Part 6 (1/2)
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.