The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 2

[Ill.u.s.tration: A globe.]

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.

[Sidenote: Forme, Fygure.] _A forme, figure, or shape_, is that thyng that is inclosed within one bond or manie bondes, so that you vnderstand that shape, that the eye doth discerne, and not the substance of the bodie.

Of _figures_ there be manie sortes, for either thei be made of p.r.i.c.kes, 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 p.r.i.c.kes.

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 p.r.i.c.kes, there maie be made diuerse formes of them, as partely here doeth folowe.

[Ill.u.s.tration: A lynearic numbre.

Trianguler numbres Longsquare nubre.

Iust square numbres a threcornered spire.

A square spire.]

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

But yet one name of a p.r.i.c.ke, whiche he taketh rather of his place then of his fourme, maie I not ouerpa.s.se. And that is, when a p.r.i.c.ke standeth in the middell of a circle (as no circle can be made by cpa.s.se 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 p.r.i.c.ke 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 _plume lyne_, as in this example you may see. where .A.B. is the grounde line, and C.D. the plumbe line.

[Ill.u.s.tration]

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 th? to the other, and lieth against the right corner in such a figure whiche is here .C.B.

[Ill.u.s.tration]

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.

[Ill.u.s.tration: tortuouse paralleles.]

[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.

[Ill.u.s.tration: parallelis.]

[Ill.u.s.tration: bought lines]

[Ill.u.s.tration: parallelis: circular. Concentrikes.]

I haue added also _paralleles tortuouse_, whiche bowe ctrarie waies with their two endes: and _paralleles circular_, whiche be lyke vnperfecte compa.s.ses: for if they bee whole circles, [Sidenote: Concentrikes] then are they called _ccentrikes_, that is to saie, circles draw? 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 pa.s.se 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 fas.h.i.+oned 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.

[Ill.u.s.tration: A twiste lyne.]

[Ill.u.s.tration: A spirail lyne]

[Ill.u.s.tration: A touche lyne.]

[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 circ.u.mference of it, as in this exaumple you maie see.

[Sidenote: A corde,]

And when that a line doth crosse the edg of the circle, th? 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.

[Ill.u.s.tration: Matche corner. Matche corner.]

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 p.r.i.c.ke or centre, from whiche all the lines that be drawen to the circ.u.mference are equall all in length, as here you see.

[Ill.u.s.tration]

[Sidenote: Circ.u.mference.] And the line that encloseth the whole compa.s.se, is called the _circ.u.mference_.

[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 circ.u.mference any waie, [Sidenote: Semidiameter.] is called the _semidiameter_, or _halfe diameter_.

[Ill.u.s.tration]

But and if the line goe crosse the circle, and pa.s.se 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 compa.s.sed 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.

[Ill.u.s.tration]

But and if that part be separate from the rest of the circle (as in this exple you see) then ar both partes called ctelles, [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 compa.s.se_.

[Ill.u.s.tration]

Sometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circ.u.mference (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 compa.s.sed lyne in it is called an _arche lyne_, as the exaumple here doeth shewe.