The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 3

[Ill.u.s.tration: An arche.]

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.

[Ill.u.s.tration: An egge forme]

[Ill.u.s.tration: A tunne forme.]

[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 pa.s.seth 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.

[Ill.u.s.tration]

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_.

[Ill.u.s.tration]

[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.

[Ill.u.s.tration: A]

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

[Ill.u.s.tration: B]

An other hath two compa.s.sed lines and one right lyne, and is as the porti of halfe a globe, example of B.

[Ill.u.s.tration: C]

An other hath but one compa.s.sed 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.

[Ill.u.s.tration: D]

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.

[Ill.u.s.tration: E]

[Ill.u.s.tration: F]

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: ?s?p?e???.] 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: ?s?s?e?es.] 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.

[Ill.u.s.tration: G]

[Ill.u.s.tration: H]

[Ill.u.s.tration: 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: s?a?e??.] 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 lg, to ani other in the same figur. Here it is to be noted, that in a trigle al the angles bee called _innergles_ except ani side bee drawenne forth in lengthe, for then is that fourthe corner caled an _vtter corner_, as in this exple because A.B, is drawen in length, therfore the gle C, is called an vtter gle.

[Ill.u.s.tration: M]

[Ill.u.s.tration: N]

[Ill.u.s.tration: O]

[Ill.u.s.tration]

[Ill.u.s.tration: Q]

[Sidenote: Quadrgle] And thus haue I done with triguled 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.

[Ill.u.s.tration: R]

[Sidenote: A losenge] The thyrd kind is called _losenges_ [Sidenote: A diamd.] 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.

[Ill.u.s.tration: 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 _diamdlike_, 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.

[Ill.u.s.tration: T]

[Ill.u.s.tration]

The fift sorte doth containe all other fashions of foure cornered figurs, and ar called of the Grekes _trapezia_, of Latin m? _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_.

[Ill.u.s.tration: A]

[Ill.u.s.tration: B]

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.

[Ill.u.s.tration]

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 roud as a _globe_ hath, or ended long as in an _egge_, and a _tunne fourme_, whose pictures are these.