The pathvvay to knowledg containing the first principles of geometrie, as they may moste aptly be applied vnto practise, bothe for vse of instrumentes geometricall, and astronomicall and also for proiection of plattes in euerye kinde, and therefore much necessary for all sortes of men.

THE PRACTIKE WORKINGE OF sondry conclusions Geometrical.

THE FYRST CONCLVSION.

To make a threlike triangle or any lyne measurable.

TAKE THE IVSTE lēgth of the lyne with your cōpasse, and stay the one foot of the compas in one of the endes of that line, tur¦ning the other vp or doun at your will, drawyng the arche of a circle against the midle of the line, and doo likewise with the same

[illustration] diagram

cōpasse vnaltered, at the other end of the line and wher these ij. croked ly∣nes 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

[illustration] diagram

is long, and draw two arch lines that mete in C, then from C. I draw ij other lines one to A, another to B, and than I haue my purpose.

THE .II. CONCLVSION. If you wil make a twileke or a nouelike triangle on ani cer¦taine line.

Consider fyrst the length that yow will haue the other si∣des to containe, and to that length open your compasse, and

then worke as you did in the threleke triangle, remembryng this, that in a nouelike triangle you must take ij. lengthes be∣syde the fyrste lyne, and draw an arche lyne with one of thē at the one ende, and with the other at the other end, the exāple is as in the o∣ther before.

[illustration] diagram

THE III. CONCL. To diuide an angle of right lines into ij. equal partes.

First open your compasseas largely as you can, so that it do not excede the length of the shortest line y^t incloseth the an∣gle. Then set one foote of the compasse in the verye point of the angle and with the other fote draw a compassed arch frō the one lyne of the angle to the other,

[illustration] diagram

that arch shall you deuide in halfe, and thē draw a line frō the āgle to y^e mid∣dle of y^t arch, and so y^e angle is diuided into ij. equall partes. Example. Let the triāgle be A.B.C, thē set I one foot of y^e cōpasse in B, and with the o∣ther I draw y^e arch D.E, which I part into ij. equall parts in F, and thē draw a line frō B, to F, & so I haue mine intēt

THE IIII. CONCL. To deuide any measurable line into ij. equall partes.

[illustration] diagram

Open your compasse to the iust lēgth of y^e line. And thē set one foote steddely at the one ende of the line, & w^t the other fote draw an arch of a circle against y^e midle of the line, both ouer it, and al∣so vnder it, then doo lyke waise

at the other ende of the line. And marke where those arche lines do meet crossewaies, and betwene those ij. pricks draw a line, and it shall cut the first line in two equall portions.

Example.

The lyne is A. B. accordyng to which I open the compasse and make .iiij. arche lines, whiche meete in C. and D, then drawe I a lyne from C, so haue I my purpose.

This conclusion serueth for makyng of quadrates and squi∣res, 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 firste 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

[illustration] diagram

lynes, as you did in the fyrst conclusion, for making of a threlyke triāgle. then if 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.

Example

The lyne is A.B. the prick on whiche I shoulde make the plumme lyne, is C. then open I the compasse as wyde as A, C, and sette one foote in C. and with the other doo I marke out C.A. and C.B, then open I the compasse as wide as A.B, and make ij. arch lines which do crosse in D, and so haue I doone.

Howe bee it, it happeneth so sommetymes, that the

pricke 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 frō that end, then doth this conclusion require a newe ayde, for the last de∣uise 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 portiōs: 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.

[illustration] diagram

The line is A. B. and A. is the prick, on whi∣che the perpendicular line must light. Therfore I deuide A. B. into fiue partes equall, then do I open the compas to the widenesse of three par∣tes (that is A. D.) and let one foote staie in A. and with the other I make an arche line in C. Afterwarde I open the compas as wide as A.B.

(that is as wide as all fiue partes) and set one foote in the .iiij. pricke, which is E, drawyng an arch line with the other foote in C. also. Then do I draw thence a line vnto A, 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 line F. G, and then make D. an other line (as is K. L.) whiche I deuide into .v. suche diuisions, as F. G. containeth .iij, then o∣pen I the compaas as wide as .iiij. partes (whiche is K. M.) and so set I one foote of the compas in F, and with the other I drawe an arch lyne toward H, then open I the cōpas as wide as K. L. (that is all .v. partes) and set one foote in G, (that is the iij. pricke) and with the other I draw an arch line toward H. also: and where those .ij. arch lines do crosse (whiche is by H.) thence draw I a line vnto F, and that maketh a very plumbe line to F. G, as my desire was. The maner of workyng of this conclusion, is like to the second conclusion, but the reason of it doth depēd of the .xlvi. proposiciō of y^e first boke of Euclide. An other waie yet. set one 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 circularline did begin, and extend thother in the circular line, settyng a marke where it doth light, then take half that quan¦titie more therevnto, and by that prick that endeth the last part, draw a line to the pricke assigned, and it shall be a perpendicular.

Example.

A. B. is the line appointed, to whi∣che

[illustration] diagram

I must make a perpendicular line to light in the pricke assigned, which is A. Therfore doo I set one foote of the compas in A, and extend the o∣ther vnto D. makyng a part of a cir∣cle

more then a quarter, that is D. E. Then do I set one foote of the compas vnaltered in D, and stretch the other in the cir∣cular line, and it doth light in F, this space betwene D. and F. I deuide into halfe in the pricke G, whiche halfe I take with the compas, and set it beyond F. vnto H, and therfore is H. the point, by whiche the perpendicular line must be drawen, so say I that the line H. A, is a plumbe line to A. B, as the conclu∣sion would.

THE .VI. CONCLVSION. To drawe a streight line from any pricke that is not in a line, and to make it perpendi∣cular to an other line.

Open your compas so wide that it may extend somewhat far∣ther, thē from the prick to the line, then sette the one foote of

[illustration] diagram

the compas in the pricke, and with the other shall you draw a cōpassed 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 ther∣of 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 line A. B. I must draw a perpēdicular. Therfore I open the cōpas so wide, that it may haue one foote in C, and thother to reach ouer the line, and with y^t foote I draw an arch line as you see, betwene A. and B, which arch line I deuide in the middell in the point D. Then drawe I a line from C. to D, and it is perpendicu∣lar to the line A. B, accordyng as my desire was.

THE .VII. CONCLVSION. To make a plumbe lyne or any porcion of a circle, and that on the vtter or inner bughte.

Mark first the prick where y^e plūbe line shal lyght: and prick out on ech side of it .ij. other poinctes equally distant from that first pricke. Then set the one foote of the cōpas 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 o∣ther arche line, that shall cut that first arche, and from the ve∣ry 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.

Example.

The arche of the circle on whiche I would erect a plumbe line, is A. B. C. and B. is the pricke where I would haue the

[illustration] diagram

plumbe line to light. Ther∣fore I meate out two equall distaunces on eche side of that pricke B. and they are A. C. Then open I the com∣pas as wide as A. C. and set∣tyng one of the feete in A. with the other I drawe an arch line which goeth by G. Likewaies I set one foote of the compas steddily in C. and with the other I drawe an arche line, goyng by G. also▪ Now consideryng that G. is the pricke of their meetyng, it shall be also the poinct from whiche I must drawe the plūbe line. Then draw I aright line from G. to B. and so haue mine intent. Now as A. B. C. hath a plumbe line erected on his

vtter bought, so may I erect a plumbe line on the inner bught of D. E. F, doynge with it as I did with the other, that is to saye, fyrste settyng for the the pricke where the plumbe line shall light, which is E, and then markyng one other on eche syde, as are D. and F. And then proceding as I dyd in the ex∣ample before.

THE VIII. CONCLVSYON. How to deuide the arche of a circle into two equall partes, without measuring the arche.

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

Example.

The arch to be diuided ys

[illustration] diagram

A.D.C, the corde is A, B. C, this corde is diuided in the middle with B, from which prick if I erecte a plum line as B.D, thē will it diuide the arch in the middle, that is to say, in D.

THE IX. 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 a perpendicular to the former line.

Example.

A.B. is the line, on which I

[illustration] diagram

wold make the plumme line, or perpendicular. And there∣fore I marke the prick, from which the plumbe lyne muste rise, which here is C. Then do I sette one edg of my squyre (that is B.C.) to the line A.B, so that the corner of the squyre do touche C. iustly. And from C. I drawe a line by the other edge of the squire, (which is C. D.) And so haue I made the plumme line D.C, which I sought for.

THE X. CONCLVSION. How to do the same thinge an other way yet

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 y^e midle 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 o∣uer the middle of the corde,

[illustration] diagram

that you didde diuide before, and then the line doth shewe you the middle of the arche.

Example.

The arch is A.D.B, of which I trye the midle thus. I draw a corde from one syde to the other (as here is A.B,) which I diuide in the middle in C. Thē take I a line with a plum¦met (that is D.E,) and so hold I the line that the plummet E, dooth hange ouer C, And

then I say that D. is the middle of the arche. And to thenien•• 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.

THE XI. 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. Thē 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, lykewise 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.

A.B, is the line vnto which

[illustration] diagram

I must draw an other gemow line, which muste passe by the prick C, first I meate with my compasse the smallest distance that is from C. to the line, and that is C. F, wherfore staying the compasse at that distaunce, I sette the one foote in A, and with the other foot I make a bowe lyne, which is D, thē like wise set I the one foote of the compasse in B, and with the o∣ther I make the second bow line, which is E. And then draw I a line, so that it toucheth the vttermost edge of bothe these bowe lines, and that lyne passeth by the pricke C, end is a ge∣mowe line to A. 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 lon∣ger then the thirde. For this rule is generall, that any two sides of euerie triangle taken to∣gether, are longer then the other side that re∣maineth.

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 cōsiuer the .iij. lines that you must take, and set one of thē for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.

Example.

[illustration] diagram

I haue .iij. lynes. A. B. and C. D. and E. 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 in C, and draw an arch line with the other foote. Likewaies I take the lēgth of E. F, and set one foote in D, and with the other foote I make an arch line crosse the other arche, and the pricke of their me∣tyng (whiche is G.) 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 candes of the lyne that is drawen.

THE XIII. 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 an∣gle, 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 assi∣gned, 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 inoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accor∣dinge to the laste conclusion, makynge a triangle of thre like lynes to that assigned triangle. If it bee not longe inoughe, thenne lengthen it fyrste, and afterwarde doo as I haue sayde beefore.

Example.

Lette the angle appoynted

[illustration] diagram

bee A. B. C, and the corner assi¦gned, B. Farthermore let the lymited line bee D. G, and the pricke assigned D.

Fyrste therefore by drawinge the line A.C, I make the tri∣angle A.B.C.

Then consideringe that D. G, is longer thanne A.B, you shall cut out a line frō D. to∣ward

[illustration] diagram

G, equ••l to A. B, as for ex∣āple D, F. Thē measure oute the other ij. lines and worke with thē according as the conclusion with the fyrste also and the se∣cond teacheth yow, and then haue you done.

THE XIIII. 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 to the fifth con∣clusion, and let it be of like length as your first line is, then opē 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 fo∣werth corner shall be, after that set the one foote of the same compasse vnsturred, in the cande of the other line, and drawe an other arche line crosse the first arche line, 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.

Example.

A. B. is the line

[illustration] diagram

proposed, of whi¦che I shall make a square quadrate, therefore firste I make a plūbe line vnto it, whiche shall lighte in A, and that plūb line is A. B, then open I my compasse as wide as the length of A. B, or B. C, (for they must be bothe e∣quall) and I set the one foote of thend in C, and with the other I make an arche line nigh vnto D, afterward I set the com∣pas again with one foote in B, A and with the other foote I make an arche line crosse the first arche line in D, and from the prick of their crossyng I draw .ij. lines, one to B, A and an other to C, and so haue I made the square quadrate that I entended.

THE .XV. CONCLVSION. To make a likeiāme equall to a triangle ap∣pointed, and that in a right lined āgle 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 ge∣mowe 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 whiche 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 o∣ther line parallele to that former line, and continewe it vnto the first gemowe line, and thē 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.

Example.

[illustration] diagram

B. C. G, is the tri∣angle appoincted vnto, whiche I muste make an e∣quall likeiamme. And D, is the an∣gle that the like∣iamme must haue. Therfore first en∣tendyng to erecte the likeiāme on the one side, that the ground line of the trian∣gle (whiche is B. G.) I do draw a gemow line by C, and make it parallele to the ground line B. G, and that new gemow line is A. H. Then do I raise a line from B. vnto the gemowe line, (whiche line is A. B) and make an angle equall to D, that is the appointed angle (accordyng as the .viij. cōclusion teacheth, and that angle is B. A. E. Then to procede, I doo parte in y^e middle the said groūd line B. G, in the prick F, frō which prick I draw

to the first gemowe line (A. H.) an other line that is parallale to A. B, and that line is E. F. Now saie I that the likeiāme B. A. E. F, is equall to the triangle B. C. G. And also that it hath one angle (that is B. A. E. like to D. the angle that was limitted. And so haue I mine intent. The prose of the equal∣nes of those two figures doeth depend of the .xli. proposition of Euclides first boke, and is the .xxxi. proposition of this se∣cond 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 fi∣gures 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 y^e .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 likeiāme. 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, whiche better shall be

perceaued by a shorte exaumple, then by a greate numbre of wordes, only without example, therefore I wyl by example sette forth the whole worke.

Example.

Fyrst, according to the last

[illustration] diagram

conclusion, I make the like∣iamme E. F. C. G, equal to the triangle D, in the appoynted angle whiche is E. Then take I the lengthe of the assigned line (which is A. B,) and with my compas I sette forthe the same lēgth in the ij. gemow li¦nes N. F. and H. G, setting one foot in E, and the other in N, and againe settyng one foote in C, and the other in H. Af∣terward I draw a line from N. to H, whiche is a gemow lyne, to ij. sydes of the likeiamme. thenne drawe I a line also from N. vnto C, and extend it vntyll it crosse the lines, E. L. and F. G, which both must be drawen forth longer then the si∣des of the likeiamme. and where that lyne doeth crosse F. G, there I sette M. Nowe to make an ende, I make an other ge∣mowe line, whiche is parallel to N. F. and H. G, and that ge∣mowe line doth passe by the pricke M, and then haue I done. Now say I that H. C. K. L, is a likeiamme equall to the trian∣gle appointed, whiche was D, and is made of a line assigned that is A. B, for H. C, is equall vnto A. B, and so is K. L, The prose of y^e equalnes of this likeiam vnto the triāgle, depēdeth of the thirty and two Theoreme: as in the hoke of Theoremes doth appear, where it is declared, that in al likeiammes, whē there are more then one made about one bias line, the filsqua∣res of euery of them muste needes be equall.

THE XVII. 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 right lined figure into triangles, and then for euery triangle to gether an equal likeiamme, according vnto the eleuen cōclusi¦on, and then to ioine al those likeiammes into one, if their si∣des happen to be equal, which thing is euer certain, when al the triangles happē iustly betwene one pair of gemow lines. but and if they will not frame so, then after that you haue for the firste triangle made his likeiamme, you shall take the lēgth of one of his sides, and set that as a line assigned, on whiche you shal make al the other likeiams, according to the twelft cō¦clusion,

[illustration] diagram

and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so y^t you mai¦easily ioyne thē into one figure.

Example.

[illustration] diagram

If the right lined fi∣gure be like vnto A, thē may it be turned into tri¦angles that wil stād be∣twene ij. parallels anye ways, as you maise by C and D, for ij. sides of both

[illustration] diagram

the triāgls ar parallels. Also if the right lined fi¦gure be like vnto E, thē wil it be turned into triā¦gles, liyng betwene two parallels also, as y^e other did before. as in the exā∣ple of F. G. But and if y^e

right lined figure be like vnto H, and so turned into triāgles as you se in K. L. M, wher it is parted into iij triāgles, thē wil not all those triangles lye betwen one pair of parallels or ge∣mow 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 sower triangles, ij. of theym maye hap∣pen 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.

[illustration] diagram

Further explicacion you shal not greatly neede, if you re∣membre what hath ben taught before, and then diligētly be∣hold how these sundry figures be turned into triāgles. In the fyrst you se I haue made v. triangles, and four paralleles. in the seconde vij. triangles and foure paralleles. in the thirde thre triāgles, and fiue parallels, in the iiij. you se fiue triāgles & four parallels. in the fift, iiij. triāgles and .iiij. parallels, & in y^e sixt ther ar fiue triāgles & iiij. paralels. Howbeit a mā maye at liberty alter them into diuers formes of triāgles & therefore I

leue 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.

THE XVIII. 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 line assigned. then adde a bias line, and make thereof a trian∣gle, this done if you take from this bias line the halfe lengthe ol your line appointed, which is the iuste length of your per∣pendicular, that part of the bias line whiche dothe remayne, is the greater portion of the deuision that you seke for, there∣fore if you cut your line according to the lengthe of it, then will the square of that greater portior be equall to the square that is made of the whole line and his lesser portion. And con¦trary wise, the square of the whole line and his lesser parte, wyll be equall to the square of the greater parte.

Example.

A.B, is the lyne assigned. E. is the

[illustration] diagram

middle pricke of A. B, B. C. is the plumb line or perpendicular, made of the halfe of A. B, equall to A.E, other B.E, the byas line is C. A, from whiche I cut a peece, that is C. D, equall to C. B, and accordyng to the lengthe lothe peece that remaineth (whiche is D. A,) I doo deuide the line A.B, at whiche diuision I set F. Now say I, that this line A, B, (w^ch was assigned vnto me) is so diuided in this point F, y^t y^t square of y^e hole line A.B, & of the one portiō (y^t is F.B, the

lesser part) is equall to the square of the other parte, whiche is F.A, and is the greater part of the first line. The profe of this equalitie shall you learne by the .xl. Theoreme.

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 .xv. conclusion, then consider the likeiamme, whether it haue all his sides equall, or not: for yf they be all equall, then haue you doone your conclusion. but and 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 cir∣cle, 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 lyke∣iamme, and also to the right lined figure appointed, as the con∣clusion willed.

Example.

[illustration] diagram

[illustration] diagram

K, is the right lined figure appointed, and B. C. D. E, is the likeiāme, with right angles equall vnto K, 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 like∣iāme, that line here is F.G, whiche is equall to B. C, and C. E. Then part I that line in the middle in the

pricke M, and on that pricke I make halfe a circle, accordyng to the length of the diameter F. G. Afterward I cut awaie a peece from F. G, equall to C. E, markyng that point with H. And on that pricke I erecte a perpendicular H. K, whiche is the iust side to the square quadrate that I seke for, therfore ac∣cordyng to the doctrine of the .x. conclusion, of that 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 qua∣drates: 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, e∣quall to the other .ij. quadrates appointed.

Example.

A.B. and C.D, are the two

[illustration] diagram

square quadrates appointed, vnto which I must make one equall square quadrate. First therfore I dooe make a righte line E.F, equall to one of the sides of the square quadrate A.B. And on the one end of it I make a plumbe line E. G, e∣quall to the side of the other quadrate D. C. Then drawe I a byas line G.F, whiche be∣yng made the side of a qua∣drate

(accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrate H. is as muche iust as the other two. I meane A. B. and D. C.

THE XXI. CONCLVSION. when any two quadrates be set forth, howe to make a squire about the one quadrate, whi∣che 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 qua∣drate, 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 tri∣angle, accordyng to the worke in the laste conclusion, by ma∣kyng 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 far∣thest angle of the first quadrate (whiche is the one ende of the groundline) and extend the other foote on the same line, accor∣dyng to the measure of the byas line, and of that line make a quadrate, enclosyng y^e first qua¦drate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.

[illustration] diagram

Example.

The first square quadrate is A. B. C. D, and the seconde is E. Now would I maked squire about the quadrate A. B. C. D, whiche shall bee equall vnto the quadrate E.

[illustration] diagram

Therfore first I draw the line A.D, more in length, accordyng to the measure of the side of E, as you see, from D. vnto F, and so the hole line of bothe these seuerall sides is A. F, thē make I a byas line from C, to F, whiche byas line is the measure of this woorke. wherefore I open my compas accordyng to the length of that byas line C. F, and set the one compas foote in A, and extend thother foote of the compas toward F, makyng this pricke G, from whiche I erect a plumbe line G. H, and so make out the square quadrate A. G. H. K, whose sides are e∣quall eche of them to A. G. And this square doth contain the first quadrate A. B. C. D, and also a squire G.H.K, whiche is equall to the second quadrate E, for as the last conclusion de∣clareth, the quadrate A. G.H. K, is equall to bothe the other quadrates proposed, that is A. B.C.D, and E. Then muste the squire G.H.K, needes be equall to E, 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 .XXI. CONCLVSION. To find out the cētre of any circle assigned.

Draw a corde or stryng line crosse the circle, then deuide in∣to .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 pro∣posed.

Example.

Let the circle be A.B.C.D, whose centre I shall seke. First therfore I draw a corde crosse the circle, that is A. C. Then do I deuide that corde in the middle, in E, and likewaies also do I deuide his arche line A.B.C, in the middle, in the pointe B. Afterward I drawe a line from B. to E, and so crosse the

circle, whiche line is B. D, in

[illustration] diagram

which line is the centre that I seeke for. Therefore if I parte that line B.D, in the middle in to two equall portions, that middle pricke (whiche here is F) 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 ma∣kinge thre prickes in the circū¦ference of the circle, at liberty where you wyll, and then fin∣dinge the centre to those thre prickes, Which worke bicause it serueth for sondry vses, I thinke meet to make it a seuerall con¦clusion by it selfe.

THE XXIII. CONCLVSION. To find the commen centre belongyng to anye three prickes appointed, if they be not in an ex¦acte 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 alwaies 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 cōmon centre. Opē your compas so wide, that it be some what more then the

halfe 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 syne toward the other pricke, Then againe 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 prickes, 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 cen∣tre that you seeke for.

Example

The ij. prickes I haue set

[illustration] diagram

to be A. B, and C, whiche I wold bring into the edg of one common circle, by finding a centre cōmen to them all, fyrst therefore I open my cōpas, so that thei occupye more then y^e halfe distance betwene ij. pricks (as are A. B.) and so set∣tinge one foote in A. and extendinge the other to∣ward B, I make the arche line D.E. Likewise settig one foot in B, and turninge the other toward A, I draw an other arche line that crosseth the first in D. and E. Then from D. to E, I draw a right lyne D.H. After this I open my cōpasse to a new distance, and make ij. arche lines betwene B. and C, whiche crosse one the other in F. and G, by whiche two pointes I draw an other line, that is F. H. And bycause that the lyne D. H. and the lyne F. H. doo meete in H, I saye that H. is the centre that serueth to those iij. prickes. Now therfore if you set one foot of your compas in H, and extend the other to any of the iij. pricks, you may draw a circle w^ch shal enclose those iij. pricks in the edg of his circūferēce, & thus haue you attained y^e vse of this cōclusiō

THE XXIIII. CONCLVSION. To drawe a touche line vnto 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 cen∣tre: and it shall gouerne you certainly in makyng the said tou∣che line. For if you draw a line frō the pricke appointed vn∣to 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 cir∣cle, and from that place drawe an other line to the centre, ta∣kyng heede where it crosseth the lesser circle, if you drawe a plumbe line from that pricke vnto the edge of the greatter circle, that line I say is a touthe line, drawen from the point assigned, according to the meaning of this conclusion.

Example.

Let the circle be called B.C. D, and his cētre E, and y^e prick

[illustration] diagram

assigned A, opē your cōpas now of such widenes, y^t the one foote may be set in E, w^ch is y^e cētre of y^e circle, & y^e other in A, w^ch is y^e pointe assigned, & so make an o∣ther greter circle (as here is A. F G) thē draw a line from A. vnto E, and wher that line doth cross y^e inner circle (w^ch heere is in the prick B.) there erect a plūb line vnto the line. A.E. and let that plumb line touch the vtter circle, as it doth here in the point F, so shall B.F. bee that plumbe lyne. Then from F. vnto E.

drawe an other line whiche shal be F.E, and it will cutte the inner circle, as it doth here in the point C, from which pointe C. if you erect a plumb line vnto A, then is that line A.C, the touche line, whiche you shoulde finde. Notwithstandinge that this is a certaine waye to fynde any touche line, and a demonstrable forme, yet more easyly by many folde may you fynde and make any suche line with a true ruler, layinge the edge of the ruler to the edge of the circle and to the pricke, and so drawing a right line, as this example sheweth, where the circle is E, the pricke assi∣gned

[illustration] diagram

is A. and the ruler C. D. by which the touch line is dra¦wen, and that is A, B, and as this way is light to doo, so is it certaine inoughe for any kinde B of workinge.

THE XXV. CONCLVSION. when you haue any peece of the circumference of a circle assigned, howe you may make oute the whole circle agreynge therevnto.

First seeke out the centre of that arche, according to the doe trine of the seuententh conclusion, and then setting one foote of your compas in the centre, and extending the other foot vn to the edge of the arche or peece of the circumference, it is ea∣sy to drawe the whole circle.

Example.

A peece of an olde piller was found, like informe to thys figure A.D.B. Now to knowe howe muche the cōpasse of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make in A table the like draught of y^t circūference by the self patrō, vsing it as it wer a croked ruler

[illustration] diagram

Then make .iij. prickes in that arche line, as I haue made, C. D. and E. And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that cētre is here F, nowe settyng one foote of your compas in F, and the other in C. D, other in E, and so makyng a com∣passe, you haue youre whole intent.

THE XXVI. CONCLVSION. To finde the centre to any arche of a circle.

If so be it that you desire to find the centre by any other way then by those .iij. prickes, consideryng that sometimes you can not haue so muche space in the thyng where the arche is dra∣wen, as should serue to make those .iiij. bowe lines, then shall you do thus: Parte that arche line into two partes, equall o∣ther vnequall, it maketh no force, and vnto ech portion draw a corde, other a string line. And then accordyng as you dyd in one arche in the .xvi. conclusion, so doe in bothe those arches here, that is to saie, deuide the arche in the middle, and also the corde, and drawe then a line by those two deuisions, so then are you sure that that line goeth by the centre. Afterward do lykewaies with the other arche and his corde, and where those .ij. lines do crosse, there is the centre, that you seke for.

Example.

The arche of the circle is A. B. C, vnto whiche I must seke

a centre, therfore firste I do

[illustration] diagram

deuide it into .ij. partes, the one of them is A. B, and the other is B. C. Then doe I cut euery arche in the middle, so is E. the middle of A.B, and G. is the middle of B.C. Like∣waies, I take the middle of their cordes, whiche I mark with F. and H, settyng F. by E, and H. by G. Then drawe I a line from E. to F, and from G. to H, and they do crosse in D, wherefore saie I, that D. is the centre, that I seke for.

THE XXVII. CONCLVSION. To drawe a circle within a triangle ap∣poincted.

For this conclusion and all other lyke, you muste vnder∣stande, that when one figure is named to be within an other, that is not other waies to be vnderstande, but that eyther eue∣ry 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 do∣eth touche iustely euery side of the triangle, and yet dooeth not crosse ouer any side of it. And so that quadrate is called properly to be drawen in a circle, when all his fower angles doeth touche the edge of the circle, And that circle is drawen in a quadrate, whose circumference doeth touche euery side of the quadrate, and lykewaies of other figures.

Examples are these. A. B. C. D. E. F.

[illustration] diagram

A. is a circle in a triangle.

[illustration] diagram

B. a triangle in a circle.

[illustration] diagram

C. a quadrate in a circle.

[illustration] diagram

D. a circle in a quadrate.

[illustration] diagram

[illustration] diagram

In these .ij. last figures E. and F, the circle is not named to be drawen in a triangle, because it doth not touche the sides of the triangle, neither is the triangle coūted to be drawen in the circle, because one of his corners doth not touche the circum∣ference of the circle, yet (as you see) the circle is within the tri∣angle, and the triangle within the circle, but nother of them is properly named to be in the other. Now to come to the con∣clusion. 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 mid∣dle pricke of any of the sides, and so drawe a compas, whiche shall touche euery side of the triangle, but shall not passe with out any of them.

Example.

The triangle is A. B. C, whose sides I do part into .ij. equall partes, eche by it selfe in these pointes D. E. F, puttyng F. be∣twene A. B, and D. betwene B. C, and E. betwene A. C. Then draw I a line from C. to F, and an other from A. to D, and the third from B. to E.

And where all those lines do mete (that is to saie 〈◊〉〈◊〉 G,) I set the one foote of my com∣passe,

[illustration] diagram

because it is the com∣mon centre, and so drawe a circle accordyng to the di∣staunce 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 conclu∣sion 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 an∣gle in the middle, as the third conclusion teacheth you. And so drawe lines frō 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 lēgth

[illustration] diagram

of the perpendicular, and so drawe your circle.

Example.

The triangle is A. B. C, whose corners I haue diui∣ded in the middle with D. E.F, and haue drawen the li∣nes of diuision A. D. B. E. and C. F, whiche crosse in G, therfore shall G. be the common centre. Then make I one perpēdicular from G. vnto the side A. C, and that

is G.H. Now sette I one fote of the compas in G, and extend the other foote vnto H. and so drawe a compas, whiche wyll iustly answere to that triāgle according to the meaning of the conclusion.

THE XXVIII. CONCLVSION. To drawe a circle about any triāgle 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 cen¦tre of the circle that must be drawen, therefore sette one foote of the compasse in that pointe, and extend the other foote to one corner of the triangle, and so make a circle, and it shall touche all iij. corners of the triangle.

Example.

A.B.C. is the triangle, whose two sides A.C. and B, C. are diuided into two e∣quall partes in D. and E, settyng D. be∣twene

[illustration] diagram

B. and C, and E. betwene A. and C. And from eche of those two pointes is ther erected a perpendicular (as you se D.F, and E.F.) which mete, and crosse in F, and stretche forth the other foot of any corner of the triangle, and so make a circle, that circle shal touch euery cor¦ner of the triangle, and shal enclose the whole triangle, accor¦dinge, as the conclusion willeth.

An other waye to do the same.

And yet an other waye may you doo it, accordinge as you learned in the seuententh conclusion, for if you call the three

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

Example.

A. B. C. is the triangle, whose iij. cor∣ners I count to be iij. pointes. Then (as the seuentene conclusion doth teache) I

[illustration] diagram

seeke a common centre, on which I may make a circle, that shall enclose those iij prickes that centre. as you se is D, for in D. doth the right lines, that passe by the angles of the arche lines, meete and crosse. And on that centre as you se, haue I made a circle, which doth inclose the iij. angles of the triāgle, and consequent lye the triangle it selfe, as the conclusion dydde intende.

THE XXIX. CONCLVSION. To make a triangle in a circle appoynted whose corners shalbe 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 a∣ny triangle assigned, then wust I first draw a tuche lyne vn¦to 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 cir∣cle: likewise in the same pricke muste I make an other angle w^t the other halfe of the touche line, equall to an other corner of the triangle appointed, and then betwen those two corners

will there resulte a third angle, equall to the third corner of that triangle. Nowe where those two lines that entre into the circle, doo touche the circumference (beside the touche line) there set I two prickes, and betwene them I drawe a thyrde line. And so haue I made a triangle in a circle appointed, whose corners bee equall to the corners of the triangle assi∣gned.

Example.

A. B. C, is the triangle

[illustration] diagram

appointed, and F.G.H. is the circle, in which I muste make an other triangle, with lyke

[illustration] diagram

angles to the angles of A.B.C. the triangle ap¦pointed. Therefore fyrst I make the touch lyne D.F.E. And then make I an angle in F, equall to A, whiche is one of the angles of the triangle. And the lyne that maketh that angle with the touche line, is F. H, whiche I drawe in lengthe vntill it touche the edge of the circle. Then againe in the same point F, I make an other corner equall to the angle C. and the line that maketh that corner with the touche line, is F.G. whiche also I drawe foorthe vntill it touche the edge of the circle. And then haue I made three angles v∣pon that one touch line, and in y^t one point F, and those iij. an∣gles be equal to the iij. angles of the triangle assigned, whiche thinge doth plainely appeare, in so muche as they bee equall

to ij. right angles, as you may gesse by the sixt theoreme. And the thre angles of euerye triangle are equall 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 cōmon sentence saith. Thē do I draw a line frome G. to H, and that line maketh a triangle F.G.H. whose angles be equall to the angles of the triangle appoin∣ted. And this triangle is drawen in a circle, as the conclusion didde wyll. The proofe of this conclusion doth appeare in the seuenty and iiij. Theoreme.

THE XXX. CONCLVSION. To make a triangle about a circle assigned whiche shall haue corners, equall to the cor∣ners 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, thē 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 tri¦angle is drawē about a circle apointed, as the cōclusiō did wil.

Example.

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

Then drawe I in the

[illustration] diagram

circle appointed a se∣midiameter, whiche is here H.F, for F. is the cētre of the circle G. H.K. Then make I on that centre an angle e∣quall

[illustration] diagram

to the vtter an∣gle B. A. D, and that angle is H.F. K. Like¦waies on the same cē∣tre by drawyng an o∣ther semidiameter, I make an other angle H. F. G, equall to the second vtter angle of the triangle, whiche is B. C. E. And thus haue I made .iij. se∣midiameters 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 triangle L. M. N, whiche is the tri∣angle that I should make, for it is drawen about a circle assig∣ned, and hath corners equall to the corners of the triangle ap∣pointed, for the corner M. is equall to C. Likewaies L. to A, and N. to B, whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.

THE XXXI. CONCLVSION. To make a portion of a circle on any right line assigned, whiche shall conteine an angle e∣quall to a right lined angle appointed.

The angle appointed, maie be a sharpe angle, a right angle, o∣ther a blunte angle, so that the worke must be diuersely han∣deled

according to the diuersities of the angles, but conside∣ringe the hardenes of those seuerall woorkes, I wyll omitte them for a more meter time, and at this tyme wyll she, we you one light waye which serueth for all kindes of angles, and that is this. When the line is proposed, and the angle assigned, you shall ioyne that line proposed so to the other twoo lines con∣tayninge 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, nye of the two lynes contayninge the angle appointed. And when you haue made a triangle of those iij. lines, then accordinge to the doctrine of the seuē and twety coclusiō, make acircle 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 triā∣gle be equal to

[illustration] diagram

y^e āgleassigned as youre selfe mai easily gesse

Example.

First for exam∣ple of a sharpe āgle let A. stād & B.C. shal be y^t lyne assigned. Thē do I make a triangle, by adding B. C, as a thirde side to those other ij. which doo in∣clude the āgle assigned, and that triāgle is D E.F, so y^t E. F.

is the line appointed, and D. is the angle assigned. Then doe I drawe a portion of a circle about that triangle, from the one ende of that line assigned vnto the other, that is to saie, from E. a long by D. vnto F, whiche portion is euermore greatter then the halfe of the circle, by reason that the angle is a sharpe angle. But if the angle be right (as in the second exaumple you see it) then shall the portion of the circle that containeth that angle, euer more be the iuste halfe of a circle. And when the angle is a blunte angle, as the thirde exaumpse dooeth pro∣pounde, then shall the portion of the circle euermore be lesse then the halfe circle. So in the seconde example, G. is the right angle assigned, and H. K. is the lyne appointed, and L.M.N. the portion of the circle aunsweryng thereto. In the third ex∣aumple, O. is the blunte corner assigned, P. Q. is the line, and R. S. T. is the portion of the circle, that containeth that blūt corner, and is drawen on R. T. the lyne appointed.

THE XXXII. CONCLVSION. To cutte of from any circle appoineed, 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 scke for.

Example.

A. is the angle appointed, and D. E. F. is the circle assigned, frō which I must cut away a portiō that doth contain an angle

equall to this angle A.

[illustration] diagram

Therfore first I do draw a touche line to the cir∣cle assigned, and that touch line is B. C, the ve∣ry pricke of the touche is D, from whiche D. J. drawe a lyne D. E, so that the angle made of those two lines be equall to the angle appointed. Then say I, that the arch of the circle D. F. E, is the arche that I seke af∣ter. For if I doo deuide that arche in the middle (as here it is done in F.) and so draw thence two lines, one to A, and the other to E, then will the angle F, be equall to the angle assigned.

THE XXXIII. CONCLVSION. To make a square quadrate in a circle assigned

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

Example.

A. B. C. D. is the circle assig∣ned, and A. C. and B. D. are the

[illustration] diagram

two diameters whiche crosse in the centre E, and make .iiij. right corners. Then do I make fowre other lines, that is A. B, B. C, C. D, and D. A, which do ioyne together the fowre endes of the ij. diameters. And so is the square

quadrate made in the circle assigned, as the conclusion wil∣leth.

THE XXXIIII. CONCLVSION. To make a square quadrate aboute annye circle assigned.

Drawe two diameters in crosse waies, so that they make foure righte angles in the centre. Then with your compasse take the length of the halfe diameter, and set one foote of the compas in eche end of those diameters, drawing twoo arche lines at euery pitchinge of the compas, so shall you haue viij. arche lines. Then yf you marke the prickes wherin those arch lines do crosse, and draw betwene those iiij. prickes iiij right lines, then haue you made the square quadrate accordinge to the request of the conclusion.

Example.

A.B.C. is the circle assigned

[illustration] diagram

in which first I draw two diameters, in crosse waies, making iiij. righte angles, and those ij. diameters are A.C. and B.D. Then sette I my compasse (whiche is o∣pened according to the se∣midiameter of the said cir∣cle) fixing one foote in the end of euery semidiameter, and drawe with the other foote twoo arche lines, one on euery side. As firste, when I sette the one foote in A,

then with the other foote I doo make twoo arche lines, one in E, and an other in F. Then sette I the one foote of the com∣passe in B, and drawe twoo arche lines F. and G. Like wise settinge the compasse foote in C, I drawe twoo other arche lines, G. and H, and on D. I make twoo other, H. and E. Then frome the crossinges of those eighte arche lines I drawe iiij. straighte lynes, that is to saye, E.F, and F. G. also G.H, and H.E, whiche iiij. straight lines do make the square quadrate that I should draw about the circle assigned.

THE XXXV. CONCLVSION. To drawe 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 one foote of the compasse in that point, and stretch forth the other foot, according to the length to halfe one of those lines, and so make a compas in the square quadrate assigned.

Example.

A, B.C.D. is the quadrate appoin∣ted, in whiche I muste make a circle.

[illustration] diagram

Therfore first I do deuide euery side in ij. equal partes, and draw ij. lines acrosse, betwene eche ij. cōtrary pric∣kes, as you se E. G, and F. H, whiche mete in K, and therfore shal K, be the centre of the circle. Then do I set one foote of the compas in K. and opē the other as wide as K.E, and so draw a circle, whiche is made ancordinge to the conclusion.

THE XXXVI. CONCLVSION. To draw a circle about a square quadrate.

Draw ij. lines betwene the iiij. corners of the quadrate, and where they mete in crosse, ther is the centre of the circle that you seeke for. Thē 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 qua∣drate proposed.

Example.

A. B. C. D. is the square quadrate pro¦posed, about which I must make a cir∣cle. Therefore do I draw ij. lines crosse the square quadrate from angle to an¦gle,

[illustration] diagram

as you se A. C. & B. D. And where they ij. do crosse (that is to say in E.) there set I the one foote of the compas as in the centre, and the other foote I do extend vnto one angle of the qua∣drate, as for exāpse to A, and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.

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

Fyrste make a circle, and deuide the circumference of it into fyue equall partes. And thenne drawe frome one pricke (which you will) two lines to ij. other prickes, that is to say to the iij. and iiij. pricke, counting that for the first, wherhence you drewe both those lines, Then drawe the thyrde lyne to make a triangle with those other twoo, and you haue doone according to the conclusion, and haue made a twelike triāgle,

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

Example.

A, B. C. is the circle, whiche I

[illustration] diagram

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

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

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

Example.

Yow se here this circle A. B. C. D. E. deuided into fiue e∣quall portions. And from eche pricke ij. lines drawen to the other ij. nexte prickes, so from A. are drawen ij. lines, one to B, and the other to E, and so from C. one to B. and an other

to D, and likewise of

[illustration] diagram

the reste. So that you haue not on¦ly 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 cir∣cle for the intente, readylye to make the other figure (I meane the cinkangle) thereby.

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

Deuide firste the circle as you did in the laste conclusion in∣to 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 semidia∣meters. And those fiue touche lines will make a cinkangle of equall sides and equall angles.

Example.

[illustration] diagram

A. B. C. D. E. is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is drawē a semidiameter, as you see. Then doo I make a touche line in the pricke B, whiche is F. G, makinge ij. right an∣gles

with the semidiameter B, and lyke waies on C. is made G. H, on D. standeth H. K, and on E, is set K. L, so that of those .v. touche lynes are made the .v. sides of a cinkeangle, accordyng to the conclusion.

An other waie.

An other 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 drawyng .v. touche lines whiche shall be iuste paral∣leles to the .v. sides of the cinkeangle in the circle, forseeyng that one of them do not crosse ouerthwarte an other and then haue you done. The exaumple of this (because it is easie) I leaue to your owne exercise.

THE XL. CONCLVSION. To make a circle in any appointed cinke∣angle 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 ex∣tend the other to the ende 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, accordyng to the conclu∣sion.

Example.

The cinkeangle assigned is A. B. C. D. E, in whiche I muste

make a circle, wherfore I draw a right line from the one an∣gle (as from B,) to the middle of the contrary side (whiche is E. D,) and that middle pricke is F. Then lykewaies from an other corner (as from E) I drawe a right line to the middle of the side that lieth against it (whiche is B. C.) and that pricke is

[illustration] diagram

G. Nowe because that these two lines do crosse in H, I saie that H. is the centre of the circle, whi∣che I would make. Ther∣fore I set one foote of the compasse in H, and extend the other foote vnto G, or F. (whiche are the endes of the lynes that lighte in the middle of the side of that cinkeangle) and so make I a circle in the cinkangle, right as the cōclusion meaneth.

THE XLI. CONCLVSION. To make a circle about any assigned cinke∣angle of equall sides, and equall corners.

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

Example.

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

meete in H, I saye that H. is the centre of the circle that I

[illustration] diagram

woulde haue, wher∣fore I sette one foote of the compasse in H. and extende the other to one corner (whiche happeneth fyrste, for all are like distaunte from H.) and so make I a circle aboute the cinkeangle assigned.

An other waye also.

An other 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 cir∣cle touchynge 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 cir¦cle as the samc conclusions do teache, and this forty conclusi¦on also.

THE XLII. CONCLVSION. To make a siseangle of equall sides, and equall angles, in any circle assigned.

Yf the centre of the circle be not knowen, then seeke oute the centre according to the doctrine of the sixetenth conclu∣sion. And with your compas take the quantitee of the semidi∣ameter iustly. And then sette one foote in one pricke of the

circūference 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 newe prickes, and point out two other prickes. And if you haue done well, you shal peceaue that there will be but euen sixe such diuisions in the circumference. Whereby it dothe well appeare, that the side of anye siseangle made in a circle, is equalle to the semidiameter of the same circle.

Example.

[illustration] diagram

The circle is B. C. D. E. F. G, whose centre I finde to bee A. Therefore I sette one foote of the compas in A, and do extēd the o∣ther foote to B, thereby takinge the semidiameter. Then sette I one foote of the compas vnremo∣ued in B, and marke with the o∣ther foote on eche side C. and G. Then from C. I marke D, and frō D, E: from E. marke I F. And then haue I but one space iuste vnto G. and so haue I made a iuste siseangle of equall sides and equall angles, in a circle appointed.

THE XLIII. CONCLVSION.

To make a siseangle of equall sides, and e∣quall angles about any circle assigned.

THE XLIIII. CONCLVSION.

To make a circle in any siseangle appoin∣ted, of equall sides and equal angles.

THE XLV. CONCLVSION.

To make a circle about any siseangle limi∣ted 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 conside∣ringe that there is a difference in the numbre of the 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 per∣ceaue what difference there is betwene eche twoo of those conclusions. For thoughe it seeme one thing to make a sisean∣gle in a circ e, and to make a circle about a siseangle, yet shall you perceaue, that it is not one thinge, nother are those twoo conclusions wrought one way. Like waise 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 xxx vij. xxx viij 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 workyng, that when you shall seeke the centre in a sise angle (whether it be to make a cir••le in it other about it) you shall drawe the two crosse lines, from one angle to the other angle that lieth againste it, and not to the middle of any side, as you did in the cinkangle.

THE XLVI. CONCLVSION.

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

This rule is generall, that how many sides the figure shall

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

FINIS.