Part 5

Example.

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

Consider fyrst the length that yow will haue the other sides to containe, and to that length open your compasse, 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 them at the one ende, and with the other at the other end, the example is as in the other before.

THE III. CONCL.

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

First open your compasse 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 compasse in the verye point of the angle, and with the other fote draw a compassed arch from the one lyne of the angle to the other, that arch shall you deuide in halfe, and then draw a line from the angle to y^e middle of y^e arch, and so y^e angle is diuided into ij. equall partes.

Example.

Let the triangle be A.B.C, then set I one foot of y^e compasse in B, and with the other I draw y^e arch D.E, which I part into ij. equall parts in F, and then draw a line from B, to F, & so I haue mine intent.

THE IIII. CONCL.

To deuide any measurable line into ij. equall partes.

Open your compasse to the iust length of y^e line. And then 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. pricks 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 compasse 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 pricke that you will in any right lyne appointed.

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

Example.

The lyne is A.B. the prick on whiche I shoulde make the plumme lyne, is C. then open I the compasse 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 compasse 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 pricke 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 from 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 portions: 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 pricke, where you would haue the plumme line to lighte (whiche I call the first pricke,) and with the other foote drawe an arche line righte ouer the pricke, as you can ayme it: then open youre compas as wide as all fiue measures be, and set the one foote in the fourth pricke, 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.

The line is A.B. and A. is the prick, 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. pricke, 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 compass 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 compas as wide as K.L. (that is all .v. partes) and set one foote in G, (that is the iij. pricke) 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 depend of the .xlvi. proposicion of y^e first boke of Euclide. An other waie yet. set one foote of the compas in the prick, 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 quantitie more there vnto, and by that prick that endeth the last part, draw a line to the pricke assigned, and it shall be a perpendicular.

Example.

A.B. is the line appointed, to whiche I must make a perpendicular line to light in the pricke assigned, which is A. Therfore doo I set one foote of the compas in A, and extend the other vnto D. makyng a part of a circle, more then a quarter, that is D.E. Then do I set one foote of the compas vnaltered in D, and stretch the other in the circular line, and it doth light in F, this space betwene D. and F. I deuide into halfe in the pricke G, whiche halfe I take with the compas, and set it beyond F. vnto H, and thefore is H. the point, by whiche the perpendicular line must be drawn, so say I that the line H.A, is a plumbe line to A.B, as the conclusion would.

THE .VI. CONCLVSION.

To drawe a streight line from any pricke that is not in a line, and to make it perpendicular to an other line.