owe adde togyther all the partes of the two lesser
squares, that is to saye, sixtene and nyne, and you perceyue
that they make twenty and fiue, whyche is an equall numbre to
the summe of the greatter square.

By this theoreme you may vnderstand a redy way to know the syde
of any ryght anguled triangle that is vnknowen, so that you
knowe the lengthe of any two sydes of it. For by tournynge the
two sydes certayne into theyr squares, and so addynge them
togyther, other subtractynge the one from the other (accordyng
as in the vse of these theoremes I haue sette foorthe) and then
fyndynge the roote of the square that remayneth, which roote
(I meane the syde of the square) is the iuste length of the
vnknowen syde, whyche is sought for. But this appertaineth to
the thyrde booke, and therefore I wyll speake no more of it at
this tyme.

_The xxxiiij. Theoreme._

If so be it, that in any triangle, the square of the one
syde be equall to the .ij. squares of the other .ij. sides,
than must nedes that corner be a right corner, which is
conteined betwene those two lesser sydes.

_Example._

As in the figure of the laste Theoreme, bicause A.C, made in
square, is asmuch as the square of A.B, and also as the square
of B.C. ioyned bothe togyther, therefore the angle that is
inclosed betwene those .ij. lesser lynes, A.B. and B.C. (that is
to say) the angle B. whiche lieth against the line A.C, must
nedes be a ryght angle. This theoreme dothe so depende of the
truthe of the laste, that whan you perceaue the truthe of the
one, you can not iustly doubt of the others truthe, for they
conteine one sentence, contrary waies pronounced.

_The .xxxv. theoreme._

If there be set forth .ij. right lines, and one of them
parted into sundry partes, how many or few so euer they be,
the square that is made of those ij. right lines proposed,
is equal to all the squares, that are made of the vndiuided
line, and euery parte of the diuided line.

[Illustration]

_Example._

The ij. lines proposed ar A.B. and C.D, and the lyne A.B. is
deuided into thre partes by E. and F. Now saith this theoreme,
that the square that is made of those two whole lines A.B. and
C.D, so that the line A.B. standeth for the length of the square,
and the other line C.D. for the bredth of the same. That square
(I say) wil be equall to all the squares that be made, of the
vndiueded lyne (which is C.D.) and euery portion of the diuided
line. An