Chapter II: Part 2

[Sidenote: A tunne or barrel form] For if it be lyke the figure of a circle pressed in length, and bothe endes lyke bygge, then is it called a _tunne forme_, or _barrell forme_, the right makyng of whiche figures, I wyll declare hereafter in the thirde booke.

An other forme there is, whiche you maie call a nutte forme, and is made of one lyne muche lyke an egge forme, saue that it hath a sharpe angle.

And it chaunceth sometyme that there is a right line drawen crosse these figures, [Sidenote: An axtre or axe lyne.] and that is called an _axelyne_, or _axtre_. Howe be it properly that line that is called an _axtre_, whiche gooeth throughe the myddell of a Globe, for as a diameter is in a circle, so is an axe lyne or axtre in a Globe, that lyne that goeth from side to syde, and passeth by the middell of it. And the two poyntes that suche a lyne maketh in the vtter bounde or platte of the globe, are named _polis_, w^{ch} you may call aptly in englysh, _tourne pointes_: of whiche I do more largely intreate, in the booke that I haue written of the vse of the globe.

But to returne to the diuersityes of figures that remayne vndeclared, the most simple of them ar such ones as be made but of two lynes, as are the _cantle of a circle_, and the _halfe circle_, of which I haue spoken allready. Likewyse the _halfe of an egge forme_, the _cantle of an egge forme_, the _halfe of a tunne fourme_, and the _cantle of a tunne fourme_, and besyde these a figure moche like to a tunne fourne, saue that it is sharp couered at both the endes, and therfore doth consist of twoo lynes, where a tunne forme is made of one lyne, [Sidenote: An yey fourme] and that figure is named an _yey fourme_.

[Sidenote: A triangle]

The nexte kynd of figures are those that be made of .iij. lynes other be all right lynes, all crooked lynes, other some right and some crooked. But what fourme so euer they be of, they are named generally triangles. for _a triangle_ is nothinge els to say, but a figure of three corners. And thys is a generall rule, looke how many lynes any figure hath, so mannye corners it hath also, yf it bee a platte forme, and not a bodye. For a bodye hath dyuers lynes metyng sometime in one corner.

Now to geue you example of triangles, there is one whiche is all of croked lynes, and may be taken fur a portion of a globe as the figur marked w^t A.

An other hath two compassed lines and one right lyne, and is as the portion of halfe a globe, example of B.

An other hath but one compassed lyne, and is the quarter of a circle, named a quadrate, and the ryght lynes make a right corner, as you se in C. Otherlesse then it as you se D, whose right lines make a sharpe corner, or greater then a quadrate, as is F, and then the right lynes of it do make a blunt corner.

Also some triangles haue all righte lynes and they be distincted in sonder by their angles, or corners. for other their corners bee all sharpe, as you see in the figure, E. other ij. sharpe and one blunt, as is the figure G. other ij. sharp and one blunt as in the figure H.

There is also an other distinction of the names of triangles, according to their sides, whiche other be all equal as in the figure E, and that the Greekes doo call _Isopleuron_, [Sidenote: #isopleurom#.] and Latine men _aeequilaterum_: and in english it may be called a _threlike triangle_, other els two sydes bee equall and the thyrd vnequall, which the Greekes call _Isosceles_, [Sidenote: #isoskeles#.] the Latine men _aequicurio_, and in english _tweyleke_ may they be called, as in G, H, and K. For, they may be of iij. kinds that is to say, with one square angle, as is G, or with a blunte corner as H, or with all in sharpe korners, as you see in K.

Further more it may be y^t they haue neuer a one syde equall to an other, and they be in iij kyndes also distinct lyke the twilekes, as you maye perceaue by these examples .M. N, and O. where M. hath a right angle, N, a blunte angle, and O, all sharpe angles [Sidenote: #skalenom#.] these the Greekes and latine men do cal _scalena_ and in englishe theye may be called _nouelekes_, for thei haue no side equall, or like long, to ani other in the same figur. Here it is to be noted, that in a triangle al the angles bee called _innerangles_ except ani side bee drawenne forth in lengthe, for then is that fourthe corner caled an _vtter corner_, as in this example because A.B, is drawen in length, therfore the angle C, is called an vtter angle.

[Sidenote: Quadrangle] And thus haue I done with trianguled figures, and nowe foloweth _quadrangles_, which are figures of iiij. corners and of iiij. lines also, of whiche there be diuers kindes, but chiefely v. that is to say, [Sidenote: A square quadrate.] a _square quadrate_, whose sides bee all equall, and al the angles square, as you se here in this figure Q. [Sidenote: A longe square.] The second kind is called a long square, whose foure corners be all square, but the sides are not equall eche to other, yet is euery side equall to that other that is against it, as you maye perceaue in this figure. R.

[Sidenote: A losenge] The thyrd kind is called _losenges_ [Sidenote: A diamond.] or _diamondes_, whose sides bee all equall, but it hath neuer a square corner, for two of them be sharpe, and the other two be blunt, as appeareth in .S.

The iiij. sorte are like vnto losenges, saue that they are longer one waye, and their sides be not equal, yet ther corners are like the corners of a losing, and therfore ar they named [Sidenote: A losenge lyke.] _losengelike_ or _diamondlike_, whose figur is noted with T. Here shal you marke that al those squares which haue their sides al equal, may be called also for easy vnderstandinge, _likesides_, as Q. and S. and those that haue only the contrary sydes equal, as R. and T. haue, those wyll I call _likeiammys_, for a difference.

The fift sorte doth containe all other fashions of foure cornered figurs, and ar called of the Grekes _trapezia_, of Latin men _mensulae_ and of Arabitians, _helmuariphe_, they may be called in englishe _borde formes_, [Sidenote: Borde formes.] they haue no syde equall to an other as these examples shew, neither keepe they any rate in their corners, and therfore are they counted _vnruled formes_, and the other foure kindes onely are counted _ruled formes_, in the kynde of quadrangles. Of these vnruled formes ther is no numbre, they are so mannye and so dyuers, yet by arte they may be changed into other kindes of figures, and therby be brought to measure and proportion, as in the thirtene conclusion is partly taught, but more plainly in my booke of measuring you may see it.

And nowe to make an eande of the dyuers kyndes of figures, there dothe folowe now figures of .v. sydes, other .v. corners, which we may call _cink-angles_, whose sydes partlye are all equall as in A, and those are counted _ruled cinkeangles_, and partlye vnequall, as in B, and they are called _vnruled_.

Likewyse shall you iudge of _siseangles_, which haue sixe corners, _septangles_, whiche haue seuen angles, and so forth, for as mannye numbres as there maye be of sydes and angles, so manye diuers kindes be there of figures, vnto which yow shall geue names according to the numbre of their sides and angles, of whiche for this tyme I wyll make an ende, [Sidenote: A squyre.] and wyll sette forthe on example of a syseangle, whiche I had almost forgotten, and that is it, whose vse commeth often in Geometry, and is called a _squire_, is made of two long squares ioyned togither, as this example sheweth.

And thus I make an eand to speake of platte formes, and will briefelye saye somwhat touching the figures of _bodeis_ which partly haue one platte forme for their bound, and y^t iust round as a _globe_ hath, or ended long as in an _egge_, and a _tunne fourme_, whose pictures are these.

Howe be it you must marke that I meane not the very figure of a tunne, when I saye tunne form, but a figure like a tunne, for a _tune fourme_, hath but one plat forme, and therfore must needs be round at the endes, where as a _tunne_ hath thre platte formes, and is flatte at eche end, as partly these pictures do shewe.

_Bodies of two plattes_, are other cantles or halues of those other bodies, that haue but one platte forme, or els they are lyke in foorme to two such cantles ioyned togither as this A. doth partly eppresse: or els it is called a _rounde spire_, or _stiple fourme_, as in this figure is some what expressed.

[Sidenote: A rounde spier.]

Nowe of three plattes there are made certain figures of bodyes, as the cantels and halues of all bodyes that haue but ij. plattys, and also the halues of halfe globys and canteles of a globe. Lykewyse a rounde piller, and a spyre made of a rounde spyre, slytte in ij. partes long ways.

But as these formes be harde to be iudged by their pycturs, so I doe entende to passe them ouer with a great number of other formes of bodyes, which afterwarde shall be set forth in the boke of Perspectiue, bicause that without perspectiue knowledge, it is not easy to iudge truly the formes of them in flatte protacture.

And thus I made an ende for this tyme, of the definitions Geometricall, appertayning to this parte of practise, and the rest wil I prosecute as cause shall serue.

THE PRACTIKE WORKINGE OF
+sondry conclusions geometrical.+

THE FYRST CONCLVSION.

To make a threlike triangle on any lyne measurable.

Take the iuste length of the lyne with your compasse, 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 compasse 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 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
certaine line.

Consider fyrst the length that yow will haue the other sides 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 besyde the fyrste lyne, and draw an arche lyne with one of them at the one ende, and with the other at the other end, the example is as in the other before.

THE III. CONCL.

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

First open your compasse as largely as you can, so that it do not excede the length of the shortest line y^t incloseth 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 from the one lyne of the angle to the other, that arch shall you deuide in halfe, and then draw a line from the angle to y^e middle of y^e arch, and so y^e angle is diuided into ij. equall partes.

_Example._

Let the triangle be A.B.C, then set I one foot of y^e compasse in B, and with the other I draw y^e arch D.E, which I part into ij. equall parts in F, and then draw a line from B, to F, & so I haue mine intent.

THE IIII. CONCL.

To deuide any measurable line into ij. equall partes.

Open your compasse to the iust length of y^e line. And then 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 also vnder it, then doo lykewaise at 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 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 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 threlyke triangle. 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 foot 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 from 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 portions: 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._

The line is A.B. and A. is the prick, on whiche 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 partes (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 liue 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 open I the compass 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 compas 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 conlusion, but the reason of it doth depend of the .xlvi. proposicion 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 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.

_Example._

A.B. is the line appointed, to whiche 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 other vnto D. makyng a part of a circle, 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 circular 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 thefore is H. the point, by whiche the perpendicular line must be drawn, so say I that the line H.A, is a plumbe line to A.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.

Open your compas as so wide that it may extend somewhat farther, then 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 compassed 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 line A.B. I must draw a perpendicular. Thefore I open the compas 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 perpendicular to the line A.B, accordyng as my desire was.

THE .VII. CONCLVSION.

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

Mark first the prick where y^e plumbe 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 compas 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.

_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 plumbe line to light. Therfore I meate out two equall distaunces on eche side of that pricke B. and they are A.C. Then open I the compas as wide as A.C. and settyng one of the feete in A. with the other I drawe an arche line which goeth by G. Like waies 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 fro whiche I must drawe the plumbe line. Then draw I a right 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 forthe 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 example 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 info 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 A.D.C, the corde is A.B.C, this corde is diuided in the middle with B, from which prick if I erect a plum line as B.D, then 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 perpendicular to the former line.

_Example._

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 is C. Then do I sette one edg of my squyre (that is B.C.) to the line A.B, so at 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 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 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. Then take I a line with a plummet (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 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.

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. Then 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._

A.B, is the line vnto which 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 seete the one foote in A, and with the other foot I make a bowe lyne, which is D, then like wise set I the one foote of the compasse in B, and with the other 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 gemowe 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 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 consider the .iij. lines that you must take, and set one of them for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.

_Example._

I haue .iij. 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 length 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 metyng (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 eandes 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 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.

_Example._

Lette the angle appoynted bee A.B.C, and the corner assigned, 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 triangle A.B.C.

Then consideringe that D.G, is longer thanne A.B, you shall cut out a line from D. toward G, equal to A.B, as for example D.F. Then measure oute the other ij. lines and worke with them according as the conclusion with the fyrste also and the second 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 conclusion, and let it be of like length as your first line is, then open 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.

_Example._

A.B. is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plumbe line vnto it, whiche shall lighte in A, and that plumb line is A.C, then open I my compasse as wide as the length of A.B, or A.C, (for they must be bothe equall) 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 compas again with one foote in B, 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, and an other to C, and so haue I made the square quadrate that I entended.

THE .XV. CONCLVSION.

To make a likeiamme equall to a triangle appointed, and that
in a right lined angle 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 then 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._

B.C.G, is the triangle appoincted vnto, whiche I muste make an equall likeiamme. And D, is the angle that the likeiamme must haue. Therfore first entendyng to erecte the likeiamme on the one side, that the ground line of the triangle (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. conclusion teacheth) and that angle is B.A.E. Then to procede, I doo parte in y^e middle the said ground line B.G, in the prick F, from which prick I draw to the first gemowe line (A.H.) an other line that is parallele to A.B, and that line is E.F. Now saie I that the likeiamme 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 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 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 likeiamme. 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 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 conclusion, I make the likeiamme 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 length in the ij. gemow lines 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. Afterward 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 sides of the likeiamme. and where that lyne doeth crosse F.G, there I sette M. Nowe to make an ende, I make an other gemowe line, whiche is parallel to N.F. and H.G, and that gemowe 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 triangle 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 profe of y^e equalnes of this likeiam vnto the triangle, dependeth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, when there are more then one made about one bias line, the filsquares 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 rightlined figure into triangles, and then for euery triangle together an equal likeiamme, according vnto the eleuen conclusion, 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 happen 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 length of one of his sides, and set that as a line assigned, on whiche you shal make the other likeiams, according to the twelft conclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so y^t you mai easily ioyne them into one figure.

_Example._

If the right lined figure be like vnto A, then may it be turned into triangles that wil stand betwene ij. parallels anye ways, as you mai se by C. and D, for ij. sides of both the trianngles ar parallels. Also if the right lined figure be like vnto E, then wil it be turned into triangles, liyng betwene two parallels also, as y^e other did before, as in the example of F.G. But and if y^e right lined figure be like vnto H, and so turned into triangles as you se in K.L.M, wher it is parted into iij triangles, then 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.

Further explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then diligently behold how these sundry figures be turned into triangles. 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 triangles, and fiue parallels, in the iiij. you se fiue triangles & four parallels. in the fift, iiij. triangles and .iiij. parallels, & in y^e sixt ther ar fiue triangles & iiij. paralels. Howbeit a man maye at liberty alter them into diuers formes of triangles & 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 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 greater portion be 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.

_Example._

A.B, is the lyne assigned. E. is the 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 lo the 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^e square of y^e hole line A.B, & of the one portion (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.

[Transcriber's Note:
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 ("at whiche diuision I
set..."), _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 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 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._

K, is the right lined figure appointed, and B.C.D.E, is the likeiamme, 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 likeiamme, 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 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.

_Example._

A.B. and C.D, are the two 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, equall to the side of the other quadrate D.C. Then drawe I a byas line G.F, which beyng made the side of a quadrate (accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrate H. is 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, 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 y^e first quadrate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.

_Example._

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

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, then 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 plumbeline G.H, and so make out the square quadrate A.G.H.K, whose sides are equall 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 declareth, 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 .XXII. CONCLVSION.

To find out the centre 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._

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 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 (which 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 makinge thre prickes in the circumference of the circle, at liberty where you wyll, and then findinge the centre to those thre pricks, Which worke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.

THE XXIII. 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 common centre. Open your compas so wide, that it be somewhat 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 lyne 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 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.

_Example_

The iij. prickes I haue set to be A.B, and C, whiche I wold bring into the edg of one common circle, by finding a centre commen to them all, fyrst therefore I open my compas, so that thei occupye more then y^e halfe distance betwene ij. pricks (as are A.B.) and so settinge one foote in A. and extendinge the other toward B, I make the arche line D.E. Likewise setting 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 compasse 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 circumference & thus haue you attained y^e vse of this conclusion.

THE XXIIII. CONCLVSION.

To drawe a touche line onto a circle, from any poincte
assigned.