<P_1.B1R>

<font> A touche lyne </font> , is a line that runneth a long by the
edge of a circle , onely touching it , but doth not crosse the
circumference of it , as in this exaumple you maie see .
(RECORD-E1-H,1.B1R.2)

And when that a line doth crosse the edg of the circle , the~ is it
called <font> a cord </font> , as you shall see anon in the speakynge
of circles . (RECORD-E1-H,1.B1R.3)

In the meane season must I not omit to declare what angles bee called
<font> matche corners </font> , that is to saie , suche as stande
directly one against the other , when twoo lines be drawen acrosse , as
here appereth . (RECORD-E1-H,1.B1R.4)

Where A. and B. are matche corners , so are C. and D. but not A. and C.
nother D. and A . (RECORD-E1-H,1.B1R.5)

Nowe will I beginne to speak of figures , that be properly so called ,
of whiche all be made of diuerse lines , except onely a circle , an
egge forme , and a tunne forme , which .iij. haue no angle and haue but
one line for their bounde , and an eye fourme whiche is made of one
lyne , and hath an angle onely . (RECORD-E1-H,1.B1R.6)

<font> A circle </font> is a figure made and enclosed with one line ,
(RECORD-E1-H,1.B1R.7)

and hath in the middell of it a pricke or centre , from whiche all the
lines that be drawen to the circumfernece are equall all in length , as
here you see . (RECORD-E1-H,1.B1R.8)

And the line that encloseth the whole compasse , is called the <font>
circumference </font> . (RECORD-E1-H,1.B1R.9)

And all the lines that bee drawen crosse the circle , and goe by the
centre , are named <font> diameters </font> , whose halfe , I meane
from the center to the circumference <P_1.B1V> any waie , is called the
<font> semidiameter </font> , or <font> halfe diameter </font> .
(RECORD-E1-H,1.B1V.10)

But and if the line goe crosse the circle , and passe beside the centre
, then is it called <font> a corde </font> , or <font> a stryng line
</font> , as I said before , and as this exaumple sheweth : where A. is
the corde . (RECORD-E1-H,1.B1V.11)

And the compassed line that aunswereth to it , is called <font> an
arche lyne </font> , or <font> a bowe lyne </font> , whiche here is
marked with B. and the diameter with C . (RECORD-E1-H,1.B1V.12)

But and if that part be separate from the rest of the circle <paren> as
in this exa~ple you see </paren> then ar both partes called ca~telles ,
the one the <font> greatter cantle </font> , as E. and the other the
<font> lesser cantle </font> , as D . (RECORD-E1-H,1.B1V.13)

And if it be parted iuste by the centre <paren> as you see in F.
</paren> then is it called a <font> semicircle </font> , or <font>
halfe compasse </font> . (RECORD-E1-H,1.B1V.14)

Sometimes it happeneth that a cantle is cutte out with two lynes drawen
from the centre to the circumference <paren> as G. is </paren>
(RECORD-E1-H,1.B1V.15)

and then maie it be called a <font> nooke cantle </font> ,
(RECORD-E1-H,1.B1V.16)

and if it be not parted from the reste of the circle <paren> as you see
in H. </paren> then is it called a <font> nooke </font> plainely
without any addicion . (RECORD-E1-H,1.B1V.17)

And the compassed lyne in it is called an <font> arche lyne </font> ,
as the exaumple here doeth shewe . (RECORD-E1-H,1.B1V.18)

<P_1.B2R>

Nowe haue you heard as touchyng circles , meetely sufficient
instruction , so that it should seme nedeles to speake any more of
figures in that kynde , saue that there doeth yet remaine ij. formes of
an imperfecte circle , (RECORD-E1-H,1.B2R.20)

for it is lyke a circle that were brused , and thereby did runne out
endelong one waie , whiche forme Geometricians dooe call an <font> egge
forme </font> , because it doeth represent the figure and shape of an
egge duely proportioned <paren> as this figure sheweth </paren> hauyng
the one ende greater then the other . (RECORD-E1-H,1.B2R.21)

For if it be lyke the figure of a circle pressed in length , and bothe
endes lyke bygge , then is it called a <font> tunne forme </font> , or
<font> barrell forme </font> , the right makyng of whiche figures , I
wyll declare hereafter in the thirde booke . (RECORD-E1-H,1.B2R.22)

An other forme there is , whiche you maie call a nutte forme ,
(RECORD-E1-H,1.B2R.23)

and is made of one lyne muche lyke an egge forme , saue that it hath a
sharpe angle . (RECORD-E1-H,1.B2R.24)

And it chaunceth sometyme that there is a right line drawen crosse
these figures , (RECORD-E1-H,1.B2R.25)

and that is called an <font> axelyne </font> , or <font> axtre </font>
. (RECORD-E1-H,1.B2R.26)

Howebeit properly that line that is called an <font> axtre </font> ,
whiche gooeth thoroughe the myddell of a Globe , (RECORD-E1-H,1.B2R.27)

for as a diameter is in a circle , so is an axe lyne or axtre in a
Globe , <P_1.B2V> that lyne that goeth from side to syde , and passeth
by the middell of it . (RECORD-E1-H,1.B2V.28)

And the two poyntes that suche a lyne maketh in the vtter bounde or
platte of the globe , are named <font> polis </font> , w=ch= you may
call aptly in englysh , <font> tourne pointes </font> : of whiche I do
more largely intreate , in the booke that I haue written of the vse of
the globe . (RECORD-E1-H,1.B2V.29)

But to returne to the diuersityes of figures that remayne vndeclared ,
the most simple of them ar such ones as be made but-3 of two lynes , as
are <font> the cantle of a circle </font> , and the <font> halfe circle
</font> , of which I haue spoken allready . (RECORD-E1-H,1.B2V.30)

Likewyse the <font> halfe of an egge forme </font> , the <font> cantle
of an egge forme </font> , the <font> halfe of a tunne fourme </font> ,
and the <font> cantle of a tunne fourme </font> , and besyde these a
figure moche like to a tunne fourme , saue that it is sharp couered at
both the endes , and therfore doth consist of twoo lynes , where a
tunne forme is made of one lyne , (RECORD-E1-H,1.B2V.31)

and that figure is named an <font> yey fourme </font> .
(RECORD-E1-H,1.B2V.32)

The nexte kynd of figures are those that be made of .iij. lynes
(RECORD-E1-H,1.B2V.33)

other be all right lynes , all crooked lynes , other some right and
some crooked . (RECORD-E1-H,1.B2V.34)

But what fourme so euer they be of , they are named generally triangles
. (RECORD-E1-H,1.B2V.35)

for <font> a triangle </font> is nothinge els to say , but a figure of
three corners . (RECORD-E1-H,1.B2V.36)

And thys is a generall rule , (RECORD-E1-H,1.B2V.37)

looke how many lynes any figure hath , (RECORD-E1-H,1.B2V.38)

so mannye corners it hath also , yf it bee a platte forme , and not a
bodye . (RECORD-E1-H,1.B2V.39)

For a bodye hath dyuers lynes metyng sometime in one corner .
(RECORD-E1-H,1.B2V.40)

Now to geue you example of triangles , there is one whiche is all of
croked lynes , and may be taken fur a portio~ of a globe as the figur
marked w=t= A (RECORD-E1-H,1.B2V.41)

An other hath two compassed lines and one right lyne ,
(RECORD-E1-H,1.B2V.42)

and is as the portion of halfe a globe , example of B.
(RECORD-E1-H,1.B2V.43)

An other hatht but one compassed <P_1.B3R_misnumbered_as_1.B1R> lyne ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.44)

and is the quarter of a circle , named a quadrate ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.45)

and the ryght lynes make a right corner , as you se in C . Other lesse
then it as you se $in {TEXT:'in'_missing} D , whose right lines make a
sharpe corner , or greater then a quadrate , as is F ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.46)

and then the right lynes of it do make a blunt corner .
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.47)

Also some triangles haue all righte lynes
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.48)

and they be distincted in sonder by their angles , or corners .
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.49)

for other their corners bee all sharpe , as you see in the figure , E.
other ij. sharpe and one right square , as is the figure G other ij.
sharp and one blunt as in the figure H
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.50)

There is also an other distinction of the names of triangles ,
according to their sides , whiche other be all equal as in the figure E
, (RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.51)

and that the Greekes doo call Isopleuron , and Latine men aequilaterium
: (RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.52)

and in english it may be called a <font> threlike triangle </font> ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.53)

other els two sydes bee equall and the thyrd vnequall , which the
Greekes call Isosceles , the Latine men aequicurio ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.54)

and in english <font> tweyleke </font> may they be called , as in G , H
, and K . (RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.55)

For , they may be of iij. kinds that is to say , with one square angle
, as is G , or with a blunte corner as H , or with all in sharpe
korners , as you see in K . (RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.56)

Further more it may be y=t= they haue neuer a one syde equall to an
other , (RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.57)

and they be in iij kyndes also distinct lyke the twilekes , as you maye
perceaue by these examples .
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1R.58)

M. N , and O where M. hath a right angle , N , A , blunte angle , and O
, all sharpe angles these the Greekes and latine men do
<P_1.B3R_misnumbered_as_1.B1V> cal scalena
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.59)

and in englishe theye may be called <font> nouelekes </font> ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.60)

for thei haue no side equall , or like lo~g , to ani other in the same
figur . (RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.61)

Here it is to be noted , that in a tria~gle al the angles bee called
<font> innera~gles </font> except ani side bee drawenne forth in
lengthe , (RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.62)

for then is that fourthe corner caled an <font> vtter corner </font> ,
as in this exa~ple because A , B , is drawen in length ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.63)

therfore the a~gle C , is called an vtter a~gle
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.64)

And thus haue I done with tria~guled figures ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.65)

and nowe foloweth <font> quadrangles </font> , which are figures of
iiij. corners and of iiij. lines also , of whiche there be diuers
kindes , but chiefely v . that is to say , a <font> square quadrate
</font> , whose sides bee all equall , and al the angles square , as
you se here in this figure Q .
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.66)

The second kind is called a long square , whose foure corners be all
square , (RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.67)

but the sides are not equall eche to other ,
(RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.68)

yet is euery side equall to that other that is against it , as you maye
perceaue in this figure R . (RECORD-E1-H,1.B3R_misnumbered_as_1.B1V.69)

<P_1.B4R>

The thyrd kind is called <font> losenges </font> or <font> diamondes
</font> , whose sides bee all equall , (RECORD-E1-H,1.B4R.71)

but it hath neuer a square corner , (RECORD-E1-H,1.B4R.72)

for two of them be sharpe , (RECORD-E1-H,1.B4R.73)

and the other two be blunt , as appeareth in , S .
(RECORD-E1-H,1.B4R.74)

The iiij. sorte are like vnto losenges , saue that they are longer one
waye , (RECORD-E1-H,1.B4R.75)

and their sides be not equal , (RECORD-E1-H,1.B4R.76)

yet ther corners are like the corners of a losing ,
(RECORD-E1-H,1.B4R.77)

and therfore ar they named <font> losengelike </font> or <font>
diamo~dlike </font> , whose figur is noted with T
(RECORD-E1-H,1.B4R.78)

Here shal you marke that al those squares which haue their sides al
equal , may be called also for easy vnderstandinge , <font> likesides
</font> , as Q. and S . (RECORD-E1-H,1.B4R.79)

and those that haue only the contrary sydes equal , as R. and T. haue ,
those wyll I call <font> likeiammys </font> , for a difference .
(RECORD-E1-H,1.B4R.80)

The fift sorte doth containe all other fashions of foure cornered
figurs , (RECORD-E1-H,1.B4R.81)

and ar called of the Grekes trapezia , of Latin me~ mensulae and of
Arabitians , helmuariphe , (RECORD-E1-H,1.B4R.82)

they may be called in englishe <font> borde formes </font> ,
(RECORD-E1-H,1.B4R.83)

they haue no syde equall to an other as these examples shew ,
(RECORD-E1-H,1.B4R.84)

neither keepe they any rate in their corners , (RECORD-E1-H,1.B4R.85)

and therfore are they counted <font> vnruled formes </font> ,
(RECORD-E1-H,1.B4R.86)

and the other foure kindes onely are counted <font> ruled formes
</font> , in the kynde of quadrangles . (RECORD-E1-H,1.B4R.87)

Of these vnruled formes ther is no numbre , they are so mannye and so
dyuers , (RECORD-E1-H,1.B4R.88)

yet by arte they may be changed into other kindes of figures , and
therby be brought to measure and proportion , as in the thirtene
conclusion is partly taught , (RECORD-E1-H,1.B4R.89)

but more plainly in my booke of measuring you may see it .
(RECORD-E1-H,1.B4R.90)

<P_1.B4V>

And nowe to make an eande of the dyuers kyndes of figures , there dothe
folowe now figures of .v. sydes , other v. corners , which we may call
<font> cinkangles </font> , whose sydes partlye are all equall as in A
, and those are counted <font> ruled cinkeangles </font> . and partlye
vnequall as in , B and they are called <font> vnruled </font> .
(RECORD-E1-H,1.B4V.92)

Likewyse shall you iudge of <font> siseangles </font> , which haue sixe
corners , <font> septangles </font> , which haue seuen angles , and so
forth , (RECORD-E1-H,1.B4V.93)

for as mannye numbres as there maye be of sydes and angles , so manye
diuers kindes be there of figures , vnto which yow shall geue names
according to the numbre of their sides and angles , of whiche for this
tyme I wyll make an ende , (RECORD-E1-H,1.B4V.94)

and wyll sette forthe on example of a syseangle , which I had almost
forgotten , (RECORD-E1-H,1.B4V.95)

and that is it , whose vse commeth often in Geometry ,
(RECORD-E1-H,1.B4V.96)

and is called a <font> squire </font> , (RECORD-E1-H,1.B4V.97)

is made of two long squares ioyned togither , as this example sheweth .
(RECORD-E1-H,1.B4V.98)

And thus I make an eand to speake of platte formes ,
(RECORD-E1-H,1.B4V.99)

and will briefelye saye somwhat touching the figures of <font> bodeis
</font> which partly haue one platte forme for their bound , and y=t=
iust rou~d as a <font> globe </font> hath , or ended long as in an
<font> egge </font> , and a <font> tunne fourme </font> , whose
pictures are these . (RECORD-E1-H,1.B4V.100)

Howebeit you must marke that I meane not the very figure of a tunne ,
when I saye tunne form , but a figure like a tunne ,
(RECORD-E1-H,1.B4V.101)

for a <font> tune fourme </font> , <P_1.C1R> hath but one plat forme ,
(RECORD-E1-H,1.C1R.102)

and therfore must needs be round at the endes , where as <font> a tunne
</font> hath thre platte formes , and is flatte at eche end , as partly
these pictures do shewe . (RECORD-E1-H,1.C1R.103)

<font> Bodies of two plattes </font> are other cantles or halues of
those other bodies , that haue but one platte forme ,
(RECORD-E1-H,1.C1R.104)

or els they are lyke in fvorme to two such cantles ioyned togither as
this A doth partly eppresse : (RECORD-E1-H,1.C1R.105)

or els it is called a <font> rounde spire </font> , or <font> triple
fourme </font> , as in this figure is some what expressed
(RECORD-E1-H,1.C1R.106)

Nowe of three plattes there are made certain figures of bodyes , as the
cantels and halues of all bodyes that haue but ij. plattys , and also
the halues of halfe globys and canteles of a globe .
(RECORD-E1-H,1.C1R.107)

Lykewyse a rounde piller , and a spyre made of a rounde spyre , slytte
in ij. partes long ways . (RECORD-E1-H,1.C1R.108)

But as these formes be harde to be iudged by their pycturs , so I doe
entende to passe them ouer with a great number of other formes of
bodyes , which afterwarde shall be set forth in the boke of Perspectiue
, bicause that without perspectiue knowledge , it is not easy to iudge
truly the formes of them in flatte protacture . (RECORD-E1-H,1.C1R.109)

And thus I make an ende for this tyme , of the definitions Geometricall
, appertayning to this parte of practise , (RECORD-E1-H,1.C1R.110)

and the rest wil I prosecute as cause shall serue .
(RECORD-E1-H,1.C1R.111)

<P_1.C1V>

<heading>

SONDRY CONCLUSIONS GEOMETRICAL . (RECORD-E1-H,1.C1V.114)

THE FYRST CONCLVSION . (RECORD-E1-H,1.C1V.115)

TO MAKE A THRELIKE TRIANGLE OR ANY LYNE MEASURABLE .
(RECORD-E1-H,1.C1V.116)

</heading>

Take the iuste le~gth of the lyne with your co~passe ,
(RECORD-E1-H,1.C1V.118)

and stay the one foot of the compas in one of the endes of that line ,
turning the other vp or doun at your will , drawyng the arche of a
circle against the midle of the line , (RECORD-E1-H,1.C1V.119)

and doo like wise with the same co~passe vnaltered , at the other end
of the line , (RECORD-E1-H,1.C1V.120)

and wher these ij. croked lynes doth crosse , frome thence drawe a lyne
to echend of your first line , (RECORD-E1-H,1.C1V.121)

and there shall appear a threlike triangle drawen on that line .
(RECORD-E1-H,1.C1V.122)

<font>

Example . (RECORD-E1-H,1.C1V.124)

</font>

A. B. is the first line , on which I wold make the threlike triangle ,
(RECORD-E1-H,1.C1V.126)

therfore I open the compasse as wyde as that line is long ,
(RECORD-E1-H,1.C1V.127)

and draw two arch lines that mete in C , (RECORD-E1-H,1.C1V.128)

then from C. I draw ij other lines one to A , another to B ,
(RECORD-E1-H,1.C1V.129)

and than I haue my purpose . (RECORD-E1-H,1.C1V.130)

<heading>

THE .II CONCLVSION . (RECORD-E1-H,1.C1V.132)

IF YOU WIL MAKE A TWILIKE OR A NOUELIKE TRIANGLE ON ANI CERTAINE LINE .
(RECORD-E1-H,1.C1V.133)

</heading>

Consider fyrst the length that yow will haue the other sides to
containe , (RECORD-E1-H,1.C1V.135)

and to that length open your compasse , (RECORD-E1-H,1.C1V.136)

and <P_1.C2R> then worke as you did in the threleke triangle ,
remembryng this , that in a nouelike triangle you must take ij.
lengthes besyde the fyrste lyne , and draw an arche lyne with one of
the~ at the one ende , and with the other at the other end ,
(RECORD-E1-H,1.C2R.137)

the exa~ple is as in the other before . (RECORD-E1-H,1.C2R.138)

<heading>

THE III. CONCL. (RECORD-E1-H,1.C2R.140)

TO DIUIDE AN ANGLE OF RIGHT LINES INTO IJ. = PARTES .
(RECORD-E1-H,1.C2R.141)

</heading>

First open your compasse as largely as you can , so that it do not
excede the length of the shortest line y=t= incloseth the angle .
(RECORD-E1-H,1.C2R.143)

Then set one foote of the compasse in the verye point of the angle
(RECORD-E1-H,1.C2R.144)

and with the other fote draw a compassed arch fro~ the one lyne of the
angle to the other , (RECORD-E1-H,1.C2R.145)

that arch shall you deuide in halfe , (RECORD-E1-H,1.C2R.146)

and the~ draw a line fro~ the a~gle to y=e= middle of y=e= arch ,
(RECORD-E1-H,1.C2R.147)

and so y=e= angle is diuided into ij. equall partes .
(RECORD-E1-H,1.C2R.148)

<font>

Example . (RECORD-E1-H,1.C2R.150)

</font>

Let the tria~gle be A. B. C , (RECORD-E1-H,1.C2R.152)

the~ set I one foot of y=e= co~passe in B , (RECORD-E1-H,1.C2R.153)

and with the other I draw y=e= arch D. E , which I part into ij. equall
parts in F , and the~ draw a line fro~ B , to F ,
(RECORD-E1-H,1.C2R.154)

& so I haue mine inte~t (RECORD-E1-H,1.C2R.155)

<heading>

THE IIII. CONCL. (RECORD-E1-H,1.C2R.157)

TO DEUIDE ANY MEASURABLE LINE INTO IJ. =L PARTES .
(RECORD-E1-H,1.C2R.158)

</heading>

Open your compasse to the iust le~gth of y=e= line .
(RECORD-E1-H,1.C2R.160)

And the~ set one foote steddely at the one ende of the line ,
(RECORD-E1-H,1.C2R.161)

& w=t= the other fote draw an arch of a circle against y=e= midle of
the line , both ouer it , and also vnder it , (RECORD-E1-H,1.C2R.162)

then doo lykewaise <P_1.C2V> at the other ende of the line .
(RECORD-E1-H,1.C2V.163)

And marke where those arche lines do meet crosse waies ,
(RECORD-E1-H,1.C2V.164)

and betwene those ij. pricks draw a line , (RECORD-E1-H,1.C2V.165)

and it shall cut the first line in two equall portions .
(RECORD-E1-H,1.C2V.166)

<font>

Example . (RECORD-E1-H,1.C2V.168)

</font>

The lyne is A. B. accordyng to which I open the compasse and make
.iiij. arche lines , whiche meete in C. and D , (RECORD-E1-H,1.C2V.170)

then drawe I a lyne from C , (RECORD-E1-H,1.C2V.171)

so haue I my purpose . (RECORD-E1-H,1.C2V.172)

This conclusion serueth for makyng of quadrates and squires , beside
many other commodities , (RECORD-E1-H,1.C2V.173)

howebeit it maye bee don more readylye by this conclusion that foloweth
nexte . (RECORD-E1-H,1.C2V.174)

<heading>

THE FIFT CONCLVSION . (RECORD-E1-H,1.C2V.176)

TO MAKE A PLUMME LINE OR ANY PRICKE THAT YOU WILL IN ANY RIGHT LYNE
APPOINTED . (RECORD-E1-H,1.C2V.177)

</heading>

Open youre compas so that it be not wyder then from the pricke
appoynted in the line to the shortest ende of the line , but rather
shorter . (RECORD-E1-H,1.C2V.179)

Then sette the one foote of the compasse in the firste pricke appointed
, (RECORD-E1-H,1.C2V.180)

and with the other fote marke ij. other prickes , one of eche syde of
that fyrste , (RECORD-E1-H,1.C2V.181)

afterwarde open your compasse to the wydenes of those ij. new prickes ,
(RECORD-E1-H,1.C2V.182)

and draw from them ij. arch lynes , as you did in the fyrst conclusion
, for making of a threlyke tria~gle . (RECORD-E1-H,1.C2V.183)

then if you do mark their crossing , and from it drawe a line to your
fyrste pricke , it shall bee a iust plum lyne on that place .
(RECORD-E1-H,1.C2V.184)

<font>

Example . (RECORD-E1-H,1.C2V.186)

</font>

The lyne is A. B. (RECORD-E1-H,1.C2V.188)

the prick on whiche I shoulde make the plumme lyne , is C .
(RECORD-E1-H,1.C2V.189)

then open I the compasse as wyde as A , C , (RECORD-E1-H,1.C2V.190)

and sette one foote in C. (RECORD-E1-H,1.C2V.191)

and with the other doo I marke out C. A. and C. B ,
(RECORD-E1-H,1.C2V.192)

then open I the compasse as wide as A. B , (RECORD-E1-H,1.C2V.193)

and make ij. arch lines which do crosse in D , (RECORD-E1-H,1.C2V.194)

and so haue I doone . (RECORD-E1-H,1.C2V.195)

Howebeeit , it happeneth so sommetymes , that the <P_1.C3R> pricke on
whiche you would make the perpendicular or plum line , is so nere the
eand of your line , that you can not extende any notable length from it
to thone end of the line , (RECORD-E1-H,1.C3R.196)

and if so be it then that you maie not drawe your line lenger fro~ that
end , then doth this conclusion require a newe ayde ,
(RECORD-E1-H,1.C3R.197)

for the last deuise will not serue . (RECORD-E1-H,1.C3R.198)

In suche case therfore shall you dooe thus : (RECORD-E1-H,1.C3R.199)

If your line be of any notable length , deuide it into fiue partes .
(RECORD-E1-H,1.C3R.200)

And if it be not so long that it maie yelde fiue notable partes , then
make an other line at will , (RECORD-E1-H,1.C3R.201)

and parte it into fiue equall portio~s : so that thre of those partes
maie be found in your line . (RECORD-E1-H,1.C3R.202)

Then open your compas as wide as thre of these fiue measures be ,
(RECORD-E1-H,1.C3R.203)

and sette the one foote of the compas in the pricke , where you would
haue the plumme line to lighte <paren> whiche I call the first pricke ,
</paren> (RECORD-E1-H,1.C3R.204)

and with the other foote drawe an arche line righte ouer the pricke ,
as you can ayme it : (RECORD-E1-H,1.C3R.205)

then open youre compas as wide as all fiue measures be ,
(RECORD-E1-H,1.C3R.206)

and set the one foote in the fourth pricke , (RECORD-E1-H,1.C3R.207)

and with the other foote draw an other arch line crosse the first ,
(RECORD-E1-H,1.C3R.208)

and where thei two do crosse , thense draw a line to the poinct where
you woulde haue the perpendicular line to light ,
(RECORD-E1-H,1.C3R.209)

and you haue doone . (RECORD-E1-H,1.C3R.210)

<font>

Example . (RECORD-E1-H,1.C3R.212)

</font>

The line is A. B. (RECORD-E1-H,1.C3R.214)

and A. is the prick , on whiche the perpendicular line must light .
(RECORD-E1-H,1.C3R.215)

Therfore I deuide A. B. into fiue partes equall ,
(RECORD-E1-H,1.C3R.216)

then do I open the compas to the widenesse of three partes <paren> that
is A. D. </paren> and let one foote staie in A. (RECORD-E1-H,1.C3R.217)

and with the other I make an arche line in C . (RECORD-E1-H,1.C3R.218)

Afterwarde I open the compas as wide as A. B. <P_1.C3V> <paren> that is
as wide as all fiue partes </paren> (RECORD-E1-H,1.C3V.219)

and set one foote in the .iiij. pricke , which is E , drawyng an arch
line with the other foote in C. also . (RECORD-E1-H,1.C3V.220)

Then do I draw thence a line vnto A , (RECORD-E1-H,1.C3V.221)

and so haue I doone . (RECORD-E1-H,1.C3V.222)

But and if the line be to shorte to be parted into fiue partes , I
shall deuide it into iij. partes only , as you see the line F. G ,
(RECORD-E1-H,1.C3V.223)

and then make D. and other line <paren> as is K. L. </paren> whiche I
deuide into .v. suche diuisions , as F. G. containeth .iij {COM:sic} ,
(RECORD-E1-H,1.C3V.224)

then open I the compaas as wide as .iiij. partes <paren> whiche is K.
M. </paren> (RECORD-E1-H,1.C3V.225)

and so set I one foote of the compas in F , (RECORD-E1-H,1.C3V.226)

and with the other I drawe an arch lyne toward H ,
(RECORD-E1-H,1.C3V.227)

then open I the co~pas as wide as K. L. <paren> that is all .v. partes
</paren> (RECORD-E1-H,1.C3V.228)

and set one foote in G , <paren> that is the iij. pricke </paren>
(RECORD-E1-H,1.C3V.229)

and with the other I draw an arch line toward H. also :
(RECORD-E1-H,1.C3V.230)

and where those .ij. arch lines do crosse <paren> whiche is by H.
</paren> thence draw I a line vnto F , (RECORD-E1-H,1.C3V.231)

and that maketh a very plumbe line to F. G , as my desire was .
(RECORD-E1-H,1.C3V.232)

The maner of workyng of this conclusion , is like to the second
conclusion , (RECORD-E1-H,1.C3V.233)

but the reason of it doth depe~d of the .xlvi. prorosicio~ of y=e=
first boke of Euclide . (RECORD-E1-H,1.C3V.234)

An other waie yet . (RECORD-E1-H,1.C3V.235)

set one foote of the compas in the prick , on whiche you would haue the
plumbe line to light , (RECORD-E1-H,1.C3V.236)

and stretche forth thother foote toward the longest end of the line ,
as wide as you can for the length of the line , (RECORD-E1-H,1.C3V.237)

and so draw a quarter of a compas or more , (RECORD-E1-H,1.C3V.238)

then without stirring of the compas , set one foote of it in the same
line , where as the circular line did begin , (RECORD-E1-H,1.C3V.239)

and extend thother in the circular line , settyng a marke where it doth
light , (RECORD-E1-H,1.C3V.240)

then take half that quantitie more there vnto , (RECORD-E1-H,1.C3V.241)

and by that prick that endeth the last part , draw a line to the pricke
assigned , (RECORD-E1-H,1.C3V.242)

and it shall be a perpendicular . (RECORD-E1-H,1.C3V.243)

<font>

Example . (RECORD-E1-H,1.C3V.245)

</font>

A. B. is the line appointed , to whiche I must make a perpendicular
line to light in the pricke assigned , which is A .
(RECORD-E1-H,1.C3V.247)

Therfore doo I set one foote of the compas in A , and extend the other
vnto D. makyng a part of a circle , <P_1.C4R> more then a quarter ,
that is D. E. (RECORD-E1-H,1.C4R.248)

Then do I set one foote of the compas vnaltered in D , and stretch the
other in the circular line , (RECORD-E1-H,1.C4R.249)

and it doth light in F , (RECORD-E1-H,1.C4R.250)

this space betwene D. and F. I deuide into halfe in the pricke G ,
whiche halfe I take with the compas , and set it beyond F. vnto H ,
(RECORD-E1-H,1.C4R.251)

and therfore is H. the point , by whiche the perpendicular line must be
drawen , (RECORD-E1-H,1.C4R.252)

so say I that the line H. A , is a plumbe line to A. B , as the
conclusion would . (RECORD-E1-H,1.C4R.253)

<heading>

THE .VI. CONCLVSION . (RECORD-E1-H,1.C4R.255)

TO DRAWE A STREIGHT LINE FROM ANY PRICKE THAT IS NOT IN A LINE , AND TO
MAKE IT PERPENDICULAR TO AN OTHER LINE . (RECORD-E1-H,1.C4R.256)

</heading>

Open your compas so wide that it may extend somewhat farther , the~
from the prick to the line , (RECORD-E1-H,1.C4R.258)

then sette the one foote of the compas in the pricke ,
(RECORD-E1-H,1.C4R.259)

and with the other shall you draw a co~passed line , that shall crosse
that other first line in .ij. places (RECORD-E1-H,1.C4R.260)

Now if you deuide that arch line into .ij. equall partes , and from the
middell pricke therof vnto the prick without the line you drawe a
streight line , it $shall $be {TEXT:shalbe} a plumbe line to that
firste lyne , accordyng to the conclusion . (RECORD-E1-H,1.C4R.261)

<font>

Example . (RECORD-E1-H,1.C4R.263)

</font>

C. is the appointed pricke , from whiche vnto the line A. B. I must
draw a perpe~dicular . (RECORD-E1-H,1.C4R.265)

Therfore I open the co~pas so wide , that it may haue one foote in C ,
and thother to reach ouer the line , (RECORD-E1-H,1.C4R.266)

and with y=t= foote I draw an arch line as you see , betwene A. and B ,
which arch line I deuide in the middell in the point D .
(RECORD-E1-H,1.C4R.267)

Then drawe I a line from C. to D , (RECORD-E1-H,1.C4R.268)

and it is perpendicular to the line A. B , accordyng as my desire was .
(RECORD-E1-H,1.C4R.269)

<P_2.E4R>

<heading>

THE XXXIIJ. THEOREME . (RECORD-E1-H,2.E4R.272)

IN ALL RIGHT ANGULED TRIANGLES , THE SQUARE OF THAT SIDE WHICHE LIETH
AGAINST THE RIGHT ANGLE , IS =L TO THE .IJ. SQUARES OF BOTH THE OTHER
SIDES (RECORD-E1-H,2.E4R.273)

</heading>

<font> Example </font> . (RECORD-E1-H,2.E4R.275)

A. B. C. is a triangle , hauing a ryght angle in B. Wherfore it
foloweth , that the square of A. C , <paren> whiche is the side that
lyeth agaynst the right angle </paren> shall be as muche as the two
squares of A. B. and B. C. which are the other .ij. sides .
(RECORD-E1-H,2.E4R.276)

By the square of any lyne , you muste vnderstande a figure made iuste
square , hauyng all his iiij. sydes equall to that line , whereof it is
the square , (RECORD-E1-H,2.E4R.277)

so is A. C. F , the square of A. C. (RECORD-E1-H,2.E4R.278)

Lykewais A. B. D. is the square of A. B. (RECORD-E1-H,2.E4R.279)

And B. C. E. is the square of B. C. (RECORD-E1-H,2.E4R.280)

Now by the numbre of the diuisions in eche of these squares , may you
perceaue not onely what the square of any line is called , but also
that the theoreme is true , and expressed playnly bothe-3 by lines and
numbre . (RECORD-E1-H,2.E4R.281)

For as you see , the greatter square <paren> that is A. C. F. </paren>
hath fiue diuisions on eche syde , all equall togyther ,
(RECORD-E1-H,2.E4R.282)

and those in the whole square are twenty and fiue .
(RECORD-E1-H,2.E4R.283)

Nowe in the left square , whiche is A. B. D. there are but .iij. of
those diuisions in one syde , (RECORD-E1-H,2.E4R.284)

and that yeldeth nyne in the whole . (RECORD-E1-H,2.E4R.285)

So lykeways you see in the meane square A. C. E. in euery syde .iiij.
partes , whiche in the whole amount vnto sixtene .
(RECORD-E1-H,2.E4R.286)

Nowe adde togyther all the partes of the two lesser squares , that is
to saye , sixtene and nyne , (RECORD-E1-H,2.E4R.287)

and you perceyue that they make twenty and fiue , whyche is an equall
numbre to the summe of the greatter square . (RECORD-E1-H,2.E4R.288)

<P_2.E4V>

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 . (RECORD-E1-H,2.E4V.290)

For by tournynge the two sydes certayne into theyr squares , and so
addynge them togyther , other subtractynge the one from the other
<paren> accordyng as in the vse of these theoremes I haue sette foorthe
</paren> and then fyndynge the roote of the square that remayneth ,
which roote <paren> I meane the syde of the square </paren> is the
iuste length of the unknowen syde , whyche is sought for .
(RECORD-E1-H,2.E4V.291)

But this appertaineth to the thyrde booke , (RECORD-E1-H,2.E4V.292)

and therefore I wyll speake no more of it at this tyme .
(RECORD-E1-H,2.E4V.293)

<heading>

THE XXXIIIJ. THEOREME . (RECORD-E1-H,2.E4V.295)

IF SO BE IT , THAT IN ANY TRIANGLE , THE SQUARE OF THE ONE SYDE BE =L
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 . (RECORD-E1-H,2.E4V.296)

</heading>

<font> Example </font> . (RECORD-E1-H,2.E4V.298)

As in the figure of the laste Theoreme , bicause A. C , made in square
, is as much 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. <paren> that is to say
</paren> the angle B. whiche lieth against the line A. C , must nedes
be a ryght angle . (RECORD-E1-H,2.E4V.299)

This teoreme 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 , (RECORD-E1-H,2.E4V.300)

for they conteine one sentence , contrary waies pronounced .
(RECORD-E1-H,2.E4V.301)

<heading>

THE .NUM. THEOREME . (RECORD-E1-H,2.E4V.303)

IF THERE BE SET FORTH .IJ. RIGHT LINES , AND ONE OF THEM PARTED INTO
SUNDRY PARTES , HOW MANY <P_2.F1R> OR FEW SO EUER THEY BE , THE SQUARE
THAT IS MADE OF THOSE IJ. RIGHT LINES PROPOSED , IS =L TO ALL THE
SQUARES , THAT ARE MADE OF THE VNDIUIDED LINE , AND EUERY PARTE OF THE
DIUIDED LINE . (RECORD-E1-H,2.F1R.304)

</heading>

<font> Example </font> . (RECORD-E1-H,2.F1R.306)

The ij. lines proposed ar A B. and C. D , (RECORD-E1-H,2.F1R.307)

and the lyne A. B. is deuided into thre partes by E. and F .
(RECORD-E1-H,2.F1R.308)

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. sta~deth for the
le~gth of the square , and the other line C. D. for the bredth of the
same . That square <paren> I say </paren> will be equall to all the
squares that be made , of the vndiueded lyne <paren> which is C. D.
</paren> and euery portion of the diueded line .
(RECORD-E1-H,2.F1R.309)

And to declare that particularly , Fyrst I make an other line G. K ,
equall to the line C. D , and the line G. H. to be equal to the line A.
B , and to bee diuided into iij. like partes , so that G. M. is equall
to A. E , and M. N. equal to E. F , (RECORD-E1-H,2.F1R.310)

and then muste N. H. nedes remaine equall to F. B .
(RECORD-E1-H,2.F1R.311)

Then of those ij. lines G. K , vndeuided , and G. H. which is deuided ,
I make a square , that is G. H. K. L , In which square if I drawe
crosse lines frome one side to the other , according to the diuisions
of the line G. H , then will it appear plaine , that the theoreme doth
affirme . (RECORD-E1-H,2.F1R.312)

For the first square G. M. O. K , must needes be equal to the square of
the line C. D , and the first portio~ of the diuided line , which is A.
E , for bicause their sides are equall . (RECORD-E1-H,2.F1R.313)

And so the seconde <P_2.F1V> square that is M. N. P. O , shall be
equall to the square of C. D , and the second part of A. B , that is E.
F . (RECORD-E1-H,2.F1V.314)

Also the third square which is N. H. L. P_N , must of necessitee be
equal to the square of C. D , and F. B , bicause those lines be so
coupeled that euery couple are equall in the seuerall figures .
(RECORD-E1-H,2.F1V.315)

And so shal you not only in this example , but in all other finde it
true , that if one line be deuided into sondry partes , and another
line whole and vndiuided , matched with him in a square , that square
which is made of these two whole lines , is as muche iuste and equally
, as all the seuerall squares , whiche bee made of the whole line
vndiuided , and euery part seuerally of the diuided line .
(RECORD-E1-H,2.F1V.316)

<heading>

THE XXXVI. THEOREME . (RECORD-E1-H,2.F1V.318)

IF A RIGHT LINE BE PARTED INTO IJ. PARTES , AS CHAUNCE MAY HAPPE , THE
SQUARE THAT IS MADE OF THAT WHOLE LINE , IS =L TO BOTHE THE SQUARES
THAT ARE MADE OF THE SAME LINE , AND THE TWOO PARTES OF IT SEUERALLY .
(RECORD-E1-H,2.F1V.319)

</heading>

<font> Example </font> . (RECORD-E1-H,2.F1V.321)

The line propouned beyng A. B. and deuided , as chaunce happeneth , in
C. into ij. unequall partes , I say that the square made of the hole
line A. B , is equal to the two squares made of the same line with the
twoo partes of it selfe , as with A. C , and with C. B ,
(RECORD-E1-H,2.F1V.322)

for the square D , E. F. G. is equal to the two other partial squares
of D. H. K. G and H. E. F. K , but that the greater square is equall to
the square of the whole line A. B , and the <P_2.F2R> partiall squares
equall to the squares of the second partes of the same line ioyned with
the whole line , (RECORD-E1-H,2.F2R.323)

your eye may iudg without muche declaracion , so that I shall not neede
to make more exposition therof , but that you may examine it , as you
did in the laste Theoreme . (RECORD-E1-H,2.F2R.324)

<heading>

THE XXXVIJ THEOREME . (RECORD-E1-H,2.F2R.326)

IF A RIGHT LINE BE DEUIDED BY CHAUNCE , AS IT MAYE HAPPEN , THE SQUARE
THAT IS MADE OF THE WHOLE LINE , AND ONE OF THE PARTES OF IT WHICH
SOEUER IT BE , SHAL BE =L TO THAT SQUARE THAT IS MADE OF THE IJ. PARTES
IOYNED TOGITHER , AND TO AN OTHER SQUARE MADE OF THAT PART , WHICH WAS
BEFORE IOYNED WITH THE WHOLE LINE . (RECORD-E1-H,2.F2R.327)

</heading>

<font> Example </font> . (RECORD-E1-H,2.F2R.329)

The line A. B. is deuided in C. into twoo partes , though not equally ,
of which two partes for an example I take the first , that is A. C ,
(RECORD-E1-H,2.F2R.330)

and of it I make one side of a square , as for example D. G.
accomptinge those two lines to be equall , (RECORD-E1-H,2.F2R.331)

the other side of the square is D. E , whiche is equall to the whole
line A. B . (RECORD-E1-H,2.F2R.332)

Now may it appeare , to your eye , that the great square made of the
whole line A. B , and of one of his partes that is A. C , <P_2.F2V>
<paren> which is equall with D. G. </paren> is equal to two partiall
squares , wherof the one is made of the saide greatter portion A. C ,
in as muche as not only D. G , beynge one of his sides , but also D. H.
beinge the other side , are eche of them equall to A. C .
(RECORD-E1-H,2.F2V.333)

The second square is H. E. F. K , in which the one side H. E , is equal
to C. B , being the lesser parte of the line , A. B , and E. F. is
equall to A. C. which is the greater parte of the same line . So that
those two squares D. H. K. G , and H , E , F , K , bee bothe of them no
more then the greate square D. E , F , G , accordinge to the wordes of
the Theoreme afore saide . (RECORD-E1-H,2.F2V.334)

<heading>

THE XXXVIIJ. THEOREME . (RECORD-E1-H,2.F2V.336)

IF A RIGHTE LINE BE DEUIDED BY CHAUNCE , INTO PARTES , THE SQUARE THAT
IS MADE OF THAT WHOLE LINE , IS =L TO BOTH THE SQUARES THAT AR MADE OF
ECHE PARTE OF THE LINE , AND MOREOUER TO TWO SQUARES MADE OF THE ONE
PORTION OF THE DIUIDED LINE IOYNED WITH THE OTHER IN SQUARE .
(RECORD-E1-H,2.F2V.337)

</heading>

<font> Example </font> . (RECORD-E1-H,2.F2V.339)

Lette the diuided line bee A , B , and parted in C , into twoo partes :
(RECORD-E1-H,2.F2V.340)

Nowe saithe the Theoreme , that the square of the whole lyne A , B , is
as mouche iuste as the square of A. C , and the square of C. B. , eche
by it selfe , and more ouer as muche twise , as A. C. and C. B.
<P_2.F3R> ioyned in one square will make . (RECORD-E1-H,2.F3R.341)

For as you se , the great square D. E. F. G , conteyneth in hym foure
lesser squares , of whiche the first and the greatest is N. M. F. K ,
and is equall to the square of the lyne A. C . (RECORD-E1-H,2.F3R.342)

The second square is the lest of them all , that is D. H. L. N ,
(RECORD-E1-H,2.F3R.343)

and it is equall to the square of the line B. C .
(RECORD-E1-H,2.F3R.344)

Then are there two other longe squares both of one bygnes , that is H.
E. N. M. and L. N. G. K , eche of them both hauyng .ij. sides equall to
A. C , the longer parte of the diuided line , and there other two sides
equall to C. B , beeyng the shorter parte of the said line A. B .
(RECORD-E1-H,2.F3R.345)

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 . (RECORD-E1-H,2.F3R.346)

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. (RECORD-E1-H,2.F3R.347)

This theoreme hath great vse not only in geometrie , but also in
arithmetike , as herafter I will declare in conuenient place
(RECORD-E1-H,2.F3R.348)

<heading>

THE .XXXIX. THEOREME . (RECORD-E1-H,2.F3R.350)

IF A RIGHT LINE BE DEUIDED INTO TWO =L 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 =L TO THE SQUARE OF HALFE THE FIRSTE LINE .
(RECORD-E1-H,2.F3R.351)

</heading>

<P_2.F3V>

<font> Example </font> . (RECORD-E1-H,2.F3V.354)

The line A. B. is diuided into ij. equal partes in C ,
(RECORD-E1-H,2.F3V.355)

and that parte C. B. is diuided agayne as hapneth in D . Wherfere saith
the Theorem that the long square made of D. B. and A. D , with the
square of C. D. <paren> which is the mydle portion </paren> 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 .
(RECORD-E1-H,2.F3V.356)

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 . (RECORD-E1-H,2.F3V.357)

The square of the myddle portion is L. M. O. P_N .
(RECORD-E1-H,2.F3V.358)

And the square of the halfe of the fyrste lyne is E. K. Q. L .
(RECORD-E1-H,2.F3V.359)

Nowe by the theoreme , that longe square F. G. M. O , with the iuste
square L. M. O. P_N , 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 .
(RECORD-E1-H,2.F3V.360)

And fyrst I say , that where the comparyson of equalitee is made
betweene the greate square <paren> whiche is made of halfe the line A.
B. </paren> and two other , where of the fyrst is the longe square F.
G. N. O , and the seconde is the full square L. M. O. P_N , which is
one portion of the great square allredye , 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 : (RECORD-E1-H,2.F3V.361)

Nowe the reste of the great square , those <P_2.F4R> 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 .
(RECORD-E1-H,2.F4R.362)

And that they two be equall , theyr sydes doo declare .
(RECORD-E1-H,2.F4R.363)

For the longest lynes that is F. K and E. Q are equall ,
(RECORD-E1-H,2.F4R.364)

and so are the shorter lynes , F. G , and E. N ,
(RECORD-E1-H,2.F4R.365)

and so appereth the truthe of the Theoreme . (RECORD-E1-H,2.F4R.366)

<heading>

THE .XL. THEOREME . (RECORD-E1-H,2.F4R.368)

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 =L TO THE SQUARE OF THAT LINE , WHICHE IS
CONPOUNDED OF HALFE THE FIRSTE LINE , AND THE PARTE NEWLY ADDED .
(RECORD-E1-H,2.F4R.369)

</heading>

<font> Example </font> . (RECORD-E1-H,2.F4R.371)

The fyrst lyne propouned is A. B , (RECORD-E1-H,2.F4R.372)

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 .
(RECORD-E1-H,2.F4R.373)

Nowe say I , that the long square A. D. M. K , is made of the whole
lyne so augme~ted , that is A. D , and the portio~ annexed , y=t= is D.
M . (RECORD-E1-H,2.F4R.374)

for D. M is equall to B. D , wherfore y=t= long square A. D. M. K ,
with the <P_2.F4V> 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 .
(RECORD-E1-H,2.F4V.375)

And it is easyly perceaued , if you consyder that the longe square A.
C. L. K. <paren> whiche onely is lefte out of the great square </paren>
hath another longe square equall to hym , and to supply his steede in
the great square , and that is G , F. M. H . (RECORD-E1-H,2.F4V.376)

For they sydes be of lyke lines in length . (RECORD-E1-H,2.F4V.377)

<heading>

THE XLI. THEOREME . (RECORD-E1-H,2.F4V.379)

IF A RIGHT LINE BE DIUIDED BY CHAUNCE , THE SQUARE OF THE SAME WHOLE
LINE , AND THE SQUARE OF ONE OF HIS PARTES ARE IUSTE =L TO THE LO~G
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 .
(RECORD-E1-H,2.F4V.380)

</heading>

<font> Example </font> . (RECORD-E1-H,2.F4V.382)

A. B. is the line diuided in C . (RECORD-E1-H,2.F4V.383)

And D. E. F. G , is the square of the whole line ,
(RECORD-E1-H,2.F4V.384)

D. H. K. M. is the square of the lesser portion <paren> whyche I take
for an example </paren> (RECORD-E1-H,2.F4V.385)

and therfore must bee twise reckened . (RECORD-E1-H,2.F4V.386)

Nowe I saye that those ij. squares are equall to two longe squares of
the whole line A. B , and his sayd portion A. C , and also to the
square of the other portion of the sayd first line , whiche portion is
C. B , and his square K. N. F. L (RECORD-E1-H,2.F4V.387)

In this theoreme there is no difficultie , if you co~syder that the
litle square D. H. K. M. is iiij. tymes reckened , that is to say ,
fyrst of all as a parte of the greatest square , whiche is D. E. F. G .
(RECORD-E1-H,2.F4V.388)

Secondly he is rekned <P_2.G1R> by him selfe . (RECORD-E1-H,2.G1R.389)

Thirdely he is accompted as parcell of the long square D. E. N. M ,
(RECORD-E1-H,2.G1R.390)

And fourthly he is taken as a part of the other long square D. H. L. G
, so that in as muche as he is twise reckened in one part of the
compariso~ of equalitee , and twise also in the second parte , there
can rise none occasion of errour or doubtfulnes therby .
(RECORD-E1-H,2.G1R.391)