Part 4

_Example._

C. is the appointed pricke, from whiche vnto the line A.B. I must draw a perpendicular. Thefore I open the compas 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 prick where y^e plumbe line shal lyght: and prick out of ech side of it .ij. other poinctes equally distant from that first pricke. Then set the one foote of the compas in one of those side prickes, and the other foote in the other side pricke, and first moue one of the feete and drawe an arche line ouer the middell pricke, then set the compas steddie with the one foote in the other side pricke, 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 pricke, where you do minde that the plumbe line shall lyghte. And so haue you performed thintent of this conclusion.

_Example._

The arche of the circle on whiche I would erect a plumbe line, is A.B.C. and B. is the pricke where I would haue the plumbe line to light. Therfore I meate out two equall distaunces on eche side of that pricke 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 pricke of their meetyng, it shall be also the poinct fro whiche I must drawe the plumbe 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 pricke 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.

_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 prick if I erect a plum line as B.D, then 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 prick is, and foresee the very corner of the squyre doo touche the pricke. 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.

_Example._

A.B. is the line, on which I wold make the plumme line, or perpendicular. And therefore I marke the prick, 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.

_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. Then 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 pricke, whereby a parallel must be drawen howe you shall doo it.

Take the iuste measure beetwene the line and the pricke, accordinge to which you shal open your compasse. Then pitch one foote of your compasse at the one ende of the line, and with the other foote draw a bowe line right ouer the pytche of the compasse, 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._

A.B, is the line vnto which I must draw an other gemow line, which muste passe by the prick C, first I meate with my compasse the smallest distance that is from C. to the line, and that is C.F, wherfore staying the compasse at that distaunce, I seete the one foote in A, and with the other foot I make a bowe lyne, which is D, then like wise set I the one foote of the compasse 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 passeth by the pricke 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 consider the .iij. lines that you must take, and set one of them for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.

_Example._

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 length 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 pricke 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.

THE XIII. CONCLVSION.

If you haue a line appointed, and a pointe in it limited, howe you maye make on it a righte lined angle, equall to an other right lined angle, all ready assigned.

Fyrste draw a line against the corner assigned, and so is it a triangle, then take heede to the line and the pointe in it assigned, and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned, and if it bee longe enoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accordinge to the laste conlusion, makynge a triangle of thre like lynes to that assigned triangle. If it bee not longe inoughe, thenn lengthen it fyrste, and afterwarde doo as I haue sayde beefore.

_Example._

Lette the angle appoynted bee A.B.C, and the corner assigned, B. Farthermore let the lymited line bee D.G, and the pricke assigned D.

Fyrste therefore by drawinge the line A.C, I make the triangle A.B.C.

Then consideringe that D.G, is longer thanne A.B, you shall cut out a line from D. toward G, equal to A.B, as for example D.F. Then measure oute the other ij. lines and worke with them according as the conclusion with the fyrste also and the second teacheth yow, and then haue you done.

THE XIIII. CONCLVSION.

To make a square quadrate of any righte lyne appoincted.

First make a plumbe line vnto your line appointed, whiche shall light at one of the endes of it, accordyng to the fifth conclusion, and let it be of like length as your first line is, then open your compasse to the iuste length of one of them, and sette one foote of the compasse in the ende of the one line, and with the other foote draw an arche line, there as you thinke that the fowerth corner shall be, after that set the one foote of the same compasse vnsturred, in the eande of the other line, and drawe an other arche line crosse the first archeline, and the poincte that they do crosse in, is the pricke of the fourth corner of the square quadrate which you seke for, therfore draw a line from that pricke to the eande of eche line, and you shall therby haue made a square quadrate.

_Example._

A.B. is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plumbe line vnto it, whiche shall lighte in A, and that plumb line is A.C, then open I my compasse as wide as the length of A.B, or A.C, (for they must be bothe equall) and I set the one foote of thend in C, and with the other I make an arche line nigh vnto D, afterward I set the compas again with one foote in B, and with the other foote I make an arche line crosse the first arche line in D, and from the prick of their crossyng I draw .ij. lines, one to B, and an other to C, and so haue I made the square quadrate that I entended.

THE .XV. CONCLVSION.

To make a likeiamme equall to a triangle appointed, and that in a right lined angle limited.

First from one of the angles of the triangle, you shall drawe a gemowe line, whiche shall be a parallele to that syde of the triangle, on whiche you will make that likeiamme. Then on one end of the side of the triangle, whiche lieth against the gemowe lyne, you shall draw forth a line vnto the gemow line, so that one angle that commeth of those .ij. lines be like to the angle which is limited vnto you. Then shall you deuide into ij. equall partes that side of the triangle whiche beareth that line, and from the pricke of that deuision, you shall raise an other line parallele to that former line, and continewe it vnto the first gemowe line, and then of those .ij. last gemowe lynes, and the first gemowe line, with the halfe side of the triangle, is made a lykeiamme equall to the triangle appointed, and hath an angle lyke to an angle limited, accordyng to the conclusion.

_Example._

B.C.G, is the triangle appoincted vnto, whiche I muste make an equall likeiamme. And D, is the angle that the likeiamme must haue. Therfore first entendyng to erecte the likeiamme on the one side, that the ground line of the triangle (whiche is B.G.) I do draw a gemow line by C, and make it parallele to the ground line B.G, and that new gemow line is A.H. Then do I raise a line from B. vnto the gemowe line, (whiche line is A.B) and make an angle equall to D, that is the appointed angle (accordyng as the .viij. conclusion teacheth) and that angle is B.A.E. Then to procede, I doo parte in y^e middle the said ground line B.G, in the prick F, from which prick I draw to the first gemowe line (A.H.) an other line that is parallele to A.B, and that line is E.F. Now saie I that the likeiamme B.A.E.F, is equall to the triangle B.C.G. And also that it hath one angle (that is B.A.E.) like to D. the angle that was limitted. And so haue I mine intent. The profe of the equalnes of those two figures doeth depend of the .xli. proposition of Euclides first boke, and is the .xxxi. proposition of this second boke of Theoremis, whiche saieth, that whan a tryangle and a likeiamme be made betwene .ij. selfe same gemow lines, and haue their ground line of one length, then is the likeiamme double to the triangle, wherof it foloweth, that if .ij. suche figures so drawen differ in their ground line onely, so that the ground line of the likeiamme be but halfe the ground line of the triangle, then be those .ij. figures equall, as you shall more at large perceiue by the boke of Theoremis, in y^e .xxxi. theoreme.

THE .XVI. CONCLVSION.

To make a likeiamme equall to a triangle appoincted, accordyng to an angle limitted, and on a line also assigned.

In the last conclusion the sides of your likeiamme wer left to your libertie, though you had an angle appoincted. Nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted, which must be the side of the likeiamme. Therfore thus shall you procede. Firste accordyng to the laste conclusion, make a likeiamme in the angle appoincted, equall to the triangle that is assigned. Then with your compasse take the length of your line appointed, and set out two lines of the same length in the second gemowe lines, beginnyng at the one side of the likeiamme, and by those two prickes shall you draw an other gemowe line, whiche shall be parallele to two sides of the likeiamme. Afterward shall you draw .ij. lines more for the accomplishement of your worke, which better shall be perceaued by a shorte exaumple, then by a greate numbre of wordes, only without example, therefore I wyl by example sette forth the whole worke.

_Example._

Fyrst, according to the last conclusion, I make the likeiamme E.F.C.G, equal to the triangle D, in the appoynted angle whiche is E. Then take I the lengthe of the assigned line (which is A.B,) and with my compas I sette forthe the same length in the ij. gemow lines N.F. and H.G, setting one foot in E, and the other in N, and againe settyng one foote in C, and the other in H. Afterward I draw a line from N. to H, whiche is a gemow lyne, to ij. sydes of the likeiamme. thenne drawe I a line also from N. vnto C. and extend it vntyll it crosse the lines, E.L. and F.G, which both must be drawen forth longer then the sides of the likeiamme. and where that lyne doeth crosse F.G, there I sette M. Nowe to make an ende, I make an other gemowe line, whiche is parallel to N.F. and H.G, and that gemowe line doth passe by the pricke M, and then haue I done. Now say I that H.C.K.L, is a likeiamme equall to the triangle appointed, whiche was D, and is made of a line assigned that is A.B, for H.C, is equall vnto A.B, and so is K.L. The profe of y^e equalnes of this likeiam vnto the triangle, dependeth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, when there are more then one made about one bias line, the filsquares of euery of them muste needes be equall.

THE XVII. CONCLVSION.

To make a likeiamme equal to any right lined figure, and that on an angle appointed.

The readiest waye to worke this conclusion, is to tourn that rightlined figure into triangles, and then for euery triangle together an equal likeiamme, according vnto the eleuen conclusion, and then to ioine al those likeiammes into one, if their sides happen to be equal, which thing is euer certain, when al the triangles happen iustly betwene one pair of gemow lines. but and if they will not frame so, then after that you haue for the firste triangle made his likeiamme, you shall take the length of one of his sides, and set that as a line assigned, on whiche you shal make the other likeiams, according to the twelft conclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so y^t you mai easily ioyne them into one figure.

_Example._

If the right lined figure be like vnto A, then may it be turned into triangles that wil stand betwene ij. parallels anye ways, as you mai se by C. and D, for ij. sides of both the trianngles ar parallels. Also if the right lined figure be like vnto E, then wil it be turned into triangles, liyng betwene two parallels also, as y^e other did before, as in the example of F.G. But and if y^e right lined figure be like vnto H, and so turned into triangles as you se in K.L.M, wher it is parted into iij triangles, then wil not all those triangles lye betwen one pair of parallels or gemow lines, but must haue many, for euery triangle must haue one paire of parallels seuerall, yet it maye happen that when there bee three or fower triangles, ij. of theym maye happen to agre to one pair of parallels, whiche thinge I remit to euery honest witte to serche, for the manner of their draught wil declare, how many paires of parallels they shall neede, of which varietee bicause the examples ar infinite, I haue set forth these few, that by them you may coniecture duly of all other like.

Further explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then diligently behold how these sundry figures be turned into triangles. In the fyrst you se I haue made v. triangles, and four paralleles. in the seconde vij. triangles and foure paralleles. in the thirde thre triangles, and fiue parallels, in the iiij. you se fiue triangles & four parallels. in the fift, iiij. triangles and .iiij. parallels, & in y^e sixt ther ar fiue triangles & iiij. paralels. Howbeit a man maye at liberty alter them into diuers formes of triangles & therefore I leue it to the discretion of the woorkmaister, to do in al suche cases as he shal thinke best, for by these examples (if they bee well marked) may all other like conclusions be wrought.

THE XVIII. CONCLVSION.

To parte a line assigned after suche a sorte, that the square that is made of the whole line and one of his parts, shal be equal to the squar that cometh of the other parte alone.

First deuide your lyne into ij. equal parts, and of the length of one part make a perpendicular to light at one end of your line assigned. then adde a bias line, and make thereof a triangle, this done if you take from this bias line the halfe lengthe of your line appointed, which is the iuste length of your perpendicular, that part of the bias line whiche dothe remayne, is the greater portion of the deuision that you seke for, therefore if you cut your line according to the lengthe of it, then will the square of that greater portion be equall to the square that is made of the whole line and his lesser portion. And contrary wise, the square of the whole line and his lesser parte, wyll be equall to the square of the greater parte.

_Example._

A.B, is the lyne assigned. E. is the middle pricke of A.B, B.C. is the plumb line or perpendicular, made of the halfe of A.B, equall to A.E, other B.E, the byas line is C.A, from whiche I cut a peece, that is C.D, equall to C.B, and accordyng to the lengthe lo the peece that remaineth (whiche is D.A,) I doo deuide the line A.B, at whiche diuision I set F. Now say I, that this line A.B, (w^{ch} was assigned vnto me) is so diuided in this point F, y^t y^e square of y^e hole line A.B, & of the one portion (y^t is F.B, the lesser part) is equall to the square of the other parte, whiche is F.A, and is the greater part of the first line. The profe of this equalitie shall you learne by the .xl. Theoreme.

[Transcriber's Note: There are two ways to make this Example work: --transpose E and F in the illustration, and change one occurrence of E to F in the text ("at whiche diuision I set..."), _or_: --keep the illustration as printed, and transpose all other occurrences of E and F in the text.]

THE .XIX. CONCLVSION.

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 pricke 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._

K, is the right lined figure appointed, and B.C.D.E, is the likeiamme, 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 likeiamme, 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 pricke M, and on that pricke 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 pricke 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.

_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 compasse, 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.

_Example._