The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 5

A.B. is the line appointed, to whiche I must make a perpendicular line to light in the p.r.i.c.ke a.s.signed, 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 s.p.a.ce betwene D. and F. I deuide into halfe in the p.r.i.c.ke 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 p.r.i.c.ke that is not in a line, and to make it perpendicular to an other line.

Open your compas as so wide that it may extend somewhat farther, th? from the p.r.i.c.k to the line, then sette the one foote of the compas in the p.r.i.c.ke, and with the other shall you draw a cpa.s.sed line, that shall crosse that other first line in .ij.

places. Now if you deuide that arch line into .ij. equall partes, and from the middell p.r.i.c.ke therof vnto the p.r.i.c.k without the line you drawe a streight line, it shalbe a plumbe line to that firste lyne, accordyng to the conclusion.

[Ill.u.s.tration]

_Example._

C. is the appointed p.r.i.c.ke, from whiche vnto the line A.B. I must draw a perp?dicular. Thefore I open the cpas so wide, that it may haue one foote in C, and thother to reach ouer the line, and with y^t foote I draw an arch line as you see, betwene A.

and B, which arch line I deuide in the middell in the point D.

Then drawe I a line from C. to D, and it is perpendicular to the line A.B, accordyng as my desire was.

THE .VII. CONCLVSION.

To make a plumbe lyne on any porcion of a circle, and that on the vtter or inner bughte.

Mark first the p.r.i.c.k where y^e plube line shal lyght: and p.r.i.c.k out of ech side of it .ij. other poinctes equally distant from that first p.r.i.c.ke. Then set the one foote of the cpas in one of those side p.r.i.c.kes, and the other foote in the other side p.r.i.c.ke, and first moue one of the feete and drawe an arche line ouer the middell p.r.i.c.ke, then set the compas steddie with the one foote in the other side p.r.i.c.ke, and with the other foote drawe an other arche line, that shall cut that first arche, and from the very poincte of their meetyng, drawe a right line vnto the firste p.r.i.c.ke, where you do minde that the plumbe line shall lyghte. And so haue you performed thintent of this conclusion.

_Example._

[Ill.u.s.tration]

The arche of the circle on whiche I would erect a plumbe line, is A.B.C. and B. is the p.r.i.c.ke where I would haue the plumbe line to light. Therfore I meate out two equall distaunces on eche side of that p.r.i.c.ke B. and they are A.C. Then open I the compas as wide as A.C. and settyng one of the feete in A. with the other I drawe an arche line which goeth by G. Like waies I set one foote of the compas steddily in C. and with the other I drawe an arche line, goyng by G. also. Now consideryng that G.

is the p.r.i.c.ke of their meetyng, it shall be also the poinct fro whiche I must drawe the plube line. Then draw I a right line from G. to B. and so haue mine intent. Now as A.B.C. hath a plumbe line erected on his vtter bought, so may I erect a plumbe line on the inner bught of D.E.F, doynge with it as I did with the other, that is to saye, fyrste settyng forthe the p.r.i.c.ke where the plumbe line shall light, which is E, and then markyng one other on eche syde, as are D. and F. And then proceding as I dyd in the example before.

THE VIII. CONCLVSYON.

How to deuide the arche of a circle into two equall partes, without measuring the arche.

Deuide the corde of that line info ij. equall portions, and then from the middle prycke erecte a plumbe line, and it shal parte that arche in the middle.

[Ill.u.s.tration]

_Example._

The arch to be diuided ys A.D.C, the corde is A.B.C, this corde is diuided in the middle with B, from which p.r.i.c.k if I erect a plum line as B.D, th? will it diuide the arch in the middle, that is to say, in D.

THE IX. CONCLVSION.

To do the same thynge other wise. And for shortenes of worke, if you wyl make a plumbe line without much labour, you may do it with your squyre, so that it be iustly made, for yf you applye the edge of the squyre to the line in which the p.r.i.c.k is, and foresee the very corner of the squyre doo touche the p.r.i.c.ke. And than frome that corner if you drawe a lyne by the other edge of the squyre, yt will be perpendicular to the former line.

[Ill.u.s.tration]

_Example._

A.B. is the line, on which I wold make the plumme line, or perpendicular. And therefore I marke the p.r.i.c.k, from which the plumbe lyne muste rise, which here is C. Then do I sette one edg of my squyre (that is B.C.) to the line A.B, so at the corner of the squyre do touche C. iustly. And from C. I drawe a line by the other edge of the squire, (which is C.D.) And so haue I made the plumme line D.C, which I sought for.

THE X. CONCLVSION.

How to do the same thinge an other way yet

If so be it that you haue an arche of suche greatnes, that your squyre wyll not suffice therto, as the arche of a brydge or of a house or window, then may you do this. Mete vnderneth the arch where y^e midle of his cord wyl be, and ther set a mark. Then take a long line with a plummet, and holde the line in suche a place of the arch, that the plummet do hang iustely ouer the middle of the corde, that you didde diuide before, and then the line doth shewe you the middle of the arche.

[Ill.u.s.tration]

_Example._

The arch is A.D.B, of which I trye the midle thus. I draw a corde from one syde to the other (as here is A.B,) which I diuide in the middle in C. Th? take I a line with a plummet (that is D.E,) and so hold I the line that the plummet E, dooth hange ouer C, And then I say that D. is the middle of the arche.

And to thentent that my plummet shall point the more iustely, I doo make it sharpe at the nether ende, and so may I trust this woorke for certaine.

THE XI. CONCLVSION.

When any line is appointed and without it a p.r.i.c.ke, whereby a parallel must be drawen howe you shall doo it.

Take the iuste measure beetwene the line and the p.r.i.c.ke, accordinge to which you shal open your compa.s.se. Th? pitch one foote of your compa.s.se at the one ende of the line, and with the other foote draw a bowe line right ouer the pytche of the compa.s.se, lyke-wise doo at the other ende of the lyne, then draw a line that shall touche the vttermoste edge of bothe those bowe lines, and it will bee a true parallele to the fyrste lyne appointed.

_Example._

[Ill.u.s.tration]

A.B, is the line vnto which I must draw an other gemow line, which muste pa.s.se by the p.r.i.c.k C, first I meate with my compa.s.se the smallest distance that is from C. to the line, and that is C.F, wherfore staying the compa.s.se at that distaunce, I seete the one foote in A, and with the other foot I make a bowe lyne, which is D, th? like wise set I the one foote of the compa.s.se in B, and with the other I make the second bow line, which is E.

And then draw I a line, so that it toucheth the vttermost edge of bothe these bowe lines, and that lyne pa.s.seth by the p.r.i.c.ke C, end is a gemowe line to A.B, as my sekyng was.

THE .XII. CONCLVSION.

To make a triangle of any .iij. lines, so that the lines be suche, that any .ij. of them be longer then the thirde. For this rule is generall, that any two sides of euerie triangle taken together, are longer then the other side that remaineth.

If you do remember the first and seconde conclusions, then is there no difficultie in this, for it is in maner the same woorke. First csider the .iij. lines that you must take, and set one of th? for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.

_Example._

[Ill.u.s.tration]

I haue .iij. A.B. and C.D. and E.F. of whiche I put .C.D. for my ground line, then with my compas I take the length of .A.B. and set the one foote of my compas in C, and draw an arch line with the other foote. Likewaies I take the l?gth of E.F, and set one foote in D, and with the other foote I make an arch line crosse the other arche, and the p.r.i.c.ke of their metyng (whiche is G.) shall be the thirde corner of the triangle, for in all suche kyndes of woorkynge to make a tryangle, if you haue one line drawen, there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee, for two of them must needes be at the two eandes of the lyne that is drawen.