of the
diuided line, and there other two sides equall to C.B, beeyng he
shorter parte of the said line A.B.

So is that greatest square, beeyng made of the hole lyne A.B,
equal to the ij. squares of eche of his partes seuerally, and
more by as muche iust as .ij. longe squares, made of the longer
portion of the diuided lyne ioyned in square with the shorter
parte of the same diuided line, as the theoreme wold. And as
here I haue put an example of a lyne diuided into .ij. partes,
so the theoreme is true of all diuided lines, of what number so
euer the partes be, foure, fyue, or syxe. etc.

This theoreme hath great vse, not only in geometrie, but also in
arithmetike, as herafter I will declare in conuenient place.

_The .xxxix. theoreme._

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._

[Illustration]

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.M.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