isE.Then take I the lengthe of the assigned line (which isA.B,) and with my compas I sette forthe the same lẽgth in the ij. gemow linesN.F.andH.G,setting one foot inE,and the other inN,and againe settyng one foote inC,and the other inH.Afterward I draw a line fromN.toH,whiche is a gemow lyne, to ij. sydes of thelikeiamme. thennedrawe I a line also fromN.vntoC.and extend it vntyll it crosse the lines,E.L.andF.G,which both must be drawen forth longer then the sides of thelikeiamme. andwhere that lyne doeth crosseF.G,there I setteM.Nowe to make an ende, I make an other gemowe line, whiche is parallel toN.F.andH.G,and that gemowe line doth passe by the prickeM,and then haue I done. Now say I thatH.C.K.L,is a likeiamme equall to the triangle appointed, whiche wasD,and is made of a line assigned that isA.B,forH.C,is equall vntoA.B,and so isK.L.The profe of yeequalnes of this likeiam vnto the triãgle, depẽdeth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, whẽ there are more then one made about one bias line, the filsquares of euery of them muste needes be equall.THEXVII. CONCLVSION.To make a likeiamme equal to any right lined figure, and that on an angle appointed.The readiest waye to worke this conclusion, is to tourn that rightlined figure into triangles, and then for euery triangletogetheran equal likeiamme, according vnto the eleuen cõclusion, and then to ioine al those likeiammes into one, if their sides happen to be equal, which thing is euer certain, when al the triangles happẽ iustly betwene one pair of gemowlines. butand if they will not frame so, then after that you haue for the firste triangle made his likeiamme, you shall take the lẽgth of one of his sides, and set that as a line assigned, on whiche you shal make the other likeiams, according to the twelft cõclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so ytyou mai easily ioyne thẽ into one figure.see textExample.If the right lined figure be like vntoA,thẽ may it be turned into triangles that wil stãd betwene ij. parallels anye ways, as youmai sebyC.andD,for ij. sides of both thetriãnglesar parallels. Also if the right lined figure be like vntoE,thẽ wil it be turned into triãgles, liyng betwene two parallels also, as yeother did before, as in the exãple ofF.G.But and if yeright lined figure be like vntoH,and so turned into triãgles as you se inK.L.M,wher it is parted into iij triãgles, thẽ wil not all those triangles lye betwen one pair of parallels or gemow lines, but must haue many, for euery triangle must haue one paire of parallels seuerall, yet it maye happen that when there bee three or fower triangles, ij. of theym maye happen to agre to one pair of parallels, whiche thinge I remit to euery honest witte to serche, for the manner of their draught wil declare, how many paires of parallels they shall neede, of which varietee bicause the examples ar infinite, I haue set forth these few, that by them you may coniecture duly of all other like.see textFurther explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then diligẽtly behold how these sundry figures be turned into triãgles. In the fyrst you se I haue made v. triangles, and fourparalleles. inthe seconde vij. triangles and foureparalleles. inthe thirde thre triãgles, and fiue parallels, in the iiij. you se fiue triãgles & fourparallels. inthe fift, iiij. triãgles and .iiij. parallels, & in yesixt ther ar fiue triãgles & iiij. paralels. Howbeit a mã maye at liberty alter them into diuers formes of triãgles & therefore Ileue it to the discretion of the woorkmaister, to do in al suche cases as he shal thinke best, for by these examples (if they bee well marked) may all other like conclusions be wrought.THEXVIII. CONCLVSION.To parte a line assigned after suche a sorte, that the square that is made of the whole line and one of his parts, shal be equal to the squar that cometh of the other parte alone.First deuide your lyne into ij. equal parts, and of the length of one part make a perpendicular to light at one end of your lineassigned. thenadde a bias line, and make thereof a triangle, this done if you take from this bias line the halfe lengthe of your line appointed, which is the iuste length of your perpendicular, that part of the bias line whiche dothe remayne, is the greater portion of the deuision that you seke for, therefore if you cut your line according to the lengthe of it, then will the square of that greaterportionbe equall to the square that is made of the whole line and his lesser portion. And contrary wise, the square of the whole line and his lesser parte, wyll be equall to the square of the greater parte.see textExample.A.B,is the lyne assigned.E.is the middle pricke ofA.B, B.C.is the plumb line or perpendicular, made of the halfe ofA.B,equall toA.E,otherB.E,the byas line isC.A,from whiche I cut a peece, that isC.D,equall toC.B,and accordyng to the lengthelothepeece that remaineth (whiche isD.A,) I doo deuide the lineA.B,at whiche diuision I setF.Now say I, that this lineA.B,(wchwas assigned vnto me) is so diuided in this pointF,ytyesquare of yehole lineA.B,& of the one portiõ (ytisF.B,thelesser part) is equall to the square of the other parte, whiche isF.A,and is the greater part of the first line. The profe of this equalitie shall you learne by the .xl. Theoreme.There are two ways to make this Example work:—transpose E and F in the illustration, and change one occurrence of E to F in the text,or:—keep the illustration as printed, and transpose all other occurrences of E and F in the text.THE.XIX. CONCLVSION.To make a square quadrate equall to any right lined figure appoincted.First make a likeiamme equall to that right lined figure, with a right angle, accordyng to the .xi. conclusion, then consider the likeiamme, whether it haue all his sides equall, or not: for yf they be all equall, then haue you doone yourconclusion. butand if the sides be not all equall, then shall you make one right line iuste as long as two of those vnequall sides, that line shall you deuide in the middle, and on that pricke drawe half a circle, then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme, and from that pointe of diuision shall you erecte a perpendicular, which shall touche the edge of the circle. And that perpendicular shall be the iuste side of the square quadrate, equall both to the lykeiamme, and also to the right lined figure appointed, as the conclusion willed.Example.see textK,is the right lined figure appointed, andB.C.D.E,is the likeiãme, with right angles equall vntoK,but because that this likeiamme is not a square quadrate, I must turne it into such one after this sort, I shall make one right line, as long as .ij. vnequall sides of the likeiãme, that line here isF.G,whiche is equall toB.C,andC.E.Then part I that line in the middle in theprickeM,and on that pricke I make halfe a circle, accordyng to the length of the diameterF.G.Afterward I cut awaie a peece fromF.G,equall toC.E,markyng that point withH.And on that pricke I erecte a perpendicularH.K,whiche is the iust side to the square quadrate that I seke for, therfore accordyng to the doctrine of the .x. conclusion, of the lyne I doe make a square quadrate, and so haue I attained the practise of this conclusion.THE.XX. CONCLVSION.When any .ij. square quadrates are set forth, how you maie make one equall to them bothe.First drawe a right line equall to the side of one of the quadrates: and on the ende of it make a perpendicular, equall in length to the side of the other quadrate, then drawe a byas line betwene those .ij. other lines, makyng thereof a right angeled triangle. And that byas lyne wyll make a square quadrate, equall to the other .ij. quadrates appointed.see textExample.A.B.andC.D,are the two square quadrates appointed, vnto which I must make one equall square quadrate. First therfore I dooe make a righte lineE.F,equall to one of the sides of the square quadrateA.B.And on the one end of it I make a plumbe lineE.G,equall to the side of the other quadrateD.C.Then drawe I a byas lineG.F,which beyng made the side of a quadrate(accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrateH.is muche iust as the other two. I meaneA.B.andD.C.THE.XXI. CONCLVSION.When any two quadrates be set forth, howe to make a squire about the one quadrate, whiche shall be equall to the other quadrate.Determine with your selfe about whiche quadrate you wil make the squire, and drawe one side of that quadrate forth in lengte, accordyng to the measure of the side of the other quadrate, whiche line you maie call the grounde line, and then haue you a right angle made on this line by an other side of the same quadrate: Therfore turne that into a right cornered triangle, accordyng to the worke in the laste conclusion, by makyng of a byas line, and that byas lyne will performe the worke of your desire. For if you take the length of that byas line with your compasse, and then set one foote of the compas in the farthest angle of the first quadrate (whiche is the one ende of the groundline) and extend the other foote on the same line, accordyng to the measure of the byas line, and of that line make a quadrate, enclosyng yefirst quadrate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.see textExample.The first square quadrate isA.B.C.D,and the seconde isE.Now would I make a squire about the quadrateA.B.C.D,whiche shall bee equall vnto the quadrateE.Therforefirst I draw the lineA.D,more in length, accordyng to the measure of the side ofE,as you see, fromD.vntoF,and so the hole line of bothe these seuerall sides isA.F,thẽ make I a byas line fromC,toF,whiche byas line is the measure of thiswoorke. whereforeI open my compas accordyng to the length of that byas lineC.F,and set the one compas foote inA,and extend thother foote of the compas towardF,makyng this prickeG,from whiche I erect a plumbelineG.H,and so make out the square quadrateA.G.H.K,whose sides are equall eche of them toA.G.And this square doth contain the first quadrateA.B.C.D,and also a squireG.H.K,whiche is equall to the second quadrateE,for as the last conclusion declareth, the quadrateA.G.H.K,is equall to bothe the other quadrates proposed, that isA.B.C.D,andE.Then muste the squireG.H.K,needes be equall toE,consideryng that all the rest of that great quadrate is nothyng els but the quadrate self,A.B.C.D,and so haue I thintent of this conclusion.THE.XXII.CONCLVSION.To find out the cẽtre of any circle assigned.Draw a corde or stryngline crosse the circle, then deuide into .ij. equall partes, both that corde, and also the bowe line, or arche line, that serueth to that corde, and from the prickes of those diuisions, if you drawe an other line crosse the circle, it must nedes passe by the centre. Therfore deuide that line in the middle, and that middle pricke is the centre of the circle proposed.Example.see textLet the circle beA.B.C.D,whose centre I shall seke. First therfore I draw a corde crosse the circle, that isA.C.Then do I deuide that corde in the middle, inE,and likewaies also do I deuide his arche lineA.B.C,in the middle, in the pointeB.Afterward I drawe a line fromB.toE,and so crosse thecircle, whiche line isB.D,in which line is the centre that I seeke for. Therefore if I parte that lineB.D,in the middle in to two equall portions, that middle pricke (which here isF) is the verye centre of the sayde circle that I seke. This conclusion may other waies be wrought, as the moste part of conclusions haue sondry formes of practise, and that is, by makinge thre prickes in the circũference of the circle, at liberty where you wyll, and then findinge the centre to those threpricks, Whichworke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.THEXXIII. CONCLVSION.To find the commen centre belongyng to anye three prickes 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 circumference of al circles is compassed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. prickes in an exact right line can not bee brought into the circumference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their cõmon centre. Opẽ your compas so wide, that it be somewhat more then thehalfe distance of two of those prickes. Then sette the one foote of the compas in the one pricke, and with the other foot draw an arche lyne toward the otherpricke, Thenagaine putte the foot of your compas in the second pricke, 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 pricks, so do with the middle pricke, 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.see textExampleThe iij. prickes I haue set to beA.B,andC,whiche I wold bring into the edg of one common circle, by finding a centre cõmen to them all, fyrst therefore I open my cõpas, so that thei occupye more then yehalfe distance betwene ij. pricks (as areA.B.) and so settinge one foote inA.and extendinge the other towardB,I make the arche lineD.E.Likewise settĩg one foot inB,and turninge the other towardA,I draw an other arche line that crosseth the first inD.andE.Then fromD.toE,I draw a right lyneD.H.After this I open my cõpasse to a new distance, and make ij. arche lines betweneB.andC,whiche crosse one the other inF.andG,by whiche two pointes I draw an other line, that isF.H.And bycause that the lyneD.H.and the lyneF.H.doo meete inH,I saye thatH.is the centre that serueth to those iij. prickes. Now therfore if you set one foot of your compas inH,and extend the other to any of the iij. pricks, you may draw a circle wchshal enclose those iij. pricks in the edg of his circũferẽce & thus haue you attained yevse of this cõclusiõ.THEXXIIII. CONCLVSION.To drawe a touche line onto a circle, from any poincte assigned.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 frõ 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 atoucheline, drawen from the point assigned, according to the meaning of this conclusion.see textsee textExample.Let the circle be calledB.C.D,and his cẽtreE,and yeprick assignedA,opẽ your cõpas now of such widenes, ytthe one foote may be set inE,wchis yecẽtre of yecircle, & yeother inA,wchis yepointe assigned, & so make an other greter circle (as here isA.F.G) thẽ draw a line fromA.vntoE,and wher that line doth cross yeinner circle (wchheere is in the prickB.) there erect a plũb line vnto the line.A.E.and let that plumb line touch the vtter circle, as it doth here in the pointF,so shallB.F.bee that plumbe lyne. Then fromF.vntoE.drawe an other line whiche shal beF.E,and it will cutte the inner circle, as it doth here in the pointC,from which pointeC.if you erect a plumb line vntoA,then is that lineA.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 isE,the pricke assigned isA.and the rulerC.D.by which the touch line is drawen, and that isA.B,and as this way is light to doo, so is it certaine inoughe for any kinde of workinge.THEXXV. CONCLVSION.When you haue any peece of the circumference of a circle assigned, howe you may make oute the whole circle agreyngetherevnto.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.see textExample.A peece of an olde pillar was found, like in forme to thys figureA.D.B.Now to knowe howe muche the cõpasse of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make inatable the like draught of ytcircũference by the self patrõ, vsing it as it wer a croked ruler.Then make .iij. prickes in that arche line, as I haue made,C. D.andE.And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that cẽtre is hereF,nowe settyng one foote of your compas inF,and the other inC. D,other inE,and so makyng a compasse, you haue youre whole intent.THEXXVI. 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.see textThe arche of the circle isA.B.C,vnto whiche I must sekea centre, therfore firste I do deuide it into .ij. partes, the one of them isA.B,and the other isB.C.Then doe I cut euery arche in the middle, so isE.the middle ofA.B,andG.is the middle ofB.C.Likewaies, I take the middle of their cordes, whiche I mark withF.andH,settyngF.byE,andH.byG.Then drawe I a line fromE.toF,and fromG.toH,and they do crosse inD,wherefore saie I, thatD.is the centre, that I seke for.THEXXVII. 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 thecircle, Andthat 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.A. is a circlein a triangle.C. a quadratein a circle.see caption and textsee caption and textsee caption and textB. a trianglein a circle.D. a circle ina quadrate.In these .ij. last figuresE.andF,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 coũted 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.see textThe triangle isA.B.C,whose sides I do part into .ij. equall partes, eche by it selfe in these pointesD.E.F,puttyngF.betweneA.B,andD.betweneB.C,andE.betweneA.C.Then draw I a line fromC.toF,and an other fromA.toD,and the third fromB.toE.Andwhere all those lines do mete (that is to saieM.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.see textAnd so drawe lines frõ 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 lẽgth of the perpendicular, and so drawe your circle.Example.The triangle isA.B.C,whose corners I haue diuided in the middle withD.E.F,and haue drawen the lines of diuisionA.D, B.E,andC.F,which crosse inG,therfore shallG.be the common centre. Then make I one perpẽdicular fromG.vnto the sideB.C,and thatisG.H.Now sette I one fote of the compas inG,and extend the other foote vntoH.and so drawe a compas, whiche wyll iustly answere to that triãgle according to the meaning of the conclusion.THEXXVIII. CONCLVSION.To drawe a circle about any triãgle 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.see textA.B.C.is the triangle, whose two sidesA.C.andB.C.are diuided into two equall partes inD.andE,settyngD.betweneB.andC,andE.betweneA.andC.And from eche of those two pointes is ther erected a perpendicular (as you seD.F,andE.F.) which mete, and crosse inF,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 threecorners 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.see textExample.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 iijprickes. that centre asyou se isD,for inD.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 triãgle, and consequentlye the triangle itselfe, as the conclusion dydde intende.THEXXIX. 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, thenmustI 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 wtthe other halfe of the touche line, equall to an other corner of the triangle appointed, and then betwen those two cornerswill 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.see textA.B.C,is the triangle appointed, andF.G.H.is the circle, in which I muste make an other triangle, with lyke angles to the angles ofA.B.C.the triangle appointed. Therefore fyrst I make the touch lyneD.F.E.And then make I an angle inF,equall toA,whiche is one of the angles of the triangle. And the lyne that maketh that angle with the touche line, isF.H,whiche I drawe in lengthe vntill it touche the edge of the circle. Then againe in the same pointF,I make an other corner equall to the angleC.and the line that maketh that corner with the touche line, isF.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 ytone pointF,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 equallto 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 cõmon sentence saith. Thẽ do I draw a line fromeG.toH,and that line maketh a triangleF.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.THEXXX. 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, thẽ 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 drawẽ about a circle apointed, as the cõclusiõ did wil.Example.A.B.C,is the triangle assigned, andG.H.K,is the circle appointed, about which I muste make a triangle hauing equall angles to the angles of that triangleA.B.C.Fyrst therefore I drawA.C.(which is one of the sides of the triangle) in length that there may appeare two vtter angles in that triangle, as you seB.A.D,andB.C.E.see textThendrawe I in the circle appointed a semidiameter, which is hereH.F,forF.is the cẽtre of the circleG.H.K.Then make I on that centre an angle equall to the vtter angleB.A.D,and that angle isH.F.K.Like waies on the same cẽtre by drawyng an other semidiameter, I make an other angleH.F.G,equall to the second vtter angle of the triangle, whiche isB.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 trianagleL.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 cornerM.is equall toC.LikewaiesL.toA,andN.toB,whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.THEXXXI. 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 handeledaccording 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 wyllsheweyou 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,anyeof 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 seuẽ and twẽty coclusiõ, 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 triãgle be equal to yeãgle assigned as youre selfe mai easily gesse.see textExample.First for example of a sharpe ãgle letA.stãd &B.Cshal be yelyne assigned. Thẽ do I make a triangle, by addingB.C,as a thirde side to those other ij. which doo include the ãgle assigned, and that triãgle isD.E.F,so ytE.F.is the line appointed, andD.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, fromE.a long byD.vntoF,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, andH.K.is the lyne appointed, andL.M.N.the portion of the circle aunsweryng thereto. In the third exaumple,O.is the blunte corner assigned,P.Q.is the line, andR.S.T.is the portion of the circle, that containeth that blũt corner, and is drawen onR.T.the line appointed.THEXXXII. 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.see textExample.A.is the angle appointed, andD.E.F.is the circle assigned, frõ which I must cut away a portiõ that doth contain an angleequall to this angleA.Therfore first I do draw a touche line to the circle assigned, and that touch line isB.C,the very pricke of the touche isD,from whicheD.I drawe a lyneD.E,so that the angle made of those two lines be equall to the angle appointed. Then say I, that the arch of the circleD.F.E,is the arche that I seke after. For if I doo deuide that arche in the middle (as here is done inF.) and so draw thence two lines, one toD,and the other toE,then will the angleF,be equall to the angle assigned.THEXXXIII. 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.see textExample.A.B.C.D.is the circle assigned, andA.C.andB.D.are the two diameters which crosse in the centreE,and make .iiij. right corners. Then do I make fowre other lines, that isA.B, B.C, C.D,andD.A,which do ioyne together the fowre endes of the ij. diameters. And so is the squarequadrate made in the circle assigned, as the conclusion willeth.THEXXXIIII. 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.see textA.B.C.is the circle assigned in which first I draw two diameters, in crosse waies, making iiij. righte angles, and those ij. diameters areA.C.andB.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 inA,then with the other foote I doo make twoo arche lines, one inE,and an other inF.Then sette I the one foote of the compasse inB,and drawe twoo arche linesF.andG.Like wise setting the compasse foote inC,I drawe twoo other arche lines,G.andH,and onD.I make twoo other,H.andE.Then frome the crossinges of those eighte arche lines I drawe iiij. straighte lynes, that is to saye,E.F,andF.G,alsoG.H,andH.E,whiche iiij. straighte lynes do make the square quadrate that I should draw about the circle assigned.THEXXXV. 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.see textExample.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. cõtrary prickes, as you seE.G,andF.H,whiche mete inK,and therfore shalK,be the centre of the circle. Then do I set one foote of the compas inK.and opẽ the other as wide asK.E,and so draw a circle, which is madeaccordingeto the conclusion.THEXXXVI. 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. Thẽ 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.see textExample.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 seA.C.&B.D.And where they ij. do crosse (that is to say inE.) 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 exãple toA,and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.THEXXXVII. 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 thoselines, Thendrawe the thyrde lyne to make a triangle with those other twoo, and you haue doone according to the conclusion, and haue made a twelike triãgle,whose ij. corners about the grounde line, are eche of theym double to the other corner.At no point in this or the accompanying book does the author show how to divide a circle into five.see textExample.A.B.C.is the circle, whiche I haue deuided into fiue equal portions. And from one of the prickes (which isA,) I haue drawẽ ij. lines,A.B.andB.C,whiche are drawen to the third and iiij. prickes. Then draw I the third lineC.B,which is the grounde line, and maketh the triangle, that I would haue, for the ãgleC.is double to the angleA,and so is the angleB.also.THEXXXVIII.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.see textYow se here this circleA.B.C.D.E.deuided into fiue equall portions. And from eche pricke ij.lines drawento the other ij. nexte prickes, so fromA.are drawen ij. lines, one toB,and the other toE,and so fromC.one toB.and an othertoD,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.THEXXXIX. 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.see textExample.A.B.C.D.E.is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is drawẽ a semidiameter, as you see. Then doo I make a touche line in the prickeB,whiche isF.G,making ij. right angleswith the semidiameterB,and lyke waies onC.is madeG.H,onD.standethH.K,and onE,is setK.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.THEXL. 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.see textExample.The cinkeangle assigned isA.B.C.D.E,in whiche I mustemake a circle, wherefore I draw a right line from the one angle (as fromB,) to the middle of the contrary side (whiche isE.D,) and that middle pricke isF.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 isB.C.) and that pricke isG.Nowe because that these two lines do crosse inH,I saie thatH.is the centre of the circle, whiche I would make. Therfore I set one foote of the compasse inH,and extend the other foote vntoG,orF.(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 cõclusion meaneth.THEXLI. CONCLVSIONTo 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.see textA.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 frõE.toG,and the other frõC.toF,and because thei domeete inH,I saye thatH.is the centre of the circle that I woulde haue, wherfore I sette one foote of the compasse inH.and extende the other to one corner (whiche happeneth fyrste, for all are like distaunte fromH.) 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.THEXLII. 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 thecircũference 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 shalperceauethat 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.see textThe circle isB.C.D.E.F.G,whose centre I finde to beeA.Therefore I sette one foote of the compas inA,and do extẽd the other foote toB,thereby takinge the semidiameter. Then sette I one foote of the compas vnremoued inB,and marke with the other foote on eche sideC.andG.Then fromC.I markeD,and frõD,E: fromE.marke IF.And then haue I but one space iuste vntoG.and so haue I made a iuste siseangle of equall sides and equall angles, in a circle appointed.THEXLIII. CONCLVSION.To make a siseangle of equall sides, and equall angles about any circle assigned.THEXLIIII. CONCLVSION.To make a circle in any siseangle appointed, of equall sides and equal angles.THEXLV. 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.THEXLVI. 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 shallhaue, 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, andthen eche of them into three partes againe: Or elsfirst deuide it into three partes, and then echof thẽ into fiue other partes, as youlist, and canne most readilye.Then draw lines betweneeuery two prickesthat benighesttogither, andther wil appear rightly drawẽ the figure, of fiftene sides, andangles equall. And so do with any other figureof what numbre of sides so euer it bee.FINIS.

THE PRACTIKE WORKINGE OFsondry conclusions geometrical.THE FYRST CONCLVSION.To make a threlike triangleonany lyne measurable.Take the iuste lẽgth of the lyne with your cõpasse, and stay the one foot of the compas in one of the endes of that line, turning the other vp or doun at your will, drawyng the arche of a circle against the midle of the line, and doo like wise with the same cõpasse vnaltered, at the other end of the line, and wher these ij. croked lynes doth crosse, frome thence drawe a lyne to ech end of your first line, and there shall appear a threlike triangle drawen on that line.Example.A.B.is the first line, on which I wold make the threlike triangle, therfore I open the compasse as wyde as that line is long, and draw two arch lines that mete inC,then fromC,I draw ij other lines one toA,another toB,and than I haue my purpose.see textsee textsee textTHE.II. CONCLVSIONIf you wil make a twileke or a nouelike triangle on ani certaine line.Consider fyrst the length that yow will haue the other sides to containe, and to that length open your compasse, andthen worke as you did in the threleke triangle, remembryng this, that in a nouelike triangle you must take ij. lengthes besyde the fyrste lyne, and draw an arche lyne with one of thẽ at the one ende, and with the other at the other end, the exãple is as in the other before.THEIII. CONCL.To diuide an angle of right lines into ij. equal partes.see textFirst open your compasse as largely as you can, so that it do not excede the length of the shortest line ytincloseth the angle. Then set one foote of the compasse in the verye point of the angle, and with the other fote draw a compassed arch frõ the one lyne of the angle to the other, that arch shall you deuide in halfe, and thẽ draw a line frõ the ãgle to yemiddle of yearch, and so yeangle is diuided into ij. equall partes.Example.Let the triãgle beA.B.C,thẽ set I one foot of yecõpasse inB,and with the other I draw yearchD.E,which I part into ij. equall parts inF,and thẽ draw a line frõB,toF,& so I haue mine intẽt.THEIIII. CONCL.To deuide any measurable line into ij. equall partes.see textOpen your compasse to the iust lẽgth of yeline. And thẽ set one foote steddely at the one ende of the line, & wtthe other fote draw an arch of a circle against yemidle of the line, both ouer it, and also vnder it, then doo lykewaiseat the other ende of the line. And marke where those arche lines do meet crosse waies, and betwene those ij. pricks draw a line, and it shallcutthe first line in two equall portions.Example.The lyne isA.B.accordyng to which I open the compasse and make .iiij. arche lines, whiche meete inC.andD,then drawe I a lyne fromC,so haue I my purpose.This conlusion serueth for makyng of quadrates and squires, beside many other commodities, howebeit it maye bee don more readylye by this conclusion that foloweth nexte.THE FIFT CONCLVSION.To make a plumme line or any pricke that you will in any right lyne appointed.Open youre compas so that it be not wyder then from the pricke appoynted in the line to the shortest ende of the line, but rather shorter. Then sette the one foote of the compasse in the first pricke appointed, and with the other fote marke ij. other prickes, one of eche syde of that fyrste, afterwarde open your compasse to the wydenes of those ij. new prickes, and draw from them ij. arch lynes, as you did in the fyrst conclusion, for making of a threlyketriãgle. thenif you do mark their crossing, and from it drawe a line to your fyrste pricke, it shall bee a iust plum lyne on that place.see textExample.The lyne isA.B.the prick on whiche I shoulde make the plumme lyne, isC.then open I the compasse as wyde asA.C,and sette one foot inC.and with the other doo I marke outC.A.andC.B,then open I the compasse as wide asA.B,and make ij. arch lines which do crosse inD,and so haue I doone.Howe bee it, it happeneth so sommetymes, that thepricke on whiche you would make the perpendicular or plum line, is so nere the eand of your line, that you can not extende any notable length from it to thone end of the line, and if so be it then that you maie not drawe your line lenger frõ that end, then doth this conclusion require a newe ayde, for the last deuise will not serue. In suche case therfore shall you dooe thus: If your line be of any notable length, deuide it into fiue partes. And if it be not so long that it maie yelde fiue notable partes, then make an other line at will, and parte it into fiue equall portiõs: so that thre of those partes maie be found in your line. Then open your compas as wide as thre of these fiue measures be, and sette the one foote of the compas in the pricke, where you would haue the plumme line to lighte (whiche I call the first pricke,) and with the other foote drawe an arche line righte ouer the pricke, as you can ayme it: then open youre compas as wide as all fiue measures be, and set the one foote in the fourth pricke, and with the other foote draw an other arch line crosse the first, and where thei two do crosse, thense draw a line to the poinct where you woulde haue the perpendicular line to light, and you haue doone.Example.see textsee textThe line isA.B.andA.is the prick, on whiche the perpendicular line must light. Therfore I deuideA.B.into fiue partes equall, then do I open the compas to the widenesse of three partes (that isA.D.) and let one foote staie inA.and with the other I make an arche line inC.Afterwarde I open the compas as wide asA.B.(that is as wide as all fiue partes) and set one foote in the .iiij. pricke, which isE,drawyng an arch line with the other foote inC.also. Then do I draw thence a line vntoA,and so haue I doone. But and if the line be to shorte to be parted into fiue partes, I shall deuide it into iij. partes only, as you see the liueF.G,and then makeD.an other line (as isK.L.) whiche I deuide into .v. suche diuisions, asF.G.containeth .iij, then open I thecompassas wide as .iiij. partes (whiche isK.M.) and so set I one foote of the compas inF,and with the other I drawe an arch lyne towardH,then open I the cõpas as wide asK.L.(that is all .v. partes) and set one foote inG,(that is the iij. pricke) and with the other I draw an arch line towardH.also: and where those .ij. arch lines do crosse (whiche is byH.) thence draw I a line vntoF,and that maketh a very plumbe line toF.G,as my desire was. The maner of workyng of this conclusion, is like to the second conlusion, but the reason of it doth depẽd of the .xlvi. proposiciõ of yefirst boke of Euclide. An other waieyet. setone foote of the compas in the prick, on whiche you would haue the plumbe line to light, and stretche forth thother foote toward the longest end of the line, as wide as you can for the length of the line, and so draw a quarter of a compas or more, then without stirryng of the compas, set one foote of it in the same line, where as the circular line did begin, and extend thother in the circular line, settyng a marke where it doth light, then take half that quantitie more there vnto, and by that prick that endeth the last part, draw a line to the pricke assigned, and it shall be a perpendicular.see textExample.A.B.is the line appointed, to whiche I must make a perpendicular line to light in the pricke assigned, which isA.Therfore doo I set one foote of the compas inA,and extend the other vntoD.makyng a part of a circle,more then a quarter, that isD.E.Then do I set one foote of the compas vnaltered inD,and stretch the other in the circular line, and it doth light inF,this space betweneD.andF.I deuide into halfe in the prickeG,whiche halfe I take with the compas, and set it beyondF.vntoH,and thefore isH.the point, by whiche the perpendicular line must be drawn, so say I that the lineH.A,is a plumbe line toA.B,as the conclusion would.THE.VI. CONCLVSION.To drawe a streight line from any pricke that is not in a line, and to make it perpendicular to an other line.see textOpen your compas as so wide that it may extend somewhat farther, thẽ from the prick to the line, then sette the one foote of the compas in the pricke, and with the other shall you draw a cõpassed line, that shall crosse that other first line in .ij. places.Now if you deuide that arch line into .ij. equall partes, and from the middell pricke therof vnto the prick without the line you drawe a streight line, it shalbe a plumbe line to that firste lyne, accordyng to the conclusion.Example.C.is the appointed pricke, from whiche vnto the lineA.B.I must draw a perpẽdicular. Thefore I open the cõpas so wide, that it may haue one foote inC,and thother to reach ouer the line, and with ytfoote I draw an arch line as you see, betweneA.andB,which arch line I deuide in the middell in the pointD.Then drawe I a line fromC.toD,and it is perpendicular to the lineA.B,accordyng as my desire was.THE.VII. CONCLVSION.To make a plumbe lyneonany porcion of a circle, and that on the vtter or inner bughte.Mark first the prick where yeplũbe line shal lyght: and prick out of ech side of it .ij. other poinctes equally distant from that first pricke. Then set the one foote of the cõpas in one of those side prickes, and the other foote in the other side pricke, and first moue one of the feete and drawe an arche line ouer the middell pricke, then set the compas steddie with the one foote in the other side pricke, and with the other foote drawe an other arche line, that shall cut that first arche, and from the very poincte of their meetyng, drawe a right line vnto the firste pricke, where you do minde that the plumbe line shall lyghte. And so haue you performed thintent of this conclusion.see textExample.The arche of the circle on whiche I would erect a plumbe line, isA.B.C.andB.is the pricke where I would haue the plumbe line to light. Therfore I meate out two equall distaunces on eche side of that prickeB.and they areA.C.Then open I the compas as wide asA.C.and settyng one of the feete inA.with the other I drawe an arche line which goeth byG.Like waies I set one foote of the compas steddily inC.and with the other I drawe an arche line, goyng byG.also. Now consideryng thatG.is the pricke of their meetyng, it shall be also the poinct fro whiche I must drawe the plũbe line. Then draw I a right line fromG.toB.and so haue mine intent. Now asA.B.C.hath a plumbe line erected on hisvtter bought, so may I erect a plumbe line on the inner bught ofD.E.F,doynge with it as I did with the other, that is to saye, fyrste settyng forthe the pricke where the plumbe line shall light, which isE,and then markyng one other on eche syde, as areD.andF.And then proceding as I dyd in the example before.THEVIII. CONCLVSYON.How to deuide the arche of a circle into two equall partes, without measuring the arche.Deuide the corde of that line info ij. equall portions, and then from the middle prycke erecte a plumbe line, and it shal parte that arche in the middle.see textExample.The arch to be diuided ysA.D.C,the corde isA.B.C,this corde is diuided in the middle withB,from which prick if I erect a plum line asB.D,thẽ will it diuide the arch in the middle, that is to say, inD.THEIX. CONCLVSION.To do the same thynge other wise. And for shortenes of worke, if you wyl make a plumbe line without much labour, you may do it with your squyre, so that it be iustly made, for yf you applye the edge of the squyre to the line in which the prick is, and foresee the very corner of the squyre doo touche the pricke. And than frome that corner if you drawe a lyne by the other edge of the squyre, yt will be perpendicular to the former line.see textExample.A.B.is the line, on which I wold make the plumme line, or perpendicular. And therefore I marke the prick, from which the plumbe lyne muste rise, which here isC.Then do I sette one edg of my squyre (that isB.C.) to the lineA.B,so at the corner of the squyre do toucheC.iustly. And fromC.I drawe a line by the other edge of the squire, (which isC.D.) And so haue I made the plumme lineD.C,which I sought for.THEX. CONCLVSION.How to do the same thinge an other way yetsee textIf so be it that you haue an arche of suche greatnes, that your squyre wyll not suffice therto, as the arche of a brydge or of a house or window, then may you do this. Mete vnderneth the arch where yemidle of his cord wyl be, and ther set a mark.Then take a long line with a plummet, and holde the line in suche a place of the arch, that the plummet do hang iustely ouer the middle of the corde, that you didde diuide before, and then the line doth shewe you the middle of the arche.Example.The arch isA.D.B,of which I trye the midle thus. I draw a corde from one syde to the other (as here isA.B,) which I diuide in the middle inC.Thẽ take I a line with a plummet (that isD.E,) and so hold I the line that the plummetE,dooth hange ouerC,Andthen I say thatD.is the middle of the arche. And to thentent that my plummet shall point the more iustely, I doo make it sharpe at the nether ende, and so may I trust this woorke for certaine.THEXI. CONCLVSION.When any line is appointed and without it a pricke, whereby a parallel must be drawen howe you shall doo it.Take the iuste measure beetwene the line and the pricke, accordinge to which you shal open your compasse. Thẽ pitch one foote of your compasse at the one ende of the line, and with the other foote draw a bowe line right ouer the pytche of the compasse, lyke-wise doo at the other ende of the lyne, then draw a line that shall touche the vttermoste edge of bothe those bowe lines, and it will bee a true parallele to the fyrste lyne appointed.Example.see textA.B,is the line vnto which I must draw an other gemow line, which muste passe by the prickC,first I meate with my compasse the smallest distance that is fromC.to the line, and that isC.F,wherfore staying the compasse at that distaunce, I seete the one foote inA,and with the other foot I make a bowe lyne, which isD,thẽ like wise set I the one foote of the compasse inB,and with the other I make the second bow line, which isE.And then draw I a line, so that it toucheth the vttermost edge of bothe these bowe lines, and that lyne passeth by the prickeC,end is a gemowe line toA.B,as my sekyng was.THE.XII. CONCLVSION.To make a triangle of any .iij. lines, so that the lines be suche, that any .ij. of them be longer then the thirde. For this rule is generall, that any two sides of euerie triangle taken together, are longer then the other side that remaineth.If you do remember the first and seconde conclusions, then is there no difficultie in this, for it is in maner the same woorke. First cõsider the .iij. lines that you must take, and set one of thẽ for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.Example.see textI haue .iij.A.B.andC.D.andE.F.of whiche I put.C.D.for my ground line, then with my compas I take the length of.A.B.and set the one foote of my compas inC,and draw an arch line with the other foote. Likewaies I take the lẽgth ofE.F,and set one foote inD,and with the other foote I make an arch line crosse the other arche, and the pricke of their metyng (whiche isG.) shall be the thirde corner of the triangle, for in all suche kyndes of woorkynge to make a tryangle, if you haue one line drawen, there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee, for two of them must needes be at the two eandes of the lyne that is drawen.THEXIII. CONCLVSION.If you haue a line appointed, and a pointe in it limited, howe you maye make on it a righte lined angle, equall to an other right lined angle, all ready assigned.Fyrste draw a line against the corner assigned, and so is it a triangle, then take heede to the line and the pointe in it assigned, and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned, and if it bee longe enoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accordinge to the laste conlusion, makynge a triangle of thre like lynes to that assigned triangle. If it bee not longe inoughe, thenn lengthen it fyrste, and afterwarde doo as I haue sayde beefore.see textExample.Lette the angle appoynted beeA.B.C,and the corner assigned,B.Farthermore let the lymited line beeD.G,and the pricke assignedD.Fyrste therefore by drawinge the lineA.C,I make the triangleA.B.C.see textThen consideringe thatD.G,is longer thanneA.B,you shall cut out a line frõD.towardG,equal toA.B,as for exãpleD.F.Thẽ measure oute the other ij. lines and worke with thẽ according as the conclusion with the fyrste also and the second teacheth yow, and then haue you done.THEXIIII. CONCLVSION.To make a square quadrate of any righte lyne appoincted.First make a plumbe line vnto your line appointed, whiche shall light at one of the endes of it,accordyng tothe fifth conclusion, and let it be of like length as your first line is, then opẽ your compasse to the iuste length of one of them, and sette one foote of the compasse in the ende of the one line, and with the other foote draw an arche line, there as you thinke that the fowerth corner shall be, after that set the one foote of the same compasse vnsturred, in the eande of the other line, and drawe an other arche line crosse the first archeline, and the poincte that they do crosse in, is the pricke of the fourth corner of the square quadrate which you seke for, therfore draw a line from that pricke to the eande of eche line, and you shall therby haue made a square quadrate.see textExample.A.B.is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plũbe line vnto it, whiche shall lighte inA,and that plũb line isA.C,then open I my compasse as wide as the length ofA.B,orA.C,(for they must be bothe equall) and I set the one foote of thend inC,and with the other I make an arche line nigh vntoD,afterward I set the compas again with one foote inB,and with the other foote I make an arche line crosse the first arche line inD,and from the prick of their crossyng I draw .ij. lines, one toB,and an other toC,and so haue I made the square quadrate that I entended.THE.XV. CONCLVSION.To make a likeiãme equall to a triangle appointed, and that in a right lined ãgle limited.First from one of the angles of the triangle, you shall drawe a gemowe line, whiche shall be a parallele to that syde of the triangle, on whiche you will make that likeiamme. Then on one end of the side of the triangle, whiche lieth against the gemowe lyne, you shall draw forth a line vnto the gemow line, so that one angle that commeth of those .ij. lines be like to the angle which is limited vnto you. Then shall you deuide into ij. equall partes that side of the triangle whiche beareth that line, and from the pricke of that deuision, you shall raise an other line parallele to that former line, and continewe it vnto the first gemowe line, and thẽ of those .ij. last gemowe lynes, and the first gemowe line, with the halfe side of the triangle, is made a lykeiamme equall to the triangle appointed, and hath an angle lyke to an angle limited, accordyng to the conclusion.see textExample.B.C.G,is the triangle appoincted vnto, whiche I muste make an equall likeiamme. AndD,is the angle that the likeiamme must haue. Therfore first entendyng to erecte the likeiãme on the one side, that the ground line of the triangle (whiche isB.G.) I do draw a gemow line byC,and make it parallele to the ground lineB.G,and that new gemow line isA.H.Then do I raise a line fromB.vnto the gemowe line, (whiche line isA.B) and make an angle equall toD,that is the appointed angle (accordyng as the .viij. cõclusion teacheth)and that angle isB.A.E.Then to procede, I doo parte in yemiddle the said groũd lineB.G,in the prickF,frõ which prick I drawto the first gemowe line (A.H.) an other line that is parallele toA.B,and that line isE.F.Now saie I that the likeiãmeB.A.E.F,is equall to the triangleB.C.G.And also that it hath one angle (that isB.A.E.)like toD.the angle that was limitted. And so haue I mine intent. The profe of the equalnes of those two figures doeth depend of the .xli. proposition of Euclides first boke, and is the .xxxi. proposition of this second boke of Theoremis, whiche saieth, that whan a tryangle and a likeiamme be made betwene .ij. selfe same gemow lines, and haue their ground line of one length, then is the likeiamme double to the triangle, wherof it foloweth, that if .ij. suche figures so drawen differ in their ground line onely, so that the ground line of the likeiamme be but halfe the ground line of the triangle, then be those .ij. figures equall, as you shall more at large perceiue by the boke of Theoremis, in ye.xxxi. theoreme.THE.XVI. CONCLVSION.To make a likeiamme equall to a triangle appoincted, accordyng to an angle limitted, and on a line also assigned.In the last conclusion the sides of your likeiamme wer left to your libertie, though you had an angle appoincted. Nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted, which must be the side of the likeiãme. Therfore thus shall you procede. Firste accordyng to the laste conclusion, make a likeiamme in the angle appoincted, equall to the triangle that is assigned. Then with your compasse take the length of your line appointed, and set out two lines of the same length in the second gemowe lines, beginnyng at the one side of the likeiamme, and by those two prickes shall you draw an other gemowe line, whiche shall be parallele to two sides of the likeiamme. Afterward shall you draw .ij. lines more for the accomplishement of your worke, which better shall beperceaued by a shorte exaumple, then by a greate numbre of wordes, only without example,thereforeI wyl by example sette forth the whole worke.Example.see textFyrst, according to the last conclusion, I make the likeiammeE.F.C.G,equal to the triangleD,in the appoynted angle whiche isE.Then take I the lengthe of the assigned line (which isA.B,) and with my compas I sette forthe the same lẽgth in the ij. gemow linesN.F.andH.G,setting one foot inE,and the other inN,and againe settyng one foote inC,and the other inH.Afterward I draw a line fromN.toH,whiche is a gemow lyne, to ij. sydes of thelikeiamme. thennedrawe I a line also fromN.vntoC.and extend it vntyll it crosse the lines,E.L.andF.G,which both must be drawen forth longer then the sides of thelikeiamme. andwhere that lyne doeth crosseF.G,there I setteM.Nowe to make an ende, I make an other gemowe line, whiche is parallel toN.F.andH.G,and that gemowe line doth passe by the prickeM,and then haue I done. Now say I thatH.C.K.L,is a likeiamme equall to the triangle appointed, whiche wasD,and is made of a line assigned that isA.B,forH.C,is equall vntoA.B,and so isK.L.The profe of yeequalnes of this likeiam vnto the triãgle, depẽdeth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, whẽ there are more then one made about one bias line, the filsquares of euery of them muste needes be equall.THEXVII. CONCLVSION.To make a likeiamme equal to any right lined figure, and that on an angle appointed.The readiest waye to worke this conclusion, is to tourn that rightlined figure into triangles, and then for euery triangletogetheran equal likeiamme, according vnto the eleuen cõclusion, and then to ioine al those likeiammes into one, if their sides happen to be equal, which thing is euer certain, when al the triangles happẽ iustly betwene one pair of gemowlines. butand if they will not frame so, then after that you haue for the firste triangle made his likeiamme, you shall take the lẽgth of one of his sides, and set that as a line assigned, on whiche you shal make the other likeiams, according to the twelft cõclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so ytyou mai easily ioyne thẽ into one figure.see textExample.If the right lined figure be like vntoA,thẽ may it be turned into triangles that wil stãd betwene ij. parallels anye ways, as youmai sebyC.andD,for ij. sides of both thetriãnglesar parallels. Also if the right lined figure be like vntoE,thẽ wil it be turned into triãgles, liyng betwene two parallels also, as yeother did before, as in the exãple ofF.G.But and if yeright lined figure be like vntoH,and so turned into triãgles as you se inK.L.M,wher it is parted into iij triãgles, thẽ wil not all those triangles lye betwen one pair of parallels or gemow lines, but must haue many, for euery triangle must haue one paire of parallels seuerall, yet it maye happen that when there bee three or fower triangles, ij. of theym maye happen to agre to one pair of parallels, whiche thinge I remit to euery honest witte to serche, for the manner of their draught wil declare, how many paires of parallels they shall neede, of which varietee bicause the examples ar infinite, I haue set forth these few, that by them you may coniecture duly of all other like.see textFurther explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then diligẽtly behold how these sundry figures be turned into triãgles. In the fyrst you se I haue made v. triangles, and fourparalleles. inthe seconde vij. triangles and foureparalleles. inthe thirde thre triãgles, and fiue parallels, in the iiij. you se fiue triãgles & fourparallels. inthe fift, iiij. triãgles and .iiij. parallels, & in yesixt ther ar fiue triãgles & iiij. paralels. Howbeit a mã maye at liberty alter them into diuers formes of triãgles & therefore Ileue it to the discretion of the woorkmaister, to do in al suche cases as he shal thinke best, for by these examples (if they bee well marked) may all other like conclusions be wrought.THEXVIII. CONCLVSION.To parte a line assigned after suche a sorte, that the square that is made of the whole line and one of his parts, shal be equal to the squar that cometh of the other parte alone.First deuide your lyne into ij. equal parts, and of the length of one part make a perpendicular to light at one end of your lineassigned. thenadde a bias line, and make thereof a triangle, this done if you take from this bias line the halfe lengthe of your line appointed, which is the iuste length of your perpendicular, that part of the bias line whiche dothe remayne, is the greater portion of the deuision that you seke for, therefore if you cut your line according to the lengthe of it, then will the square of that greaterportionbe equall to the square that is made of the whole line and his lesser portion. And contrary wise, the square of the whole line and his lesser parte, wyll be equall to the square of the greater parte.see textExample.A.B,is the lyne assigned.E.is the middle pricke ofA.B, B.C.is the plumb line or perpendicular, made of the halfe ofA.B,equall toA.E,otherB.E,the byas line isC.A,from whiche I cut a peece, that isC.D,equall toC.B,and accordyng to the lengthelothepeece that remaineth (whiche isD.A,) I doo deuide the lineA.B,at whiche diuision I setF.Now say I, that this lineA.B,(wchwas assigned vnto me) is so diuided in this pointF,ytyesquare of yehole lineA.B,& of the one portiõ (ytisF.B,thelesser part) is equall to the square of the other parte, whiche isF.A,and is the greater part of the first line. The profe of this equalitie shall you learne by the .xl. Theoreme.There are two ways to make this Example work:—transpose E and F in the illustration, and change one occurrence of E to F in the text,or:—keep the illustration as printed, and transpose all other occurrences of E and F in the text.THE.XIX. CONCLVSION.To make a square quadrate equall to any right lined figure appoincted.First make a likeiamme equall to that right lined figure, with a right angle, accordyng to the .xi. conclusion, then consider the likeiamme, whether it haue all his sides equall, or not: for yf they be all equall, then haue you doone yourconclusion. butand if the sides be not all equall, then shall you make one right line iuste as long as two of those vnequall sides, that line shall you deuide in the middle, and on that pricke drawe half a circle, then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme, and from that pointe of diuision shall you erecte a perpendicular, which shall touche the edge of the circle. And that perpendicular shall be the iuste side of the square quadrate, equall both to the lykeiamme, and also to the right lined figure appointed, as the conclusion willed.Example.see textK,is the right lined figure appointed, andB.C.D.E,is the likeiãme, with right angles equall vntoK,but because that this likeiamme is not a square quadrate, I must turne it into such one after this sort, I shall make one right line, as long as .ij. vnequall sides of the likeiãme, that line here isF.G,whiche is equall toB.C,andC.E.Then part I that line in the middle in theprickeM,and on that pricke I make halfe a circle, accordyng to the length of the diameterF.G.Afterward I cut awaie a peece fromF.G,equall toC.E,markyng that point withH.And on that pricke I erecte a perpendicularH.K,whiche is the iust side to the square quadrate that I seke for, therfore accordyng to the doctrine of the .x. conclusion, of the lyne I doe make a square quadrate, and so haue I attained the practise of this conclusion.THE.XX. CONCLVSION.When any .ij. square quadrates are set forth, how you maie make one equall to them bothe.First drawe a right line equall to the side of one of the quadrates: and on the ende of it make a perpendicular, equall in length to the side of the other quadrate, then drawe a byas line betwene those .ij. other lines, makyng thereof a right angeled triangle. And that byas lyne wyll make a square quadrate, equall to the other .ij. quadrates appointed.see textExample.A.B.andC.D,are the two square quadrates appointed, vnto which I must make one equall square quadrate. First therfore I dooe make a righte lineE.F,equall to one of the sides of the square quadrateA.B.And on the one end of it I make a plumbe lineE.G,equall to the side of the other quadrateD.C.Then drawe I a byas lineG.F,which beyng made the side of a quadrate(accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrateH.is muche iust as the other two. I meaneA.B.andD.C.THE.XXI. CONCLVSION.When any two quadrates be set forth, howe to make a squire about the one quadrate, whiche shall be equall to the other quadrate.Determine with your selfe about whiche quadrate you wil make the squire, and drawe one side of that quadrate forth in lengte, accordyng to the measure of the side of the other quadrate, whiche line you maie call the grounde line, and then haue you a right angle made on this line by an other side of the same quadrate: Therfore turne that into a right cornered triangle, accordyng to the worke in the laste conclusion, by makyng of a byas line, and that byas lyne will performe the worke of your desire. For if you take the length of that byas line with your compasse, and then set one foote of the compas in the farthest angle of the first quadrate (whiche is the one ende of the groundline) and extend the other foote on the same line, accordyng to the measure of the byas line, and of that line make a quadrate, enclosyng yefirst quadrate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.see textExample.The first square quadrate isA.B.C.D,and the seconde isE.Now would I make a squire about the quadrateA.B.C.D,whiche shall bee equall vnto the quadrateE.Therforefirst I draw the lineA.D,more in length, accordyng to the measure of the side ofE,as you see, fromD.vntoF,and so the hole line of bothe these seuerall sides isA.F,thẽ make I a byas line fromC,toF,whiche byas line is the measure of thiswoorke. whereforeI open my compas accordyng to the length of that byas lineC.F,and set the one compas foote inA,and extend thother foote of the compas towardF,makyng this prickeG,from whiche I erect a plumbelineG.H,and so make out the square quadrateA.G.H.K,whose sides are equall eche of them toA.G.And this square doth contain the first quadrateA.B.C.D,and also a squireG.H.K,whiche is equall to the second quadrateE,for as the last conclusion declareth, the quadrateA.G.H.K,is equall to bothe the other quadrates proposed, that isA.B.C.D,andE.Then muste the squireG.H.K,needes be equall toE,consideryng that all the rest of that great quadrate is nothyng els but the quadrate self,A.B.C.D,and so haue I thintent of this conclusion.THE.XXII.CONCLVSION.To find out the cẽtre of any circle assigned.Draw a corde or stryngline crosse the circle, then deuide into .ij. equall partes, both that corde, and also the bowe line, or arche line, that serueth to that corde, and from the prickes of those diuisions, if you drawe an other line crosse the circle, it must nedes passe by the centre. Therfore deuide that line in the middle, and that middle pricke is the centre of the circle proposed.Example.see textLet the circle beA.B.C.D,whose centre I shall seke. First therfore I draw a corde crosse the circle, that isA.C.Then do I deuide that corde in the middle, inE,and likewaies also do I deuide his arche lineA.B.C,in the middle, in the pointeB.Afterward I drawe a line fromB.toE,and so crosse thecircle, whiche line isB.D,in which line is the centre that I seeke for. Therefore if I parte that lineB.D,in the middle in to two equall portions, that middle pricke (which here isF) is the verye centre of the sayde circle that I seke. This conclusion may other waies be wrought, as the moste part of conclusions haue sondry formes of practise, and that is, by makinge thre prickes in the circũference of the circle, at liberty where you wyll, and then findinge the centre to those threpricks, Whichworke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.THEXXIII. CONCLVSION.To find the commen centre belongyng to anye three prickes 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 circumference of al circles is compassed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. prickes in an exact right line can not bee brought into the circumference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their cõmon centre. Opẽ your compas so wide, that it be somewhat more then thehalfe distance of two of those prickes. Then sette the one foote of the compas in the one pricke, and with the other foot draw an arche lyne toward the otherpricke, Thenagaine putte the foot of your compas in the second pricke, 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 pricks, so do with the middle pricke, 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.see textExampleThe iij. prickes I haue set to beA.B,andC,whiche I wold bring into the edg of one common circle, by finding a centre cõmen to them all, fyrst therefore I open my cõpas, so that thei occupye more then yehalfe distance betwene ij. pricks (as areA.B.) and so settinge one foote inA.and extendinge the other towardB,I make the arche lineD.E.Likewise settĩg one foot inB,and turninge the other towardA,I draw an other arche line that crosseth the first inD.andE.Then fromD.toE,I draw a right lyneD.H.After this I open my cõpasse to a new distance, and make ij. arche lines betweneB.andC,whiche crosse one the other inF.andG,by whiche two pointes I draw an other line, that isF.H.And bycause that the lyneD.H.and the lyneF.H.doo meete inH,I saye thatH.is the centre that serueth to those iij. prickes. Now therfore if you set one foot of your compas inH,and extend the other to any of the iij. pricks, you may draw a circle wchshal enclose those iij. pricks in the edg of his circũferẽce & thus haue you attained yevse of this cõclusiõ.THEXXIIII. CONCLVSION.To drawe a touche line onto a circle, from any poincte assigned.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 frõ 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 atoucheline, drawen from the point assigned, according to the meaning of this conclusion.see textsee textExample.Let the circle be calledB.C.D,and his cẽtreE,and yeprick assignedA,opẽ your cõpas now of such widenes, ytthe one foote may be set inE,wchis yecẽtre of yecircle, & yeother inA,wchis yepointe assigned, & so make an other greter circle (as here isA.F.G) thẽ draw a line fromA.vntoE,and wher that line doth cross yeinner circle (wchheere is in the prickB.) there erect a plũb line vnto the line.A.E.and let that plumb line touch the vtter circle, as it doth here in the pointF,so shallB.F.bee that plumbe lyne. Then fromF.vntoE.drawe an other line whiche shal beF.E,and it will cutte the inner circle, as it doth here in the pointC,from which pointeC.if you erect a plumb line vntoA,then is that lineA.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 isE,the pricke assigned isA.and the rulerC.D.by which the touch line is drawen, and that isA.B,and as this way is light to doo, so is it certaine inoughe for any kinde of workinge.THEXXV. CONCLVSION.When you haue any peece of the circumference of a circle assigned, howe you may make oute the whole circle agreyngetherevnto.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.see textExample.A peece of an olde pillar was found, like in forme to thys figureA.D.B.Now to knowe howe muche the cõpasse of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make inatable the like draught of ytcircũference by the self patrõ, vsing it as it wer a croked ruler.Then make .iij. prickes in that arche line, as I haue made,C. D.andE.And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that cẽtre is hereF,nowe settyng one foote of your compas inF,and the other inC. D,other inE,and so makyng a compasse, you haue youre whole intent.THEXXVI. 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.see textThe arche of the circle isA.B.C,vnto whiche I must sekea centre, therfore firste I do deuide it into .ij. partes, the one of them isA.B,and the other isB.C.Then doe I cut euery arche in the middle, so isE.the middle ofA.B,andG.is the middle ofB.C.Likewaies, I take the middle of their cordes, whiche I mark withF.andH,settyngF.byE,andH.byG.Then drawe I a line fromE.toF,and fromG.toH,and they do crosse inD,wherefore saie I, thatD.is the centre, that I seke for.THEXXVII. 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 thecircle, Andthat 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.A. is a circlein a triangle.C. a quadratein a circle.see caption and textsee caption and textsee caption and textB. a trianglein a circle.D. a circle ina quadrate.In these .ij. last figuresE.andF,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 coũted 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.see textThe triangle isA.B.C,whose sides I do part into .ij. equall partes, eche by it selfe in these pointesD.E.F,puttyngF.betweneA.B,andD.betweneB.C,andE.betweneA.C.Then draw I a line fromC.toF,and an other fromA.toD,and the third fromB.toE.Andwhere all those lines do mete (that is to saieM.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.see textAnd so drawe lines frõ 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 lẽgth of the perpendicular, and so drawe your circle.Example.The triangle isA.B.C,whose corners I haue diuided in the middle withD.E.F,and haue drawen the lines of diuisionA.D, B.E,andC.F,which crosse inG,therfore shallG.be the common centre. Then make I one perpẽdicular fromG.vnto the sideB.C,and thatisG.H.Now sette I one fote of the compas inG,and extend the other foote vntoH.and so drawe a compas, whiche wyll iustly answere to that triãgle according to the meaning of the conclusion.THEXXVIII. CONCLVSION.To drawe a circle about any triãgle 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.see textA.B.C.is the triangle, whose two sidesA.C.andB.C.are diuided into two equall partes inD.andE,settyngD.betweneB.andC,andE.betweneA.andC.And from eche of those two pointes is ther erected a perpendicular (as you seD.F,andE.F.) which mete, and crosse inF,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 threecorners 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.see textExample.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 iijprickes. that centre asyou se isD,for inD.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 triãgle, and consequentlye the triangle itselfe, as the conclusion dydde intende.THEXXIX. 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, thenmustI 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 wtthe other halfe of the touche line, equall to an other corner of the triangle appointed, and then betwen those two cornerswill 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.see textA.B.C,is the triangle appointed, andF.G.H.is the circle, in which I muste make an other triangle, with lyke angles to the angles ofA.B.C.the triangle appointed. Therefore fyrst I make the touch lyneD.F.E.And then make I an angle inF,equall toA,whiche is one of the angles of the triangle. And the lyne that maketh that angle with the touche line, isF.H,whiche I drawe in lengthe vntill it touche the edge of the circle. Then againe in the same pointF,I make an other corner equall to the angleC.and the line that maketh that corner with the touche line, isF.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 ytone pointF,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 equallto 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 cõmon sentence saith. Thẽ do I draw a line fromeG.toH,and that line maketh a triangleF.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.THEXXX. 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, thẽ 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 drawẽ about a circle apointed, as the cõclusiõ did wil.Example.A.B.C,is the triangle assigned, andG.H.K,is the circle appointed, about which I muste make a triangle hauing equall angles to the angles of that triangleA.B.C.Fyrst therefore I drawA.C.(which is one of the sides of the triangle) in length that there may appeare two vtter angles in that triangle, as you seB.A.D,andB.C.E.see textThendrawe I in the circle appointed a semidiameter, which is hereH.F,forF.is the cẽtre of the circleG.H.K.Then make I on that centre an angle equall to the vtter angleB.A.D,and that angle isH.F.K.Like waies on the same cẽtre by drawyng an other semidiameter, I make an other angleH.F.G,equall to the second vtter angle of the triangle, whiche isB.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 trianagleL.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 cornerM.is equall toC.LikewaiesL.toA,andN.toB,whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.THEXXXI. 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 handeledaccording 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 wyllsheweyou 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,anyeof 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 seuẽ and twẽty coclusiõ, 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 triãgle be equal to yeãgle assigned as youre selfe mai easily gesse.see textExample.First for example of a sharpe ãgle letA.stãd &B.Cshal be yelyne assigned. Thẽ do I make a triangle, by addingB.C,as a thirde side to those other ij. which doo include the ãgle assigned, and that triãgle isD.E.F,so ytE.F.is the line appointed, andD.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, fromE.a long byD.vntoF,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, andH.K.is the lyne appointed, andL.M.N.the portion of the circle aunsweryng thereto. In the third exaumple,O.is the blunte corner assigned,P.Q.is the line, andR.S.T.is the portion of the circle, that containeth that blũt corner, and is drawen onR.T.the line appointed.THEXXXII. 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.see textExample.A.is the angle appointed, andD.E.F.is the circle assigned, frõ which I must cut away a portiõ that doth contain an angleequall to this angleA.Therfore first I do draw a touche line to the circle assigned, and that touch line isB.C,the very pricke of the touche isD,from whicheD.I drawe a lyneD.E,so that the angle made of those two lines be equall to the angle appointed. Then say I, that the arch of the circleD.F.E,is the arche that I seke after. For if I doo deuide that arche in the middle (as here is done inF.) and so draw thence two lines, one toD,and the other toE,then will the angleF,be equall to the angle assigned.THEXXXIII. 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.see textExample.A.B.C.D.is the circle assigned, andA.C.andB.D.are the two diameters which crosse in the centreE,and make .iiij. right corners. Then do I make fowre other lines, that isA.B, B.C, C.D,andD.A,which do ioyne together the fowre endes of the ij. diameters. And so is the squarequadrate made in the circle assigned, as the conclusion willeth.THEXXXIIII. 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.see textA.B.C.is the circle assigned in which first I draw two diameters, in crosse waies, making iiij. righte angles, and those ij. diameters areA.C.andB.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 inA,then with the other foote I doo make twoo arche lines, one inE,and an other inF.Then sette I the one foote of the compasse inB,and drawe twoo arche linesF.andG.Like wise setting the compasse foote inC,I drawe twoo other arche lines,G.andH,and onD.I make twoo other,H.andE.Then frome the crossinges of those eighte arche lines I drawe iiij. straighte lynes, that is to saye,E.F,andF.G,alsoG.H,andH.E,whiche iiij. straighte lynes do make the square quadrate that I should draw about the circle assigned.THEXXXV. 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.see textExample.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. cõtrary prickes, as you seE.G,andF.H,whiche mete inK,and therfore shalK,be the centre of the circle. Then do I set one foote of the compas inK.and opẽ the other as wide asK.E,and so draw a circle, which is madeaccordingeto the conclusion.THEXXXVI. 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. Thẽ 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.see textExample.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 seA.C.&B.D.And where they ij. do crosse (that is to say inE.) 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 exãple toA,and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.THEXXXVII. 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 thoselines, Thendrawe the thyrde lyne to make a triangle with those other twoo, and you haue doone according to the conclusion, and haue made a twelike triãgle,whose ij. corners about the grounde line, are eche of theym double to the other corner.At no point in this or the accompanying book does the author show how to divide a circle into five.see textExample.A.B.C.is the circle, whiche I haue deuided into fiue equal portions. And from one of the prickes (which isA,) I haue drawẽ ij. lines,A.B.andB.C,whiche are drawen to the third and iiij. prickes. Then draw I the third lineC.B,which is the grounde line, and maketh the triangle, that I would haue, for the ãgleC.is double to the angleA,and so is the angleB.also.THEXXXVIII.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.see textYow se here this circleA.B.C.D.E.deuided into fiue equall portions. And from eche pricke ij.lines drawento the other ij. nexte prickes, so fromA.are drawen ij. lines, one toB,and the other toE,and so fromC.one toB.and an othertoD,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.THEXXXIX. 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.see textExample.A.B.C.D.E.is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is drawẽ a semidiameter, as you see. Then doo I make a touche line in the prickeB,whiche isF.G,making ij. right angleswith the semidiameterB,and lyke waies onC.is madeG.H,onD.standethH.K,and onE,is setK.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.THEXL. 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.see textExample.The cinkeangle assigned isA.B.C.D.E,in whiche I mustemake a circle, wherefore I draw a right line from the one angle (as fromB,) to the middle of the contrary side (whiche isE.D,) and that middle pricke isF.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 isB.C.) and that pricke isG.Nowe because that these two lines do crosse inH,I saie thatH.is the centre of the circle, whiche I would make. Therfore I set one foote of the compasse inH,and extend the other foote vntoG,orF.(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 cõclusion meaneth.THEXLI. CONCLVSIONTo 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.see textA.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 frõE.toG,and the other frõC.toF,and because thei domeete inH,I saye thatH.is the centre of the circle that I woulde haue, wherfore I sette one foote of the compasse inH.and extende the other to one corner (whiche happeneth fyrste, for all are like distaunte fromH.) 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.THEXLII. 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 thecircũference 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 shalperceauethat 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.see textThe circle isB.C.D.E.F.G,whose centre I finde to beeA.Therefore I sette one foote of the compas inA,and do extẽd the other foote toB,thereby takinge the semidiameter. Then sette I one foote of the compas vnremoued inB,and marke with the other foote on eche sideC.andG.Then fromC.I markeD,and frõD,E: fromE.marke IF.And then haue I but one space iuste vntoG.and so haue I made a iuste siseangle of equall sides and equall angles, in a circle appointed.THEXLIII. CONCLVSION.To make a siseangle of equall sides, and equall angles about any circle assigned.THEXLIIII. CONCLVSION.To make a circle in any siseangle appointed, of equall sides and equal angles.THEXLV. 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.THEXLVI. 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 shallhaue, 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, andthen eche of them into three partes againe: Or elsfirst deuide it into three partes, and then echof thẽ into fiue other partes, as youlist, and canne most readilye.Then draw lines betweneeuery two prickesthat benighesttogither, andther wil appear rightly drawẽ the figure, of fiftene sides, andangles equall. And so do with any other figureof what numbre of sides so euer it bee.FINIS.

THE PRACTIKE WORKINGE OFsondry conclusions geometrical.THE FYRST CONCLVSION.To make a threlike triangleonany lyne measurable.Take the iuste lẽgth of the lyne with your cõpasse, and stay the one foot of the compas in one of the endes of that line, turning the other vp or doun at your will, drawyng the arche of a circle against the midle of the line, and doo like wise with the same cõpasse vnaltered, at the other end of the line, and wher these ij. croked lynes doth crosse, frome thence drawe a lyne to ech end of your first line, and there shall appear a threlike triangle drawen on that line.Example.A.B.is the first line, on which I wold make the threlike triangle, therfore I open the compasse as wyde as that line is long, and draw two arch lines that mete inC,then fromC,I draw ij other lines one toA,another toB,and than I haue my purpose.see textsee textsee textTHE.II. CONCLVSIONIf you wil make a twileke or a nouelike triangle on ani certaine line.Consider fyrst the length that yow will haue the other sides to containe, and to that length open your compasse, andthen worke as you did in the threleke triangle, remembryng this, that in a nouelike triangle you must take ij. lengthes besyde the fyrste lyne, and draw an arche lyne with one of thẽ at the one ende, and with the other at the other end, the exãple is as in the other before.THEIII. CONCL.To diuide an angle of right lines into ij. equal partes.see textFirst open your compasse as largely as you can, so that it do not excede the length of the shortest line ytincloseth the angle. Then set one foote of the compasse in the verye point of the angle, and with the other fote draw a compassed arch frõ the one lyne of the angle to the other, that arch shall you deuide in halfe, and thẽ draw a line frõ the ãgle to yemiddle of yearch, and so yeangle is diuided into ij. equall partes.Example.Let the triãgle beA.B.C,thẽ set I one foot of yecõpasse inB,and with the other I draw yearchD.E,which I part into ij. equall parts inF,and thẽ draw a line frõB,toF,& so I haue mine intẽt.THEIIII. CONCL.To deuide any measurable line into ij. equall partes.see textOpen your compasse to the iust lẽgth of yeline. And thẽ set one foote steddely at the one ende of the line, & wtthe other fote draw an arch of a circle against yemidle of the line, both ouer it, and also vnder it, then doo lykewaiseat the other ende of the line. And marke where those arche lines do meet crosse waies, and betwene those ij. pricks draw a line, and it shallcutthe first line in two equall portions.Example.The lyne isA.B.accordyng to which I open the compasse and make .iiij. arche lines, whiche meete inC.andD,then drawe I a lyne fromC,so haue I my purpose.This conlusion serueth for makyng of quadrates and squires, beside many other commodities, howebeit it maye bee don more readylye by this conclusion that foloweth nexte.THE FIFT CONCLVSION.To make a plumme line or any pricke that you will in any right lyne appointed.Open youre compas so that it be not wyder then from the pricke appoynted in the line to the shortest ende of the line, but rather shorter. Then sette the one foote of the compasse in the first pricke appointed, and with the other fote marke ij. other prickes, one of eche syde of that fyrste, afterwarde open your compasse to the wydenes of those ij. new prickes, and draw from them ij. arch lynes, as you did in the fyrst conclusion, for making of a threlyketriãgle. thenif you do mark their crossing, and from it drawe a line to your fyrste pricke, it shall bee a iust plum lyne on that place.see textExample.The lyne isA.B.the prick on whiche I shoulde make the plumme lyne, isC.then open I the compasse as wyde asA.C,and sette one foot inC.and with the other doo I marke outC.A.andC.B,then open I the compasse as wide asA.B,and make ij. arch lines which do crosse inD,and so haue I doone.Howe bee it, it happeneth so sommetymes, that thepricke on whiche you would make the perpendicular or plum line, is so nere the eand of your line, that you can not extende any notable length from it to thone end of the line, and if so be it then that you maie not drawe your line lenger frõ that end, then doth this conclusion require a newe ayde, for the last deuise will not serue. In suche case therfore shall you dooe thus: If your line be of any notable length, deuide it into fiue partes. And if it be not so long that it maie yelde fiue notable partes, then make an other line at will, and parte it into fiue equall portiõs: so that thre of those partes maie be found in your line. Then open your compas as wide as thre of these fiue measures be, and sette the one foote of the compas in the pricke, where you would haue the plumme line to lighte (whiche I call the first pricke,) and with the other foote drawe an arche line righte ouer the pricke, as you can ayme it: then open youre compas as wide as all fiue measures be, and set the one foote in the fourth pricke, and with the other foote draw an other arch line crosse the first, and where thei two do crosse, thense draw a line to the poinct where you woulde haue the perpendicular line to light, and you haue doone.Example.see textsee textThe line isA.B.andA.is the prick, on whiche the perpendicular line must light. Therfore I deuideA.B.into fiue partes equall, then do I open the compas to the widenesse of three partes (that isA.D.) and let one foote staie inA.and with the other I make an arche line inC.Afterwarde I open the compas as wide asA.B.(that is as wide as all fiue partes) and set one foote in the .iiij. pricke, which isE,drawyng an arch line with the other foote inC.also. Then do I draw thence a line vntoA,and so haue I doone. But and if the line be to shorte to be parted into fiue partes, I shall deuide it into iij. partes only, as you see the liueF.G,and then makeD.an other line (as isK.L.) whiche I deuide into .v. suche diuisions, asF.G.containeth .iij, then open I thecompassas wide as .iiij. partes (whiche isK.M.) and so set I one foote of the compas inF,and with the other I drawe an arch lyne towardH,then open I the cõpas as wide asK.L.(that is all .v. partes) and set one foote inG,(that is the iij. pricke) and with the other I draw an arch line towardH.also: and where those .ij. arch lines do crosse (whiche is byH.) thence draw I a line vntoF,and that maketh a very plumbe line toF.G,as my desire was. The maner of workyng of this conclusion, is like to the second conlusion, but the reason of it doth depẽd of the .xlvi. proposiciõ of yefirst boke of Euclide. An other waieyet. setone foote of the compas in the prick, on whiche you would haue the plumbe line to light, and stretche forth thother foote toward the longest end of the line, as wide as you can for the length of the line, and so draw a quarter of a compas or more, then without stirryng of the compas, set one foote of it in the same line, where as the circular line did begin, and extend thother in the circular line, settyng a marke where it doth light, then take half that quantitie more there vnto, and by that prick that endeth the last part, draw a line to the pricke assigned, and it shall be a perpendicular.see textExample.A.B.is the line appointed, to whiche I must make a perpendicular line to light in the pricke assigned, which isA.Therfore doo I set one foote of the compas inA,and extend the other vntoD.makyng a part of a circle,more then a quarter, that isD.E.Then do I set one foote of the compas vnaltered inD,and stretch the other in the circular line, and it doth light inF,this space betweneD.andF.I deuide into halfe in the prickeG,whiche halfe I take with the compas, and set it beyondF.vntoH,and thefore isH.the point, by whiche the perpendicular line must be drawn, so say I that the lineH.A,is a plumbe line toA.B,as the conclusion would.THE.VI. CONCLVSION.To drawe a streight line from any pricke that is not in a line, and to make it perpendicular to an other line.see textOpen your compas as so wide that it may extend somewhat farther, thẽ from the prick to the line, then sette the one foote of the compas in the pricke, and with the other shall you draw a cõpassed line, that shall crosse that other first line in .ij. places.Now if you deuide that arch line into .ij. equall partes, and from the middell pricke therof vnto the prick without the line you drawe a streight line, it shalbe a plumbe line to that firste lyne, accordyng to the conclusion.Example.C.is the appointed pricke, from whiche vnto the lineA.B.I must draw a perpẽdicular. Thefore I open the cõpas so wide, that it may haue one foote inC,and thother to reach ouer the line, and with ytfoote I draw an arch line as you see, betweneA.andB,which arch line I deuide in the middell in the pointD.Then drawe I a line fromC.toD,and it is perpendicular to the lineA.B,accordyng as my desire was.THE.VII. CONCLVSION.To make a plumbe lyneonany porcion of a circle, and that on the vtter or inner bughte.Mark first the prick where yeplũbe line shal lyght: and prick out of ech side of it .ij. other poinctes equally distant from that first pricke. Then set the one foote of the cõpas in one of those side prickes, and the other foote in the other side pricke, and first moue one of the feete and drawe an arche line ouer the middell pricke, then set the compas steddie with the one foote in the other side pricke, and with the other foote drawe an other arche line, that shall cut that first arche, and from the very poincte of their meetyng, drawe a right line vnto the firste pricke, where you do minde that the plumbe line shall lyghte. And so haue you performed thintent of this conclusion.see textExample.The arche of the circle on whiche I would erect a plumbe line, isA.B.C.andB.is the pricke where I would haue the plumbe line to light. Therfore I meate out two equall distaunces on eche side of that prickeB.and they areA.C.Then open I the compas as wide asA.C.and settyng one of the feete inA.with the other I drawe an arche line which goeth byG.Like waies I set one foote of the compas steddily inC.and with the other I drawe an arche line, goyng byG.also. Now consideryng thatG.is the pricke of their meetyng, it shall be also the poinct fro whiche I must drawe the plũbe line. Then draw I a right line fromG.toB.and so haue mine intent. Now asA.B.C.hath a plumbe line erected on hisvtter bought, so may I erect a plumbe line on the inner bught ofD.E.F,doynge with it as I did with the other, that is to saye, fyrste settyng forthe the pricke where the plumbe line shall light, which isE,and then markyng one other on eche syde, as areD.andF.And then proceding as I dyd in the example before.THEVIII. CONCLVSYON.How to deuide the arche of a circle into two equall partes, without measuring the arche.Deuide the corde of that line info ij. equall portions, and then from the middle prycke erecte a plumbe line, and it shal parte that arche in the middle.see textExample.The arch to be diuided ysA.D.C,the corde isA.B.C,this corde is diuided in the middle withB,from which prick if I erect a plum line asB.D,thẽ will it diuide the arch in the middle, that is to say, inD.THEIX. CONCLVSION.To do the same thynge other wise. And for shortenes of worke, if you wyl make a plumbe line without much labour, you may do it with your squyre, so that it be iustly made, for yf you applye the edge of the squyre to the line in which the prick is, and foresee the very corner of the squyre doo touche the pricke. And than frome that corner if you drawe a lyne by the other edge of the squyre, yt will be perpendicular to the former line.see textExample.A.B.is the line, on which I wold make the plumme line, or perpendicular. And therefore I marke the prick, from which the plumbe lyne muste rise, which here isC.Then do I sette one edg of my squyre (that isB.C.) to the lineA.B,so at the corner of the squyre do toucheC.iustly. And fromC.I drawe a line by the other edge of the squire, (which isC.D.) And so haue I made the plumme lineD.C,which I sought for.THEX. CONCLVSION.How to do the same thinge an other way yetsee textIf so be it that you haue an arche of suche greatnes, that your squyre wyll not suffice therto, as the arche of a brydge or of a house or window, then may you do this. Mete vnderneth the arch where yemidle of his cord wyl be, and ther set a mark.Then take a long line with a plummet, and holde the line in suche a place of the arch, that the plummet do hang iustely ouer the middle of the corde, that you didde diuide before, and then the line doth shewe you the middle of the arche.Example.The arch isA.D.B,of which I trye the midle thus. I draw a corde from one syde to the other (as here isA.B,) which I diuide in the middle inC.Thẽ take I a line with a plummet (that isD.E,) and so hold I the line that the plummetE,dooth hange ouerC,Andthen I say thatD.is the middle of the arche. And to thentent that my plummet shall point the more iustely, I doo make it sharpe at the nether ende, and so may I trust this woorke for certaine.THEXI. CONCLVSION.When any line is appointed and without it a pricke, whereby a parallel must be drawen howe you shall doo it.Take the iuste measure beetwene the line and the pricke, accordinge to which you shal open your compasse. Thẽ pitch one foote of your compasse at the one ende of the line, and with the other foote draw a bowe line right ouer the pytche of the compasse, lyke-wise doo at the other ende of the lyne, then draw a line that shall touche the vttermoste edge of bothe those bowe lines, and it will bee a true parallele to the fyrste lyne appointed.Example.see textA.B,is the line vnto which I must draw an other gemow line, which muste passe by the prickC,first I meate with my compasse the smallest distance that is fromC.to the line, and that isC.F,wherfore staying the compasse at that distaunce, I seete the one foote inA,and with the other foot I make a bowe lyne, which isD,thẽ like wise set I the one foote of the compasse inB,and with the other I make the second bow line, which isE.And then draw I a line, so that it toucheth the vttermost edge of bothe these bowe lines, and that lyne passeth by the prickeC,end is a gemowe line toA.B,as my sekyng was.THE.XII. CONCLVSION.To make a triangle of any .iij. lines, so that the lines be suche, that any .ij. of them be longer then the thirde. For this rule is generall, that any two sides of euerie triangle taken together, are longer then the other side that remaineth.If you do remember the first and seconde conclusions, then is there no difficultie in this, for it is in maner the same woorke. First cõsider the .iij. lines that you must take, and set one of thẽ for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.Example.see textI haue .iij.A.B.andC.D.andE.F.of whiche I put.C.D.for my ground line, then with my compas I take the length of.A.B.and set the one foote of my compas inC,and draw an arch line with the other foote. Likewaies I take the lẽgth ofE.F,and set one foote inD,and with the other foote I make an arch line crosse the other arche, and the pricke of their metyng (whiche isG.) shall be the thirde corner of the triangle, for in all suche kyndes of woorkynge to make a tryangle, if you haue one line drawen, there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee, for two of them must needes be at the two eandes of the lyne that is drawen.THEXIII. CONCLVSION.If you haue a line appointed, and a pointe in it limited, howe you maye make on it a righte lined angle, equall to an other right lined angle, all ready assigned.Fyrste draw a line against the corner assigned, and so is it a triangle, then take heede to the line and the pointe in it assigned, and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned, and if it bee longe enoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accordinge to the laste conlusion, makynge a triangle of thre like lynes to that assigned triangle. If it bee not longe inoughe, thenn lengthen it fyrste, and afterwarde doo as I haue sayde beefore.see textExample.Lette the angle appoynted beeA.B.C,and the corner assigned,B.Farthermore let the lymited line beeD.G,and the pricke assignedD.Fyrste therefore by drawinge the lineA.C,I make the triangleA.B.C.see textThen consideringe thatD.G,is longer thanneA.B,you shall cut out a line frõD.towardG,equal toA.B,as for exãpleD.F.Thẽ measure oute the other ij. lines and worke with thẽ according as the conclusion with the fyrste also and the second teacheth yow, and then haue you done.THEXIIII. CONCLVSION.To make a square quadrate of any righte lyne appoincted.First make a plumbe line vnto your line appointed, whiche shall light at one of the endes of it,accordyng tothe fifth conclusion, and let it be of like length as your first line is, then opẽ your compasse to the iuste length of one of them, and sette one foote of the compasse in the ende of the one line, and with the other foote draw an arche line, there as you thinke that the fowerth corner shall be, after that set the one foote of the same compasse vnsturred, in the eande of the other line, and drawe an other arche line crosse the first archeline, and the poincte that they do crosse in, is the pricke of the fourth corner of the square quadrate which you seke for, therfore draw a line from that pricke to the eande of eche line, and you shall therby haue made a square quadrate.see textExample.A.B.is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plũbe line vnto it, whiche shall lighte inA,and that plũb line isA.C,then open I my compasse as wide as the length ofA.B,orA.C,(for they must be bothe equall) and I set the one foote of thend inC,and with the other I make an arche line nigh vntoD,afterward I set the compas again with one foote inB,and with the other foote I make an arche line crosse the first arche line inD,and from the prick of their crossyng I draw .ij. lines, one toB,and an other toC,and so haue I made the square quadrate that I entended.THE.XV. CONCLVSION.To make a likeiãme equall to a triangle appointed, and that in a right lined ãgle limited.First from one of the angles of the triangle, you shall drawe a gemowe line, whiche shall be a parallele to that syde of the triangle, on whiche you will make that likeiamme. Then on one end of the side of the triangle, whiche lieth against the gemowe lyne, you shall draw forth a line vnto the gemow line, so that one angle that commeth of those .ij. lines be like to the angle which is limited vnto you. Then shall you deuide into ij. equall partes that side of the triangle whiche beareth that line, and from the pricke of that deuision, you shall raise an other line parallele to that former line, and continewe it vnto the first gemowe line, and thẽ of those .ij. last gemowe lynes, and the first gemowe line, with the halfe side of the triangle, is made a lykeiamme equall to the triangle appointed, and hath an angle lyke to an angle limited, accordyng to the conclusion.see textExample.B.C.G,is the triangle appoincted vnto, whiche I muste make an equall likeiamme. AndD,is the angle that the likeiamme must haue. Therfore first entendyng to erecte the likeiãme on the one side, that the ground line of the triangle (whiche isB.G.) I do draw a gemow line byC,and make it parallele to the ground lineB.G,and that new gemow line isA.H.Then do I raise a line fromB.vnto the gemowe line, (whiche line isA.B) and make an angle equall toD,that is the appointed angle (accordyng as the .viij. cõclusion teacheth)and that angle isB.A.E.Then to procede, I doo parte in yemiddle the said groũd lineB.G,in the prickF,frõ which prick I drawto the first gemowe line (A.H.) an other line that is parallele toA.B,and that line isE.F.Now saie I that the likeiãmeB.A.E.F,is equall to the triangleB.C.G.And also that it hath one angle (that isB.A.E.)like toD.the angle that was limitted. And so haue I mine intent. The profe of the equalnes of those two figures doeth depend of the .xli. proposition of Euclides first boke, and is the .xxxi. proposition of this second boke of Theoremis, whiche saieth, that whan a tryangle and a likeiamme be made betwene .ij. selfe same gemow lines, and haue their ground line of one length, then is the likeiamme double to the triangle, wherof it foloweth, that if .ij. suche figures so drawen differ in their ground line onely, so that the ground line of the likeiamme be but halfe the ground line of the triangle, then be those .ij. figures equall, as you shall more at large perceiue by the boke of Theoremis, in ye.xxxi. theoreme.THE.XVI. CONCLVSION.To make a likeiamme equall to a triangle appoincted, accordyng to an angle limitted, and on a line also assigned.In the last conclusion the sides of your likeiamme wer left to your libertie, though you had an angle appoincted. Nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted, which must be the side of the likeiãme. Therfore thus shall you procede. Firste accordyng to the laste conclusion, make a likeiamme in the angle appoincted, equall to the triangle that is assigned. Then with your compasse take the length of your line appointed, and set out two lines of the same length in the second gemowe lines, beginnyng at the one side of the likeiamme, and by those two prickes shall you draw an other gemowe line, whiche shall be parallele to two sides of the likeiamme. Afterward shall you draw .ij. lines more for the accomplishement of your worke, which better shall beperceaued by a shorte exaumple, then by a greate numbre of wordes, only without example,thereforeI wyl by example sette forth the whole worke.Example.see textFyrst, according to the last conclusion, I make the likeiammeE.F.C.G,equal to the triangleD,in the appoynted angle whiche isE.Then take I the lengthe of the assigned line (which isA.B,) and with my compas I sette forthe the same lẽgth in the ij. gemow linesN.F.andH.G,setting one foot inE,and the other inN,and againe settyng one foote inC,and the other inH.Afterward I draw a line fromN.toH,whiche is a gemow lyne, to ij. sydes of thelikeiamme. thennedrawe I a line also fromN.vntoC.and extend it vntyll it crosse the lines,E.L.andF.G,which both must be drawen forth longer then the sides of thelikeiamme. andwhere that lyne doeth crosseF.G,there I setteM.Nowe to make an ende, I make an other gemowe line, whiche is parallel toN.F.andH.G,and that gemowe line doth passe by the prickeM,and then haue I done. Now say I thatH.C.K.L,is a likeiamme equall to the triangle appointed, whiche wasD,and is made of a line assigned that isA.B,forH.C,is equall vntoA.B,and so isK.L.The profe of yeequalnes of this likeiam vnto the triãgle, depẽdeth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, whẽ there are more then one made about one bias line, the filsquares of euery of them muste needes be equall.THEXVII. CONCLVSION.To make a likeiamme equal to any right lined figure, and that on an angle appointed.The readiest waye to worke this conclusion, is to tourn that rightlined figure into triangles, and then for euery triangletogetheran equal likeiamme, according vnto the eleuen cõclusion, and then to ioine al those likeiammes into one, if their sides happen to be equal, which thing is euer certain, when al the triangles happẽ iustly betwene one pair of gemowlines. butand if they will not frame so, then after that you haue for the firste triangle made his likeiamme, you shall take the lẽgth of one of his sides, and set that as a line assigned, on whiche you shal make the other likeiams, according to the twelft cõclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so ytyou mai easily ioyne thẽ into one figure.see textExample.If the right lined figure be like vntoA,thẽ may it be turned into triangles that wil stãd betwene ij. parallels anye ways, as youmai sebyC.andD,for ij. sides of both thetriãnglesar parallels. Also if the right lined figure be like vntoE,thẽ wil it be turned into triãgles, liyng betwene two parallels also, as yeother did before, as in the exãple ofF.G.But and if yeright lined figure be like vntoH,and so turned into triãgles as you se inK.L.M,wher it is parted into iij triãgles, thẽ wil not all those triangles lye betwen one pair of parallels or gemow lines, but must haue many, for euery triangle must haue one paire of parallels seuerall, yet it maye happen that when there bee three or fower triangles, ij. of theym maye happen to agre to one pair of parallels, whiche thinge I remit to euery honest witte to serche, for the manner of their draught wil declare, how many paires of parallels they shall neede, of which varietee bicause the examples ar infinite, I haue set forth these few, that by them you may coniecture duly of all other like.see textFurther explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then diligẽtly behold how these sundry figures be turned into triãgles. In the fyrst you se I haue made v. triangles, and fourparalleles. inthe seconde vij. triangles and foureparalleles. inthe thirde thre triãgles, and fiue parallels, in the iiij. you se fiue triãgles & fourparallels. inthe fift, iiij. triãgles and .iiij. parallels, & in yesixt ther ar fiue triãgles & iiij. paralels. Howbeit a mã maye at liberty alter them into diuers formes of triãgles & therefore Ileue it to the discretion of the woorkmaister, to do in al suche cases as he shal thinke best, for by these examples (if they bee well marked) may all other like conclusions be wrought.THEXVIII. CONCLVSION.To parte a line assigned after suche a sorte, that the square that is made of the whole line and one of his parts, shal be equal to the squar that cometh of the other parte alone.First deuide your lyne into ij. equal parts, and of the length of one part make a perpendicular to light at one end of your lineassigned. thenadde a bias line, and make thereof a triangle, this done if you take from this bias line the halfe lengthe of your line appointed, which is the iuste length of your perpendicular, that part of the bias line whiche dothe remayne, is the greater portion of the deuision that you seke for, therefore if you cut your line according to the lengthe of it, then will the square of that greaterportionbe equall to the square that is made of the whole line and his lesser portion. And contrary wise, the square of the whole line and his lesser parte, wyll be equall to the square of the greater parte.see textExample.A.B,is the lyne assigned.E.is the middle pricke ofA.B, B.C.is the plumb line or perpendicular, made of the halfe ofA.B,equall toA.E,otherB.E,the byas line isC.A,from whiche I cut a peece, that isC.D,equall toC.B,and accordyng to the lengthelothepeece that remaineth (whiche isD.A,) I doo deuide the lineA.B,at whiche diuision I setF.Now say I, that this lineA.B,(wchwas assigned vnto me) is so diuided in this pointF,ytyesquare of yehole lineA.B,& of the one portiõ (ytisF.B,thelesser part) is equall to the square of the other parte, whiche isF.A,and is the greater part of the first line. The profe of this equalitie shall you learne by the .xl. Theoreme.There are two ways to make this Example work:—transpose E and F in the illustration, and change one occurrence of E to F in the text,or:—keep the illustration as printed, and transpose all other occurrences of E and F in the text.THE.XIX. CONCLVSION.To make a square quadrate equall to any right lined figure appoincted.First make a likeiamme equall to that right lined figure, with a right angle, accordyng to the .xi. conclusion, then consider the likeiamme, whether it haue all his sides equall, or not: for yf they be all equall, then haue you doone yourconclusion. butand if the sides be not all equall, then shall you make one right line iuste as long as two of those vnequall sides, that line shall you deuide in the middle, and on that pricke drawe half a circle, then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme, and from that pointe of diuision shall you erecte a perpendicular, which shall touche the edge of the circle. And that perpendicular shall be the iuste side of the square quadrate, equall both to the lykeiamme, and also to the right lined figure appointed, as the conclusion willed.Example.see textK,is the right lined figure appointed, andB.C.D.E,is the likeiãme, with right angles equall vntoK,but because that this likeiamme is not a square quadrate, I must turne it into such one after this sort, I shall make one right line, as long as .ij. vnequall sides of the likeiãme, that line here isF.G,whiche is equall toB.C,andC.E.Then part I that line in the middle in theprickeM,and on that pricke I make halfe a circle, accordyng to the length of the diameterF.G.Afterward I cut awaie a peece fromF.G,equall toC.E,markyng that point withH.And on that pricke I erecte a perpendicularH.K,whiche is the iust side to the square quadrate that I seke for, therfore accordyng to the doctrine of the .x. conclusion, of the lyne I doe make a square quadrate, and so haue I attained the practise of this conclusion.THE.XX. CONCLVSION.When any .ij. square quadrates are set forth, how you maie make one equall to them bothe.First drawe a right line equall to the side of one of the quadrates: and on the ende of it make a perpendicular, equall in length to the side of the other quadrate, then drawe a byas line betwene those .ij. other lines, makyng thereof a right angeled triangle. And that byas lyne wyll make a square quadrate, equall to the other .ij. quadrates appointed.see textExample.A.B.andC.D,are the two square quadrates appointed, vnto which I must make one equall square quadrate. First therfore I dooe make a righte lineE.F,equall to one of the sides of the square quadrateA.B.And on the one end of it I make a plumbe lineE.G,equall to the side of the other quadrateD.C.Then drawe I a byas lineG.F,which beyng made the side of a quadrate(accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrateH.is muche iust as the other two. I meaneA.B.andD.C.THE.XXI. CONCLVSION.When any two quadrates be set forth, howe to make a squire about the one quadrate, whiche shall be equall to the other quadrate.Determine with your selfe about whiche quadrate you wil make the squire, and drawe one side of that quadrate forth in lengte, accordyng to the measure of the side of the other quadrate, whiche line you maie call the grounde line, and then haue you a right angle made on this line by an other side of the same quadrate: Therfore turne that into a right cornered triangle, accordyng to the worke in the laste conclusion, by makyng of a byas line, and that byas lyne will performe the worke of your desire. For if you take the length of that byas line with your compasse, and then set one foote of the compas in the farthest angle of the first quadrate (whiche is the one ende of the groundline) and extend the other foote on the same line, accordyng to the measure of the byas line, and of that line make a quadrate, enclosyng yefirst quadrate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.see textExample.The first square quadrate isA.B.C.D,and the seconde isE.Now would I make a squire about the quadrateA.B.C.D,whiche shall bee equall vnto the quadrateE.Therforefirst I draw the lineA.D,more in length, accordyng to the measure of the side ofE,as you see, fromD.vntoF,and so the hole line of bothe these seuerall sides isA.F,thẽ make I a byas line fromC,toF,whiche byas line is the measure of thiswoorke. whereforeI open my compas accordyng to the length of that byas lineC.F,and set the one compas foote inA,and extend thother foote of the compas towardF,makyng this prickeG,from whiche I erect a plumbelineG.H,and so make out the square quadrateA.G.H.K,whose sides are equall eche of them toA.G.And this square doth contain the first quadrateA.B.C.D,and also a squireG.H.K,whiche is equall to the second quadrateE,for as the last conclusion declareth, the quadrateA.G.H.K,is equall to bothe the other quadrates proposed, that isA.B.C.D,andE.Then muste the squireG.H.K,needes be equall toE,consideryng that all the rest of that great quadrate is nothyng els but the quadrate self,A.B.C.D,and so haue I thintent of this conclusion.THE.XXII.CONCLVSION.To find out the cẽtre of any circle assigned.Draw a corde or stryngline crosse the circle, then deuide into .ij. equall partes, both that corde, and also the bowe line, or arche line, that serueth to that corde, and from the prickes of those diuisions, if you drawe an other line crosse the circle, it must nedes passe by the centre. Therfore deuide that line in the middle, and that middle pricke is the centre of the circle proposed.Example.see textLet the circle beA.B.C.D,whose centre I shall seke. First therfore I draw a corde crosse the circle, that isA.C.Then do I deuide that corde in the middle, inE,and likewaies also do I deuide his arche lineA.B.C,in the middle, in the pointeB.Afterward I drawe a line fromB.toE,and so crosse thecircle, whiche line isB.D,in which line is the centre that I seeke for. Therefore if I parte that lineB.D,in the middle in to two equall portions, that middle pricke (which here isF) is the verye centre of the sayde circle that I seke. This conclusion may other waies be wrought, as the moste part of conclusions haue sondry formes of practise, and that is, by makinge thre prickes in the circũference of the circle, at liberty where you wyll, and then findinge the centre to those threpricks, Whichworke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.THEXXIII. CONCLVSION.To find the commen centre belongyng to anye three prickes 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 circumference of al circles is compassed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. prickes in an exact right line can not bee brought into the circumference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their cõmon centre. Opẽ your compas so wide, that it be somewhat more then thehalfe distance of two of those prickes. Then sette the one foote of the compas in the one pricke, and with the other foot draw an arche lyne toward the otherpricke, Thenagaine putte the foot of your compas in the second pricke, 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 pricks, so do with the middle pricke, 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.see textExampleThe iij. prickes I haue set to beA.B,andC,whiche I wold bring into the edg of one common circle, by finding a centre cõmen to them all, fyrst therefore I open my cõpas, so that thei occupye more then yehalfe distance betwene ij. pricks (as areA.B.) and so settinge one foote inA.and extendinge the other towardB,I make the arche lineD.E.Likewise settĩg one foot inB,and turninge the other towardA,I draw an other arche line that crosseth the first inD.andE.Then fromD.toE,I draw a right lyneD.H.After this I open my cõpasse to a new distance, and make ij. arche lines betweneB.andC,whiche crosse one the other inF.andG,by whiche two pointes I draw an other line, that isF.H.And bycause that the lyneD.H.and the lyneF.H.doo meete inH,I saye thatH.is the centre that serueth to those iij. prickes. Now therfore if you set one foot of your compas inH,and extend the other to any of the iij. pricks, you may draw a circle wchshal enclose those iij. pricks in the edg of his circũferẽce & thus haue you attained yevse of this cõclusiõ.THEXXIIII. CONCLVSION.To drawe a touche line onto a circle, from any poincte assigned.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 frõ 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 atoucheline, drawen from the point assigned, according to the meaning of this conclusion.see textsee textExample.Let the circle be calledB.C.D,and his cẽtreE,and yeprick assignedA,opẽ your cõpas now of such widenes, ytthe one foote may be set inE,wchis yecẽtre of yecircle, & yeother inA,wchis yepointe assigned, & so make an other greter circle (as here isA.F.G) thẽ draw a line fromA.vntoE,and wher that line doth cross yeinner circle (wchheere is in the prickB.) there erect a plũb line vnto the line.A.E.and let that plumb line touch the vtter circle, as it doth here in the pointF,so shallB.F.bee that plumbe lyne. Then fromF.vntoE.drawe an other line whiche shal beF.E,and it will cutte the inner circle, as it doth here in the pointC,from which pointeC.if you erect a plumb line vntoA,then is that lineA.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 isE,the pricke assigned isA.and the rulerC.D.by which the touch line is drawen, and that isA.B,and as this way is light to doo, so is it certaine inoughe for any kinde of workinge.THEXXV. CONCLVSION.When you haue any peece of the circumference of a circle assigned, howe you may make oute the whole circle agreyngetherevnto.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.see textExample.A peece of an olde pillar was found, like in forme to thys figureA.D.B.Now to knowe howe muche the cõpasse of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make inatable the like draught of ytcircũference by the self patrõ, vsing it as it wer a croked ruler.Then make .iij. prickes in that arche line, as I haue made,C. D.andE.And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that cẽtre is hereF,nowe settyng one foote of your compas inF,and the other inC. D,other inE,and so makyng a compasse, you haue youre whole intent.THEXXVI. 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.see textThe arche of the circle isA.B.C,vnto whiche I must sekea centre, therfore firste I do deuide it into .ij. partes, the one of them isA.B,and the other isB.C.Then doe I cut euery arche in the middle, so isE.the middle ofA.B,andG.is the middle ofB.C.Likewaies, I take the middle of their cordes, whiche I mark withF.andH,settyngF.byE,andH.byG.Then drawe I a line fromE.toF,and fromG.toH,and they do crosse inD,wherefore saie I, thatD.is the centre, that I seke for.THEXXVII. 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 thecircle, Andthat 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.A. is a circlein a triangle.C. a quadratein a circle.see caption and textsee caption and textsee caption and textB. a trianglein a circle.D. a circle ina quadrate.In these .ij. last figuresE.andF,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 coũted 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.see textThe triangle isA.B.C,whose sides I do part into .ij. equall partes, eche by it selfe in these pointesD.E.F,puttyngF.betweneA.B,andD.betweneB.C,andE.betweneA.C.Then draw I a line fromC.toF,and an other fromA.toD,and the third fromB.toE.Andwhere all those lines do mete (that is to saieM.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.see textAnd so drawe lines frõ 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 lẽgth of the perpendicular, and so drawe your circle.Example.The triangle isA.B.C,whose corners I haue diuided in the middle withD.E.F,and haue drawen the lines of diuisionA.D, B.E,andC.F,which crosse inG,therfore shallG.be the common centre. Then make I one perpẽdicular fromG.vnto the sideB.C,and thatisG.H.Now sette I one fote of the compas inG,and extend the other foote vntoH.and so drawe a compas, whiche wyll iustly answere to that triãgle according to the meaning of the conclusion.THEXXVIII. CONCLVSION.To drawe a circle about any triãgle 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.see textA.B.C.is the triangle, whose two sidesA.C.andB.C.are diuided into two equall partes inD.andE,settyngD.betweneB.andC,andE.betweneA.andC.And from eche of those two pointes is ther erected a perpendicular (as you seD.F,andE.F.) which mete, and crosse inF,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 threecorners 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.see textExample.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 iijprickes. that centre asyou se isD,for inD.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 triãgle, and consequentlye the triangle itselfe, as the conclusion dydde intende.THEXXIX. 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, thenmustI 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 wtthe other halfe of the touche line, equall to an other corner of the triangle appointed, and then betwen those two cornerswill 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.see textA.B.C,is the triangle appointed, andF.G.H.is the circle, in which I muste make an other triangle, with lyke angles to the angles ofA.B.C.the triangle appointed. Therefore fyrst I make the touch lyneD.F.E.And then make I an angle inF,equall toA,whiche is one of the angles of the triangle. And the lyne that maketh that angle with the touche line, isF.H,whiche I drawe in lengthe vntill it touche the edge of the circle. Then againe in the same pointF,I make an other corner equall to the angleC.and the line that maketh that corner with the touche line, isF.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 ytone pointF,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 equallto 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 cõmon sentence saith. Thẽ do I draw a line fromeG.toH,and that line maketh a triangleF.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.THEXXX. 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, thẽ 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 drawẽ about a circle apointed, as the cõclusiõ did wil.Example.A.B.C,is the triangle assigned, andG.H.K,is the circle appointed, about which I muste make a triangle hauing equall angles to the angles of that triangleA.B.C.Fyrst therefore I drawA.C.(which is one of the sides of the triangle) in length that there may appeare two vtter angles in that triangle, as you seB.A.D,andB.C.E.see textThendrawe I in the circle appointed a semidiameter, which is hereH.F,forF.is the cẽtre of the circleG.H.K.Then make I on that centre an angle equall to the vtter angleB.A.D,and that angle isH.F.K.Like waies on the same cẽtre by drawyng an other semidiameter, I make an other angleH.F.G,equall to the second vtter angle of the triangle, whiche isB.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 trianagleL.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 cornerM.is equall toC.LikewaiesL.toA,andN.toB,whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.THEXXXI. 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 handeledaccording 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 wyllsheweyou 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,anyeof 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 seuẽ and twẽty coclusiõ, 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 triãgle be equal to yeãgle assigned as youre selfe mai easily gesse.see textExample.First for example of a sharpe ãgle letA.stãd &B.Cshal be yelyne assigned. Thẽ do I make a triangle, by addingB.C,as a thirde side to those other ij. which doo include the ãgle assigned, and that triãgle isD.E.F,so ytE.F.is the line appointed, andD.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, fromE.a long byD.vntoF,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, andH.K.is the lyne appointed, andL.M.N.the portion of the circle aunsweryng thereto. In the third exaumple,O.is the blunte corner assigned,P.Q.is the line, andR.S.T.is the portion of the circle, that containeth that blũt corner, and is drawen onR.T.the line appointed.THEXXXII. 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.see textExample.A.is the angle appointed, andD.E.F.is the circle assigned, frõ which I must cut away a portiõ that doth contain an angleequall to this angleA.Therfore first I do draw a touche line to the circle assigned, and that touch line isB.C,the very pricke of the touche isD,from whicheD.I drawe a lyneD.E,so that the angle made of those two lines be equall to the angle appointed. Then say I, that the arch of the circleD.F.E,is the arche that I seke after. For if I doo deuide that arche in the middle (as here is done inF.) and so draw thence two lines, one toD,and the other toE,then will the angleF,be equall to the angle assigned.THEXXXIII. 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.see textExample.A.B.C.D.is the circle assigned, andA.C.andB.D.are the two diameters which crosse in the centreE,and make .iiij. right corners. Then do I make fowre other lines, that isA.B, B.C, C.D,andD.A,which do ioyne together the fowre endes of the ij. diameters. And so is the squarequadrate made in the circle assigned, as the conclusion willeth.THEXXXIIII. 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.see textA.B.C.is the circle assigned in which first I draw two diameters, in crosse waies, making iiij. righte angles, and those ij. diameters areA.C.andB.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 inA,then with the other foote I doo make twoo arche lines, one inE,and an other inF.Then sette I the one foote of the compasse inB,and drawe twoo arche linesF.andG.Like wise setting the compasse foote inC,I drawe twoo other arche lines,G.andH,and onD.I make twoo other,H.andE.Then frome the crossinges of those eighte arche lines I drawe iiij. straighte lynes, that is to saye,E.F,andF.G,alsoG.H,andH.E,whiche iiij. straighte lynes do make the square quadrate that I should draw about the circle assigned.THEXXXV. 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.see textExample.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. cõtrary prickes, as you seE.G,andF.H,whiche mete inK,and therfore shalK,be the centre of the circle. Then do I set one foote of the compas inK.and opẽ the other as wide asK.E,and so draw a circle, which is madeaccordingeto the conclusion.THEXXXVI. 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. Thẽ 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.see textExample.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 seA.C.&B.D.And where they ij. do crosse (that is to say inE.) 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 exãple toA,and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.THEXXXVII. 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 thoselines, Thendrawe the thyrde lyne to make a triangle with those other twoo, and you haue doone according to the conclusion, and haue made a twelike triãgle,whose ij. corners about the grounde line, are eche of theym double to the other corner.At no point in this or the accompanying book does the author show how to divide a circle into five.see textExample.A.B.C.is the circle, whiche I haue deuided into fiue equal portions. And from one of the prickes (which isA,) I haue drawẽ ij. lines,A.B.andB.C,whiche are drawen to the third and iiij. prickes. Then draw I the third lineC.B,which is the grounde line, and maketh the triangle, that I would haue, for the ãgleC.is double to the angleA,and so is the angleB.also.THEXXXVIII.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.see textYow se here this circleA.B.C.D.E.deuided into fiue equall portions. And from eche pricke ij.lines drawento the other ij. nexte prickes, so fromA.are drawen ij. lines, one toB,and the other toE,and so fromC.one toB.and an othertoD,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.THEXXXIX. 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.see textExample.A.B.C.D.E.is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is drawẽ a semidiameter, as you see. Then doo I make a touche line in the prickeB,whiche isF.G,making ij. right angleswith the semidiameterB,and lyke waies onC.is madeG.H,onD.standethH.K,and onE,is setK.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.THEXL. 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.see textExample.The cinkeangle assigned isA.B.C.D.E,in whiche I mustemake a circle, wherefore I draw a right line from the one angle (as fromB,) to the middle of the contrary side (whiche isE.D,) and that middle pricke isF.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 isB.C.) and that pricke isG.Nowe because that these two lines do crosse inH,I saie thatH.is the centre of the circle, whiche I would make. Therfore I set one foote of the compasse inH,and extend the other foote vntoG,orF.(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 cõclusion meaneth.THEXLI. CONCLVSIONTo 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.see textA.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 frõE.toG,and the other frõC.toF,and because thei domeete inH,I saye thatH.is the centre of the circle that I woulde haue, wherfore I sette one foote of the compasse inH.and extende the other to one corner (whiche happeneth fyrste, for all are like distaunte fromH.) 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.THEXLII. 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 thecircũference 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 shalperceauethat 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.see textThe circle isB.C.D.E.F.G,whose centre I finde to beeA.Therefore I sette one foote of the compas inA,and do extẽd the other foote toB,thereby takinge the semidiameter. Then sette I one foote of the compas vnremoued inB,and marke with the other foote on eche sideC.andG.Then fromC.I markeD,and frõD,E: fromE.marke IF.And then haue I but one space iuste vntoG.and so haue I made a iuste siseangle of equall sides and equall angles, in a circle appointed.THEXLIII. CONCLVSION.To make a siseangle of equall sides, and equall angles about any circle assigned.THEXLIIII. CONCLVSION.To make a circle in any siseangle appointed, of equall sides and equal angles.THEXLV. 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.THEXLVI. 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 shallhaue, 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, andthen eche of them into three partes againe: Or elsfirst deuide it into three partes, and then echof thẽ into fiue other partes, as youlist, and canne most readilye.Then draw lines betweneeuery two prickesthat benighesttogither, andther wil appear rightly drawẽ the figure, of fiftene sides, andangles equall. And so do with any other figureof what numbre of sides so euer it bee.FINIS.

Take the iuste lẽgth of the lyne with your cõpasse, and stay the one foot of the compas in one of the endes of that line, turning the other vp or doun at your will, drawyng the arche of a circle against the midle of the line, and doo like wise with the same cõpasse vnaltered, at the other end of the line, and wher these ij. croked lynes doth crosse, frome thence drawe a lyne to ech end of your first line, and there shall appear a threlike triangle drawen on that line.

A.B.is the first line, on which I wold make the threlike triangle, therfore I open the compasse as wyde as that line is long, and draw two arch lines that mete inC,then fromC,I draw ij other lines one toA,another toB,and than I haue my purpose.

Consider fyrst the length that yow will haue the other sides to containe, and to that length open your compasse, andthen worke as you did in the threleke triangle, remembryng this, that in a nouelike triangle you must take ij. lengthes besyde the fyrste lyne, and draw an arche lyne with one of thẽ at the one ende, and with the other at the other end, the exãple is as in the other before.

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First open your compasse as largely as you can, so that it do not excede the length of the shortest line ytincloseth the angle. Then set one foote of the compasse in the verye point of the angle, and with the other fote draw a compassed arch frõ the one lyne of the angle to the other, that arch shall you deuide in halfe, and thẽ draw a line frõ the ãgle to yemiddle of yearch, and so yeangle is diuided into ij. equall partes.

Let the triãgle beA.B.C,thẽ set I one foot of yecõpasse inB,and with the other I draw yearchD.E,which I part into ij. equall parts inF,and thẽ draw a line frõB,toF,& so I haue mine intẽt.

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Open your compasse to the iust lẽgth of yeline. And thẽ set one foote steddely at the one ende of the line, & wtthe other fote draw an arch of a circle against yemidle of the line, both ouer it, and also vnder it, then doo lykewaiseat the other ende of the line. And marke where those arche lines do meet crosse waies, and betwene those ij. pricks draw a line, and it shallcutthe first line in two equall portions.

The lyne isA.B.accordyng to which I open the compasse and make .iiij. arche lines, whiche meete inC.andD,then drawe I a lyne fromC,so haue I my purpose.

This conlusion serueth for makyng of quadrates and squires, beside many other commodities, howebeit it maye bee don more readylye by this conclusion that foloweth nexte.

Open youre compas so that it be not wyder then from the pricke appoynted in the line to the shortest ende of the line, but rather shorter. Then sette the one foote of the compasse in the first pricke appointed, and with the other fote marke ij. other prickes, one of eche syde of that fyrste, afterwarde open your compasse to the wydenes of those ij. new prickes, and draw from them ij. arch lynes, as you did in the fyrst conclusion, for making of a threlyketriãgle. thenif you do mark their crossing, and from it drawe a line to your fyrste pricke, it shall bee a iust plum lyne on that place.

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The lyne isA.B.the prick on whiche I shoulde make the plumme lyne, isC.then open I the compasse as wyde asA.C,and sette one foot inC.and with the other doo I marke outC.A.andC.B,then open I the compasse as wide asA.B,and make ij. arch lines which do crosse inD,and so haue I doone.

Howe bee it, it happeneth so sommetymes, that thepricke on whiche you would make the perpendicular or plum line, is so nere the eand of your line, that you can not extende any notable length from it to thone end of the line, and if so be it then that you maie not drawe your line lenger frõ that end, then doth this conclusion require a newe ayde, for the last deuise will not serue. In suche case therfore shall you dooe thus: If your line be of any notable length, deuide it into fiue partes. And if it be not so long that it maie yelde fiue notable partes, then make an other line at will, and parte it into fiue equall portiõs: so that thre of those partes maie be found in your line. Then open your compas as wide as thre of these fiue measures be, and sette the one foote of the compas in the pricke, where you would haue the plumme line to lighte (whiche I call the first pricke,) and with the other foote drawe an arche line righte ouer the pricke, as you can ayme it: then open youre compas as wide as all fiue measures be, and set the one foote in the fourth pricke, and with the other foote draw an other arch line crosse the first, and where thei two do crosse, thense draw a line to the poinct where you woulde haue the perpendicular line to light, and you haue doone.

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The line isA.B.andA.is the prick, on whiche the perpendicular line must light. Therfore I deuideA.B.into fiue partes equall, then do I open the compas to the widenesse of three partes (that isA.D.) and let one foote staie inA.and with the other I make an arche line inC.Afterwarde I open the compas as wide asA.B.(that is as wide as all fiue partes) and set one foote in the .iiij. pricke, which isE,drawyng an arch line with the other foote inC.also. Then do I draw thence a line vntoA,and so haue I doone. But and if the line be to shorte to be parted into fiue partes, I shall deuide it into iij. partes only, as you see the liueF.G,and then makeD.an other line (as isK.L.) whiche I deuide into .v. suche diuisions, asF.G.containeth .iij, then open I thecompassas wide as .iiij. partes (whiche isK.M.) and so set I one foote of the compas inF,and with the other I drawe an arch lyne towardH,then open I the cõpas as wide asK.L.(that is all .v. partes) and set one foote inG,(that is the iij. pricke) and with the other I draw an arch line towardH.also: and where those .ij. arch lines do crosse (whiche is byH.) thence draw I a line vntoF,and that maketh a very plumbe line toF.G,as my desire was. The maner of workyng of this conclusion, is like to the second conlusion, but the reason of it doth depẽd of the .xlvi. proposiciõ of yefirst boke of Euclide. An other waieyet. setone foote of the compas in the prick, on whiche you would haue the plumbe line to light, and stretche forth thother foote toward the longest end of the line, as wide as you can for the length of the line, and so draw a quarter of a compas or more, then without stirryng of the compas, set one foote of it in the same line, where as the circular line did begin, and extend thother in the circular line, settyng a marke where it doth light, then take half that quantitie more there vnto, and by that prick that endeth the last part, draw a line to the pricke assigned, and it shall be a perpendicular.

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A.B.is the line appointed, to whiche I must make a perpendicular line to light in the pricke assigned, which isA.Therfore doo I set one foote of the compas inA,and extend the other vntoD.makyng a part of a circle,more then a quarter, that isD.E.Then do I set one foote of the compas vnaltered inD,and stretch the other in the circular line, and it doth light inF,this space betweneD.andF.I deuide into halfe in the prickeG,whiche halfe I take with the compas, and set it beyondF.vntoH,and thefore isH.the point, by whiche the perpendicular line must be drawn, so say I that the lineH.A,is a plumbe line toA.B,as the conclusion would.

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Open your compas as so wide that it may extend somewhat farther, thẽ from the prick to the line, then sette the one foote of the compas in the pricke, and with the other shall you draw a cõpassed line, that shall crosse that other first line in .ij. places.Now if you deuide that arch line into .ij. equall partes, and from the middell pricke therof vnto the prick without the line you drawe a streight line, it shalbe a plumbe line to that firste lyne, accordyng to the conclusion.

C.is the appointed pricke, from whiche vnto the lineA.B.I must draw a perpẽdicular. Thefore I open the cõpas so wide, that it may haue one foote inC,and thother to reach ouer the line, and with ytfoote I draw an arch line as you see, betweneA.andB,which arch line I deuide in the middell in the pointD.Then drawe I a line fromC.toD,and it is perpendicular to the lineA.B,accordyng as my desire was.

Mark first the prick where yeplũbe line shal lyght: and prick out of ech side of it .ij. other poinctes equally distant from that first pricke. Then set the one foote of the cõpas in one of those side prickes, and the other foote in the other side pricke, and first moue one of the feete and drawe an arche line ouer the middell pricke, then set the compas steddie with the one foote in the other side pricke, and with the other foote drawe an other arche line, that shall cut that first arche, and from the very poincte of their meetyng, drawe a right line vnto the firste pricke, where you do minde that the plumbe line shall lyghte. And so haue you performed thintent of this conclusion.

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The arche of the circle on whiche I would erect a plumbe line, isA.B.C.andB.is the pricke where I would haue the plumbe line to light. Therfore I meate out two equall distaunces on eche side of that prickeB.and they areA.C.Then open I the compas as wide asA.C.and settyng one of the feete inA.with the other I drawe an arche line which goeth byG.Like waies I set one foote of the compas steddily inC.and with the other I drawe an arche line, goyng byG.also. Now consideryng thatG.is the pricke of their meetyng, it shall be also the poinct fro whiche I must drawe the plũbe line. Then draw I a right line fromG.toB.and so haue mine intent. Now asA.B.C.hath a plumbe line erected on hisvtter bought, so may I erect a plumbe line on the inner bught ofD.E.F,doynge with it as I did with the other, that is to saye, fyrste settyng forthe the pricke where the plumbe line shall light, which isE,and then markyng one other on eche syde, as areD.andF.And then proceding as I dyd in the example before.

Deuide the corde of that line info ij. equall portions, and then from the middle prycke erecte a plumbe line, and it shal parte that arche in the middle.

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The arch to be diuided ysA.D.C,the corde isA.B.C,this corde is diuided in the middle withB,from which prick if I erect a plum line asB.D,thẽ will it diuide the arch in the middle, that is to say, inD.

To do the same thynge other wise. And for shortenes of worke, if you wyl make a plumbe line without much labour, you may do it with your squyre, so that it be iustly made, for yf you applye the edge of the squyre to the line in which the prick is, and foresee the very corner of the squyre doo touche the pricke. And than frome that corner if you drawe a lyne by the other edge of the squyre, yt will be perpendicular to the former line.

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A.B.is the line, on which I wold make the plumme line, or perpendicular. And therefore I marke the prick, from which the plumbe lyne muste rise, which here isC.Then do I sette one edg of my squyre (that isB.C.) to the lineA.B,so at the corner of the squyre do toucheC.iustly. And fromC.I drawe a line by the other edge of the squire, (which isC.D.) And so haue I made the plumme lineD.C,which I sought for.

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If so be it that you haue an arche of suche greatnes, that your squyre wyll not suffice therto, as the arche of a brydge or of a house or window, then may you do this. Mete vnderneth the arch where yemidle of his cord wyl be, and ther set a mark.Then take a long line with a plummet, and holde the line in suche a place of the arch, that the plummet do hang iustely ouer the middle of the corde, that you didde diuide before, and then the line doth shewe you the middle of the arche.

The arch isA.D.B,of which I trye the midle thus. I draw a corde from one syde to the other (as here isA.B,) which I diuide in the middle inC.Thẽ take I a line with a plummet (that isD.E,) and so hold I the line that the plummetE,dooth hange ouerC,Andthen I say thatD.is the middle of the arche. And to thentent that my plummet shall point the more iustely, I doo make it sharpe at the nether ende, and so may I trust this woorke for certaine.

Take the iuste measure beetwene the line and the pricke, accordinge to which you shal open your compasse. Thẽ pitch one foote of your compasse at the one ende of the line, and with the other foote draw a bowe line right ouer the pytche of the compasse, lyke-wise doo at the other ende of the lyne, then draw a line that shall touche the vttermoste edge of bothe those bowe lines, and it will bee a true parallele to the fyrste lyne appointed.

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A.B,is the line vnto which I must draw an other gemow line, which muste passe by the prickC,first I meate with my compasse the smallest distance that is fromC.to the line, and that isC.F,wherfore staying the compasse at that distaunce, I seete the one foote inA,and with the other foot I make a bowe lyne, which isD,thẽ like wise set I the one foote of the compasse inB,and with the other I make the second bow line, which isE.And then draw I a line, so that it toucheth the vttermost edge of bothe these bowe lines, and that lyne passeth by the prickeC,end is a gemowe line toA.B,as my sekyng was.

If you do remember the first and seconde conclusions, then is there no difficultie in this, for it is in maner the same woorke. First cõsider the .iij. lines that you must take, and set one of thẽ for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.

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I haue .iij.A.B.andC.D.andE.F.of whiche I put.C.D.for my ground line, then with my compas I take the length of.A.B.and set the one foote of my compas inC,and draw an arch line with the other foote. Likewaies I take the lẽgth ofE.F,and set one foote inD,and with the other foote I make an arch line crosse the other arche, and the pricke of their metyng (whiche isG.) shall be the thirde corner of the triangle, for in all suche kyndes of woorkynge to make a tryangle, if you haue one line drawen, there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee, for two of them must needes be at the two eandes of the lyne that is drawen.

Fyrste draw a line against the corner assigned, and so is it a triangle, then take heede to the line and the pointe in it assigned, and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned, and if it bee longe enoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accordinge to the laste conlusion, makynge a triangle of thre like lynes to that assigned triangle. If it bee not longe inoughe, thenn lengthen it fyrste, and afterwarde doo as I haue sayde beefore.

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Lette the angle appoynted beeA.B.C,and the corner assigned,B.Farthermore let the lymited line beeD.G,and the pricke assignedD.

Fyrste therefore by drawinge the lineA.C,I make the triangleA.B.C.

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Then consideringe thatD.G,is longer thanneA.B,you shall cut out a line frõD.towardG,equal toA.B,as for exãpleD.F.Thẽ measure oute the other ij. lines and worke with thẽ according as the conclusion with the fyrste also and the second teacheth yow, and then haue you done.

First make a plumbe line vnto your line appointed, whiche shall light at one of the endes of it,accordyng tothe fifth conclusion, and let it be of like length as your first line is, then opẽ your compasse to the iuste length of one of them, and sette one foote of the compasse in the ende of the one line, and with the other foote draw an arche line, there as you thinke that the fowerth corner shall be, after that set the one foote of the same compasse vnsturred, in the eande of the other line, and drawe an other arche line crosse the first archeline, and the poincte that they do crosse in, is the pricke of the fourth corner of the square quadrate which you seke for, therfore draw a line from that pricke to the eande of eche line, and you shall therby haue made a square quadrate.

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A.B.is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plũbe line vnto it, whiche shall lighte inA,and that plũb line isA.C,then open I my compasse as wide as the length ofA.B,orA.C,(for they must be bothe equall) and I set the one foote of thend inC,and with the other I make an arche line nigh vntoD,afterward I set the compas again with one foote inB,and with the other foote I make an arche line crosse the first arche line inD,and from the prick of their crossyng I draw .ij. lines, one toB,and an other toC,and so haue I made the square quadrate that I entended.

First from one of the angles of the triangle, you shall drawe a gemowe line, whiche shall be a parallele to that syde of the triangle, on whiche you will make that likeiamme. Then on one end of the side of the triangle, whiche lieth against the gemowe lyne, you shall draw forth a line vnto the gemow line, so that one angle that commeth of those .ij. lines be like to the angle which is limited vnto you. Then shall you deuide into ij. equall partes that side of the triangle whiche beareth that line, and from the pricke of that deuision, you shall raise an other line parallele to that former line, and continewe it vnto the first gemowe line, and thẽ of those .ij. last gemowe lynes, and the first gemowe line, with the halfe side of the triangle, is made a lykeiamme equall to the triangle appointed, and hath an angle lyke to an angle limited, accordyng to the conclusion.

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B.C.G,is the triangle appoincted vnto, whiche I muste make an equall likeiamme. AndD,is the angle that the likeiamme must haue. Therfore first entendyng to erecte the likeiãme on the one side, that the ground line of the triangle (whiche isB.G.) I do draw a gemow line byC,and make it parallele to the ground lineB.G,and that new gemow line isA.H.Then do I raise a line fromB.vnto the gemowe line, (whiche line isA.B) and make an angle equall toD,that is the appointed angle (accordyng as the .viij. cõclusion teacheth)and that angle isB.A.E.Then to procede, I doo parte in yemiddle the said groũd lineB.G,in the prickF,frõ which prick I drawto the first gemowe line (A.H.) an other line that is parallele toA.B,and that line isE.F.Now saie I that the likeiãmeB.A.E.F,is equall to the triangleB.C.G.And also that it hath one angle (that isB.A.E.)like toD.the angle that was limitted. And so haue I mine intent. The profe of the equalnes of those two figures doeth depend of the .xli. proposition of Euclides first boke, and is the .xxxi. proposition of this second boke of Theoremis, whiche saieth, that whan a tryangle and a likeiamme be made betwene .ij. selfe same gemow lines, and haue their ground line of one length, then is the likeiamme double to the triangle, wherof it foloweth, that if .ij. suche figures so drawen differ in their ground line onely, so that the ground line of the likeiamme be but halfe the ground line of the triangle, then be those .ij. figures equall, as you shall more at large perceiue by the boke of Theoremis, in ye.xxxi. theoreme.

In the last conclusion the sides of your likeiamme wer left to your libertie, though you had an angle appoincted. Nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted, which must be the side of the likeiãme. Therfore thus shall you procede. Firste accordyng to the laste conclusion, make a likeiamme in the angle appoincted, equall to the triangle that is assigned. Then with your compasse take the length of your line appointed, and set out two lines of the same length in the second gemowe lines, beginnyng at the one side of the likeiamme, and by those two prickes shall you draw an other gemowe line, whiche shall be parallele to two sides of the likeiamme. Afterward shall you draw .ij. lines more for the accomplishement of your worke, which better shall beperceaued by a shorte exaumple, then by a greate numbre of wordes, only without example,thereforeI wyl by example sette forth the whole worke.

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Fyrst, according to the last conclusion, I make the likeiammeE.F.C.G,equal to the triangleD,in the appoynted angle whiche isE.Then take I the lengthe of the assigned line (which isA.B,) and with my compas I sette forthe the same lẽgth in the ij. gemow linesN.F.andH.G,setting one foot inE,and the other inN,and againe settyng one foote inC,and the other inH.Afterward I draw a line fromN.toH,whiche is a gemow lyne, to ij. sydes of thelikeiamme. thennedrawe I a line also fromN.vntoC.and extend it vntyll it crosse the lines,E.L.andF.G,which both must be drawen forth longer then the sides of thelikeiamme. andwhere that lyne doeth crosseF.G,there I setteM.Nowe to make an ende, I make an other gemowe line, whiche is parallel toN.F.andH.G,and that gemowe line doth passe by the prickeM,and then haue I done. Now say I thatH.C.K.L,is a likeiamme equall to the triangle appointed, whiche wasD,and is made of a line assigned that isA.B,forH.C,is equall vntoA.B,and so isK.L.The profe of yeequalnes of this likeiam vnto the triãgle, depẽdeth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, whẽ there are more then one made about one bias line, the filsquares of euery of them muste needes be equall.

The readiest waye to worke this conclusion, is to tourn that rightlined figure into triangles, and then for euery triangletogetheran equal likeiamme, according vnto the eleuen cõclusion, and then to ioine al those likeiammes into one, if their sides happen to be equal, which thing is euer certain, when al the triangles happẽ iustly betwene one pair of gemowlines. butand if they will not frame so, then after that you haue for the firste triangle made his likeiamme, you shall take the lẽgth of one of his sides, and set that as a line assigned, on whiche you shal make the other likeiams, according to the twelft cõclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so ytyou mai easily ioyne thẽ into one figure.

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If the right lined figure be like vntoA,thẽ may it be turned into triangles that wil stãd betwene ij. parallels anye ways, as youmai sebyC.andD,for ij. sides of both thetriãnglesar parallels. Also if the right lined figure be like vntoE,thẽ wil it be turned into triãgles, liyng betwene two parallels also, as yeother did before, as in the exãple ofF.G.But and if yeright lined figure be like vntoH,and so turned into triãgles as you se inK.L.M,wher it is parted into iij triãgles, thẽ wil not all those triangles lye betwen one pair of parallels or gemow lines, but must haue many, for euery triangle must haue one paire of parallels seuerall, yet it maye happen that when there bee three or fower triangles, ij. of theym maye happen to agre to one pair of parallels, whiche thinge I remit to euery honest witte to serche, for the manner of their draught wil declare, how many paires of parallels they shall neede, of which varietee bicause the examples ar infinite, I haue set forth these few, that by them you may coniecture duly of all other like.

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Further explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then diligẽtly behold how these sundry figures be turned into triãgles. In the fyrst you se I haue made v. triangles, and fourparalleles. inthe seconde vij. triangles and foureparalleles. inthe thirde thre triãgles, and fiue parallels, in the iiij. you se fiue triãgles & fourparallels. inthe fift, iiij. triãgles and .iiij. parallels, & in yesixt ther ar fiue triãgles & iiij. paralels. Howbeit a mã maye at liberty alter them into diuers formes of triãgles & therefore Ileue it to the discretion of the woorkmaister, to do in al suche cases as he shal thinke best, for by these examples (if they bee well marked) may all other like conclusions be wrought.

First deuide your lyne into ij. equal parts, and of the length of one part make a perpendicular to light at one end of your lineassigned. thenadde a bias line, and make thereof a triangle, this done if you take from this bias line the halfe lengthe of your line appointed, which is the iuste length of your perpendicular, that part of the bias line whiche dothe remayne, is the greater portion of the deuision that you seke for, therefore if you cut your line according to the lengthe of it, then will the square of that greaterportionbe equall to the square that is made of the whole line and his lesser portion. And contrary wise, the square of the whole line and his lesser parte, wyll be equall to the square of the greater parte.

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A.B,is the lyne assigned.E.is the middle pricke ofA.B, B.C.is the plumb line or perpendicular, made of the halfe ofA.B,equall toA.E,otherB.E,the byas line isC.A,from whiche I cut a peece, that isC.D,equall toC.B,and accordyng to the lengthelothepeece that remaineth (whiche isD.A,) I doo deuide the lineA.B,at whiche diuision I setF.Now say I, that this lineA.B,(wchwas assigned vnto me) is so diuided in this pointF,ytyesquare of yehole lineA.B,& of the one portiõ (ytisF.B,thelesser part) is equall to the square of the other parte, whiche isF.A,and is the greater part of the first line. The profe of this equalitie shall you learne by the .xl. Theoreme.

There are two ways to make this Example work:—transpose E and F in the illustration, and change one occurrence of E to F in the text,or:—keep the illustration as printed, and transpose all other occurrences of E and F in the text.

First make a likeiamme equall to that right lined figure, with a right angle, accordyng to the .xi. conclusion, then consider the likeiamme, whether it haue all his sides equall, or not: for yf they be all equall, then haue you doone yourconclusion. butand if the sides be not all equall, then shall you make one right line iuste as long as two of those vnequall sides, that line shall you deuide in the middle, and on that pricke drawe half a circle, then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme, and from that pointe of diuision shall you erecte a perpendicular, which shall touche the edge of the circle. And that perpendicular shall be the iuste side of the square quadrate, equall both to the lykeiamme, and also to the right lined figure appointed, as the conclusion willed.

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K,is the right lined figure appointed, andB.C.D.E,is the likeiãme, with right angles equall vntoK,but because that this likeiamme is not a square quadrate, I must turne it into such one after this sort, I shall make one right line, as long as .ij. vnequall sides of the likeiãme, that line here isF.G,whiche is equall toB.C,andC.E.Then part I that line in the middle in theprickeM,and on that pricke I make halfe a circle, accordyng to the length of the diameterF.G.Afterward I cut awaie a peece fromF.G,equall toC.E,markyng that point withH.And on that pricke I erecte a perpendicularH.K,whiche is the iust side to the square quadrate that I seke for, therfore accordyng to the doctrine of the .x. conclusion, of the lyne I doe make a square quadrate, and so haue I attained the practise of this conclusion.

First drawe a right line equall to the side of one of the quadrates: and on the ende of it make a perpendicular, equall in length to the side of the other quadrate, then drawe a byas line betwene those .ij. other lines, makyng thereof a right angeled triangle. And that byas lyne wyll make a square quadrate, equall to the other .ij. quadrates appointed.

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A.B.andC.D,are the two square quadrates appointed, vnto which I must make one equall square quadrate. First therfore I dooe make a righte lineE.F,equall to one of the sides of the square quadrateA.B.And on the one end of it I make a plumbe lineE.G,equall to the side of the other quadrateD.C.Then drawe I a byas lineG.F,which beyng made the side of a quadrate(accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrateH.is muche iust as the other two. I meaneA.B.andD.C.

Determine with your selfe about whiche quadrate you wil make the squire, and drawe one side of that quadrate forth in lengte, accordyng to the measure of the side of the other quadrate, whiche line you maie call the grounde line, and then haue you a right angle made on this line by an other side of the same quadrate: Therfore turne that into a right cornered triangle, accordyng to the worke in the laste conclusion, by makyng of a byas line, and that byas lyne will performe the worke of your desire. For if you take the length of that byas line with your compasse, and then set one foote of the compas in the farthest angle of the first quadrate (whiche is the one ende of the groundline) and extend the other foote on the same line, accordyng to the measure of the byas line, and of that line make a quadrate, enclosyng yefirst quadrate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.

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The first square quadrate isA.B.C.D,and the seconde isE.Now would I make a squire about the quadrateA.B.C.D,whiche shall bee equall vnto the quadrateE.

Therforefirst I draw the lineA.D,more in length, accordyng to the measure of the side ofE,as you see, fromD.vntoF,and so the hole line of bothe these seuerall sides isA.F,thẽ make I a byas line fromC,toF,whiche byas line is the measure of thiswoorke. whereforeI open my compas accordyng to the length of that byas lineC.F,and set the one compas foote inA,and extend thother foote of the compas towardF,makyng this prickeG,from whiche I erect a plumbelineG.H,and so make out the square quadrateA.G.H.K,whose sides are equall eche of them toA.G.And this square doth contain the first quadrateA.B.C.D,and also a squireG.H.K,whiche is equall to the second quadrateE,for as the last conclusion declareth, the quadrateA.G.H.K,is equall to bothe the other quadrates proposed, that isA.B.C.D,andE.Then muste the squireG.H.K,needes be equall toE,consideryng that all the rest of that great quadrate is nothyng els but the quadrate self,A.B.C.D,and so haue I thintent of this conclusion.

Draw a corde or stryngline crosse the circle, then deuide into .ij. equall partes, both that corde, and also the bowe line, or arche line, that serueth to that corde, and from the prickes of those diuisions, if you drawe an other line crosse the circle, it must nedes passe by the centre. Therfore deuide that line in the middle, and that middle pricke is the centre of the circle proposed.

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Let the circle beA.B.C.D,whose centre I shall seke. First therfore I draw a corde crosse the circle, that isA.C.Then do I deuide that corde in the middle, inE,and likewaies also do I deuide his arche lineA.B.C,in the middle, in the pointeB.Afterward I drawe a line fromB.toE,and so crosse thecircle, whiche line isB.D,in which line is the centre that I seeke for. Therefore if I parte that lineB.D,in the middle in to two equall portions, that middle pricke (which here isF) is the verye centre of the sayde circle that I seke. This conclusion may other waies be wrought, as the moste part of conclusions haue sondry formes of practise, and that is, by makinge thre prickes in the circũference of the circle, at liberty where you wyll, and then findinge the centre to those threpricks, Whichworke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.

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 circumference of al circles is compassed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. prickes in an exact right line can not bee brought into the circumference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their cõmon centre. Opẽ your compas so wide, that it be somewhat more then thehalfe distance of two of those prickes. Then sette the one foote of the compas in the one pricke, and with the other foot draw an arche lyne toward the otherpricke, Thenagaine putte the foot of your compas in the second pricke, 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 pricks, so do with the middle pricke, 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.

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The iij. prickes I haue set to beA.B,andC,whiche I wold bring into the edg of one common circle, by finding a centre cõmen to them all, fyrst therefore I open my cõpas, so that thei occupye more then yehalfe distance betwene ij. pricks (as areA.B.) and so settinge one foote inA.and extendinge the other towardB,I make the arche lineD.E.Likewise settĩg one foot inB,and turninge the other towardA,I draw an other arche line that crosseth the first inD.andE.Then fromD.toE,I draw a right lyneD.H.After this I open my cõpasse to a new distance, and make ij. arche lines betweneB.andC,whiche crosse one the other inF.andG,by whiche two pointes I draw an other line, that isF.H.And bycause that the lyneD.H.and the lyneF.H.doo meete inH,I saye thatH.is the centre that serueth to those iij. prickes. Now therfore if you set one foot of your compas inH,and extend the other to any of the iij. pricks, you may draw a circle wchshal enclose those iij. pricks in the edg of his circũferẽce & thus haue you attained yevse of this cõclusiõ.

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 frõ 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 atoucheline, drawen from the point assigned, according to the meaning of this conclusion.

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Let the circle be calledB.C.D,and his cẽtreE,and yeprick assignedA,opẽ your cõpas now of such widenes, ytthe one foote may be set inE,wchis yecẽtre of yecircle, & yeother inA,wchis yepointe assigned, & so make an other greter circle (as here isA.F.G) thẽ draw a line fromA.vntoE,and wher that line doth cross yeinner circle (wchheere is in the prickB.) there erect a plũb line vnto the line.A.E.and let that plumb line touch the vtter circle, as it doth here in the pointF,so shallB.F.bee that plumbe lyne. Then fromF.vntoE.drawe an other line whiche shal beF.E,and it will cutte the inner circle, as it doth here in the pointC,from which pointeC.if you erect a plumb line vntoA,then is that lineA.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 isE,the pricke assigned isA.and the rulerC.D.by which the touch line is drawen, and that isA.B,and as this way is light to doo, so is it certaine inoughe for any kinde of workinge.

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.

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A peece of an olde pillar was found, like in forme to thys figureA.D.B.Now to knowe howe muche the cõpasse of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make inatable the like draught of ytcircũference by the self patrõ, vsing it as it wer a croked ruler.Then make .iij. prickes in that arche line, as I haue made,C. D.andE.And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that cẽtre is hereF,nowe settyng one foote of your compas inF,and the other inC. D,other inE,and so makyng a compasse, you haue youre whole intent.

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.

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The arche of the circle isA.B.C,vnto whiche I must sekea centre, therfore firste I do deuide it into .ij. partes, the one of them isA.B,and the other isB.C.Then doe I cut euery arche in the middle, so isE.the middle ofA.B,andG.is the middle ofB.C.Likewaies, I take the middle of their cordes, whiche I mark withF.andH,settyngF.byE,andH.byG.Then drawe I a line fromE.toF,and fromG.toH,and they do crosse inD,wherefore saie I, thatD.is the centre, that I seke for.

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 thecircle, Andthat circle is drawen in a quadrate, whose circumference doeth touche euery side of the quadrate, and lykewaies of other figures.

In these .ij. last figuresE.andF,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 coũted 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.

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The triangle isA.B.C,whose sides I do part into .ij. equall partes, eche by it selfe in these pointesD.E.F,puttyngF.betweneA.B,andD.betweneB.C,andE.betweneA.C.Then draw I a line fromC.toF,and an other fromA.toD,and the third fromB.toE.

Andwhere all those lines do mete (that is to saieM.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.see textAnd so drawe lines frõ 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 lẽgth of the perpendicular, and so drawe your circle.

The triangle isA.B.C,whose corners I haue diuided in the middle withD.E.F,and haue drawen the lines of diuisionA.D, B.E,andC.F,which crosse inG,therfore shallG.be the common centre. Then make I one perpẽdicular fromG.vnto the sideB.C,and thatisG.H.Now sette I one fote of the compas inG,and extend the other foote vntoH.and so drawe a compas, whiche wyll iustly answere to that triãgle according to the meaning of the conclusion.

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.

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A.B.C.is the triangle, whose two sidesA.C.andB.C.are diuided into two equall partes inD.andE,settyngD.betweneB.andC,andE.betweneA.andC.And from eche of those two pointes is ther erected a perpendicular (as you seD.F,andE.F.) which mete, and crosse inF,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 threecorners 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.

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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 iijprickes. that centre asyou se isD,for inD.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 triãgle, and consequentlye the triangle itselfe, as the conclusion dydde intende.

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, thenmustI 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 wtthe other halfe of the touche line, equall to an other corner of the triangle appointed, and then betwen those two cornerswill 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.

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A.B.C,is the triangle appointed, andF.G.H.is the circle, in which I muste make an other triangle, with lyke angles to the angles ofA.B.C.the triangle appointed. Therefore fyrst I make the touch lyneD.F.E.And then make I an angle inF,equall toA,whiche is one of the angles of the triangle. And the lyne that maketh that angle with the touche line, isF.H,whiche I drawe in lengthe vntill it touche the edge of the circle. Then againe in the same pointF,I make an other corner equall to the angleC.and the line that maketh that corner with the touche line, isF.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 ytone pointF,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 equallto 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 cõmon sentence saith. Thẽ do I draw a line fromeG.toH,and that line maketh a triangleF.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.

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, thẽ 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 drawẽ about a circle apointed, as the cõclusiõ did wil.

A.B.C,is the triangle assigned, andG.H.K,is the circle appointed, about which I muste make a triangle hauing equall angles to the angles of that triangleA.B.C.Fyrst therefore I drawA.C.(which is one of the sides of the triangle) in length that there may appeare two vtter angles in that triangle, as you seB.A.D,andB.C.E.

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Thendrawe I in the circle appointed a semidiameter, which is hereH.F,forF.is the cẽtre of the circleG.H.K.Then make I on that centre an angle equall to the vtter angleB.A.D,and that angle isH.F.K.Like waies on the same cẽtre by drawyng an other semidiameter, I make an other angleH.F.G,equall to the second vtter angle of the triangle, whiche isB.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 trianagleL.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 cornerM.is equall toC.LikewaiesL.toA,andN.toB,whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.

The angle appointed, maie be a sharpe angle, a right angle, other a blunte angle, so that the worke must be diuersely handeledaccording 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 wyllsheweyou 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,anyeof 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 seuẽ and twẽty coclusiõ, 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 triãgle be equal to yeãgle assigned as youre selfe mai easily gesse.

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First for example of a sharpe ãgle letA.stãd &B.Cshal be yelyne assigned. Thẽ do I make a triangle, by addingB.C,as a thirde side to those other ij. which doo include the ãgle assigned, and that triãgle isD.E.F,so ytE.F.is the line appointed, andD.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, fromE.a long byD.vntoF,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, andH.K.is the lyne appointed, andL.M.N.the portion of the circle aunsweryng thereto. In the third exaumple,O.is the blunte corner assigned,P.Q.is the line, andR.S.T.is the portion of the circle, that containeth that blũt corner, and is drawen onR.T.the line appointed.

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.

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A.is the angle appointed, andD.E.F.is the circle assigned, frõ which I must cut away a portiõ that doth contain an angleequall to this angleA.Therfore first I do draw a touche line to the circle assigned, and that touch line isB.C,the very pricke of the touche isD,from whicheD.I drawe a lyneD.E,so that the angle made of those two lines be equall to the angle appointed. Then say I, that the arch of the circleD.F.E,is the arche that I seke after. For if I doo deuide that arche in the middle (as here is done inF.) and so draw thence two lines, one toD,and the other toE,then will the angleF,be equall to the angle 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.

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A.B.C.D.is the circle assigned, andA.C.andB.D.are the two diameters which crosse in the centreE,and make .iiij. right corners. Then do I make fowre other lines, that isA.B, B.C, C.D,andD.A,which do ioyne together the fowre endes of the ij. diameters. And so is the squarequadrate made in the circle assigned, as the conclusion willeth.

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.

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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 areA.C.andB.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 inA,then with the other foote I doo make twoo arche lines, one inE,and an other inF.Then sette I the one foote of the compasse inB,and drawe twoo arche linesF.andG.Like wise setting the compasse foote inC,I drawe twoo other arche lines,G.andH,and onD.I make twoo other,H.andE.Then frome the crossinges of those eighte arche lines I drawe iiij. straighte lynes, that is to saye,E.F,andF.G,alsoG.H,andH.E,whiche iiij. straighte lynes do make the square quadrate that I should draw about the circle assigned.

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.

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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. cõtrary prickes, as you seE.G,andF.H,whiche mete inK,and therfore shalK,be the centre of the circle. Then do I set one foote of the compas inK.and opẽ the other as wide asK.E,and so draw a circle, which is madeaccordingeto the conclusion.

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. Thẽ 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.

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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 seA.C.&B.D.And where they ij. do crosse (that is to say inE.) 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 exãple toA,and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.

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 thoselines, Thendrawe the thyrde lyne to make a triangle with those other twoo, and you haue doone according to the conclusion, and haue made a twelike triãgle,whose ij. corners about the grounde line, are eche of theym double to the other corner.

At no point in this or the accompanying book does the author show how to divide a circle into five.

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

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.

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Yow se here this circleA.B.C.D.E.deuided into fiue equall portions. And from eche pricke ij.lines drawento the other ij. nexte prickes, so fromA.are drawen ij. lines, one toB,and the other toE,and so fromC.one toB.and an othertoD,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.

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.

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A.B.C.D.E.is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is drawẽ a semidiameter, as you see. Then doo I make a touche line in the prickeB,whiche isF.G,making ij. right angleswith the semidiameterB,and lyke waies onC.is madeG.H,onD.standethH.K,and onE,is setK.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.

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.

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The cinkeangle assigned isA.B.C.D.E,in whiche I mustemake a circle, wherefore I draw a right line from the one angle (as fromB,) to the middle of the contrary side (whiche isE.D,) and that middle pricke isF.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 isB.C.) and that pricke isG.Nowe because that these two lines do crosse inH,I saie thatH.is the centre of the circle, whiche I would make. Therfore I set one foote of the compasse inH,and extend the other foote vntoG,orF.(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 cõclusion meaneth.

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.

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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 frõE.toG,and the other frõC.toF,and because thei domeete inH,I saye thatH.is the centre of the circle that I woulde haue, wherfore I sette one foote of the compasse inH.and extende the other to one corner (whiche happeneth fyrste, for all are like distaunte fromH.) 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.

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 thecircũference 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 shalperceauethat 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.

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The circle isB.C.D.E.F.G,whose centre I finde to beeA.Therefore I sette one foote of the compas inA,and do extẽd the other foote toB,thereby takinge the semidiameter. Then sette I one foote of the compas vnremoued inB,and marke with the other foote on eche sideC.andG.Then fromC.I markeD,and frõD,E: fromE.marke IF.And then haue I but one space iuste vntoG.and so haue I made a iuste siseangle of equall sides and equall angles, in a circle appointed.

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.

This rule is generall, that how many sides the figure shallhaue, 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, andthen eche of them into three partes againe: Or elsfirst deuide it into three partes, and then echof thẽ into fiue other partes, as youlist, and canne most readilye.Then draw lines betweneeuery two prickesthat benighesttogither, andther wil appear rightly drawẽ the figure, of fiftene sides, andangles equall. And so do with any other figureof what numbre of sides so euer it bee.

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