<P_2,B1R>

<heading>

THE THEOREMES OF GEOMETRY , BEFORE <font> WHICHE ARE SET FORTHE </font>
<font> CERTAINE GRAUNTABLE REQUESTES WHICHE SERUE FOR DEMONSTRATIONS
</font> MATHEMATICALL . (RECORD-E1-P2,2,B1R.3)

</heading>

That fro~ any pricke to one other , there may be drawen a right line .
As for example . {COM:figure_omitted} (RECORD-E1-P2,2,B1R.5)

A being the one pricke , and B. the other , you maye drawe betwene them
from the one to the other , that is to say , from A. vnto B , and from
B. to A. (RECORD-E1-P2,2,B1R.6)

That any right line of measurable length may be drawen forth longer ,
and straight . (RECORD-E1-P2,2,B1R.7)

Example of A.B , which as it is a line of measurable lengthe , so may
it be drawen forth longer , as for example vnto C ,
{COM:figure_omitted} and that in true streightenes without crokinge .
(RECORD-E1-P2,2,B1R.8)

That vpon any centre , there may be made a circle of anye qua~titee
that a man wyll . (RECORD-E1-P2,2,B1R.9)

Let the centre be set to A , (RECORD-E1-P2,2,B1R.10)

{COM:figure_omitted}

what shal hinder a man to drawe a circle about it , of what quantitee
that he lusteth , as you se the forme here : other bygger or lesse , as
it shall lyke him to doo ? (RECORD-E1-P2,2,B1R.12)

<P_2,B1V>

That all right angles be equall eche to other . (RECORD-E1-P2,2,B1V.14)

Set for an example A. and B , {COM:figure_omitted} of which two though
A. seme the greater angle to some men of small experience , it
happeneth only bicause that the lines aboute A , are longer the~ the
lines about B , as you may proue by drawing them longer ,
(RECORD-E1-P2,2,B1V.15)

for so shal B. seme the greater angle yf you make his lines longer then
the lines that make the angle A. (RECORD-E1-P2,2,B1V.16)

And to proue it by demonstration , I say thus . (RECORD-E1-P2,2,B1V.17)

If any ij. right corners be not equal , then one right corner is
greater then an other , (RECORD-E1-P2,2,B1V.18)

but that corner which is greater then a right angle , is a blunt corner
<paren> by his definition </paren> (RECORD-E1-P2,2,B1V.19)

so must one corner be both a right corner and a blunt corner also ,
whiche is not possible : (RECORD-E1-P2,2,B1V.20)

And againe : the lesser right corner must be a sharpe corner , by his
definition , bicause it is lesse then a right angle , which thing is
impossible . (RECORD-E1-P2,2,B1V.21)

Therefore I conclude that all right angles be equall .
(RECORD-E1-P2,2,B1V.22)

Yf one right line do crosse two other right lines , and make ij. inner
corners of one side lesser the~ ij. righte corners , it is certaine ,
that if those two lines be drawen forth right on that side that the
sharpe inner corners be , they wil at le~gth mete togither , and crosse
on an other . (RECORD-E1-P2,2,B1V.23)

The ij. lines beinge as A.B. and C.D , and the third line crossing them
as dooth heere E.F , making ij inner $corners {TEXT:cornes} <paren> as
ar G.H. </paren> lesser then two right corners , {COM:figure_omitted}
sith ech of <P_2,B2R> them is lesse then a right corner , as your eyes
maye iudge , then say I , if those ij. lines A.B. and C.D. be drawen in
lengthe on that side that G. and H. are , the {COM:sic} will at length
meet and crosse one an other . (RECORD-E1-P2,2,B2R.24)

Two right lines make no platte forme . (RECORD-E1-P2,2,B2R.25)

A platte forme , as you harde before , hath bothe length and bredthe ,
(RECORD-E1-P2,2,B2R.26)

and is inclosed with lines as with his boundes ,
(RECORD-E1-P2,2,B2R.27)

but ij. right lines $can $not {TEXT:cannot} inclose al the bondes of
any platte forme . (RECORD-E1-P2,2,B2R.28)

Take for an example firste these two right lines AB. and A.C. whiche
meete togither in A , {COM:figure_omitted} but yet $can $not
{TEXT:cannot} be called a platte forme , bicause there is no bond from
B. to C , (RECORD-E1-P2,2,B2R.29)

but if you will drawe a line betwene them twoo , that is frome B. to C
, then will it be a platte forme , that is to say , a triangle ,
(RECORD-E1-P2,2,B2R.30)

but then are there iij. lines , and not only ij .
(RECORD-E1-P2,2,B2R.31)

Likewise may you say of D.E. and F.G , {COM:figure_omitted} whiche doo
$not
{COM:'not'_missing_in_text,_but_required_since_the_lines_de_and_fg_are_
parallel_in_the_figure} make a platte forme , (RECORD-E1-P2,2,B2R.32)

nother yet can they make any without helpe of two lines more , whereof
the one must be drawen from D. to F , and the other frome E. to G ,
(RECORD-E1-P2,2,B2R.33)

and then will it be a longe square . (RECORD-E1-P2,2,B2R.34)

So then of two right lines can bee made no platte forme .
(RECORD-E1-P2,2,B2R.35)

But of ij. croked lines be made a platte forme , as you se in the eye
form . (RECORD-E1-P2,2,B2R.36)

And also of one rightline , & one croked line , maye a platte fourme
bee made , as the semicircle F. doothe sette forth .
(RECORD-E1-P2,2,B2R.37)

{COM:figure_omitted}

<P_2,B2V>

<heading>

CERTAYN COMMON SENTENCES MANIFEST TO <font> SENCE , AND ACKNOWLEDGED OF
ALL MEN </font> . (RECORD-E1-P2,2,B2V.41)

THE FIRSTE COMMON SENTENCE . (RECORD-E1-P2,2,B2V.42)

</heading>

What so euer things be equal to one other thinge , those same bee
equall betwene them selues . (RECORD-E1-P2,2,B2V.44)

Examples therof you may take both in greatnes and also in numbre .
(RECORD-E1-P2,2,B2V.45)

First <paren> though it pertaine not proprely to geometry , but to
helpe the vnderstandinge of the rules , whiche may bee wrought by both
artes </paren> thus may you perceaue . (RECORD-E1-P2,2,B2V.46)

If the summe of mounye in my purse , and the mony in your purse be
equall eche of them to the mony that any other man hathe , then must
needes your mony and mine be equall togyther . (RECORD-E1-P2,2,B2V.47)

Likewise , if anye ij. quantities , as A and B , be equal to an other ,
as vn to C , then muste nedes A. and B. be equall eche to other , as A.
equall to B , and B. equall to A , whiche thinge the better to peceaue
, tourne these quantities into numbre , (RECORD-E1-P2,2,B2V.48)

so shall A. and B. make sixteene , and C. as many .
{COM:figure_omitted} As you may perceaue by multipliyng the numbre of
their sides togither . (RECORD-E1-P2,2,B2V.49)

<heading>

THE SECONDE COMMON SENTENCE . (RECORD-E1-P2,2,B2V.51)

</heading>

And if you adde equall portions to thinges that be equall , what so
amounteth of them <P_2,B3R> shall be equall . (RECORD-E1-P2,2,B3R.53)

Example , (RECORD-E1-P2,2,B3R.54)

Yf you and I haue like summes of mony , and then receaue eche of vs
like summes more , then our summes wil be like styll .
(RECORD-E1-P2,2,B3R.55)

Also if A. and B. <paren> as in the former example </paren> bee equall
, then by adding an equal portion to them both , as to ech of them ,
the quarter of A. <paren> that is foure </paren> they will be equall
still . (RECORD-E1-P2,2,B3R.56)

<heading>

THE THIRDE COMMON SENTENCE . (RECORD-E1-P2,2,B3R.58)

</heading>

And if you abate euen portions from things that are equal , those
partes that remain shall be equall also . (RECORD-E1-P2,2,B3R.60)

This you may perceaue by the laste example . For that that was added
there , is subtracted heere . (RECORD-E1-P2,2,B3R.61)

and so the one doothe approue the other . (RECORD-E1-P2,2,B3R.62)

<heading>

THE FOURTH COMMON SENTENCE . (RECORD-E1-P2,2,B3R.64)

</heading>

If you abate equalle partes from vnequal thinges , the remainers shall
be vnequall . As bicause that a hundreth and eight and forty be vnequal
if I take tenne from them both , there will remayne nynetye and eight
and thirty , which are also vnequall , (RECORD-E1-P2,2,B3R.66)

and likewise in quantities it is to be iudged . (RECORD-E1-P2,2,B3R.67)

<heading>

THE FIFTE COMMON SENTENCE . (RECORD-E1-P2,2,B3R.69)

</heading>

when euen portions are added to vnequalle thinges , those that amounte
$shall $be {TEXT:shalbe} vnequall . (RECORD-E1-P2,2,B3R.71)

<P_2,B3V>

So if you adde twenty to fifty , and lyke ways to nynty , you shall
make seuenty , and a hundred and ten whiche are no lesse vnequall ,
than were fifty and nynty . (RECORD-E1-P2,2,B3V.73)

<heading>

THE SYXT COMMON SENTENCE . (RECORD-E1-P2,2,B3V.75)

</heading>

If two thinges be double to any other , those same two thinges are
equal togither . (RECORD-E1-P2,2,B3V.77)

{COM:figures_omitted}

Bicause A. and B. are eche of them double to C , therefore must A. and
B. nedes be equall togither . (RECORD-E1-P2,2,B3V.79)

For as v. times viij. maketh xl. which is double to iiij. times v ,
that is xx so iiij. times x , likewise is double to xx .
(RECORD-E1-P2,2,B3V.80)

<paren> for it maketh fortie </paren> (RECORD-E1-P2,2,B3V.81)

and therefore must neades be equall to forty . (RECORD-E1-P2,2,B3V.82)

<heading>

THE SEUENTH COMMON SENTENCE . (RECORD-E1-P2,2,B3V.84)

</heading>

If any two thinges be the halfes of one other thing , than are thei
.ij. equall togither . (RECORD-E1-P2,2,B3V.86)

So are D. and C. in the laste example equal togyther , bicause they are
eche of them the halfe of A , other of B , as their numbre declareth .
(RECORD-E1-P2,2,B3V.87)

<heading>

THE EYGHT COMMON SENTENCE . (RECORD-E1-P2,2,B3V.89)

</heading>

If any one quantitee be laide on an other , and thei agree , so that
the one <P_2,B4R> excedeth not the other , then are they equall
togither . (RECORD-E1-P2,2,B4R.91)

As if this figure A.B.C. {COM:figure_omitted} be layed on that other
D.E.F , {COM:figure_omitted} so that A. be layed to D , B. to E , and
C. to F , you shall see them agre in sides exactlye and the one not to
excede the other , (RECORD-E1-P2,2,B4R.92)

for the line A.B. is equall to D.E , (RECORD-E1-P2,2,B4R.93)

and the third lyne C.A , is equal to F.D so that euery side in the one
is equall to some one side of the other . wherfore it is playne , that
the two triangles are equall togither . (RECORD-E1-P2,2,B4R.94)

<heading>

THE NYNTH COMMON SENTENCE . (RECORD-E1-P2,2,B4R.96)

</heading>

Euery whole thing is greater than any of his partes .
(RECORD-E1-P2,2,B4R.98)

This sentence nedeth none example . (RECORD-E1-P2,2,B4R.99)

For the thyng is more playner then any declaration ,
(RECORD-E1-P2,2,B4R.100)

yet considering that other commen sentence that foloweth nexte that .
<heading> THE TENTHE COMMON SENTENCE . </heading> Euery whole thinge is
equall to all his partes taken togither . It shall be mete to expresse
both w=t= one example , (RECORD-E1-P2,2,B4R.101)

for of thys last se~tence many me~ at the first hearing do make a doubt
. (RECORD-E1-P2,2,B4R.102)

Ther fore as in this example of the circle deuided into su~dry partes
<P_2,B4V> {COM:figure_omitted} it doeth appere that no parte can be so
great as the whole circle , <paren> accordyng to the meanyng of the
eight sentence </paren> so yet it is certain , that all those eight
partes together be equall vnto the whole circle .
(RECORD-E1-P2,2,B4V.103)

And this is the meanyng of that common sentence <paren> whiche many vse
, and fewe do rightly understand </paren> that is , that <font> All the
partes of any thing are nothing els , but the whole . </font> And
contrary waies : <font> The whole is nothing els , but all his partes
taken togither . </font> whiche saiynges some haue vnderstand to meane
thus : that all the partes are of the same kind that the whole thyng is
: (RECORD-E1-P2,2,B4V.104)

but that that meanyng is false , it doth plainly appere by this figure
A.B , {COM:figure_omitted} whose partes A. and B , are triangles , and
the whole figure is a square , (RECORD-E1-P2,2,B4V.105)

and so they are not of one kind . (RECORD-E1-P2,2,B4V.106)

But and if they applie it to the matter or substance of thinges <paren>
as some do </paren> then it is moste false , (RECORD-E1-P2,2,B4V.107)

for euery compound thyng is made of partes of diuerse matter and
substance . (RECORD-E1-P2,2,B4V.108)

Take for example a man , a house , a boke , and all other compound
thinges . (RECORD-E1-P2,2,B4V.109)

Some vnderstand it thus , that the partes all together can make none
other forme , but that that the whole doth shewe , whiche is also false
, (RECORD-E1-P2,2,B4V.110)

for I make fiue hundred diuerse figures of the partes of some one
figure , as you shall better perceiue in the third boke .
(RECORD-E1-P2,2,B4V.111)

And in the meane seaso~ take for an exa~ple this square figure folowing
A.B.C.D , {COM:figures_on_next_page_omitted} w=ch= is deuided but into
two parts , (RECORD-E1-P2,2,B4V.112)

and yet <paren> as you se </paren> I haue made fiue figures more beside
the firste , with onely diuerse ioynyng of those two partes .
(RECORD-E1-P2,2,B4V.113)

But of this shall I speake more largely in an other place ,
(RECORD-E1-P2,2,B4V.114)

in the mean season content your self with these principles , whiche are
certain of the chiefe groundes wheron all demonstrations mathematical
are fourmed of which though the moste parte seeme so plaine , that no
childe doth doubte of them , thinke not therfore that the art vnto
whiche they serue , is simple , other childishe ,
(RECORD-E1-P2,2,B4V.115)

but rather consider , howe certayne <P_2,C1R> the profes of that arte
is , y=t= hath for his grou~des soche playne truthes , & as I may say ,
such vndowbtfull and sensible principles , (RECORD-E1-P2,2,C1R.116)

And this is the cause why all learned menne dooth approue the certenty
of geometry , and co~sequently of the other artes mathematical , which
haue the grounds <paren> as Arithmetike , musike and astronomy </paren>
aboue all other artes and sciences , that be vsed amo~gest men .
(RECORD-E1-P2,2,C1R.117)

This muche haue I sayd of the first principles ,
(RECORD-E1-P2,2,C1R.118)

and now will I go on with the theoremes , whiche I do only by examples
declae {COM:sic} , minding to reserue the proofes to a peculiar boke
which I will then set forth , when I perceaue this to be thankfully
taken of the readers of it . (RECORD-E1-P2,2,C1R.119)

<heading>

THE THEOREMES OF GEOMETRY BRIEFLYE DECLARED BY SHORTE EXAMPLES .
(RECORD-E1-P2,2,C1R.121)

</heading>

<heading>

THE FIRSTE THEOREME . (RECORD-E1-P2,2,C1R.124)

</heading>

When .ij. triangles be so drawen , that the one of the~ hath ij. sides
equal to ij sides of the <P_2,C1V> other triangle , and that the angles
enclosed with those sides , bee equal also in bothe triangles , then is
the thirde side likewise equall in them . (RECORD-E1-P2,2,C1V.126)

And the whole triangles be of one greatnes , and euery angle in the one
equall to his matche angle in the other , I meane those angles that be
inclosed with like sides . (RECORD-E1-P2,2,C1V.127)

<font>

Example . (RECORD-E1-P2,2,C1V.129)

</font>

This triangle A.B.C. hath ij. sides <paren> that is to say </paren>
C.A. and C.B , equal to ij. sides of the other triangle F.G.H ,
(RECORD-E1-P2,2,C1V.131)

{COM:figures_omitted}

for A.C. is equall to F.G , (RECORD-E1-P2,2,C1V.133)

and B.C. is equall to G.H. And also the angle C. contayned beetweene
F.G , and G.H , (RECORD-E1-P2,2,C1V.134)

for both of them answere to the eight parte of a circle .
(RECORD-E1-P2,2,C1V.135)

Ther fore doth it remayne that A.B. whiche is the thirde lyne in the
firste triangle , doth agre in lengthe with F.H , w=ch= is the third
line in $y=e= $seco~d {TEXT:y=e=se_co~d} tria~gle & y=t= hole tria~gle
A.B.C. must nedes be equal to y=e= hole triangle F.G.H. And euery
corner equall to his match , that is to say , A. equall to F , B. to H
, and C. to G , (RECORD-E1-P2,2,C1V.136)

for those bee called <font> match corners </font> , which are inclosed
with like sides , other els do lye against like sides .
(RECORD-E1-P2,2,C1V.137)

<heading>

THE SECOND THEOREME . (RECORD-E1-P2,2,C1V.139)

</heading>

In twileke triangles the ij. corners that be <P_2,C2R> about the groud
{COM:sic} line , are equal togither . (RECORD-E1-P2,2,C2R.141)

And if the sides that be equal , be drawe~ out in le~gth the~ wil the
corners that are vnder the ground lines , be equal also togither .
(RECORD-E1-P2,2,C2R.142)

<font>

Example (RECORD-E1-P2,2,C2R.144)

</font>

A.B.C. is a twileke triangle , (RECORD-E1-P2,2,C2R.146)

for the one side A.C , is equal to the other side B.C.
(RECORD-E1-P2,2,C2R.147)

{COM:figure_omitted}

And therfore I saye that the inner corners A. and B , which are about
the ground lines , <paren> that is A.B. </paren> be equall to gither .
(RECORD-E1-P2,2,C2R.149)

And farther if C.A. and C.B. bee drawen forthe vnto D and E. as you se
that I haue drawen them , then saye I that the two vtter angles vnder
A. and B , are equal also togither : as the theorem said . The profe
wherof , as of al the rest , shal apeare in Euclide , whome I intende
to set foorth in english with sondry new additions , if I may perceaue
that it wil be thankfully taken . (RECORD-E1-P2,2,C2R.150)

<heading>

THE THIRDE THEOREME . (RECORD-E1-P2,2,C2R.152)

</heading>

If in annye triangle there bee twoo angles equall togither , then shall
the sides , that lie against those angles , be equal also .
(RECORD-E1-P2,2,C2R.154)

<font>

Example (RECORD-E1-P2,2,C2R.156)

</font>

This triangle A.B.C. {COM:figure_omitted} hath two corners equal eche
to other , that is A. and B , as I do by supposition limite , wherfore
it foloweth that the side A.C , is equal to that other side B.C ,
(RECORD-E1-P2,2,C2R.158)

for the side A.C , lieth againste the angle B ,
(RECORD-E1-P2,2,C2R.159)

and the side B.C , lieth against the angle A . (RECORD-E1-P2,2,C2R.160)

<P_2,C2V>

<heading>

THE FOURTH THEOREME . (RECORD-E1-P2,2,C2V.163)

</heading>

when two lines are drawen fro~ the endes of anie one line , and meet in
anie pointe , it is not possible to draw two other lines of like
lengthe ech to his match that shal begi~ at the same pointes , and end
in anie other pointe then the twoo first did . (RECORD-E1-P2,2,C2V.165)

<font>

Example . (RECORD-E1-P2,2,C2V.167)

</font>

{COM:figure_omitted}

The first line is A.B , on which I haue erected two other lines A.C ,
and B.C , that meete in the pricke C , wherefore I say , it is not
possible to draw ij. other lines from A. and B. which shal mete in one
point <paren> as you se A.D. and B.D. mete in D. </paren> but that the
match lines $shall $be {TEXT:shalbe} vnequal , (RECORD-E1-P2,2,C2V.170)

I mean by <font> match lines </font> , the two lines on one side ,
(RECORD-E1-P2,2,C2V.171)

that is the ij. on the right hand , or the ij. on the lefte hand ,
(RECORD-E1-P2,2,C2V.172)

for as you se in this example A.D. is longer the~ A.C.
(RECORD-E1-P2,2,C2V.173)

and A.D. shall bee of one lengthe , if B.D. and B.C. bee like longe .
(RECORD-E1-P2,2,C2V.174)

For if one couple of matche lines be equall <paren> as the same example
A.E. is equall to A.C. in length </paren> then must B.E. needes be
vnequall to B.C . (RECORD-E1-P2,2,C2V.175)

as you see , it is here shorter . (RECORD-E1-P2,2,C2V.176)

<heading>

THE FIFTE THEOREME . (RECORD-E1-P2,2,C2V.178)

<heading>

If two tria~gles haue there ij. sides equal one to an other , and their
grou~d lines equal also , then <P_2,C3R> shall their corners , whiche
are contained betwene like sides , be equall one to the other .
(RECORD-E1-P2,2,C3R.180)

<font>

Example . (RECORD-E1-P2,2,C3R.182)

</font>

Because these two triangles A.B.C , and D.E.F. haue two sides equall
one to an other . {COM:figure_omitted} For A.C. is equall to D.F , and
B.C. is equall to E.F , and again their grou~d lines A.B. and D.E. are
lyke in length , therfore is eche angle of the one triangle equall to
ech angle of the other , comparyng together those angles that are
contained within lyke sides , (RECORD-E1-P2,2,C3R.184)

so is A. equall to D , B. to E , and C. to F , (RECORD-E1-P2,2,C3R.185)

for they are contayned within like sides , as before is said .
(RECORD-E1-P2,2,C3R.186)

<heading>

THE SIXT THEOREME . (RECORD-E1-P2,2,C3R.188)

</heading>

when any right line standeth on an other , the ij. angles that thei
make , other are both right angles , or els equall to .ij. righte
angles . (RECORD-E1-P2,2,C3R.190)

<font>

Example . (RECORD-E1-P2,2,C3R.192)

</font>

A.B. is a right line , (RECORD-E1-P2,2,C3R.194)

and on it there doth light another right line , drawen from C.
perpendicularly on it , (RECORD-E1-P2,2,C3R.195)

{COM:figure_omitted}

therefore saie I , that the .ij. angles that thei do make , are .ij.
right angles as maie be iudged by the definition of a right angle .
(RECORD-E1-P2,2,C3R.197)

But in the second part of the example , where A.B. beyng still the
right line , on whiche D. standeth <P_2,C3V> in slope wayes , but yet
they are equall to two righte angles , (RECORD-E1-P2,2,C3V.198)

for so muche as the one is to greate , more then a righte angle , so
muche iuste is the other to little , so that bothe togither are equall
to two right angles , as you may perceiue . (RECORD-E1-P2,2,C3V.199)

<heading>

THE SEUENTH THEOREME . (RECORD-E1-P2,2,C3V.201)

</heading>

If .ij. lines be drawen to any one pricke in an other lyne , and those
.ij. lines do make with the fyrst lyne , two right angles , other suche
as be equall to two right angles , and that towarde one hande , than
those two lines doo make one streyght lyne . (RECORD-E1-P2,2,C3V.203)

<font>

Example . (RECORD-E1-P2,2,C3V.205)

</font>

A.B. is a streyght lyne , on which there doth lyght two other lines one
frome D , and the other frome C , (RECORD-E1-P2,2,C3V.207)

{COM:figure_omitted}

but considerynge that they meete in one pricke E , and that the angles
on one hand be equal to two right corners <paren> as the laste theoreme
dothe declare </paren> therfore maye D.E. and E.C. be counted for one
ryght lyne . (RECORD-E1-P2,2,C3V.209)

<heading>

THE EIGHT THEOREME . (RECORD-E1-P2,2,C3V.211)

</heading>

when two lines do cut one an other crosseways they do make their matche
angles equall . (RECORD-E1-P2,2,C3V.213)

<P_2,C4R>

<font>

Example . (RECORD-E1-P2,2,C4R.216)

</font>

What matche angles are , I haue tolde you in the definition of the
termes . (RECORD-E1-P2,2,C4R.218)

And here A , and B. are matche corners in this example , as are also C.
and D , so that the corner A , is equall to B , and the angle C , is
equall to D . (RECORD-E1-P2,2,C4R.219)

{COM:figure_omitted}

<heading>

THE NYNTH THEOREME . (RECORD-E1-P2,2,C4R.222)

</heading>

whan so euer in any triangle the line of one side is drawen forthe in
lengthe , that vtter angle is greater than any of the two inner corners
, that ioyne not with it . (RECORD-E1-P2,2,C4R.224)

<font>

Example . (RECORD-E1-P2,2,C4R.226)

</font>

The triangle A.D.C {COM:figure_omitted} hathe hys grounde lyne A.C.
drawen forthe in lengthe vnto B , so that the vtter corenr that it
maketh at C , is greater then any of the two inner corners that lye
againste it , and ioyne not wyth it , whyche are A. and D ,
(RECORD-E1-P2,2,C4R.228)

for they both are lesser then a ryght angle , (RECORD-E1-P2,2,C4R.229)

and be sharpe angles , (RECORD-E1-P2,2,C4R.230)

but C. is a blonte angle , and therfore greater then a ryght angle .
(RECORD-E1-P2,2,C4R.231)

<heading>

THE TENTH THEOREME . (RECORD-E1-P2,2,C4R.233)

</heading>

In euery triangle any .ij. corners , how so euer you take the~ , ar
lesse the~ ij. right corners . (RECORD-E1-P2,2,C4R.235)

<P_2,C4V>

<font>

Example . (RECORD-E1-P2,2,C4V.238)

</font>

In the first triangle E , {COM:figure_omitted} whiche is a threlyke ,
and therfore hath all his angles sharpe , take anie twoo corners that
you will , (RECORD-E1-P2,2,C4V.240)

and you shall perceiue that they be lesser then ij. right corners ,
(RECORD-E1-P2,2,C4V.241)

for in euery triangle that hath all sharpe corners <paren> as you see
it to be in this example </paren> euery corner is lesse then a right
corner . (RECORD-E1-P2,2,C4V.242)

And therfore also euery two corners must nedes be lesse then two right
corners . (RECORD-E1-P2,2,C4V.243)

Furthermore in that other triangle marked with M , {COM:figure_omitted}
whiche hath .ij. sharpe corners and one right , any .ij. of them also
are lesse then two ryght angles . (RECORD-E1-P2,2,C4V.244)

For though you take the right corner for one , yet the other whiche is
a sharpe corner , is lesse then a right corner .
(RECORD-E1-P2,2,C4V.245)

And so it is true in all kindes of triangles , as you maie perceiue
more plainly by the .xxij. Theoreme . (RECORD-E1-P2,2,C4V.246)

<heading>

THE .XI. THEOREME . (RECORD-E1-P2,2,C4V.248)

</heading>

In euery triangle , the greattest side lieth against the greattest
angle . (RECORD-E1-P2,2,C4V.250)

<font>

Example . (RECORD-E1-P2,2,C4V.252)

</font>

As in this triangle A.B.C , {COM:figure_omitted} the greatest angle is
C. (RECORD-E1-P2,2,C4V.254)

And A.B. <paren> whiche is the side that lieth against it </paren> is
the greatest and longest side . (RECORD-E1-P2,2,C4V.255)

And contrary waies , as A.C. is the shortest side , so B. <paren>
whiche is the angle liyng against it </paren> is the <P_2,D1R> smallest
and sharpest angle , (RECORD-E1-P2,2,D1R.256)

for this doth folow also , that as the longest side lyeth against the
greatest angle , so it that foloweth (RECORD-E1-P2,2,D1R.257)

<heading>

THE TWELFT THEOREME . (RECORD-E1-P2,2,D1R.259)

</heading>

In euery triangle the greattest angle lieth against the longest side .
(RECORD-E1-P2,2,D1R.261)

For these ij. theoremes are one in truthe . (RECORD-E1-P2,2,D1R.262)

<heading>

THE THIRTENTH THEOREME . (RECORD-E1-P2,2,D1R.264)

</heading>

In euerie triangle anie ij. sides togither how so euer you take them ,
are longer the~ the thirde . (RECORD-E1-P2,2,D1R.266)

For example you shal take this triangle A.B. {COM:figure_omitted} which
hath a veery blunt corner , and therfore one of his sides greater a
good deale then any of the other , (RECORD-E1-P2,2,D1R.267)

and yet the ij. lesser sides togither ar $greate {COM:sic} then it .
(RECORD-E1-P2,2,D1R.268)

And if it bee so in a blunte angeled triangle , it must nedes be true
in all other , (RECORD-E1-P2,2,D1R.269)

for there is no other kinde of triangles that hathe the one side so
greate aboue the other sids {COM:sic} , as they y=t= haue blunt corners
. (RECORD-E1-P2,2,D1R.270)

<heading>

THE FOURTENTH THEOREME . (RECORD-E1-P2,2,D1R.272)

</heading>

If there be drawen from the endes of anie side of a triangle .ij. lines
metinge within the triangle , those two lines shall be lesse then the
other twoo sides of the triangle , (RECORD-E1-P2,2,D1R.274)

but yet the <P_2,D1V> corner that thei make , shall bee greater then
that corner of the triangle , whiche standeth ouer it .
(RECORD-E1-P2,2,D1V.275)

<font>

Example . (RECORD-E1-P2,2,D1V.277)

</font>

A.B.C. is a triangle . {COM:figure_omitted} on whose ground line A.B.
there is drawen ij. lines , from the ij. endes of it , I say from A.
and B , (RECORD-E1-P2,2,D1V.279)

and they meete within the triangle in the pointe D , wherfore I say ,
that as those two lynes A.D. and B.D , are lesser then A.C. and B.C ,
so the angle D , is greatter then the angle C , which is the angle
against it . (RECORD-E1-P2,2,D1V.280)

<heading>

THE FIFTENTH THEOREME . (RECORD-E1-P2,2,D1V.282)

</heading>

If a triangle haue two sides equall to the two sides of an other
triangle , but yet the a~gle that is contained betwene those sides ,
greater then the like angle in the other triangle , then is his grounde
line greater then the grounde line of the other triangle .
(RECORD-E1-P2,2,D1V.284)

<font>

Example . (RECORD-E1-P2,2,D1V.286)

</font>

A.B.C. is a triangle , {COM:figure_omitted} whose sides A.C. and B.C ,
ar equall to E.D. and D.F , the two sides of the triangle D.E.F ,
(RECORD-E1-P2,2,D1V.288)

{COM:figure_on_next_page_omitted}

but bicause the angle in D , is greatter then the angle C. whiche are
the ij. angles contayned betwene the equal lynes </paren>
{COM:no_matching_open_paren} <P_2,D2R> therfore muste the ground line
E.F. nedes bee greatter thenne the grounde line A.B , as you se
plainely . (RECORD-E1-P2,2,D2R.290)

<heading>

THE XVI. THEOREME . (RECORD-E1-P2,2,D2R.292)

</heading>

If a triangle haue twoo sides equalle to the two sides of an other
triangle , but yet hathe a longer ground line the~ that other triangle
, then is his angle that lieth betwene the equall sides , greater the~
the like corner in the other triangle . (RECORD-E1-P2,2,D2R.294)

<font>

Example . (RECORD-E1-P2,2,D2R.296)

</font>

This Theoreme is nothing els , but the sentence of the first Theoreme
turned backward , (RECORD-E1-P2,2,D2R.298)

and therefore nedeth none other profe nother declaration , then the
other example . (RECORD-E1-P2,2,D2R.299)

<heading>

THE SEUENTENTH THEOREME . (RECORD-E1-P2,2,D2R.301)

</heading>

If two triangles be of such sort , that two angles of the one be equal
to ij. angles of the other , and that one side of the one be equal to
on side of the other , whether that side do adioyne to one of the
equall corners , or els lye againste <P_2,D2V> one of them , then shall
the other twoo sides of those triangles bee equalle togither ,
(RECORD-E1-P2,2,D2V.303)

and the thirde corner also shall be equall in those two triangles .
(RECORD-E1-P2,2,D2V.304)

<font>

Example . (RECORD-E1-P2,2,D2V.306)

</font>

Bicause that A.B.C , the one triangle {COM:figure_omitted} hath two
corners A. and B , equal to D. E , that are twoo corners of the other
triangle . D.E.F. {COM:figure_omitted} and that they haue one side in
theym bothe equall , that is A.B , which is equall to D.E , therefore
shall both the other ij. sides be equall one to an other , as A.C. and
B.C. equall to D.F and E.F , (RECORD-E1-P2,2,D2V.308)

and also the thirde angle in them both $shall $be {TEXT:shalbe} equall
, (RECORD-E1-P2,2,D2V.309)

that is , the angle C. shal be equall to the angle F .
(RECORD-E1-P2,2,D2V.310)

<heading>

THE EIGHTENTH THEOREME . (RECORD-E1-P2,2,D2V.312)

</heading>

when on .ij. right lines ther is drawen a third right line crosse waies
, and maketh .ij. matche corners of the one line equall to the like
twoo matche corners of the other line , then ar those two lines gemmow
lines , or paralleles . (RECORD-E1-P2,2,D2V.314)

<font>

Example . (RECORD-E1-P2,2,D2V.316)

</font>

<P_2,D3R>

The .ij. fyrst lynes are A.B. and C.D , (RECORD-E1-P2,2,D3R.319)

the thyrd lyne that crosseth them is E.F . (RECORD-E1-P2,2,D3R.320)

{COM:figure_omitted}

And bycause that E.F. maketh ij. matche angles with A.B , equall to
.ij. other lyke matche angles on C.D , <paren> that is to say E.G ,
equall to K.F , and M.N. equall also to H , L. </paren> therfore are
those ij. lynes A.B. and C.D. gemow lynes , (RECORD-E1-P2,2,D3R.322)

vnderstand here by <font> lyke matche corners </font> , those that go
one way as doth E.G , and K.F , lyke ways N.M , and H.L ,
(RECORD-E1-P2,2,D3R.323)

for as E.G. and H.L , other N.M. and K.F. go not one waie , so be not
they lyke match corners . (RECORD-E1-P2,2,D3R.324)

<heading>

THE NYNTENTH THEOREME . (RECORD-E1-P2,2,D3R.326)

</heading>

when on two right lines there is drawen a thirde right line crossewaies
, and maketh the ij. ouer corners towarde one hande equall togither ,
then ar those .ij. lines paralleles . And in like maner if two inner
corners toward one hande , be equall to .ii. right angles .
(RECORD-E1-P2,2,D3R.328)

<font>

Example . (RECORD-E1-P2,2,D3R.330)

</font>

As the Theoreme dothe speake of .ij. ouer angles , so muste you
vnderstande also of .ij. nether angles , (RECORD-E1-P2,2,D3R.332)

for the iudgement is lyke in bothe . (RECORD-E1-P2,2,D3R.333)

Take for an example the figure of the last theoreme , where A.B , and
C.D , be called paralleles also , bicause E. and K , <paren> whiche are
.ij. ouer corners </paren> are equall , and lyke waies L. and M .
(RECORD-E1-P2,2,D3R.334)

And so are in lyke maner the nether corners N. and H , and G. and F .
(RECORD-E1-P2,2,D3R.335)

Nowe to the seconde parte of the theoreme , (RECORD-E1-P2,2,D3R.336)

those .ij. lynes A.B. and C.D , shall be called paralleles , bicause
the ij. inner corners . As for example those two that bee toward the
right hande <paren> that is G. and L. </paren> are equall <P_2,D3V>
<paren> by the fyrst parte of this nyntenth theoreme </paren>
(RECORD-E1-P2,2,D3V.337)

therfore muste G. and L. be equall to two ryght angles .
(RECORD-E1-P2,2,D3V.338)

<heading>

THE XX. THEOREME . (RECORD-E1-P2,2,D3V.340)

</heading>

when a right line is drawen crosse ouer .ij. right gemow lines , it
maketh .ij. matche corners of the one line , equall to two matche
corners of the other line , and also bothe ouer corners of one hande
equall togither , and bothe nether corners likewaies , and more ouer
two inner corners , and two vtter corners also towarde one hande ,
equall to two right angles . (RECORD-E1-P2,2,D3V.342)

<font>

Example . (RECORD-E1-P2,2,D3V.344)

</font>

Bycause A.B. and C.D , <paren> in the last figure </paren> are
paralleles , therefore the two matche corners of the one lyne , as E.G.
be equall vnto the .ij. matche corners of the other line , that is K.F
, and lykewaies M.N , equall to H.L. And also E. and K. bothe ouer
corners of the lefte hande equall togyther , (RECORD-E1-P2,2,D3V.346)

and so are M. and L , the two ouer corners on the ryghte hande , in
lyke maner N. and H , the two nether corners on the lefte hande ,
equall eche to other , and G. and F. the two nether angles on the right
hande equall togither . (RECORD-E1-P2,2,D3V.347)

& Farthermore yet G. and L. the .ij. inner angles on the right hande
bee equall to two right angles , (RECORD-E1-P2,2,D3V.348)

and so are M. and F. the .ij. vtter angles on the same hande ,
(RECORD-E1-P2,2,D3V.349)

in lyke manner shall you say of N. and K. the two inner corners on the
left hand . and of E. and H. the two vtter corners on the same hande .
(RECORD-E1-P2,2,D3V.350)

And thus you see the agreable sentence of these .iij. theoremes to
tende to this purpose , to declare by the angles how to iudge
paralleles , and contrary waies howe you may by paralleles iudge the
proportion of the angles . (RECORD-E1-P2,2,D3V.351)

<P_2,D4R>

<heading>

THE XXI. THEOREME . (RECORD-E1-P2,2,D4R.354)

</heading>

what so euer lines be paralleles to any other line , those same be
paralleles togither . (RECORD-E1-P2,2,D4R.356)

<font>

Example . (RECORD-E1-P2,2,D4R.358)

</font>

A.B. is a gemow line ; or a parallele vnto C.D .
(RECORD-E1-P2,2,D4R.360)

And E.F , lykewaies is a parallele vnto C.D. {COM:figures_omitted}
Wherfore it foloweth , that A.B. must nedes bee a parallele vnto E.F .
(RECORD-E1-P2,2,D4R.361)

<heading>

THE .XXIJ. THEOREME . (RECORD-E1-P2,2,D4R.363)

</heading>

In euery triangle , when any side is drawen forth in length , the vtter
angle is equall to the ij. inner angles that lie againste it .
(RECORD-E1-P2,2,D4R.365)

And all iij. inner angles of any triangle are equall to ij. right
angles . (RECORD-E1-P2,2,D4R.366)

<font>

Example . (RECORD-E1-P2,2,D4R.368)

</font>

The triangle beeyng A.D.E. {COM:figure_omitted} and the syde A.E.
drawen foorthe vnto B , there is made an vtter corner , which is C ,
(RECORD-E1-P2,2,D4R.370)

and this vtter corner C , is equall to bothe the inner corners that lye
agaynst it , whyche are A. and D . (RECORD-E1-P2,2,D4R.371)

And all thre inner corners , that is to say , A. D. and E , are equall
to two ryght corners , whereof it foloweth , <font> that all the three
corners of any one triangle are equall to all the three corners of
euerye other triangle </font> . (RECORD-E1-P2,2,D4R.372)

For what so euer thynges are equalle to anny one thyrde thynge , those
same are <P_2,D4V> equalle togitther , by the fyrste common sentence ,
so that bycause all the .iij. angles of euery triangle are equall to
two ryghte angles , and all ryghte angles bee equall togyther <paren>
by the fourth request </paren> therfore must it nedes folow , that all
the thre corners of euery triangle <paren> accomptyng them togyther
</paren> are equall to iij. corners of any other triangle , taken all
togyther . (RECORD-E1-P2,2,D4V.373)

<heading>

THE .XXIII. THEOREME . (RECORD-E1-P2,2,D4V.375)

</heading>

when any ij. right lines doth touche and couple .ij. other righte lines
, whiche are equall in length and paralleles , and if those .ij. lines
bee drawen towarde one hande , then are thei also equall together , and
paralleles . (RECORD-E1-P2,2,D4V.377)

<font>

Example . (RECORD-E1-P2,2,D4V.379)

</font>

A.B. and C.D. {COM:figure_omitted} are ij. ryght lynes and paralleles
and equall in length , (RECORD-E1-P2,2,D4V.381)

and they ar touched and ioyned togither by ij. other lynes A.C. and B.D
, (RECORD-E1-P2,2,D4V.382)

this beyng so , and A.C. and B.D. beyng drawen towarde one syde <paren>
that is to saye , both towarde the lefte hande </paren> therefore are A
, C. and B.D. bothe equall and also paralleles .
(RECORD-E1-P2,2,D4V.383)

<heading>

THE .XXIIIJ. THEOREME . (RECORD-E1-P2,2,D4V.385)

</heading>

In any likeiamme the two contrary sides ar equall togither ,
(RECORD-E1-P2,2,D4V.387)

and so are eche .ij. contrary angles , (RECORD-E1-P2,2,D4V.388)

and the bias line that is drawen in it , doth diuide it into two equall
portions . (RECORD-E1-P2,2,D4V.389)

<P_2,E1R>

<font>

Example . (RECORD-E1-P2,2,E1R.392)

</font>

Here are two likeiammes ioyned togither , (RECORD-E1-P2,2,E1R.394)

{COM:figure_omitted}

the one is a longe square A.B.E , (RECORD-E1-P2,2,E1R.396)

and the other is a losengelike D.C.E.F. which ij. likeiammes ar proued
equall togither , bycause they haue one ground line , that is , F.E ,
And are made betwene one payre of gemow lines , I meane A.D. and E.H .
(RECORD-E1-P2,2,E1R.397)

By this Theoreme may you know the arte of the righte measuringe of
likeiammes , as in my booke of measuring I wil more plainly declare .
(RECORD-E1-P2,2,E1R.398)

<heading>

THE XXVI. THEOREME . (RECORD-E1-P2,2,E1R.400)

</heading>

All likeiammes that haue equal grounde lines and are drawen betwene one
paire of paralleles , are equal togither . (RECORD-E1-P2,2,E1R.402)

<font>

Example . (RECORD-E1-P2,2,E1R.404)

</font>

Fyrste you muste marke the difference betwene this Theoreme and the
laste , (RECORD-E1-P2,2,E1R.406)

for the laste Theoreme presupposed to the diuers likeiammes one ground
line common to them , (RECORD-E1-P2,2,E1R.407)

but this theoreme doth presuppose a diuers ground line for euery
likeiamme , only meaning them to be equal in length , though they be
diuers in numbhe {COM:sic} . (RECORD-E1-P2,2,E1R.408)

As for example . In the last figure ther are two parallels , A.D. and
E.H , (RECORD-E1-P2,2,E1R.409)

and betwene them are drawen thre likeiammes , (RECORD-E1-P2,2,E1R.410)

the firste is , A.B.E.F , (RECORD-E1-P2,2,E1R.411)

the second is E.C.D.F , (RECORD-E1-P2,2,E1R.412)

and the thirde is C.G.H.D . (RECORD-E1-P2,2,E1R.413)

The firste and the seconde haue one ground line , <paren> that is E.F.
</paren> (RECORD-E1-P2,2,E1R.414)

and therfore in so muche as they are betwene one paire of paralleles ,
they are equall accordinge to the fiue and twentye Theoreme ,
(RECORD-E1-P2,2,E1R.415)

but the thirde likeiamme that is C.G.H.D. hathe his grounde line G.H ,
seuerall frome <P_2,E1V> the other , but yet equall vnto it . wherefore
the third likeiam is equall to the other two firste likeiammes .
(RECORD-E1-P2,2,E1V.416)

And for a proofe that G.H. being the ground line of the third likeiamme
, is equall to E.F , whiche is the ground line to both the other
likeiams , that may be thus declared , (RECORD-E1-P2,2,E1V.417)

G.H. is equall to C.D , seynge they are the contrary sides of one
likeiamme <paren> by the foure and twentye theoreme </paren>
(RECORD-E1-P2,2,E1V.418)

and so are C.D. and E.F. by the same theoreme .
(RECORD-E1-P2,2,E1V.419)

Therfore seynge both those ground lines , E.F. and G.H , are equall to
one thirde line <paren> that is C.D. </paren> they must nedes be equall
togyther by the firste common sentence . (RECORD-E1-P2,2,E1V.420)

<heading>

THE XXVII. THEOREME . (RECORD-E1-P2,2,E1V.422)

</heading>

All triangles hauinge one grounde lyne , $and {TEXT:an} standing
betwene one paire of parallels , ar equall togither .
(RECORD-E1-P2,2,E1V.424)

<font>

Example . (RECORD-E1-P2,2,E1V.426)

</font>

A.B. and C.F. are twoo gemowe lines , betweene which there be made two
triangles , A.D.E. and D.E.B , so that D.E , is the common ground line
to them bothe . {COM:figure_omitted} wherfore it doth folow , that
those two triangles A.D.E. and D.E.B. are equall eche to other .
(RECORD-E1-P2,2,E1V.428)

<heading>

THE XXVIIJ. THEOREME . (RECORD-E1-P2,2,E1V.430)

</heading>

All triangles that haue like long ground lines , and bee made betweene
one paire of gemow lines , are equall togither .
(RECORD-E1-P2,2,E1V.432)

<P_2,E2R>

<font>

Example . (RECORD-E1-P2,2,E2R.435)

</font>

Example of this Theoreme you may see in the last figure , where as sixe
triangles made betwene those two gemowe lines A.B. and C.F , the first
triangle is A.C.D , the seconde is A.D.E , the thirde is A.D.B , the
fourth is A.B.E , the fifte is D.E.B , and the sixte is B.E.F , of
which sixe triangles , A.D.E. and D.E.B. are equall bicause they haue
one common grounde line . And so likewise A.B.E. and A.B.D , whose
commen grounde line is A.B , (RECORD-E1-P2,2,E2R.437)

but A.C.D. is equal to B.E.F , being both betwene one couple of
parallels , not bicause thei haue one groune line , but bicause they
haue their ground lines equall , (RECORD-E1-P2,2,E2R.438)

for C.D. is equall to E.F , as you may declare thus .
(RECORD-E1-P2,2,E2R.439)

C.D , is equall to A.B. <paren> by the foure and twenty Theoreme
</paren> (RECORD-E1-P2,2,E2R.440)

for thei are two contrary sides of one lykeiamme . A.C.D.B ,
(RECORD-E1-P2,2,E2R.441)

and E.F by the same theoreme , is equall to A.B ,
(RECORD-E1-P2,2,E2R.442)

for thei ar the two y=e= contrary sides of the likeiamme , A.E.F.B ,
wherfore C.D. must needes be equall to E.F . (RECORD-E1-P2,2,E2R.443)

like wise the triangle A.C.D , is equal to A.B.E , bicause they ar made
betwene one paire of parallels and haue their groundlines like , I
meane C.D. and A.B . (RECORD-E1-P2,2,E2R.444)

Againe A.D.E , is equal to eche of them both , (RECORD-E1-P2,2,E2R.445)

for his ground line D.E , is equall to A.B , in so muche as they are
the contrary sides of one likeiamme , (RECORD-E1-P2,2,E2R.446)

that is the long square A.B.D.E. (RECORD-E1-P2,2,E2R.447)

And thus may you proue the equalnes of all the reste .
(RECORD-E1-P2,2,E2R.448)

<heading>

THE XXIX. THEOREME . (RECORD-E1-P2,2,E2R.450)

</heading>

Al equal triangles that are made on one grounde line , and rise one
waye , must needes be betwene one paire of parallels .
(RECORD-E1-P2,2,E2R.452)

<font>

Example . (RECORD-E1-P2,2,E2R.454)

</font>

Take for example A.D.F , and D.E.B , which as the xxvij. <P_2,E2V>
conclusion dooth proue </paren> are equall togither ,
(RECORD-E1-P2,2,E2V.456)

and as you see , they haue on ground line D.E. (RECORD-E1-P2,2,E2V.457)

And againe they rise towarde one side , that is to say , vpwarde toward
the line A.B , wher fore they must needes be inclosed betweene one
paire of parallels , which are heere in this example A.B. and D.E .
(RECORD-E1-P2,2,E2V.458)

<heading>

THE THIRTY THEOREME . (RECORD-E1-P2,2,E2V.460)

</heading>

Equal triangles that haue $their $ground {TEXT:the_irground} lines
equal , and be drawe~ toward one side , are made betwene one paire of
paralleles . (RECORD-E1-P2,2,E2V.462)

<font>

Example . (RECORD-E1-P2,2,E2V.464)

</font>

The example that declared the last theoreme , maye well serue to the
declaration of this also . (RECORD-E1-P2,2,E2V.466)

For those ij. theoremes do diffre but in this one pointe , that the
laste theoreme meaneth of triangles , that haue one ground line common
to them both , (RECORD-E1-P2,2,E2V.467)

and this theoreme doth presuppose the grounde lines to bee diuers , but
yet of one length , as A.C.D , and B.E.F , (RECORD-E1-P2,2,E2V.468)

as they are ij. equall triangles approued , by the eighte and twentye
Theorem , so in the same Theorem it is declared , y=t= their grou~d
lines are equall togither , that is C.D , and E.F ,
(RECORD-E1-P2,2,E2V.469)

now this beeynge true , and considering that they are made towarde one
side , it foloweth , that they are made betwene one paire of parallels
(RECORD-E1-P2,2,E2V.470)

when I saye , drawen towarde one side , I meane that the triangles must
be drawen other both vpward frome one parallel , other els both
downward , (RECORD-E1-P2,2,E2V.471)

then are they drawen betwene two paire of parallels , presupposinge one
to bee drawen by their ground line , (RECORD-E1-P2,2,E2V.472)

and then do they ryse toward contrary sides . (RECORD-E1-P2,2,E2V.473)

<P_2,E3R>

<heading>

THE XXXI. THEOREME . (RECORD-E1-P2,2,E3R.476)

</heading>

If a likeiamme haue one ground line with a triangle , and be drawen
betwene one paire of paralleles , then shall the likeiamme be double to
the triangle . (RECORD-E1-P2,2,E3R.478)

<font>

Example . (RECORD-E1-P2,2,E3R.480)

</font>

A.H. and B.G. are .ij. gemow lines , betwene which there is made a
triangle B.CG , and a lykeiamme , A.B.G.C , whiche haue a grounde lyne
, (RECORD-E1-P2,2,E3R.482)

that is to saye , B.G . (RECORD-E1-P2,2,E3R.483)

{COM:figure_omitted}

Therfore doth it folow that the lyke iamme A.B.G.C. is double to the
triangle B.C.G . (RECORD-E1-P2,2,E3R.485)

For euery halfe of that lykeiamme is equall to the triangle , I meane
A.B.F.E. other F.E.C.G. as you may coniecture by the .xi. conclusion
geometrical . (RECORD-E1-P2,2,E3R.486)

And as this Theoreme dothe speake of a triangle and likeiamme that haue
one groundelyne , so is it true also , yf theyr groundelynes be equall
, though they bee dyuers , so that thei be made betwene one payre of
paralleles . (RECORD-E1-P2,2,E3R.487)

And hereof may you perceaue the reason , why in measuryng the platte of
a triangle , you must multiply the perpendicular lyne by halfe the
grounde lyne , or els the hole grounde lyne by halfe the perpendicular
, (RECORD-E1-P2,2,E3R.488)

for by any of these bothe {COM:sic} waies is there made a lykiamme
equall to halfe suche a one as shulde be made on the same hole grounde
lyne with the triangle , and betweene one payre of paralleles .
(RECORD-E1-P2,2,E3R.489)

Therfore as that lykeiamme is double to the triangle , so the halfe of
it , must needes be equall to the triangle . (RECORD-E1-P2,2,E3R.490)

Compare the .xi. conclusion with this theoreme .
(RECORD-E1-P2,2,E3R.491)

<heading>

THE .XXXIJ. THEOREME . (RECORD-E1-P2,2,E3R.493)

</heading>

In all likeiammes where there are more than <P_2,E3V> one made aboute
one bias line , the fill squares of euery of them must nedes be equall
. (RECORD-E1-P2,2,E3V.495)

<font>

Example . (RECORD-E1-P2,2,E3V.497)

</font>

Fyrst before I declare the examples , it shal be mete to shew the true
vndersta~dyng of this theorem . (RECORD-E1-P2,2,E3V.499)

Therfore by the <font> Bias line </font> , I meane that lyne , which in
any square figure dooth runne from corner to corner .
(RECORD-E1-P2,2,E3V.500)

And euery square which is diuided by that bias line into equall halues
from corner to corner <paren> that is to say , into .ij. equall
triangles </paren> those be counted <font> to stande aboute one bias
line </font> , (RECORD-E1-P2,2,E3V.501)

and the other squares , whiche touche that bias line , with one of
their corners onely , those doo I call <font> Fyll squares </font> ,
accordyng to the greke name , whiche is <font> anapleromata </font> ,
and called in latin <font> supplementa </font> , bycause that they make
one generall square , includyng and enclosyng the other diuers squares
, as in this exa~ple H.C.E.N. is one square likeiamme , and L.M.G.C. is
an other , {COM:figures_omitted} whiche bothe are made aboute one bias
line , (RECORD-E1-P2,2,E3V.502)

that is N.M , (RECORD-E1-P2,2,E3V.503)

than K.L.H.C. and C.E.F.G. are .ij. fyll squares ,
(RECORD-E1-P2,2,E3V.504)

for they doo fyll vp the sydes of the .ij fyrste square lykeiammes , in
suche sorte , that of all them foure is made one greate generall square
K.M.F.N . (RECORD-E1-P2,2,E3V.505)

Nowe to the sentence of the theoreme , I say , that the .ij. fill
squares , H.K.L.C. and C.E.F.G. are both equall togither , <paren> as
it shall bee declared in the booke of proofes </paren> bicause they are
the fill squares of two likeiammes made aboute one bias line , as the
exaumple sheweth . (RECORD-E1-P2,2,E3V.506)

Conferre the twelfthe conclusion with this theoreme .
(RECORD-E1-P2,2,E3V.507)

{COM:insert_helsinki_2}

<P_2,G1R>

<heading>

THE XLIJ. THEOREME . (RECORD-E1-P2,2,G1R.511)

</heading>

If a right line be deuided as chance happeneth the iiij. long squares ,
that may be made of that whole line and one of his partes with the
square of the other part , shall be equall to the square that is made
of the whole line and the saide first portion ioyned to him in lengthe
as one whole line . (RECORD-E1-P2,2,G1R.513)

<font>

Example . (RECORD-E1-P2,2,G1R.515)

</font>

The firste line is A.B , (RECORD-E1-P2,2,G1R.517)

and is deuided by C. into two vnequall partes as happeneth
(RECORD-E1-P2,2,G1R.518)

{COM:figure_omitted}

the longsquare of yt , and his lesser portion A.C , is foure times
drawen , (RECORD-E1-P2,2,G1R.520)

the first is E.G.M.K , (RECORD-E1-P2,2,G1R.521)

the seconde is K.M.Q.O , (RECORD-E1-P2,2,G1R.522)

the third is H.K.R.S , (RECORD-E1-P2,2,G1R.523)

and the fourthe is K.L.S.T. (RECORD-E1-P2,2,G1R.524)

{COM:figure_omitted}

And where as it appeareth that one of the little squares <paren> I
meane K.L.PO </paren> is reckened twise , ones as parcell of the second
long square and agayne as parte of the <P_2,G1V> thirde longsquare , to
auoide ambiguite , you may place one insteede of it , an other square
of equalitee , with it . (RECORD-E1-P2,2,G1V.526)

that is to saye , D.E.K.H , which was at no tyme accompting as percell
of any one of them , (RECORD-E1-P2,2,G1V.527)

and then haue you iiij. long squares distinctly made of the whole line
A.B , and his lesser portion A.C . (RECORD-E1-P2,2,G1V.528)

And within them is there a greate full square P.Q.T.V. whiche is the
iust square of B.C , beynge the greatter portion of the line A.B.
(RECORD-E1-P2,2,G1V.529)

And that those fiue squares doo make iuste as muche as the whole square
of that longer line D.G , <paren> whiche is as longe as A.B , and A.C.
</paren> ioyned togither </paren> it may be iudged easyly by the eye ,
sith that one greate square doth comprehe~d in it all the other fiue
squares , that is to say , foure longsqares <paren> as before mencioned
</paren> and one full square , which is the intent of the Theoreme .
(RECORD-E1-P2,2,G1V.530)