equall, and so are the shorter lynes, F.G, and E.N, and so appereth the truthe of the Theoreme. _The .xl. theoreme._ If a right line be diuided into .ij. euen partes, and an other right line annexed to one ende of that line, so that it make one righte line with the firste. The longe square that is made of this whole line so augmented, and the portion that is added, with the square of halfe the right line, shall be equall to the square of that line, whiche is compounded of halfe the firste line, and the parte newly added. _Example._ [Illustration] The fyrst lyne propounded is A.B, and it is diuided into ij. equall partes in C, and an other ryght lyne, I meane B.D annexed to one ende of the fyrste lyne. Nowe say I, that the long square A.D.M.K, is made of the whole lyne so augmẽted, that is A.D, and the portiõ annexed, y^t is D.M, for D.M is equall to B.D, wherfore y^t long square A.D.M.K, with the square of halfe the first line, that is E.G.H.L, is equall to the great square E.F.D.C. whiche square is made of the line C.D. that is to saie, of a line compounded of halfe the first line, beyng C.B, and the portion annexed, that is B.D. And it is easyly perceaued, if you consyder that the longe square A.C.L.K. (whiche onely is lefte out of the great square) hath another longe square equall to hym, and to supply his steede in the great square, and that is G.F.M.H. For their sydes be of lyke lines in length. _The xli. Theoreme._ If a right line bee diuided by chaunce, the square of the same whole line, and the square of one of his partes are iuste equall to the lõg square of the whole line, and the sayde parte twise taken, and more ouer to the square of the other parte of the sayd line. _Example._ [Illustration] A.B. is the line diuided in C. And D.E.F.G, is the square of the whole line, D.H.K.M. is the square of the lesser portion (whyche I take for an example) and therfore must bee twise reckened. Nowe I saye that those ij. squares are equall to two longe squares of the whole line A.B, and his sayd portion A.C, and also to the square of the other portion of the sayd first line, whiche portion is C.B, and his square K.N.F.L. In this theoreme there is no difficultie, if you cõsyder that the litle square D.H.K.M. is .iiij. tymes reckened, that is to say, fyrst of all as a parte of the greatest square, whiche is D.E.F.G. Secondly he is rekned by him selfe. Thirdely he is accompted as parcell of the long square D.E.N.M, And fourthly he is taken as a part of the other long square D.H.L.G, so that in as muche as he is twise reckened in one part of the comparisõ of equalitee, and twise also in the second parte, there can rise none occasion of errour or doubtfulnes therby. _The xlij. Theoreme._