[ The thirty Theoreme.]
Equal triangles that haue their ground lines equal, and be drawẽ 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, yt their groũd 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 xxxi. 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.
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.