The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 6

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 a.s.signed.

Fyrste draw a line against the corner a.s.signed, and so is it a triangle, then take heede to the line and the pointe in it a.s.signed, and consider if that line from the p.r.i.c.ke to this end bee as long as any of the sides that make the triangle a.s.signed, and if it bee longe enoughe, then p.r.i.c.k 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 a.s.signed triangle. If it bee not longe inoughe, thenn lengthen it fyrste, and afterwarde doo as I haue sayde beefore.

[Ill.u.s.tration]

_Example._

Lette the angle appoynted bee A.B.C, and the corner a.s.signed, B.

Farthermore let the lymited line bee D.G, and the p.r.i.c.ke a.s.signed D.

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

[Ill.u.s.tration]

Then consideringe that D.G, is longer thanne A.B, you shall cut out a line fr D. toward G, equal to A.B, as for exple D.F.

Th? measure oute the other ij. lines and worke with th?

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 op? your compa.s.se to the iuste length of one of them, and sette one foote of the compa.s.se 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 compa.s.se 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 p.r.i.c.ke of the fourth corner of the square quadrate which you seke for, therfore draw a line from that p.r.i.c.ke to the eande of eche line, and you shall therby haue made a square quadrate.

_Example._

[Ill.u.s.tration]

A.B. is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plube line vnto it, whiche shall lighte in A, and that plub line is A.C, then open I my compa.s.se 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 p.r.i.c.k 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 likeime equall to a triangle appointed, and that in a right lined gle 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 p.r.i.c.ke of that deuision, you shall raise an other line parallele to that former line, and continewe it vnto the first gemowe line, and th? 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.

[Ill.u.s.tration]

_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 likeime 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. cclusion teacheth) and that angle is B.A.E. Then to procede, I doo parte in y^e middle the said groud line B.G, in the p.r.i.c.k F, fr which p.r.i.c.k 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 likeime 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 .x.x.xi. 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 .x.x.xi. theoreme.

THE .XVI. CONCLVSION.

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

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 likeime.

Therfore thus shall you procede. Firste accordyng to the laste conclusion, make a likeiamme in the angle appoincted, equall to the triangle that is a.s.signed. Then with your compa.s.se 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 p.r.i.c.kes 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._

[Ill.u.s.tration]

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 a.s.signed line (which is A.B,) and with my compas I sette forthe the same l?gth 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 pa.s.se by the p.r.i.c.ke 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 a.s.signed 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 trigle, dep?deth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, wh? 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 cclusion, 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 happ? 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 l?gth of one of his sides, and set that as a line a.s.signed, on whiche you shal make the other likeiams, according to the twelft cclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so y^t you mai easily ioyne th? into one figure.

[Ill.u.s.tration]

_Example._

If the right lined figure be like vnto A, th? may it be turned into triangles that wil std betwene ij. parallels anye ways, as you mai se by C. and D, for ij. sides of both the tringles ar parallels. Also if the right lined figure be like vnto E, th?

wil it be turned into trigles, liyng betwene two parallels also, as y^e other did before, as in the exple of F.G. But and if y^e right lined figure be like vnto H, and so turned into trigles as you se in K.L.M, wher it is parted into iij trigles, th? 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.

[Ill.u.s.tration]

Further explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then dilig?tly behold how these sundry figures be turned into trigles. 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 trigles, and fiue parallels, in the iiij. you se fiue trigles & four parallels. in the fift, iiij. trigles and .iiij. parallels, & in y^e sixt ther ar fiue trigles & iiij. paralels. Howbeit a m maye at liberty alter them into diuers formes of trigles & 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 a.s.signed 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 a.s.signed. 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.

[Ill.u.s.tration]

_Example._

A.B, is the lyne a.s.signed. E. is the middle p.r.i.c.ke 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 a.s.signed vnto me) is so diuided in this point F, y^t y^e square of y^e hole line A.B, & of the one porti (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 ill.u.s.tration, and change one occurrence of E to F in the text ("at whiche diuision I set..."), _or_: --keep the ill.u.s.tration as printed, and transpose all other occurrences of E and F in the text.]

THE .XIX. CONCLVSION.