f you take from this bias line the halfe
lengthe of your line appointed, which is the iuste length of
your perpendicular, that part of the bias line whiche dothe
remayne, is the greater portion of the deuision that you seke
for, therefore if you cut your line according to the lengthe of
it, then will the square of that greater portion be equall to
the square that is made of the whole line and his lesser
portion. And contrary wise, the square of the whole line and his
lesser parte, wyll be equall to the square of the greater parte.

[Illustration]

_Example._

A.B, is the lyne assigned. E. is the middle pricke of A.B, B.C.
is the plumb line or perpendicular, made of the halfe of A.B,
equall to A.E, other B.E, the byas line is C.A, from whiche I
cut a peece, that is C.D, equall to C.B, and accordyng to the
lengthe lo the peece that remaineth (whiche is D.A,) I doo
deuide the line A.B, at whiche diuision I set F. Now say I, that
this line A.B, (w^{ch} was assigned vnto me) is so diuided in
this point F, y^t y^e square of y^e hole line A.B, & of the one
portion (y^t is F.B, the lesser part) is equall to the square of
the other parte, whiche is F.A, and is the greater part of the
first line. The profe of this equalitie shall you learne by the
.xl. Theoreme.

[Transcriber's Note:
There are two ways to make this Example work:
--transpose E and F in the illustration, and change one
occurrence of E to F in the text ("at whiche diuision I
set..."), _or_:
--keep the illustration as printed, and transpose all other
occurrences of E and F in the text.]

THE .XIX. CONCLVSION.

To make a square quadrate equall to any right lined figure
appoincted.

First make a likeiamme equall to that right lined figure, with a
right angle, accordyng to the .xi. conclusion, then consider the
likeiamme, whether it haue all his sides equall, or not: for yf
they be all equall, then haue you doone your conclusion. but and
if the sides be not all equall, then shall you make one right
line iuste as long as two of those vnequall sides, that line
shall you deuide in the middle, and on that pricke drawe half a
circle, then cutte from that diameter of the halfe circle a
certayne portion equall to the one side of the likeiamme, and
from that pointe of diuision shall you erecte a perpendicular,
which shall touche the edge of the circle. And that
perpendicular shall be the iuste side of the square quadrate,
equall both to the