her halfe ioyned in one with the annexed portion, as
one whole line.

_Example._

[Illustration]

The line is A.B, and is diuided firste into twoo equal partes in
C, and then is there annexed to it an other portion whiche is
B.D. Now saith the Theoreme, that the square of A.D, and the
square of B.D, ar double to the square of A.C, and to the square
of C.D. The line A.B. containing four partes, then must needes
his halfe containe ij. partes of such partes I suppose B.D.
(which is the annexed line) to containe thre, so shal the hole
line comprehend vij. parts, and his square xlix. parts, where
vnto if you ad y^e square of the annexed lyne, whiche maketh
nyne, than those bothe doo yelde, lviij. whyche must be double
to the square of the halfe lyne with the annexed portion. The
halfe lyne by it selfe conteyneth but .ij. partes, and therfore
his square dooth make foure. The halfe lyne with the annexed
portion conteyneth fiue, and the square of it is .xxv, now put
foure to .xxv, and it maketh iust .xxix, the euen halfe of fifty
and eight, wherby appereth the truthe of the theoreme.

_The .xlv. theoreme._

In all triangles that haue a blunt angle, the square of the
side that lieth against the blunt angle, is greater than the
two squares of the other twoo sydes, by twise as muche as is
comprehended of the one of those .ij. sides (inclosyng the
blunt corner) and the portion of the same line, beyng drawen
foorth in lengthe, which lieth betwene the said blunt corner
and a perpendicular line lightyng on it, and drawen from one
of the sharpe angles of the foresayd triangle.

_Example._

For the declaration of this theoreme and the next also, whose
vse are wonderfull in the practise of Geometrie, and in
measuryng especially, it shall be nedefull to declare that euery
triangle that hath no ryght angle as those whyche are called (as
in the boke of practise is declared) sharp cornered triangles,
and blunt cornered triangles, yet may they be brought to haue a
ryght angle, eyther by partyng them into two lesser triangles,
or els by addyng an other triangle vnto them, whiche may be a
great helpe for the ayde of measuryng, as more largely shall be
sette foorthe in the boke of measuryng. But for this present
place, this forme wyll I vse, (whiche Theon also vseth) to adde
one triangle vnto an other, to bryng the blunt cornered triangle
into a ryght angled triangle, whereby the proportion of the
squares of th