Part 4

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.

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.

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.

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.

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.