Part 3

So that whan so euer any suche meetyng of lines doth happen, the place of their metyng is called an Angle or corner.

Of angles there be three generall kindes: a sharpe angle, a square angle, and a blunte angle. [Sidenote: A righte angle.] The square angle, whiche is commonly named a right corner, is made of twoo lynes meetyng together in fourme of a squire, whiche two lines, if they be drawen forth in length, will crosse one an other: as in the examples folowyng you maie see.

And these angles (as you see) are made partly of streght lynes, partly of croken lines, and partly of both together. Howbeit in right angles I haue put none example of croked lines, because it would muche trouble a lerner to iudge them: for their true iudgment doth appertaine to arte perspectiue, and as I may say, rather to reason then to sense.

But now as of many prickes there is made one line, so of diuerse lines are there made sundry formes, figures, and shapes, whiche all yet be called by one propre name, [Sidenote: A platte forme.] Platte formes, and thei haue bothe length and bredth, but yet no depenesse.

And the boundes of euerie platte forme are lines: as by the examples you maie perceiue.

Of platte formes some be plain, and some be croked, and some parly plaine, and partlie croked.

And if it be partlie plaine, and partlie crooked, then is it called a Myxte platte, of all whiche, these are exaumples.

And as of many prickes is made a line, and of diuerse lines one platte forme, [Sidenote: A bodie.] so of manie plattes is made a bodie, whiche conteigneth Lengthe, bredth, and depenesse. [Sidenote: Depenesse.] By Depenesse I vnderstand, not as the common sort doth, the holownesse of any thing, as of a well, a diche, a potte, and suche like, but I meane the massie thicknesse of any bodie, as in exaumple of a potte: the depenesse is after the common name, the space from his brimme to his bottome. But as I take it here, the depenesse of his bodie is his thicknesse in the sides, whiche is an other thyng cleane different from the depenesse of his holownes, that the common people meaneth.

Now all bodies haue platte formes for their boundes, [Sidenote: Cubike.] so in a dye (whiche is called a cubike bodie) by geomatricians, [Sidenote: Asheler.] and an ashler of masons, there are .vi. sides, whiche are .vi. platte formes, and are the boundes of the dye.

But because you shall not muse what I dooe call a bound, [Sidenote: A bounde.] I mean therby a generall name, betokening the beginning, end and side, of any forme.

Of figures there be manie sortes, for either thei be made of prickes, lines, or platte formes. Not withstandyng to speake properlie, a figure is euer made by platte formes, and not of bare lines vnclosed, neither yet of prickes.

Yet for the lighter forme of teachyng, it shall not be vnsemely to call all suche shapes, formes and figures, whiche y^e eye maie discerne distinctly.

And first to begin with prickes, there maie be made diuerse formes of them, as partely here doeth folowe.

And so maie there be infinite formes more, whiche I omitte for this time, considering that their knowledg appertaineth more to Arithmetike figurall, than to Geometrie.

But yet one name of a pricke, whiche he taketh rather of his place then of his fourme, maie I not ouerpasse. And that is, when a pricke standeth in the middell of a circle (as no circle can be made by compasse without it) then is it called a centre. [Sidenote: A centre] And thereof doe masons, and other worke menne call that patron, a centre, whereby thei drawe the lines, for iust hewyng of stones for arches, vaultes, and chimneies, because the chefe vse of that patron is wrought by findyng that pricke or centre, from whiche all the lynes are drawen, as in the thirde booke it doeth appere.

Lynes make diuerse figures also, though properly thei maie not be called figures, as I said before (vnles the lines do close) but onely for easie maner of teachyng, all shall be called figures, that the eye can discerne, of whiche this is one, when one line lyeth flatte (whiche is named [Sidenote: A ground line.] the ground line) and an other commeth downe on it, and is called [Sidenote: A perpendicular.] [Sidenote: A plume lyne.] a perpendiculer or plumme lyne, as in this example you may see. where .A.B. is the grounde line, and C.D. the plumbe line.

And like waies in this figure there are three lines, the grounde lyne whiche is A.B. the plumme line that is A.C. and the bias line, whiche goeth from the one of them to the other, and lieth against the right corner in such a figure whiche is here .C.B.

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.

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.

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.

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.

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.

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.