Part 16 (1/2)
_Example._
Example of this Theoreme you may see in the last figure, where as sixe triangles made betwene those two gemowe lines A.B. and C.F, the first triangle is A.C.D, the seconde is A.D.E, the thirde is A.D.B, the fourth is A.B.E, the fifte is D.E.B, and the sixte is B.E.F, of which sixe triangles, A.D.E. and D.E.B.
are equall, bicause they haue one common grounde line. And so likewise A.B.E. and A.B.D, whose commen grounde line is A.B, but A.C.D. is equal to B.E.F, being both betwene one couple of parallels, not bicause thei haue one ground line, but bicause they haue their ground lines equall, for C.D. is equall to E.F, as you may declare thus. C.D, is equall to A.B. (by the foure and twenty Theoreme) for thei are two contrary sides of one lykeiamme. A.C.D.B, and E.F by the same theoreme, is equall to A.B, for thei ar the two y^e contrary sides of the likeiamme, A.E.F.B, wherfore C.D. must needes be equall to E.F. like wise the triangle A.C.D, is equal to A.B.E, bicause they ar made betwene one paire of parallels and haue their groundlines like, I meane C.D. and A.B. Againe A.D.E, is equal to eche of them both, for his ground line D.E, is equall to A.B, inso muche as they are the contrary sides of one likeiamme, that is the long square A.B.D.E. And thus may you proue the equalnes of all the reste.
_The xxix. Theoreme._
Al equal triangles that are made on one grounde line, and rise one waye, must needes be betwene one paire of parallels.
_Example._
Take for example A.D.E, and D.E.B, which (as the xxvij.
conclusion dooth proue) are equall togither, and as you see, they haue one ground line D.E. And againe they rise towarde one side, that is to say, vpwarde toward the line A.B, wherfore they must needes be inclosed betweene one paire of parallels, which are heere in this example A.B. and D.E.
_The thirty Theoreme._
Equal triangles that haue their ground lines equal, and be draw? toward one side, ar made betwene one paire of paralleles.
_Example._
The example that declared the last theoreme, maye well serue to the declaracion of this also. For those ij. theoremes do diffre but in this one pointe, that the laste theoreme meaneth of triangles, that haue one ground line common to them both, and this theoreme dothe presuppose the grounde lines to bee diuers, but yet of one length, as A.C.D, and B.E.F, as they are ij.
equall triangles approued, by the eighte and twentye Theorem, so in the same Theorem it is declared, y^t their groud lines are equall togither, that is C.D, and E.F, now this beeynge true, and considering that they are made towarde one side, it foloweth, that they are made betwene one paire of parallels when I saye, drawen towarde one side, I meane that the triangles must be drawen other both vpward frome one parallel, other els both downward, for if the one be drawen vpward and the other downward, then are they drawen betwene two paire of parallels, presupposinge one to bee drawen by their ground line, and then do they ryse toward contrary sides.
_The x.x.xi. theoreme._
If a likeiamme haue one ground line with a triangle, and be drawen betwene one paire of paralleles, then shall the likeiamme be double to the triangle.
_Example._
[Ill.u.s.tration]
A.H. and B.G. are .ij. gemow lines, betwene which there is made a triangle B.C.G, and a lykeiamme, A.B.G.C, whiche haue a grounde lyne, that is to saye, B.G. Therfore doth it folow that the lyke iamme A.B.G.C. is double to the triangle B.C.G. For euery halfe of that lykeiamme is equall to the triangle, I meane A.B.F.E. other F.E.C.G. as you may coniecture by the .xi.
conclusion geometrical.
And as this Theoreme dothe speake of a triangle and likeiamme that haue one groundelyne, so is it true also, yf theyr groundelynes bee equall, though they bee dyuers, so that thei be made betwene one payre of paralleles. And hereof may you perceaue the reason, why in measuryng the platte of a triangle, you must multiply the perpendicular lyne by halfe the grounde lyne, or els the hole grounde lyne by halfe the perpendicular, for by any of these bothe waies is there made a lykeiamme equall to halfe suche a one as shulde be made on the same hole grounde lyne with the triangle, and betweene one payre of paralleles.
Therfore as that lykeiamme is double to the triangle, so the halfe of it, must needes be equall to the triangle. Compare the .xi. conclusion with this theoreme.
_The .x.x.xij. Theoreme._
In all likeiammes where there are more than one made aboute one bias line, the fill squares of euery of them must nedes be equall.
[Ill.u.s.tration]
_Example._
Fyrst before I declare the examples, it shal be mete to shew the true vnderstdyng of this theorem. [Sidenote: _Bias lyne._]