Part 3

But consideryng that I shall haue occasion to declare sundry figures anon, I will first shew some certaine varietees of lines that close no figures, but are bare lynes, and of the other lines will I make mencion in the description of the figures.

[Sidenote: Parallelys]

[Sidenote: Gemowe lynes.]

_Paralleles_, or _gemowe lynes_ be suche lines as be drawen foorth still in one distaunce, and are no nerer in one place then in an other, for and if they be nerer at one ende then at the other, then are they no paralleles, but maie bee called _bought lynes_, and loe here exaumples of them bothe.

I haue added also _paralleles tortuouse_, whiche bowe contrarie waies with their two endes: and _paralleles circular_, whiche be lyke vnperfecte compasses: for if they bee whole circles, [Sidenote: Concentrikes] then are they called _concentrikes_, that is to saie, circles drawen on one centre.

Here might I note the error of good _Albert Durer_, which affirmeth that no perpendicular lines can be paralleles. which errour doeth spring partlie of ouersight of the difference of a streight line, and partlie of mistakyng certain principles geometrical, which al I wil let passe vntil an other tyme, and wil not blame him, which hath deserued worthyly infinite praise.

And to returne to my matter. [Sidenote: A twine line.] an other fashioned line is there, which is named a twine or twist line, and it goeth as a wreyth about some other bodie. [Sidenote: A spirall line.] And an other sorte of lines is there, that is called a _spirall line_, [Sidenote: A worme line.] or a _worm line_, whiche representeth an apparant forme of many circles, where there is not one in dede: of these .ii. kindes of lines, these be examples.

[Sidenote: A tuch line.]

_A touche lyne_, is a line that runneth a long by the edge of a circle, onely touching it, but doth not crosse the circumference of it, as in this exaumple you maie see.

[Sidenote: A corde,]

And when that a line doth crosse the edg of the circle, then is it called _a cord_, as you shall see anon in the speakynge of circles.

[Sidenote: Matche corners]

In the meane season must I not omit to declare what angles bee called _matche corners_, that is to saie, suche as stande directly one against the other, when twoo lines be drawen a crosse, as here appereth.

Where A. and B. are matche corners, so are C. and D. but not A. and C. nother D. and A.

Nowe will I beginne to speak of figures, that be properly so called, of whiche all be made of diuerse lines, except onely a circle, an egge forme, and a tunne forme, which .iij. haue no angle and haue but one line for their bounde, and an eye fourme whiche is made of one lyne, and hath an angle onely.

[Sidenote: A circle.]

_A circle_ is a figure made and enclosed with one line, and hath in the middell of it a pricke or centre, from whiche all the lines that be drawen to the circumference are equall all in length, as here you see.

[Sidenote: Circumference.] And the line that encloseth the whole compasse, is called the _circumference_.

[Sidenote: A diameter.] And all the lines that bee drawen crosse the circle, and goe by the centre, are named _diameters_, whose halfe, I meane from the center to the circumference any waie, [Sidenote: Semidiameter.] is called the _semidiameter_, or _halfe diameter_.

But and if the line goe crosse the circle, and passe beside the centre, [Sidenote: A cord, or a stringlyne.] then is it called _a corde_, or _a stryng line_, as I said before, and as this exaumple sheweth: where A. is the corde. And the compassed line that aunswereth to it, [Sidenote: An archline] is called _an arche lyne_, [Sidenote: A bowline.] or _a bowe lyne_, whiche here marked with B. and the diameter with C.

But and if that part be separate from the rest of the circle (as in this example you see) then ar both partes called cantelles, [Sidenote: A cantle] the one the _greatter cantle_ as E. and the other the _lesser cantle_, as D. And if it be parted iuste by the centre (as you see in F.) [Sidenote: A semyecircle] then is it called a _semicircle_, or _halfe compasse_.

Sometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circumference (as G. is) [Sidenote: A nooke cantle] and then maie it be called a _nooke cantle_, and if it be not parted from the reste of the circle (as you see in H.) [Sidenote: A nooke.] then is it called a _nooke_ plainely without any addicion. And the compassed lyne in it is called an _arche lyne_, as the exaumple here doeth shewe.

Nowe haue you heard as touchyng circles, meetely sufficient instruction, so that it should seme nedeles to speake any more of figures in that kynde, saue that there doeth yet remaine ij. formes of an imperfecte circle, for it is lyke a circle that were brused, and thereby did runne out endelong one waie, whiche forme Geometricians dooe call an [Sidenote: An egge fourme.] _egge forme_, because it doeth represent the figure and shape of an egge duely proportioned (as this figure sheweth) hauyng the one ende greate then the other.

[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.