<B CESCIE1B> 
<Q E1 EX SCIO RECORD> 
<N GEOMETRY> 
<A RECORD ROBERT> 
<C E1> 
<O 1500-1570> 
<M X> 
<K X> 
<D ENGLISH> 
<V PROSE> 
<T SCIENCE OTHER> 
<G X> 
<F X> 
<W WRITTEN> 
<X MALE> 
<Y 20-40> 
<H PROF> 
<U PROF> 
<E X> 
<J X> 
<I X> 
<Z EXPOS> 
<S SAMPLE X> 
 
 
[^RECORD, ROBERT. 
THE PATH-WAY TO KNOWLEDG, CONTAINING 
THE FIRST PRINCIPLES OF GEOMETRIE, 1551. 
THE ENGLISH EXPERIENCE, 687. 
AMSTERDAM: THEATRUM ORBIS TERRARUM,
LTD. AND NORWOOD, N. J.: W. J. JOHNSON,
INC., 1974 (FACSIMILE).
PP. B1R.1 - C4R.34  (SAMPLE 1) 
PP. E4R.1 - G1R.6   (SAMPLE 2)^] 
 
<S SAMPLE 1> 
<P B1R> 
   (^A touche lyne^) , 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. 
   And when that a line doth crosse the 
edg of the circle, the~ is it called (^a cord^) , 
as you shall see anon in the speakynge 
of circles. 
   In the meane season must I not omit 
to declare what angles bee called (^matche corners^) , that is  #
to 
saie, suche as stande directly one against the other, when twoo 
lines be drawen acrosse, as here 
appereth. 
   Where A. and B. are matche corners, 
so are C. and D. but not A. 
and C. nother D. and A. 
   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.
(^A circle^) is a figure made and enclosed with one line, 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.
   And the line that encloseth the
whole compasse, is called the 
(^circumference^) .
   And all the lines that bee drawen
crosse the circle, and goe by the centre,
are named (^diameters^) , whose halfe, I
meane from the center to the circumference
<P B1V>
any waie, is called the (^semidiameter^) , or (^halfe
diameter^) .
   But and if the line goe crosse the circle,
and passe beside the centre, then is
it called (^a corde^) , or (^a stryng line^) ,
as I said before, and as this exaumple
sheweth: where A. if the corde.
And the compassed line that aunswereth
to it, is called (^an arche lyne^) , or
(^a bowe lyne^) , whiche here is marked 
with B. and the diameter with C.
   But and if that 
part be separate 
from the rest of 
the circle (as in
this exa~ple you
see) then ar both
partes called ca~telles,
the one
the (^greatter cantle^) , as E. and the other the (^lesser      #
cantle^) ,
as D. And if it be parted iuste by the centre (as you see in    #
F.)
then is it called a (^semicircle^) , or (^halfe compasse^) .
   Sometimes it happeneth that a cantle is cutte out with two
lynes drawen from the centre to the circumference (as G. is)
and then maie it be called a (^nooke cantle^) ,
and if it be not parted from the reste
of the circle (as you see in H.) then is it 
called a (^nooke^) plainely without any
addicion. And the compassed lyne in it
is called an (^arche lyne^) , as the exaumple
here doeth shewe.
<P 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, for it is lyke a
circle that were brused, and thereby
did runne out endelong one waie, whiche
forme Geometricians dooe call an
(^egge forme^) , because it doeth
represent the figure and shape of
an egge duely proportioned (as
this figure sheweth) hauyng the
one ende greater then the other.
   For if it be lyke the figure of a circle pressed in length,  #
and
bothe endes lyke bygge, then is it called a (^tunne forme^) ,   #
or
(^barrell forme^) , the right makyng of whiche figures, I wyll
declare hereafter in the thirde booke.
   An other forme there is, whiche you maie call a nutte forme,
and is made of one lyne muche lyke an egge forme, saue that it 
hath a sharpe angle.
   And it chaunceth sometyme that there is a right line drawen
crosse these figures, and that is called an (^axelyne^) , or    #
(^axtre^) .
Howebeit properly that line that is called an (^axtre^) ,
whiche gooeth thoroughe the myddell of a Globe, for as a
diameter is in a circle, so is an axe lyne or axtre in a Globe,
<P B2V>
that lyne that goeth from side to syde, and passeth by the      #
middell 
of it. And the two poyntes that suche a lyne maketh in
the vtter bounde or platte of the globe, are named (^polis^) ,  #
w=ch=
you may call aptly in englysh, (^tourne pointes^) : of whiche I
do more largely intreate, in the booke that I haue written of
the vse of the globe.
   But to returne to the diuersityes of figures that remayne
vndeclared, the most simple of them ar such ones as be made
but of two lynes, as are the (^cantle of a circle^) , and the   #
(^halfe
circle^) , of which I haue spoken allready. Likewyse the        #
(^halfe
of an egge forme^) , the (^cantle of an egge forme^) , the      #
(^halfe
of a tunne fourme^) , and the (^cantle of a tunne fourme^) , 
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, and that figure is named
an (^yey fourme^) .
   The nexte kynd of figures are those
that be made of .iij. lynes other be all right lynes, all       #
crooked
lynes, other some right and some crooked. But what fourme
so euer they be of, they are named generally triangles. for     #
(^a triangle^)
is nothinge els to say, but a figure of three corners.
And thys is a generall rule, looke how
many lynes any figure hath, so mannye
corners it hath also, yf it bee a platte
forme, and not a bodye. For a bodye
hath dyuers lynes metyng sometime
in one corner.
   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
   An other hath two compassed lines
and one right lyne, and is as the portion
of halfe a globe, example of B.
   An other hatht but one compassed
<P B3R>
lyne, and is the quarter of a circle, named a
quadrate, and the ryght lynes make a right corner,
as you se in C. Other lesse then it as you
se D, whose right lines make a sharpe corner,
or greater then a quadrate, as is F, and then 
the right lynes of it do make a blunt corner.
   Also some triangles haue all righte lynes
and they be distincted in sonder by their angles,
or corners. 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
   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,
and that the Greekes doo call (\Isopleuron\) ,
and Latine men (\aequilaterium\) : and
in english it may be called a (^threlike
triangle^) , other els two sydes bee equall
and the thyrd vnequall, which
the Greekes call (\Isosceles\) , the Latine
men (\aequicurio\) , and in english
(^tweyleke^) may they be called, as in G,
H, and K. 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.
   Further more it may be y=t= they haue
neuer a one syde equall to an other,
and they be in iij kyndes also distinct
lyke the twilekes, as you maye perceaue
by these examples. 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 B3V>
cal (\scalena\)
and in englishe 
theye
may be called
(^nouelekes^) ,
for thei
haue no side
equall, or
like lo~g, to ani other in the same figur.
Here it is to be noted, that in a tria~gle
al the angles bee called (^innera~gles^)
except ani side
bee drawenne
forth in lengthe,
for then
is that fourthe
corner caled an
(^vtter corner^) ,
as in this exa~ple
because A, B, is drawen in length, therfore 
the a~gle C, is called an vtter a~gle
   And thus haue I done with tria~guled
figures, and nowe foloweth (^quadrangles^) ,
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 (^square
quadrate^) , whose sides bee
all equall, and al the angles
square, as you se here in this 
figure Q. The second kind
is called a long square, whose foure corners
be all square, but the sides are not
equall eche to other, yet is euery side
equall to that other that is against it, as
you maye perceaue in this figure R.
<P B4R>
   The thyrd kind is called (^losenges^)
or (^diamondes^) , whose sides bee all equall,
but it hath neuer a square corner,
for two of them be sharpe, and the
other two be blunt, as appeareth in, S.
   The iiij. sorte are like vnto losenges,
saue that they are longer one waye, and
their sides be not equal, yet ther corners
are like the corners of a losing, and therfore 
ar they named (^losengelike^) or (^diamo~dlike^) , 
whose figur is noted with T
Here shal you marke that al those squares
which haue their sides al equal, may
be called also for easy vnderstandinge,
(^likesides^) , as Q. and S. and those that
haue only the contrary sydes equal, as
R. and T. haue, those wyll I call (^likeiammys^) , 
for a difference.
   The fift sorte
doth containe all
other fashions of
foure cornered figurs,
and ar called
of the Grekes (\trapezia\) ,
of Latin me~
(\mensulae\) and of Arabitians, (\helmuariphe\) ,
they may be called in englishe
(^borde formes^) , they haue no syde equall
to an other as these examples shew, neither keepe they
any rate in their corners, and therfore are they counted        #
(^vnruled
formes^) , and the other foure kindes onely are counted
(^ruled formes^) , in the kynde of quadrangles. Of these        #
vnruled
formes ther is no numbre, they are so mannye and so dyuers,
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, but more plainly
in my booke of measuring you may see it.
<P 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 (^cinkangles^) , 
whose sydes partlye are all equall
as in A, and those are counted
(^ruled cinkeangles^) . and partlye vnequall
as in, B and they are called (^vnruled^) .
   Likewyse shall you iudge of (^fifeangles^) ,
which haue fixe corners, (^septangles^) ,
which haue seuen angles, and so forth, 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, and wyll sette forthe on example
of a syseangle, which I had almost forgotten,
and that is it, whose vse commeth
often in Geometry, and is called a
(^squire^) , is made of two long squares ioyned
togither, as this example sheweth.
   And thus I make an eand to speake of
platte formes, and will briefelye saye
somwhat touching the figures of (^bodeis^)
which partly haue one platte forme
for their bound, and y=t= iust rou~d
as a (^globe^) hath, or ended long
as in an (^egge^) , and a (^tunne
fourme^) , whose pictures are
these.
   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, for a (^tune fourme^) ,
<P C1R>
hath but one plat forme, and therfore must needs be round at
the endes, where as (^a tunne^) hath thre platte formes, and is
flatte at eche end, as partly these pictures do shewe.
   (^Bodies of two plattes^) are other cantles or halues of
those other bodies, that haue but one platte forme, or els
they are lyke in fvorme to two such cantles ioyned togither
as this A doth partly eppresse: or els
it is called a (^rounde spire^) , or (^triple
fourme^) , as in this figure is some 
what  expressed
   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. Lykewyse a rounde piller,
and a spyre made of a rounde spyre,
slytte in ij. partes long ways.
   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.
   And thus I make an ende for this tyme, of the definitions
Geometricall, appertayning to this
parte of practise, and the rest wil
I prosecute as cause shall
serue.

<P C1V>
[}SONDRY CONCLUSIONS GEOMETRICAL.}]
[}THE FYRST CONCLVSION.}]
[}TO MAKE A THRELIKE TRIANGLE OR ANY LYNE
MEASURABLE.}]

   Take the iuste
le~gth of the lyne with your co~passe,
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, and doo like wise with the same
co~passe vnaltered, at the other end of
the line, and wher these ij. croked lynes
doth crosse, frome thence drawe a
lyne to echend of your first line, and
there shall appear a threlike triangle
drawen on that line.
(^Example.^)
A.B. is the first line, on which I wold
make the threlike triangle, therfore I
open the compasse as wyde as that line
is long, and draw two arch lines that
mete in C, then from C. I draw ij other
lines one to A, another to B, and than
I haue my purpose.

[}THE. II CONCLVSION.}]
[}IF YOU WIL MAKE A TWILIKE OR
A NOUELIKE TRIANGLE ON ANI CERTAINE
LINE.}]
   Consider fyrst the length that yow will haue the other sides
to containe, and to that length open your compasse, and
<P 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, the exa~ple is as in the other
before.

[}THE III. CONCL.}]
[}TO DIUIDE AN ANGLE OF RIGHT
LINES INTO IJ. EQUAL PARTES.}]
   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.
Then set one foote of the compasse in the verye point of
the angle and with the other fote draw a compassed arch fro~
the one lyne of the angle to the other,
that arch shall you deuide in halfe, and
the~ draw a line fro~ the a~gle to y=e= middle
of y=e= arch, and so y=e= angle is diuided
into ij. equall partes. (^Example.^)
Let the tria~gle be A.B.C, the~ set I one
foot of y=e= co~passe in B, 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, & so I haue mine inte~t

[}THE IIII. CONCL.}]
[}TO DEUIDE ANY MEASURABLE
LINE INTO IJ. EQUALL PARTES.}]

Open your compasse to the iust le~gth of 
y=e= line. And the~ set one foote steddely at
the one ende of the line, & 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, then doo lykewaise
<P C2V>
at the other ende of the line. And marke where those arche
lines do meet crosse waies, and betwene those ij. pricks draw
a line, and it shall cut the first line in two equall portions.
(^Example.^)
   The lyne is A.B. accordyng to which I open the compasse
and make .iiij. arche lines, whiche meete in C. and D, then
drawe I a lyne from C, so haue I my purpose. 
   This conclusion serueth for makyng of quadrates and squires,
beside many other commodities, howebeit it maye bee
don more readylye by this conclusion that foloweth nexte.

[}THE FIFT CONCLVSION.}]
[}TO MAKE A PLUMME LINE OR ANY PRICKE THAT
YOU WILL IN ANY RIGHT LYNE APPOINTED.}]

   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. Then sette the one foote of the compasse
in the firste pricke appointed, and with the other fote marke
ij. other prickes, one of eche syde of that fyrste, afterwarde
open your compasse to the wydenes of those ij. new prickes,
and draw from them ij. arch
lynes, as you did in the fyrst
conclusion, for making of a 
threlyke tria~gle. 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.
(^Example.^)
The lyne is A.B. the prick on 
whiche I shoulde make the plumme lyne, is C. then open I
the compasse as wyde as A, C, and sette one foote in C.
and with the other doo I marke out C.A. and C.B, then open
I the compasse as wide as A.B, and make ij. arch lines which
do crosse in D, and so haue I doone.
   Howebeeit, it happeneth so sommetymes, that the
<P 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, 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, for the last     #
deuise
will not serue. In suche case therfore shall you dooe thus:
If your line be of any notable length, deuide it into fiue      #
partes.
And if it be not so long that it maie yelde fiue notable        #
partes,
then make an other line at will, and parte it into fiue equall
portio~s: so that thre of those partes maie be found in your    #
line.
Then open your compas as wide as thre of these fiue measures
be, and sette the one foote of the compas in the pricke, where
you would haue the plumme line to lighte (whiche I call the
first pricke,) and with the other foote drawe an arche line
righte ouer the pricke, as you can ayme it: then open youre
compas as wide as all fiue measures be, and set the one foote   #
in
the fourth pricke, and with the other foote draw an other arch
line crosse the first, and where thei two do crosse, thense     #
draw
a line to the poinct where you woulde haue the perpendicular
line to light, and you haue doone.
(^Example.^)
   The line is A.B. and
A. is the prick, on whiche
the perpendicular
line must light. Therfore
I deuide A.B. into fiue
partes equall, then do I
open the compas to the 
widenesse of three partes 
(that is A.D.) and
let one foote staie in A.
and with the other I
make an arche line in C.
Afterwarde I open the
compas as wide as A.B.
<P C3V>
(that is as wide as all fiue partes) and set one foote in the   #
 .iiij.
pricke, which is E, drawyng an arch line with the other foote
in C. also. Then do I draw thence a line vnto A, and so haue
I doone. 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, and then make D. and other line (as is K.L.) whiche I
deuide into .v. suche diuisions, as F.G. containeth .iij, then  #
open
I the compaas as wide as .iiij. partes (whiche is K.M.)
and so set I one foote of the compas in F, and with the other I
drawe an arch lyne toward H, then open I the co~pas as wide
as K.L. (that is all .v. partes) and set one foote in G, (that  #
is the
iij. pricke) and with the other I draw an arch line toward H.
also: and where those .ij. arch lines do crosse (whiche is by   #
H.)
thence draw I a line vnto F, and that maketh a very plumbe
line to F.G, as my desire was. The maner of workyng of this
conclusion, is like to the second conclusion, but the reason    #
of it
doth depe~d of the .xlvi. prorosicio~ of y=e= first boke of     #
Euclide.
An other waie yet. set one foote of the compas in the prick, on
whiche you would haue the plumbe line to light, and stretche
forth thother foote toward the longest end of the line, as wide
as you can for the length of the line, and so draw a quarter    #
of a
compas or more, then without stirring of the compas, set one
foote of it in the same line, where as the circular line did    #
begin,
and extend thother in the circular line, settyng a marke where
it doth light, then take half that quantitie
more there vnto, and by that
prick that endeth the last part, draw
a line to the pricke assigned, and it
shall be a perpendicular.
(^Example.^)
  A.B. is the line appointed, to whiche
I must make a perpendicular line
to light in the pricke assigned, which
is A. Therfore doo I set one foote of
the compas in A, and extend the other
vnto D. makyng a part of a circle, 
<P C4R>
more then a quarter, that is D.E. Then do I set one foote
of the compas vnaltered in D, and stretch the other in the      #
circular
line, and it doth light in F, 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, and therfore is H. the
point, by whiche the perpendicular line must be drawen, so say
I that the line H.A, is a plumbe line to A.B, as the conclusion
would.

[}THE .VI. CONCLVSION.}]
[}TO DRAWE A STREIGHT LINE FROM ANY PRICKE
THAT IS NOT IN A LINE, AND TO MAKE IT PERPENDICULAR
TO AN OTHER LINE.}]

Open your compas so wide that it may extend somewhat farther,
the~ from the prick to the
line, then sette the one foote of
the compas in the pricke, and
with the other shall you draw
a co~passed line, that shall crosse
that other first line in .ij. places
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 shalbe a plumbe line to that
firste lyne, accordyng to the
conclusion. (^Example.^)
   C. is the appointed pricke, from whiche vnto the line A.B. I
must draw a perpe~dicular. Therfore I open the co~pas so wide,
that it may haue one foote in C, and thother to reach ouer the
line, 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. Then drawe I a line from C. to D, and it is perpendicular
to the line A.B, accordyng as my desire was. 

<S SAMPLE 2>
<P E4R>
[}THE XXXIIJ. THEOREME.}]
[}IN ALL RIGHT ANGULED TRIANGLES, THE SQUARE OF
THAT SIDE WHICHE LIETH AGAINST THE RIGHT ANGLE, IS
EQUALL TO THE .IJ. SQUARES OF BOTH THE OTHER SIDES}]

[}EXAMPLE.}]
   
   A.B.C. is a triangle, hauing
a ryght angle in B. Wherfore
it foloweth, that the square of
A.C, (whiche is the side that
lyeth agaynst the right angle)
shall be as muche as the two
squares of A.B. and B.C.
which are the other .ij. sides.
   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, so is A.C.F, the square
of A.C. Lykewais A.B.D. is the square of A.B. And B.C.E.
is the square of B.C. 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 by lines and numbre. For as you
see, the greatter square (that is A.C.F.) hath fiue diuisions   #
on
eche syde, all equall togyther, and those in the whole square
are twenty and fiue. Nowe in the left square, whiche is
A.B.D. there are but .iij. of those diuisions in one syde, and
that yeldeth nyne in the whole. So lykeways you see in the
meane square A.C.E. in euery syde .iiij. partes, whiche in the
whole amount vnto sixtene. Nowe 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.
<P 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. 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 unknowen 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 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. (that is
to say) the angle B. whiche lieth against the line A.C, must    #
nedes
be a ryght angle. 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, 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
<P F1R>
OR FEW SO EUER THEY BE, THE SQUARE THAT IS MADE
OF THOSE IJ. RIGHT LINES PROPOSED, IS EQUALL TO ALL
THE SQUARES, THAT ARE MADE OF THE VNDIUIDED
LINE, AND EUERY PARTE OF THE DIUIDED LINE.}]

[}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. sta~deth for the le~gth 
of the square, and the other 
line C.D. for the bredth of the same. That square (I say) will  #
be
equall to all the squares that be made, of the vndiueded lyne
(which is C.D.) and euery portion of the diueded line. 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, and then muste N.H. 
nedes remaine  equall to F.B. 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. 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. And so the seconde
<P 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. Also the third square
which is N.H.L.P, 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. 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.

[}THE XXXVI. THEOREME.}]

[}IF A RIGHT LINE BE PARTED INTO IJ. PARTES, AS
CHAUNCE MAY HAPPE, THE SQUARE THAT IS MADE OF
THAT WHOLE LINE, IS EQUALL TO BOTHE THE SQUARES
THAT ARE MADE OF THE SAME LINE, AND THE TWOO
PARTES OF IT SEUERALLY.}]

[}EXAMPLE.}]

   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, 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 F2R>
partiall squares equall to the squares of the second partes of
the same line ioyned with the whole line, 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.

[}THE XXXVIJ THEOREME.}]

[}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 EQUALL 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.}]

[}EXAMPLE.}]

   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, and of it I make 
one side of a square,
as for example D.G.
accomptinge those two lines to be equall, the other side of the
square is D.E, whiche is equall to the whole line A.B.
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 F2V>
(which is equall with D.G.) 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. 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.

[}THE XXXVIIJ. THEOREME.}]

[}IF A RIGHTE LINE BE DEUIDED BY CHAUNCE, INTO
PARTES, THE SQUARE THAT IS MADE OF THAT WHOLE
LINE, IS EQUALL 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.}]

[}EXAMPLE.}]

Lette the diuided line bee A,B,
and parted in C, into twoo partes:
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 F3R>
ioyned in one square will make. 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. The second square is the lest of them 
all, that is D.H.L.N, and it is equall to the square of the
line B.C. 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.
   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.}]
<P F3V>
[}EXAMPLE.}]

   The line A.B. is diuided into
ij. equal partes in C, 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. (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 seconde is the full square L.
M.O.P, 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: Nowe the reste of the great square, those
<P 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. And that they two be equall,
theyr sydes doo declare. For the longest lynes that is F.K and
E.Q are equall, and so are the shorter lynes, F.G, and E.N, 
and so appereth the truthe of the Theoreme.

[}THE .XL. THEOREME.}]

[}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 EQUALL TO THE SQUARE OF THAT
LINE, WHICHE IS CONPOUNDED OF HALFE THE FIRSTE
LINE, AND THE PARTE NEWLY ADDED.}]

[}EXAMPLE.}]

   The fyrst lyne propouned is
A.B, 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.
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. for D.M
is equall to B.D, wherfore y=t= long square A.D.M.K, with the
<P 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. And it is
easyly perceaued, if you consyder that the longe square A.C.
L.K. (whiche onely is lefte out of the great square) hath       #
another
longe square equall to hym, and to supply his steede
in the great square, and that is G,F.M.H. For they sydes be 
of lyke lines in length.

[}THE XLI. THEOREME.}]

[}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 EQUALL 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.}]

[}EXAMPLE.}]

   A.B. is the line diuided in C. And 
D.E.F.G, is the square of the whole 
line, D.H.K.M. is the square of
the lesser portion (whyche I take
for an example) and therfore must bee
twise reckened. 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 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. Secondly he is rekned
<P G1R>
by him selfe. Thirdely he is accompted as parcell of the long
square D.E.N.M, 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.



