If a right line be deuided into two equall partes, and one of these .ij. partes diuided agayn into two other partes, as happeneth the longe square that is made of the thyrd or later part of that diuided line, with the residue of the same line, and the square of the mydlemoste parte, are bothe togither equall to the square of halfe the firste line.
Example.
The line A.B. is diuided into ij. equal partes in C, and that parte C.B. is diuided agayne as hapneth in D. Wherfore saith the Theorem that the long square made of D.B. and A.D, with the square of C.D. (which is the mydle portion) shall bothe be equall to the square of half the lyne A.B, that is to saye, to the square of A.C, or els of C.D, which make all one. The long square F.G.N.O. whiche is the longe square that the theoreme speaketh of, is made of .ij. long squares, wherof the fyrst is F.G.M.K, and the seconde is K.N.O.M. The square of the myddle portion is L.M.O.P. and the square of the halfe of the fyrste lyne is E.K.Q.L. Nowe by the theoreme, that longe square F.G.N.O, with the iuste square L.M.O.P, muste bee equall to the greate square E.K.Q.L, whyche thynge bycause it seemeth somewhat difficult to vnderstande, althoughe I intende not here to make demonstrations of the Theoremes, bycause it is appoynted to be done in the newe edition of Euclide, yet I wyll shew you brefely how the equalitee of the partes doth stande. And fyrst I say, that where the comparyson of equalitee is made betweene the greate square (whiche is made of halfe the line A.B.) and two other, where of the fyrst is the longe square F.G.N.O, and the second is the full square L.M.O.P, which is one portion of the great square all redye, and so is that longe square K.N.M.O, beynge a parcell also of the longe square F.G.N.O, Wherfore as those two partes are common to bothe partes compared in equalitee, and therfore beynge bothe abated from eche parte, if the reste of bothe the other partes bee equall, than were those whole partes equall before: Nowe the reste of the great square, those
two lesser squares beyng taken away, is that longe square E.N.P.Q, whyche is equall to the long square F.G.K.M, beyng the rest of the other parte. And that they two be equall, theyr sydes doo declare. For the longest lynes that is F.K and E.Q are 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.
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.