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._

[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 augmented, that is A.D, and the portion 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 long 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