The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 7

To make a square quadrate equall to any right lined figure appoincted.

First make a likeiamme equall to that right lined figure, with a right angle, accordyng to the .xi. conclusion, then consider the likeiamme, whether it haue all his sides equall, or not: for yf they be all equall, then haue you doone your conclusion. but and if the sides be not all equall, then shall you make one right line iuste as long as two of those vnequall sides, that line shall you deuide in the middle, and on that p.r.i.c.ke drawe half a circle, then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme, and from that pointe of diuision shall you erecte a perpendicular, which shall touche the edge of the circle. And that perpendicular shall be the iuste side of the square quadrate, equall both to the lykeiamme, and also to the right lined figure appointed, as the conclusion willed.

_Example._

[Ill.u.s.tration]

K, is the right lined figure appointed, and B.C.D.E, is the likeime, with right angles equall vnto K, but because that this likeiamme is not a square quadrate, I must turne it into such one after this sort, I shall make one right line, as long as .ij. vnequall sides of the likeime, that line here is F.G, whiche is equall to B.C, and C.E. Then part I that line in the middle in the p.r.i.c.ke M, and on that p.r.i.c.ke I make halfe a circle, accordyng to the length of the diameter F.G. Afterward I cut awaie a peece from F.G, equall to C.E, markyng that point with H. And on that p.r.i.c.ke I erecte a perpendicular H.K, whiche is the iust side to the square quadrate that I seke for, therfore accordyng to the doctrine of the .x. conclusion, of the lyne I doe make a square quadrate, and so haue I attained the practise of this conclusion.

THE .XX. CONCLVSION.

When any .ij. square quadrates are set forth, how you maie make one equall to them bothe.

First drawe a right line equall to the side of one of the quadrates: and on the ende of it make a perpendicular, equall in length to the side of the other quadrate, then drawe a byas line betwene those .ij. other lines, makyng thereof a right angeled triangle. And that byas lyne wyll make a square quadrate, equall to the other .ij. quadrates appointed.

[Ill.u.s.tration]

_Example._

A.B. and C.D, are the two square quadrates appointed, vnto which I must make one equall square quadrate. First therfore I dooe make a righte line E.F, equall to one of the sides of the square quadrate A.B. And on the one end of it I make a plumbe line E.G, equall to the side of the other quadrate D.C. Then drawe I a byas line G.F, which beyng made the side of a quadrate (accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrate H. is muche iust as the other two. I meane A.B. and D.C.

THE .XXI. CONCLVSION.

When any two quadrates be set forth, howe to make a squire about the one quadrate, whiche shall be equall to the other quadrate.

Determine with your selfe about whiche quadrate you wil make the squire, and drawe one side of that quadrate forth in lengte, accordyng to the measure of the side of the other quadrate, whiche line you maie call the grounde line, and then haue you a right angle made on this line by an other side of the same quadrate: Therfore turne that into a right cornered triangle, accordyng to the worke in the laste conclusion, by makyng of a byas line, and that byas lyne will performe the worke of your desire. For if you take the length of that byas line with your compa.s.se, and then set one foote of the compas in the farthest angle of the first quadrate (whiche is the one ende of the groundline) and extend the other foote on the same line, accordyng to the measure of the byas line, and of that line make a quadrate, enclosyng y^e first quadrate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.

[Ill.u.s.tration]

_Example._

The first square quadrate is A.B.C.D, and the seconde is E. Now would I make a squire about the quadrate A.B.C.D, whiche shall bee equall vnto the quadrate E.

Therfore first I draw the line A.D, more in length, accordyng to the measure of the side of E, as you see, from D. vnto F, and so the hole line of bothe these seuerall sides is A.F, th? make I a byas line from C, to F, whiche byas line is the measure of this woorke. wherefore I open my compas accordyng to the length of that byas line C.F, and set the one compas foote in A, and extend thother foote of the compas toward F, makyng this p.r.i.c.ke G, from whiche I erect a plumbeline G.H, and so make out the square quadrate A.G.H.K, whose sides are equall eche of them to A.G. And this square doth contain the first quadrate A.B.C.D, and also a squire G.H.K, whiche is equall to the second quadrate E, for as the last conclusion declareth, the quadrate A.G.H.K, is equall to bothe the other quadrates proposed, that is A.B.C.D, and E. Then muste the squire G.H.K, needes be equall to E, consideryng that all the rest of that great quadrate is nothyng els but the quadrate self, A.B.C.D, and so haue I thintent of this conclusion.

THE .XXII. CONCLVSION.

To find out the c?tre of any circle a.s.signed.

Draw a corde or stryngline crosse the circle, then deuide into .ij. equall partes, both that corde, and also the bowe line, or arche line, that serueth to that corde, and from the p.r.i.c.kes of those diuisions, if you drawe an other line crosse the circle, it must nedes pa.s.se by the centre. Therfore deuide that line in the middle, and that middle p.r.i.c.ke is the centre of the circle proposed.

_Example._

[Ill.u.s.tration]

Let the circle be A.B.C.D, whose centre I shall seke. First therfore I draw a corde crosse the circle, that is A.C. Then do I deuide that corde in the middle, in E, and likewaies also do I deuide his arche line A.B.C, in the middle, in the pointe B.

Afterward I drawe a line from B. to E, and so crosse the circle, whiche line is B.D, in which line is the centre that I seeke for. Therefore if I parte that line B.D, in the middle in to two equall portions, that middle p.r.i.c.ke (which here is F) is the verye centre of the sayde circle that I seke. This conclusion may other waies be wrought, as the moste part of conclusions haue sondry formes of practise, and that is, by makinge thre p.r.i.c.kes in the circuference of the circle, at liberty where you wyll, and then findinge the centre to those thre p.r.i.c.ks, Which worke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.

THE XXIII. CONCLVSION.

To find the commen centre belongyng to anye three p.r.i.c.kes appointed, if they be not in an exacte right line.

It is to be noted, that though euery small arche of a greate circle do seeme to be a right lyne, yet in very dede it is not so, for euery part of the circ.u.mference of al circles is compa.s.sed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. p.r.i.c.kes in an exact right line can not bee brought into the circ.u.mference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their cmon centre. Op? your compas so wide, that it be somewhat more then the halfe distance of two of those p.r.i.c.kes. Then sette the one foote of the compas in the one p.r.i.c.ke, and with the other foot draw an arche lyne toward the other p.r.i.c.ke, Then againe putte the foot of your compas in the second p.r.i.c.ke, and with the other foot make an arche line, that may crosse the firste arch line in ij. places. Now as you haue done with those two p.r.i.c.ks, so do with the middle p.r.i.c.ke, and the thirde that remayneth. Then draw ij. lines by the poyntes where those arche lines do crosse, and where those two lines do meete, there is the centre that you seeke for.

_Example_

[Ill.u.s.tration]

The iij. p.r.i.c.kes I haue set to be A.B, and C, whiche I wold bring into the edg of one common circle, by finding a centre cmen to them all, fyrst therefore I open my cpas, so that thei occupye more then y^e halfe distance betwene ij. p.r.i.c.ks (as are A.B.) and so settinge one foote in A. and extendinge the other toward B, I make the arche line D.E. Likewise settig one foot in B, and turninge the other toward A, I draw an other arche line that crosseth the first in D. and E. Then from D. to E, I draw a right lyne D.H. After this I open my cpa.s.se to a new distance, and make ij. arche lines betwene B. and C, whiche crosse one the other in F. and G, by whiche two pointes I draw an other line, that is F.H. And bycause that the lyne D.H. and the lyne F.H.

doo meete in H, I saye that H. is the centre that serueth to those iij. p.r.i.c.kes. Now therfore if you set one foot of your compas in H, and extend the other to any of the iij. p.r.i.c.ks, you may draw a circle w^{ch} shal enclose those iij. p.r.i.c.ks in the edg of his circufer?ce & thus haue you attained y^e vse of this cclusi.

THE XXIIII. CONCLVSION.

To drawe a touche line onto a circle, from any poincte a.s.signed.

Here must you vnderstand that the p.r.i.c.ke must be without the circle, els the conclusion is not possible. But the p.r.i.c.ke or poinct beyng without the circle, thus shall you procede: Open your compas, so that the one foote of it maie be set in the centre of the circle, and the other foote on the p.r.i.c.ke appoincted, and so draw an other circle of that largenesse about the same centre: and it shall gouerne you certainly in makyng the said touche line. For if you draw a line fr the p.r.i.c.ke appointed vnto the centre of the circle, and marke the place where it doeth crosse the lesser circle, and from that poincte erect a plumbe line that shall touche the edge of the vtter circle, and marke also the place where that plumbe line crosseth that vtter circle, and from that place drawe an other line to the centre, takyng heede where it crosseth the lesser circle, if you drawe a plumbe line from that p.r.i.c.ke vnto the edge of the greatter circle, that line I say is a touche line, drawen from the point a.s.signed, according to the meaning of this conclusion.

[Ill.u.s.tration]

_Example._

Let the circle be called B.C.D, and his c?tre E, and y^e p.r.i.c.k a.s.signed A, op? your cpas now of such widenes, y^t the one foote may be set in E, w^{ch} is y^e c?tre of y^e circle, & y^e other in A, w^{ch} is y^e pointe a.s.signed, & so make an other greter circle (as here is A.F.G) th? draw a line from A. vnto E, and wher that line doth cross y^e inner circle (w^{ch} heere is in the p.r.i.c.k B.) there erect a plub line vnto the line. A.E. and let that plumb line touch the vtter circle, as it doth here in the point F, so shall B.F. bee that plumbe lyne. Then from F.

vnto E. drawe an other line whiche shal be F.E, and it will cutte the inner circle, as it doth here in the point C, from which pointe C. if you erect a plumb line vnto A, then is that line A.C, the touche line, whiche you shoulde finde. Not withstandinge that this is a certaine waye to fynde any touche line, and a demonstrable forme, yet more easyly by many folde may you fynde and make any suche line with a true ruler, layinge the edge of the ruler to the edge of the circle and to the p.r.i.c.ke, and so drawing a right line, as this example sheweth, where the circle is E, the p.r.i.c.ke a.s.signed is A. and the ruler C.D. by which the touch line is drawen, and that is A.B, and as this way is light to doo, so is it certaine inoughe for any kinde of workinge.

[Ill.u.s.tration]

THE XXV. CONCLVSION.

When you haue any peece of the circ.u.mference of a circle a.s.signed, howe you may make oute the whole circle agreynge therevnto.

First seeke out of the centre of that arche, according to the doctrine of the seuententh conclusion, and then setting one foote of your compas in the centre, and extending the other foot vnto the edge of the arche or peece of the circ.u.mference, it is easy to drawe the whole circle.

_Example._

A peece of an olde pillar was found, like in forme to thys figure A.D.B. Now to knowe howe muche the cpa.s.se of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make in a table the like draught of y^t circuference by the self patr, vsing it as it wer a croked ruler. Then make .iij. p.r.i.c.kes in that arche line, as I haue made, C. D. and E. And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that c?tre is here F, nowe settyng one foote of your compas in F, and the other in C. D, other in E, and so makyng a compa.s.se, you haue youre whole intent.

[Ill.u.s.tration]

THE XXVI. CONCLVSION.

To finde the centre to any arche of a circle.