The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 4

[Ill.u.s.tration: The globe as is before.]

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 pa.s.se 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 l?gth of the lyne with your cpa.s.se, 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 cpa.s.se 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.

[Ill.u.s.tration]

_Example._

A.B. is the first line, on which I wold make the threlike triangle, therfore I open the compa.s.se 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.

[Ill.u.s.tration]

Consider fyrst the length that yow will haue the other sides to containe, and to that length open your compa.s.se, 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 th? at the one ende, and with the other at the other end, the exple is as in the other before.

[Ill.u.s.tration]

THE III. CONCL.

To diuide an angle of right lines into ij. equal partes.

First open your compa.s.se 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 compa.s.se in the verye point of the angle, and with the other fote draw a compa.s.sed arch fr the one lyne of the angle to the other, that arch shall you deuide in halfe, and th? draw a line fr the gle to y^e middle of y^e arch, and so y^e angle is diuided into ij. equall partes.

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_Example._

Let the trigle be A.B.C, th? set I one foot of y^e cpa.s.se in B, and with the other I draw y^e arch D.E, which I part into ij.

equall parts in F, and th? draw a line fr B, to F, & so I haue mine int?t.

THE IIII. CONCL.

To deuide any measurable line into ij. equall partes.

[Ill.u.s.tration]

Open your compa.s.se to the iust l?gth of y^e line. And th? 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. p.r.i.c.ks 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 compa.s.se 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 p.r.i.c.ke that you will in any right lyne appointed.

Open youre compas so that it be not wyder then from the p.r.i.c.ke appoynted in the line to the shortest ende of the line, but rather shorter. Then sette the one foote of the compa.s.se in the first p.r.i.c.ke appointed, and with the other fote marke ij. other p.r.i.c.kes, one of eche syde of that fyrste, afterwarde open your compa.s.se to the wydenes of those ij. new p.r.i.c.kes, and draw from them ij. arch lynes, as you did in the fyrst conclusion, for making of a threlyke trigle. then if you do mark their crossing, and from it drawe a line to your fyrste p.r.i.c.ke, it shall bee a iust plum lyne on that place.

[Ill.u.s.tration]

_Example._

The lyne is A.B. the p.r.i.c.k on whiche I shoulde make the plumme lyne, is C. then open I the compa.s.se 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 compa.s.se 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 p.r.i.c.ke 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 fr 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 portis: 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 p.r.i.c.ke, where you would haue the plumme line to lighte (whiche I call the first p.r.i.c.ke,) and with the other foote drawe an arche line righte ouer the p.r.i.c.ke, as you can ayme it: then open youre compas as wide as all fiue measures be, and set the one foote in the fourth p.r.i.c.ke, 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._

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The line is A.B. and A. is the p.r.i.c.k, 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. p.r.i.c.ke, 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 compa.s.s 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 cpas as wide as K.L. (that is all .v.

partes) and set one foote in G, (that is the iij. p.r.i.c.ke) 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 dep?d of the .xlvi. proposici of y^e first boke of Euclide. An other waie yet. set one foote of the compas in the p.r.i.c.k, 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 quant.i.tie more there vnto, and by that p.r.i.c.k that endeth the last part, draw a line to the p.r.i.c.ke a.s.signed, and it shall be a perpendicular.

[Ill.u.s.tration]

_Example._