<P_1,A1R>

<heading>

THE DEFINITIONS OF THE PRINCIPLES OF GEOMETRY . (RECORD-E1-P1,1,A1R.3)

</heading>

Geometry teacheth the drawyng , Measuring and proporcion of figures ,
(RECORD-E1-P1,1,A1R.5)

but in as muche as no figure can be drawen , but it muste haue certayne
bou~des and inclosures of lines : and euery lyne also is begon and
ended at some certaine prycke , fyrst it shall be meete to know these
smaller partes of euery figure , that therby the whole figures may the
better bee iudged , and distincte in sonder . (RECORD-E1-P1,1,A1R.6)

<font> A Poynt or a Prycke , </font> is named of Geometricans that
small and vnsensible shape , which hath in it no partes , that is to
say : nother length , breadth nor depth . (RECORD-E1-P1,1,A1R.7)

But as this exactnes of definition is more meeter for onlye Theorike
speculacion , then for practise and outwarde worke <paren> consideringe
that myne intente is to applye all these whole principles to woork
</paren> I thynke meeter for this purpose , to call <font> a poynt or
prycke , </font> that small printe of penne , pencyle , or other
instrumente , which is not moued , nor drawen from his fyrst touche ,
and therfore hath no notable length nor bredthe : as this example doeth
declare . {COM:three_points_omitted} Where I haue set .iij. prickes ,
eche of them hauyng both le~gth and bredth , thogh it be but smal , and
therfore not notable (RECORD-E1-P1,1,A1R.8)

Nowe of a great numbre of these prickes , is made <font> a Lyne ,
</font> as you may perceiue by this forme ensuyng .
{COM:series_of_points_omitted} where as I haue set a number of prickes
, so if you with your pen will set in more other prickes betweene
euerye two of these , then wil it be a lyne , as here you may see
(RECORD-E1-P1,1,A1R.9)

{COM:line_omitted}

and this <font> lyne , </font> is called of Geometricians , <font>
Lengthe withoute breadth . </font> (RECORD-E1-P1,1,A1R.11)

But as they in theyr theorikes <paren> which ar only mind workes
</paren> <P_1,A1V> do precisely vnderstand these definitions , so it
shal be sufficient for those men , whiche seke the vse of the same
thinges , as sense may duely iudge them , and applye to handy workes if
they vnderstand them so to be true , that outwarde sense canne fynde
none erroure therin . (RECORD-E1-P1,1,A1V.12)

Of lynes there bee two principall kyndes . (RECORD-E1-P1,1,A1V.13)

The one is called a right or straight lyne , and the other a croked
lyne . (RECORD-E1-P1,1,A1V.14)

<font> A straight lyne , </font> is the shortest that maye be drawenne
betweene two prickes . (RECORD-E1-P1,1,A1V.15)

And all other lines , that go not right forth from prick to prick , but
boweth any waye , such are called <font> Croke lynes </font> as in
these examples folowyng ye may se , where I haue set but one forme of a
straight lyne , (RECORD-E1-P1,1,A1V.16)

for more formes there be not , (RECORD-E1-P1,1,A1V.17)

but of crooked lynes there bee innumerable diuersities , whereof for
examples sum I haue sette here . (RECORD-E1-P1,1,A1V.18)

A right lyne . (RECORD-E1-P1,1,A1V.19)

<font> Croked lines . </font> (RECORD-E1-P1,1,A1V.20)

{COM:figures_omitted}

So now you must vnderstand , that <font> euery lyne is drawen betwene
twoo prickes , </font> wherof the one is at the beginning , and the
other at the ende . (RECORD-E1-P1,1,A1V.22)

Therfore when soeuer you do see any formes of lynes to touche at one
notable pricke , as in this example , then shall you <P_1,A2R> not call
it one croked lyne , but rather twoo lynes : in as muche as there is a
notable and sensible angle by .A. whiche euermore is made by the
meetyng of two seuerall lynes . (RECORD-E1-P1,1,A2R.23)

And likewayes shall you iudge of this figure , which is made of two
lines , and not of one onely . So that whan so euer any suche meetyng
of lines doth happen , the place of their metyng is called an <font>
Angle or corner . </font> (RECORD-E1-P1,1,A2R.24)

Of angles there be three generall kindes : a sharpe angle , a square
angle , and a blunte angle . (RECORD-E1-P1,1,A2R.25)

<font> The square angle , </font> whiche is commonly named <font> a
right corner , </font> is made of twoo lynes meetyng together in fourme
of a squire , which two lines , if they be drawen forth in length ,
will cross one an other : as in the examples folowyng you maie see .
(RECORD-E1-P1,1,A2R.26)

<font> A sharpe angle </font> is so called , because it is lesser than
is a square angle , and the lines that make it , do not open so wide in
their departynge as in a square corner , (RECORD-E1-P1,1,A2R.27)

and if thei be drawen crosse , all fower corners will not be equall .
(RECORD-E1-P1,1,A2R.28)

<font> A blunt or brode corner , </font> is greater then is a square
angle , (RECORD-E1-P1,1,A2R.29)

and his lines do parte more in sonder then in a right angle , of which
all take these examples . (RECORD-E1-P1,1,A2R.30)

<font>

Right angles . (RECORD-E1-P1,1,A2R.32)

Sharpe angles . (RECORD-E1-P1,1,A2R.33)

</font>

{COM:figures_omitted}

And these angles <paren> as you see </paren> are made partly of streght
lynes , partly of croked lines , and partly of both together .
(RECORD-E1-P1,1,A2R.36)

Howbeit in right angles I haue put none example of croked lines ,
because it would <P_1,A2V> muche trouble a lerner to iudge them :
(RECORD-E1-P1,1,A2V.37)

for their true iudgement doth appertaine to arte perspectiue , and as I
may say , rather to reason then to sense . (RECORD-E1-P1,1,A2V.38)

<font>

Blunte or brode angles . (RECORD-E1-P1,1,A2V.40)

</font>

{COM:figures_omitted}

But now as of many prickes there is made one line , so <font> of
diuerse lines are there made sundry formes , figures , and shapes ,
</font> which all yet be called by one propre name , <font> Platte
formes , </font> (RECORD-E1-P1,1,A2V.43)

and thei haue bothe <font> length and bredth , but yet no depenesse .
</font> (RECORD-E1-P1,1,A2V.44)

And <font> the boundes </font> of euerie platte forme are lines : as by
the examples you maie perceiue . (RECORD-E1-P1,1,A2V.45)

Of platte formes some be plain , and some be croked , and some partly
plaine , and partlie croked . (RECORD-E1-P1,1,A2V.46)

<font> A plaine platte </font> is that , which is made al equall in
height , so that the middle partes nother bulke vp , nother shrink down
more then the both endes . (RECORD-E1-P1,1,A2V.47)

For whan the one parte is higher then the other , then is it named a
<font> Croked platte </font> . (RECORD-E1-P1,1,A2V.48)

And if it be partlie plaine , and partlie crooked , then is it called a
<font> Myxte platte , </font> of all whiche , these are exaumples .
(RECORD-E1-P1,1,A2V.49)

<font>

A plaine platte . (RECORD-E1-P1,1,A2V.51)

A croked platte . (RECORD-E1-P1,1,A2V.52)

A myxte platte . (RECORD-E1-P1,1,A2V.53)

</font>

And as of many prickes is made a line , and of diuerse lines one platte
forme , so of manie plattes is made <font> a bodie </font> which
conteighneth <font> Lengthe , bredth , and depenesse . </font>
(RECORD-E1-P1,1,A2V.55)

By <font> Depenesse </font> I vnderstand , not as the common sort doth
, the holownesse of anything , as of a well , a diche , a potte , and
suche like , (RECORD-E1-P1,1,A2V.56)

but I meane the massie thickness <P_1,A3R> of any bodie , as in
exaumple of a potte : the depenesse is after the common name , the
space from his brimme to his bottome . (RECORD-E1-P1,1,A3R.57)

But as I take it here , the depenesse of his bodie is his thicknesse in
the sides , which is an other thyng cleane different from the depenesse
of his holownes , that the common people meaneth .
(RECORD-E1-P1,1,A3R.58)

Now all bodies haue platte formes for their boundes ,
(RECORD-E1-P1,1,A3R.59)

so in a dye <paren> which is called <font> a cubike bodie </font>
</paren> by geometricians , and an <font> ashler </font> of masons ,
there are .vi. sides , which are .vi. platte formes , and are the
boundes of the dye . (RECORD-E1-P1,1,A3R.60)

But in a <font> Globe </font> , <paren> which is a bodie rounde as a
bowle </paren> there is but one platte forme , and one bounde ,
(RECORD-E1-P1,1,A3R.61)

and these are the exaumples of them bothe . (RECORD-E1-P1,1,A3R.62)

<font>

A dye or ashler . (RECORD-E1-P1,1,A3R.64)

A globe . (RECORD-E1-P1,1,A3R.65)

</font>

{COM:figures_omitted}

But because you shall not muse what I dooe call <font> a bound ,
</font> I mean thereby a generall name , betokening the beginning , end
and side , of any forme . (RECORD-E1-P1,1,A3R.68)

<font> A forme , figure , or shape , </font> is that thyng that is
inclosed within one bond or many bondes , so that you vnderstand that
shape , that the eye doth discerne , and not the substance of the bodie
. (RECORD-E1-P1,1,A3R.69)

Of <font> figures </font> there be manie sortes ,
(RECORD-E1-P1,1,A3R.70)

for either thei be made of prickes , lines , or platte formes .
(RECORD-E1-P1,1,A3R.71)

Notwithstandyng to speake properlie , <font> a figure </font> is euer
made by platte formes , and not of bare lines vnclosed , neither yet of
prickes . (RECORD-E1-P1,1,A3R.72)

Yet for the lighter forme of teachyng , it shall not be vnsemely to
call all suche shapes , formes and figures , whiche y=e= eye maie
discerne distinctly . (RECORD-E1-P1,1,A3R.73)

And first to begin with prickes , there maie be made diuerse formes of
them , as partely here doeth folowe . (RECORD-E1-P1,1,A3R.74)

<P_1,A3V>

<font>

A lynearie number . (RECORD-E1-P1,1,A3V.77)

Trianguler numbres (RECORD-E1-P1,1,A3V.78)

Long square nu~bre (RECORD-E1-P1,1,A3V.79)

Iust square numbers (RECORD-E1-P1,1,A3V.80)

a threcornered spire . (RECORD-E1-P1,1,A3V.81)

A square spire . (RECORD-E1-P1,1,A3V.82)

</font>

{COM:figures_omitted}

And so maie there be infinite formes more , whiche I omitte for this
time , co~sidering that their knowledg appertaineth more to Arithmetike
figurall , than to Geometrie . (RECORD-E1-P1,1,A3V.85)

But yet one name of the pricke , whiche he taketh rather of his place
then of his fourme , maie I not ouerpasse . (RECORD-E1-P1,1,A3V.86)

And that is , when a pricke standeth in the middell of a circle <paren>
as no circle can be made by co~passe without it </paren> then is it
called <font> a centre </font> . (RECORD-E1-P1,1,A3V.87)

And thereof doe masons , and other worke menne call that patron , a
<font> centre </font> , whereby they drawe the lines , for iust hewyng
of stones for arches , vaultes , and chimneies , because the chefe vse
of that patron is wrought by findyng that pricke or centre , from
whiche all the lynes are drawen , as in the thirde booke it doeth
appere . (RECORD-E1-P1,1,A3V.88)

Lynes make diuerse figures also , though properly thei maie not be
called figures , as I said before <paren> vnles the lines do close
</paren> (RECORD-E1-P1,1,A3V.89)

but onely for easie maner of teachyng , all shall be called figures ,
<P_1,A4R> that the eye can discerne , of whiche this is one , when one
line lyeth flatte <paren> whiche is named the <font> ground line
</font> </paren> and an other commeth downe on it , and is called a
<font> perpendiculer </font> or <font> plu~me lyne </font> , as in this
example you may see , where .A.B. is the grounde line , and C.D. the
plumbe line . (RECORD-E1-P1,1,A4R.90)

{COM:figure_omitted}

And likewaies in this figure there are three lines , the grounde lyne
whiche is A.B. the plumme line that is A.C. and the bias line , whiche
goeth from the one of the~ to the other , and lieth against the right
corner in such a figure whiche is here .C.B. (RECORD-E1-P1,1,A4R.92)

{COM:figure_omitted}

But consideryng that I shall haue occasion to declare sundry figures
anon , I will first shew some certain varietees of lines that close no
figures , but are bare lines , (RECORD-E1-P1,1,A4R.94)

and of the other lines will I make mencion in the description of the
figures . (RECORD-E1-P1,1,A4R.95)

<font> Paralleles , </font> or <font> gemowe lynes </font> be such
lines as be drawen foorth still in one distaunce , and are not nerer in
one place then in an other , (RECORD-E1-P1,1,A4R.96)

for and if they be nerer at one ende then at the other , then are they
no parallels , (RECORD-E1-P1,1,A4R.97)

but maie bee called <font> bought lynes </font> (RECORD-E1-P1,1,A4R.98)

and soe here exaumples of the bothe . (RECORD-E1-P1,1,A4R.99)

<font>

tortuouse paralleles . (RECORD-E1-P1,1,A4R.101)

</font>

{COM:figure_omitted}

<P_1,A4V>

<font>

parallelis . (RECORD-E1-P1,1,A4V.106)

bought lines (RECORD-E1-P1,1,A4V.107)

parallelis circular : (RECORD-E1-P1,1,A4V.108)

Concentrikes . (RECORD-E1-P1,1,A4V.109)

</font>

{COM:figures_omitted}

I haue added also <font> paralleles tortuouse , </font> which bowe
co~trarie waies with their two endes : and <font> paralleles circular ,
</font> which be lyke vnperfecte compasses : (RECORD-E1-P1,1,A4V.112)

for if they bee whole circles , then are they called co~centrikes ,
(RECORD-E1-P1,1,A4V.113)

that is to saie , circles drawe~ on one centre .
(RECORD-E1-P1,1,A4V.114)

Here might I note the error of good <font> Albert Durer , </font> which
affirmeth that no perpendicular lines can be paralleles , which errour
doeth spring partlie of ouersight of the difference of a streight line
, and partlie of mistakyng certain principles geometrical , which al I
wil let passe vntil an other tyme , and wil not blame him , which hath
deserued worthyly infinite praise . (RECORD-E1-P1,1,A4V.115)

And to returne to my matter . (RECORD-E1-P1,1,A4V.116)

an other fashioned line is there , which is named a twine or twist line
, (RECORD-E1-P1,1,A4V.117)

and it goeth as a wreyth about some other bodie .
(RECORD-E1-P1,1,A4V.118)

And an other sorte of lines is there , that is called a <font> spirall
line , </font> or a <font> worm line , </font> whiche representeth an
apparant forme of many circles , where there is not one in dede :
(RECORD-E1-P1,1,A4V.119)

of these .ii. kindes of lines , these be examples .
(RECORD-E1-P1,1,A4V.120)

<font>

A twiste lyne . (RECORD-E1-P1,1,A4V.122)

A spirail lyne (RECORD-E1-P1,1,A4V.123)

</font>

{COM:figures_omitted}

{COM:insert_helsinki_sample_1_here}

<P_1,C4V>

<heading>

THE .VII. CONCLVSION . (RECORD-E1-P1,1,C4V.129)

</heading>

<heading>

TO MAKE A PLUMBE LYNE OR ANY PORCION OF A CIRCLE ,
(RECORD-E1-P1,1,C4V.132)

AND THAT ON THE VTTER OR INNER BUGHTE . (RECORD-E1-P1,1,C4V.133)

</heading>

Mark first the pricke where y=e= plu~mbe line shal lyght :
(RECORD-E1-P1,1,C4V.135)

and prick out on each side of it .ij. other poinctes equally distant
from that first pricke . (RECORD-E1-P1,1,C4V.136)

Then set the one foote of the co~pas in one of those side prickes , and
the other foote in the other side pricke , (RECORD-E1-P1,1,C4V.137)

and first moue one of the feete (RECORD-E1-P1,1,C4V.138)

and drawe an arche line ouer the middell pricke ,
(RECORD-E1-P1,1,C4V.139)

then set the compas steddie with the one foote in the other side pricke
, (RECORD-E1-P1,1,C4V.140)

and with the other foote drawe an other arche line , that shall cut
that first arche , (RECORD-E1-P1,1,C4V.141)

and from the very poincte of the meetyng , drawe a right line vnto the
firste pricke , where you do minde that the plumbe line shall lyghte .
(RECORD-E1-P1,1,C4V.142)

And so haue you performed thintent of this conclusion .
(RECORD-E1-P1,1,C4V.143)

<font>

Example . (RECORD-E1-P1,1,C4V.145)

</font>

{COM:figure_omitted}

The arche of the circle on whiche I would erect a plumbe line , is
A.B.C . (RECORD-E1-P1,1,C4V.148)

and B. is the pricke where I would haue the plumbe line to light .
(RECORD-E1-P1,1,C4V.149)

Therfore I meate out two equall distaunces on eache side of that pricke
B . (RECORD-E1-P1,1,C4V.150)

and they are A. C . (RECORD-E1-P1,1,C4V.151)

Then open I the compas as wide as A.C . (RECORD-E1-P1,1,C4V.152)

and settyng one of the feete in A. with the other I drawe an arch line
which goeth by G . (RECORD-E1-P1,1,C4V.153)

Likewaies I set one foote of the compas steddily in C .
(RECORD-E1-P1,1,C4V.154)

and with the other I draw an arche line , goyng by G. also .
(RECORD-E1-P1,1,C4V.155)

Now consideryng that G. is the pricke of their meetyng , it shall be
also the poinct from whiche I must drawe the plu~be line .
(RECORD-E1-P1,1,C4V.156)

Then draw I a right line from G. to B. (RECORD-E1-P1,1,C4V.157)

and so haue mine intent . (RECORD-E1-P1,1,C4V.158)

Now as A.B.C. hath a plumbe line erected on his <P_1,D1R> vtter bought
, so may I erect a plumbe line on the inner $bught of D.E.F , doynge
with it as I did with the other , (RECORD-E1-P1,1,D1R.159)

that is to saye , fyrste settyng forthe the pricke where the plumbe
line shall light , which is E , and then markyng one other on eache
syde , as are D. and F . And then proceding as I dyd in the example
before . (RECORD-E1-P1,1,D1R.160)

<heading>

THE VIII. CONCLVSION . (RECORD-E1-P1,1,D1R.162)

</heading>

<heading>

HOW TO DEUIDE THE ARCHE OF A CIRCLE INTO TWO =L PARTES , WITHOUT
MEASURING THE ARCHE . (RECORD-E1-P1,1,D1R.165)

</heading>

Deuide the corde of that line into ij. equall portions ,
(RECORD-E1-P1,1,D1R.167)

and then from the middle prycke erecte a plumbe line ,
(RECORD-E1-P1,1,D1R.168)

and it shal parte that arche in the middle . (RECORD-E1-P1,1,D1R.169)

<font>

Example . (RECORD-E1-P1,1,D1R.171)

</font>

{COM:figure_omitted}

The arch to be diuided ys A.D.C , (RECORD-E1-P1,1,D1R.174)

the corde is A,B.C {COM:sic} , (RECORD-E1-P1,1,D1R.175)

this corde is diuided in the middle with B , from which prick if I
erecte a plumb line as B.D , the~ will it diuide the arch in the middle
, that is to say , in D . (RECORD-E1-P1,1,D1R.176)

<heading>

THE IX. CONCLVSION . (RECORD-E1-P1,1,D1R.178)

</heading>

To do the same thynge other wise . (RECORD-E1-P1,1,D1R.180)

And for shortenes of worke , if you wyl make a plumbe line without much
labour , you may do it with your squyre , so that it be iustly made ,
(RECORD-E1-P1,1,D1R.181)

for yf you applye the edge of the squyre to the line in which the
pricke is , and foresee the very corner of the squyre doo touche the
pricke . And than frome that corner if you drawe a lyne by the other
edge of the squyre , yt will be a perpendicular to the former line .
(RECORD-E1-P1,1,D1R.182)

<P_1,D1V>

<font>

Example . (RECORD-E1-P1,1,D1V.185)

</font>

{COM:figure_omitted}

A.B. is the line , on which I would make the plumme line , or
perpendicular . (RECORD-E1-P1,1,D1V.188)

And therefore I marke the prick , from which the plumbe lyne muste rise
, which here is C . (RECORD-E1-P1,1,D1V.189)

Then do I sette one edg of my squyre <paren> that is B.C. </paren> to
the line A.B , so that the corner of the squyre do touche C. iustly .
(RECORD-E1-P1,1,D1V.190)

And from C. I drawe a line by the other edge of the squire , <paren>
which is C.D . </paren> (RECORD-E1-P1,1,D1V.191)

And so haue I made the plumme line D.C , which I sought for .
(RECORD-E1-P1,1,D1V.192)

<heading>

THE X. CONCLVSION . (RECORD-E1-P1,1,D1V.194)

</heading>

<heading>

HOW TO DO THE SAME THINGE AN OTHER WAY YET (RECORD-E1-P1,1,D1V.197)

</heading>

If so be it that you haue an arche of such greatnes , that your squyre
wyll not suffice therto , as the arche of a brydge or of a house or
window , then may you do this . (RECORD-E1-P1,1,D1V.199)

Mete vnderneth the arch where y=e= midle of his cord wyl be ,
(RECORD-E1-P1,1,D1V.200)

and there set a mark (RECORD-E1-P1,1,D1V.201)

Then take a long line with a plummet , (RECORD-E1-P1,1,D1V.202)

and holde the line in such a place of the arch that the plummet do hang
iustely ouer the middle of the corde , that you didde diuide before ,
(RECORD-E1-P1,1,D1V.203)

and then the line doth shewe you the middle of the arche .
(RECORD-E1-P1,1,D1V.204)

<font>

Example . (RECORD-E1-P1,1,D1V.206)

</font>

{COM:figure_omitted}

The arch is A.D.B , of which I trye the midle thus .
(RECORD-E1-P1,1,D1V.209)

I draw a corde from one syde to the other <paren> as here is A.B ,
</paren> which I diuide in the middle in C . (RECORD-E1-P1,1,D1V.210)

The~ take I a line with a plummet <paren> that is D.E , </paren>
(RECORD-E1-P1,1,D1V.211)

and so hold I the line that the plummet B , dooth hange ouer C ,
(RECORD-E1-P1,1,D1V.212)

And <P_1,D2R> then I say that D. is the middle of the arche .
(RECORD-E1-P1,1,D2R.213)

And to thentent that my plummet shall point the more iustely , I doo
make it sharpe at the nether ende , (RECORD-E1-P1,1,D2R.214)

and so may I trust this woorke for certaine . (RECORD-E1-P1,1,D2R.215)

<heading>

THE XI. CONCLVSION . (RECORD-E1-P1,1,D2R.217)

</heading>

<heading>

WHEN ANY LINE IS APPOINTED AND WITHOUT IT A PRICKE , WHEREBY A PARALLEL
MUST BE DRAWEN HOW YOU SHALL DOO IT . (RECORD-E1-P1,1,D2R.220)

</heading>

Take the iuste measure betweene the line and the pricke , accordinge to
which you shal open your compasse . (RECORD-E1-P1,1,D2R.222)

The~ pitch one foote of your compasse at the one ende of the line ,
(RECORD-E1-P1,1,D2R.223)

and with the other foote draw a bowe line right ouer the pytche of the
compasse , (RECORD-E1-P1,1,D2R.224)

lykewise doo at the other ende of the lyne , (RECORD-E1-P1,1,D2R.225)

then draw a line that shall touche the vttermoste edge of bothe those
bowe lines , (RECORD-E1-P1,1,D2R.226)

and it will bee a true parallele to the fyrste lyne appointed .
(RECORD-E1-P1,1,D2R.227)

<font>

Example . (RECORD-E1-P1,1,D2R.229)

</font>

{COM:figure_omitted}

A.B , is the line vnto which I must draw an other gemowe line , which
must passe by the prick C , (RECORD-E1-P1,1,D2R.232)

first I meate with my compasse the smallest distance that is from C. to
the line , (RECORD-E1-P1,1,D2R.233)

and that is C.F , wherfore staying the compasse at that distaunce , I
sette the one foote in A , (RECORD-E1-P1,1,D2R.234)

and with the other foot I make a bowe lyne , which is D ,
(RECORD-E1-P1,1,D2R.235)

the~ like wise set I the one foote of the compasse in B ,
(RECORD-E1-P1,1,D2R.236)

and with the other I make the second bow line , which is E .
(RECORD-E1-P1,1,D2R.237)

And then draw I a line , so that it toucheth the vttermost edge of
bothe these bowe lines , (RECORD-E1-P1,1,D2R.238)

and that lyne passeth by the pricke C , (RECORD-E1-P1,1,D2R.239)

and is a gemowe line to A.B , as my sekyng was .
(RECORD-E1-P1,1,D2R.240)

<P_1,D2V>

<heading>

THE .XII. CONCLVSION . (RECORD-E1-P1,1,D2V.243)

</heading>

<heading>

TO MAKE A TRIANGLE OF ANY .III. LINES , SO THAT THE LINES BE SUCHE ,
THAT ANY .IJ. OF THEM BE LONGER THEN THE THIRDE .
(RECORD-E1-P1,1,D2V.246)

FOR THIS RULE IS GENERALL , THAT ANY TWO SIDES OF EUERIE TRIANGLE TAKEN
TOGETHER , ARE LONGER THEN THE OTHER SIDE THAT REMAINETH .
(RECORD-E1-P1,1,D2V.247)

</heading>

If you do remember the first and seconde conclusions , then is there no
difficultie in this , (RECORD-E1-P1,1,D2V.249)

for it in maner the same woorke . (RECORD-E1-P1,1,D2V.250)

First co~sider the .iij. lines that you must take ,
(RECORD-E1-P1,1,D2V.251)

and set one of the~ for the ground line , (RECORD-E1-P1,1,D2V.252)

then worke with the other .ij. lines as you did in the first and second
conclusions . (RECORD-E1-P1,1,D2V.253)

<font>

Example . (RECORD-E1-P1,1,D2V.255)

</font>

{COM:figure_omitted}

I haue .iij. lynes .A.B. and C.D. and E.F. of which I put .C.D. for my
ground line , (RECORD-E1-P1,1,D2V.258)

then with my compas I take the length of .A.B. ,
(RECORD-E1-P1,1,D2V.259)

and set the one foote of my compas in C , (RECORD-E1-P1,1,D2V.260)

and draw an archline with the other foote . (RECORD-E1-P1,1,D2V.261)

Likewaies I take the le~gth of E.F , (RECORD-E1-P1,1,D2V.262)

and set one foote in D , (RECORD-E1-P1,1,D2V.263)

and with the other foote I make an arch line crosse the other arche ,
(RECORD-E1-P1,1,D2V.264)

and the pricke of their metyng <paren> whiche is G. </paren> shall be
the thirde corner of the triangle , (RECORD-E1-P1,1,D2V.265)

for in all such kyndes of woorkynge to make a tryangle , if you haue
one line drawen , there remayneth nothyng els but to fynde where the
pitche of the thirde corner shal bee , (RECORD-E1-P1,1,D2V.266)

for two of them must needes be at the two eandes of the lyne that is
drawen . (RECORD-E1-P1,1,D2V.267)

<P_1,D3R>

<heading>

THE XIII. CONCLVSION . (RECORD-E1-P1,1,D3R.270)

</heading>

<heading>

IF YOU HAUE A LINE APPOINTED , AND A POINTE IN IT LIMITED , HOWE YOU
MAYE MAKE ON IT A RIGHTE LINED ANGLE , =LY TO AN OTHER RIGHT LINED
ANGLE , ALLREADY ASSIGNED . (RECORD-E1-P1,1,D3R.273)

</heading>

Fyrste draw a line against the corner assigned ,
(RECORD-E1-P1,1,D3R.275)

and so is it a triangle , (RECORD-E1-P1,1,D3R.276)

then take heede to the line and the pointe in it assigned ,
(RECORD-E1-P1,1,D3R.277)

and consider if that line from the pricke to this end bee as long as
any of the sides that make the triangle assigned ,
(RECORD-E1-P1,1,D3R.278)

and if it bee longe inoughe , then prick out there the length of one of
the lines , (RECORD-E1-P1,1,D3R.279)

and then woorke with the other two lines , accordinge to the laste
conclusion , makynge a triangle of thre like lynes to that assigned
triangle . (RECORD-E1-P1,1,D3R.280)

If it bee not longe inoughe , thenne lengthen it fyrste ,
(RECORD-E1-P1,1,D3R.281)

and afterwarde doo as I haue sayde before . (RECORD-E1-P1,1,D3R.282)

<font>

Example . (RECORD-E1-P1,1,D3R.284)

</font>

{COM:figure_omitted}

Lette the angle appoynted bee A.B.C , and the corner assigned , B .
(RECORD-E1-P1,1,D3R.287)

Farthermore let the lymited line bee D.G , and the pricke assigned D .
(RECORD-E1-P1,1,D3R.288)

Fyrste therefore by drawinge the line A.C , I make the triangle A.B.C .
(RECORD-E1-P1,1,D3R.289)

Then consideringe that D.G , is longer thanne A.B , you shall cut out a
line fro~ D toward G , equal to A.B , as for exa~ple D,F .
(RECORD-E1-P1,1,D3R.290)

The~ measure oute the other ij. lines (RECORD-E1-P1,1,D3R.291)

and worke with the~ according as the conclusion with the fyrste also
and the second teacheth yow , (RECORD-E1-P1,1,D3R.292)

and then haue you done . (RECORD-E1-P1,1,D3R.293)

<P_1,D3V>

<heading>

THE XIIII. CONCLVSION . (RECORD-E1-P1,1,D3V.296)

</heading>

<heading>

TO MAKE A SQUARE QUADRATE OF ANY RIGHTE LYNE APPOINCTED .
(RECORD-E1-P1,1,D3V.299)

</heading>

First make a plumbe line vnto your line apointed , whiche shall light
at one of the endes of it , accordyng to the fifth conclusion ,
(RECORD-E1-P1,1,D3V.301)

and let it be of like length as your first line is ,
(RECORD-E1-P1,1,D3V.302)

then ope~ your compasse to the iuste length of one of them ,
(RECORD-E1-P1,1,D3V.303)

and sette one foote of the compasse in the ende of the one line ,
(RECORD-E1-P1,1,D3V.304)

and with the other foote draw an arche line , there as you thinke that
the fowerth corner shall be , (RECORD-E1-P1,1,D3V.305)

after that set the one foote of the same compasse vnsturred , in the
eande of the other line , (RECORD-E1-P1,1,D3V.306)

and draw an other arche line crosse the first arche line ,
(RECORD-E1-P1,1,D3V.307)

and the poincte that they do crosse in , is the pricke of the fourth
corner of the square quadrate which you seke for ,
(RECORD-E1-P1,1,D3V.308)

therfore draw a line from that pricke to the eande of eche line ,
(RECORD-E1-P1,1,D3V.309)

and you shall therby haue made a square quadrate .
(RECORD-E1-P1,1,D3V.310)

<font>

Example . (RECORD-E1-P1,1,D3V.312)

</font>

{COM:figure_omitted}

A.B. is the line proposed , of whiche I shall make a square quadrate ,
(RECORD-E1-P1,1,D3V.315)

therefore firste I make a plu~be line vnto it , which shall lighte in A
, and the plu~b line in A.C , (RECORD-E1-P1,1,D3V.316)

then open I my compasse as wide as the length of A.B , or A.C ,
(RECORD-E1-P1,1,D3V.317)

<paren> for they must be bothe equall </paren> (RECORD-E1-P1,1,D3V.318)

and I set the one foote of thend in C , (RECORD-E1-P1,1,D3V.319)

and with the other I make an arche line nigh vnto D ,
(RECORD-E1-P1,1,D3V.320)

afterward I set the compas again with one foote in B ,
(RECORD-E1-P1,1,D3V.321)

and with the other foote I make an arche line crosse the first arche
line in D , (RECORD-E1-P1,1,D3V.322)

and from the prick of their crossyng I draw .ij. lines , one to B , and
an other to C , (RECORD-E1-P1,1,D3V.323)

and so haue I made the square quadrate that I entended .
(RECORD-E1-P1,1,D3V.324)

<P_1,D4R>

<heading>

THE .XV. CONCLVSION . (RECORD-E1-P1,1,D4R.327)

</heading>

<heading>

TO MAKE A LIKEIA~ME =L TO A TRIANGLE APPOINTED ,
(RECORD-E1-P1,1,D4R.330)

AND THAT IN A RIGHT LINED A~GLE LIMITED . (RECORD-E1-P1,1,D4R.331)

</heading>

First from one of the angles of the triangle , you shall draw a gemowe
line , whiche shall be a parallel to that syde of the triangle , on
whiche you will make that likeiamme . (RECORD-E1-P1,1,D4R.333)

Then on one end of the side of the triangle , whiche lieth against the
gemowe lyne , you shall draw forth a line vnto the gemow line , so that
one angle that commeth of those .ij. lines be like to the angle whiche
is limited vnto you . (RECORD-E1-P1,1,D4R.334)

Then shall you deuide into ij. equall partes that side of the triangle
whiche beareth that line , (RECORD-E1-P1,1,D4R.335)

and from the pricke of that deuision , you shall raise an other line
parallele to that former line , and continewe it vnto the first gemowe
line , (RECORD-E1-P1,1,D4R.336)

and the~ of those .ij. last gemowe lyndes , and the first gemowe line ,
with the halfe side of the triangle , is made a lykeiamme equally to
the triangle appointed , (RECORD-E1-P1,1,D4R.337)

and hath an angle lyke to an angle limited , accordyng vnto the
conclusion . (RECORD-E1-P1,1,D4R.338)

<font>

Example . (RECORD-E1-P1,1,D4R.340)

</font>

{COM:figure_omitted}

B.C.G , is the triangle appoincted vnto , {COM:sic} which I muste make
an equall likeiamme . (RECORD-E1-P1,1,D4R.343)

And D , is the angle that the likeiamme must haue .
(RECORD-E1-P1,1,D4R.344)

Therfore first entendyng to erecte the likeia~me on the one side , that
the ground line of the triangle <paren> whiche is B.G. </paren> I do
draw a gemow line by C , and make it parallele to the ground line B.G ,
(RECORD-E1-P1,1,D4R.345)

and that new gemow line is A.H . (RECORD-E1-P1,1,D4R.346)

Then do I raise a line from B. vnto the gemowe line , <paren> which
line is A.B </paren> and make an angle equally to D ,
(RECORD-E1-P1,1,D4R.347)

that is the appointed angle <paren> {COM:no_matching_close_paren}
accordyng as the .viij. co~clusion teacheth , (RECORD-E1-P1,1,D4R.348)

and that angle is B.A.E . (RECORD-E1-P1,1,D4R.349)

Then to procede , I doo parte in y=e= middle the said grou~d line B.G ,
in the prick F , fro~ which prick I draw <P_1,D4V> to the first gemowe
line <paren> A.H. </paren> an other line that is parallel to A.B ,
(RECORD-E1-P1,1,D4V.350)

and that line is E.F . (RECORD-E1-P1,1,D4V.351)

Now saie I that the likeia~me B.A.E.F , is equall to the triangle
B.C.G. And also that it hath one angle <paren>
{COM:no_matching_close_paren} that is B.A.E. like to D. the angle that
was limitted . (RECORD-E1-P1,1,D4V.352)

And so haue I mine intent . (RECORD-E1-P1,1,D4V.353)

The profe of the equalnes of those two figures doeth depend of the .xli
proposition of Euclides first boke , (RECORD-E1-P1,1,D4V.354)

and is the .xxxi. proposition of this second boke of Theoremis , whiche
saieth , that whan a tryangle and a likeiamme be made between .ij.
selfe same gemow lines , and haue their ground line of one length ,
then is the likeiamme double to the triangle , wherof it foloweth ,
that if .ij. suche figures so drawen differ in their groundline onely ,
so that the ground line of the likeiamme be but halfe the ground line
of the triangle , then be those .ij. figures equall , as you shall more
at large perceiue by the boke of Theoremis , in y=e= .xxxi. theoreme .
(RECORD-E1-P1,1,D4V.355)

<heading>

THE .XVI. CONCLVSION . (RECORD-E1-P1,1,D4V.357)

</heading>

<heading>

TO MAKE A LIKEIAMME =L TO A TRIANGLE APPOINCTED , ACCORDYNG TO AN ANGLE
LIMITTED , AND ON A LINE ALSO ASSIGNED . (RECORD-E1-P1,1,D4V.360)

</heading>

In the last conclusion the sides of your likeiamme wer left to your
libertie , though you had an angle appoincted .
(RECORD-E1-P1,1,D4V.362)

Nowe in this conclusion you are somwhat more restrained of libertie
sith the line is limitted , which must be the side of the likeia~me .
(RECORD-E1-P1,1,D4V.363)

Therfore thus shall you procede . (RECORD-E1-P1,1,D4V.364)

Firste accordyng to the laste conclusion , make a likeiamme in the
angle appoincted , equall to the triangle that is assigned .
(RECORD-E1-P1,1,D4V.365)

Then with your compasse take the length of your line appointed ,
(RECORD-E1-P1,1,D4V.366)

and set out two lines of the same length in the second gemowe lines ,
beginnyng at the one side of the likeiamme , (RECORD-E1-P1,1,D4V.367)

and by those two prickes shall you draw an other gemowe line , which
shall be parallele to two sides of the likeiamme .
(RECORD-E1-P1,1,D4V.368)

Afterward shall you draw .ij. lines more for the accomplishement of
your worke , whiche better shall be <P_1,E1R> perceaued by a shorte
exaumple , then by a greate numbre of wordes only without example ,
(RECORD-E1-P1,1,E1R.369)

therefore I wyl by example sette forth the whole worke .
(RECORD-E1-P1,1,E1R.370)

<font>

Example . (RECORD-E1-P1,1,E1R.372)

</font>

{COM:figure_omitted}

Fyrst , according to the last conclusion , I make the likeiamme E.F.C.G
, equal to the triangle D , in the appoynted angle whiche is E .
(RECORD-E1-P1,1,E1R.375)

Then take I the lengthe of the assigned line <paren> which is A.B ,
</paren> (RECORD-E1-P1,1,E1R.376)

and with my compas I sette forthe the same le~gth in the ij. gemow
lines N.F. and H.G , setting one foot in E , and the other in N , and
againe settyng one foote in C. , and the other in H .
(RECORD-E1-P1,1,E1R.377)

Afterward I draw a line from N. to H , whiche is a gemow lyne , to ij.
sydes of the likeiamme . (RECORD-E1-P1,1,E1R.378)

thenne drawe I a line also from N. vnto C , (RECORD-E1-P1,1,E1R.379)

and extend it vntyll it crosse the lines , E.L. and F.G , which both
must be drawen forth longer then the sides of the likeiame .
(RECORD-E1-P1,1,E1R.380)

and where that lyne doeth crosse F.G , there I sette M .
(RECORD-E1-P1,1,E1R.381)

Nowe to make an ende , I make an other gemowe line , which is parallel
to N.F and H.G , (RECORD-E1-P1,1,E1R.382)

and that gemowe line doth passe by the pricke M ,
(RECORD-E1-P1,1,E1R.383)

and then haue I done . (RECORD-E1-P1,1,E1R.384)

Now say I that H.C.K.L , is a likeiamme equall to the triangle
appointed , which was D , and is made of a line assigned that is A.B ,
(RECORD-E1-P1,1,E1R.385)

for H.C , is equall vnto A.B , (RECORD-E1-P1,1,E1R.386)

and so is K.L , (RECORD-E1-P1,1,E1R.387)

The profe of y=e= equalnes of this likeiam vnto the tria~gle ,
depe~deth of the thirty and two Theoreme : as in the boke of Theoremes
doth appear , where it is declared , that in al likeiammes , whe~ there
are more then one made about one bias line , the filsquares of euery of
them muste needes be equall . (RECORD-E1-P1,1,E1R.388)

<P_1,E1V>

<heading>

THE XVII. CONCLVSION . (RECORD-E1-P1,1,E1V.391)

</heading>

<heading>

TO MAKE A LIKEIAMME = TO ANY RIGHT LINED FIGURE ,
(RECORD-E1-P1,1,E1V.394)

AND THAT ON AN ANGLE APPOINTED . (RECORD-E1-P1,1,E1V.395)

</heading>

The readiest way to worke this conclusion , is to tourn that rightlined
figure into triangles , and then for euery triangle to gether an equal
likeiamme , according vnto the eleuen co~clusion , and then to ioine al
those likeiammes into one , if their sides happen to be equal , which
thing is euer certain , when al the triangles happe~ iustly betwene one
pair of gemow lines . (RECORD-E1-P1,1,E1V.397)

but and if they will not frame so , then after that you haue for the
first triangle made his likeiamme , you shall take the le~gth of one of
his sides , and set that as a line assigned , on whiche you shal make
al the other likeiams , according to the twelft co~clusion ,
(RECORD-E1-P1,1,E1V.398)

and so shall you haue al your likeiames with ij. sides equal , and ij.
like angles , so y=t= you mai easily ioyne the~ into one figure .
(RECORD-E1-P1,1,E1V.399)

<font>

Example . (RECORD-E1-P1,1,E1V.401)

</font>

{COM:figures_omitted}

If the right lined figure be like vnto A , the~ may it be turned into
triangles that will sta~d betwene ij. parallels anye ways , as you mai
se by C and D , (RECORD-E1-P1,1,E1V.404)

for ij. sides of both the tria~gles are parallels .
(RECORD-E1-P1,1,E1V.405)

Also if the right lined figure be like vnto E , the~ wil it be turned
into tria~gles , liyng betwene two parallels also , as y=e= other did
before , as in the exa~ple of F.G . (RECORD-E1-P1,1,E1V.406)

But and if y=e= <P_1,E2R> right lined figure be like vnto H , and so
turned into tria~gles as you se in K.L.M , wher it is parted into iij
tria~gles , the~ wil not all those triangles lye between one pair of
parallels or gemow lines , (RECORD-E1-P1,1,E2R.407)

but must haue many , (RECORD-E1-P1,1,E2R.408)

for euery triangle must haue one paire of parallels seuerall ,
(RECORD-E1-P1,1,E2R.409)

yet it maye hapen that when there bee three or fower triangles , ij. of
theym maye happen to agre to one pair of parallels , whiche thinge I
remit to euery honest witte to serche , (RECORD-E1-P1,1,E2R.410)

for the manner of their draught wil declare , how many paires of
parallels they shall neede , of which varietee bicause the examples are
infinite , I haue set forth these few , that by them you may coniecture
duly of all other like . (RECORD-E1-P1,1,E2R.411)

{COM:figures_omitted}

Further explicacion you shal not greatly neede , if you remembre what
hath ben taught before , and then dilige~tly behold how these sundry
figures be turned into tria~gles . (RECORD-E1-P1,1,E2R.413)

In the fyrst you se I haue made v. triangles , and four paralleles . in
the seconde vij. triangles and foure paralleles , in the thirde thre
tria~gles , and fiue parallels , (RECORD-E1-P1,1,E2R.414)

in the iiij. you se fiue tria~gles & four parallels . in the fifth ,
iiij. tria~gles and .iiij. parallels , (RECORD-E1-P1,1,E2R.415)

& in y=e= sixt there are fiue tria~gles & iiij. paralels .
(RECORD-E1-P1,1,E2R.416)

Howbeit a ma~ maye at liberty alter them into diuers formes of
tria~gles , (RECORD-E1-P1,1,E2R.417)

& therefore I <P_1,E2V> leue it to the discretion of the woorkmaister ,
to do in al suche cases as he shall thinke best ,
(RECORD-E1-P1,1,E2V.418)

for by these examples <paren> if they bee well marked </paren> may al
other like conclusions be wrought . (RECORD-E1-P1,1,E2V.419)

<heading>

THE XVIII. CONCLVSION . (RECORD-E1-P1,1,E2V.421)

</heading>

<heading>

TO PARTE A LINE ASSIGNED AFTER SUCHE A SORTE , THAT THE SQUARE THAT IS
MADE OF THE WHOLE LINE AND ONE OF HIS PARTS , SHAL BE = TO THE SQUAR
{COM:sic} THAT COMETH OF THE OTHER PARTE ALONE .
(RECORD-E1-P1,1,E2V.424)

</heading>

First deuide your lyne into ij. equal parts , (RECORD-E1-P1,1,E2V.426)

and of the length of one part make a perpendicular to light at one end
of your line assigned . (RECORD-E1-P1,1,E2V.427)

then adde a bias line , (RECORD-E1-P1,1,E2V.428)

and make therof a triangle , (RECORD-E1-P1,1,E2V.429)

this done if you take from this bias line the halfe lengthe of your
line appointed , which is the iuste length of your perpendicular , that
part of the bias line whiche dothe remayne , is the greater portion of
the deuision that you seke for , (RECORD-E1-P1,1,E2V.430)

therefore if you cut your line according to the lengthe of it , then
will the square of that greater $portion {TEXT:portior} be equall to
the square that is made of the whole line and his lesser portion .
(RECORD-E1-P1,1,E2V.431)

And contrarywise , the square of the whole line and his lesser parte ,
wyll be equall to the square of the greater parte .
(RECORD-E1-P1,1,E2V.432)

<font>

Example . (RECORD-E1-P1,1,E2V.434)

</font>

{COM:figure_omitted}

A.B , is the lyne assigned . (RECORD-E1-P1,1,E2V.437)

E. is the middle pricke of A.B , (RECORD-E1-P1,1,E2V.438)

B.C. is the plumb line or perpendicular , made of the halfe of A.B ,
equall to A.E , other B.E , (RECORD-E1-P1,1,E2V.439)

the byas line is C.A , from whiche I cut a peece ,
(RECORD-E1-P1,1,E2V.440)

that is C.D , equall to C.B , (RECORD-E1-P1,1,E2V.441)

and accordyng to the lengthe $of {TEXT:so} the peece that remaineth
<paren> whiche is D.A , </paren> I doo deuide the line A.B , at whiche
diuision I set E . (RECORD-E1-P1,1,E2V.442)

Now say I , that this line A.B <paren> w=ch= was assigned vnto me
</paren> is so diuided in this point F , y=t= y=e= square of y=e= hole
line A.B , & of the one portio~ <paren> y=t= is F.B , the <P_1,E3R>
lesser part </paren> is equall to the square of the other parte ,
whiche is F.A , and is the greater part of the first line .
(RECORD-E1-P1,1,E3R.443)

The profe of this equalitie shall you learne by the .xl. Theoreme .
(RECORD-E1-P1,1,E3R.444)

<heading>

THE .XIX. CONCLVSION . (RECORD-E1-P1,1,E3R.446)

</heading>

<heading>

TO MAKE A SQAURE QUADRATE =L TO ANY RIGHT LINED FIGURE APPOINCTED .
(RECORD-E1-P1,1,E3R.449)

</heading>

First make a likeiamme equall to that right lined figure , with a right
angle , accordyng to the .xi. conclusion , (RECORD-E1-P1,1,E3R.451)

then consider the liekiamme , whether it haue all his sides equall , or
not : (RECORD-E1-P1,1,E3R.452)

for yf they be all equall , then haue you doone your conclusion .
(RECORD-E1-P1,1,E3R.453)

but and if the sides be not all equall , then shall you make one right
line iuste as long as two of those vnequall sides ,
(RECORD-E1-P1,1,E3R.454)

that line shall you deuide in the middle , (RECORD-E1-P1,1,E3R.455)

and on that pricke drawe half a circle , (RECORD-E1-P1,1,E3R.456)

then cutte from that diameter of the halfe circle a certayne portion
equall to the one side of the likeiamme , (RECORD-E1-P1,1,E3R.457)

and from that pointe of diuision shall you erecte a perpendicular ,
which shall touche the edge of the circle . (RECORD-E1-P1,1,E3R.458)

and that perpendicular shall be the iuste side of the square quadrate ,
equall both to the lykeiamme , and also to the right lined figure
appointed , as the conclusion willed . (RECORD-E1-P1,1,E3R.459)

<font>

Example . (RECORD-E1-P1,1,E3R.461)

</font>

{COM:figures_omitted}

K , is the right lined figure appointed , (RECORD-E1-P1,1,E3R.464)

and B.C.D.E , is the likeia~me , with right angles equall vnto K ,
(RECORD-E1-P1,1,E3R.465)

but because that this likeiamme is not a square quadrate , I must turne
it into such one after this sort , (RECORD-E1-P1,1,E3R.466)

I shall make one right line , as long as .ij. vnequall sides of the
likeia~me , (RECORD-E1-P1,1,E3R.467)

that line here is F.G , which is equall to B.C , and C.E .
(RECORD-E1-P1,1,E3R.468)

Then part I that line in the middle in the <P_1,E3V> pricke M ,
(RECORD-E1-P1,1,E3V.469)

and on that pricke I make halfe a circle , accordyng to the length of
the diameter F.G . (RECORD-E1-P1,1,E3V.470)

Afterward I cut awaie a peece from F.G , equally to C.E , markyng that
point with H . (RECORD-E1-P1,1,E3V.471)

And on that pricke I erecte a perpendicular H.K , whiche is the iust
side to the square quadrate that I seke for , (RECORD-E1-P1,1,E3V.472)

therfore accordyng to the doctrine of the .x. conclusion , of that lyne
I doe make a square quadrate , (RECORD-E1-P1,1,E3V.473)

and so haue I attained the practise of this conclusion .
(RECORD-E1-P1,1,E3V.474)

<heading>

THE .XX. CONCLVSION . (RECORD-E1-P1,1,E3V.476)

</heading>

<heading>

WHEN ANY .IJ. SQUARE QUADRATES ARE SET FORTH , HOW YOU MAIE MAKE ONE =L
TO THEM BOTHE . (RECORD-E1-P1,1,E3V.479)

</heading>

First draw a right line equall to the side of one of the quadrates :
(RECORD-E1-P1,1,E3V.481)

and on the ende of it make a perpendicular , equall in length to the
side of the other quadrate , (RECORD-E1-P1,1,E3V.482)

then draw a byas line betwene those .ij. other lines , makyng thereof a
right angeled triangle . (RECORD-E1-P1,1,E3V.483)

And that byas lyne wyll make a square quadrate , equall to the other
.ij. quadrates appointed . (RECORD-E1-P1,1,E3V.484)

<font>

Example . (RECORD-E1-P1,1,E3V.486)

</font>

{COM:figures_omitted}

A.B. and C.D , are the two square quadrates appointed , vnto which I
must make one equal square quadrate . (RECORD-E1-P1,1,E3V.489)

First therfore I dooe make a righte line E.F , equall to one of the
sides of the square quadrate A.B . (RECORD-E1-P1,1,E3V.490)

And on the one end of it I make a plumbe line E.G , equall to the side
of the other quadrate D.C . (RECORD-E1-P1,1,E3V.491)

Then drawe I a byas line G.F , whiche beyng made the side of a quadrate
<P_1,E4R> <paren> accordyng to the tenth conclusion </paren> will
accomplisshe the worke of this practise : (RECORD-E1-P1,1,E4R.492)

for the quadrate H. is as muche iust as the other two . I meane A.B.
and D.C . (RECORD-E1-P1,1,E4R.493)

<heading>

THE XXI. CONCLVSION . (RECORD-E1-P1,1,E4R.495)

</heading>

<heading>

WHEN ANY TWO QUADRATES BE SET FORTH , HOWE TO MAKE A SQUIRE ABOUT THE
ONE QUADRATE , WHICHE SHALL BE =L TO THE OTHER QUADRATE .
(RECORD-E1-P1,1,E4R.498)

</heading>

Determine with your selfe about which quadrate you wil make the squire
, (RECORD-E1-P1,1,E4R.500)

and drawe one side of that quadrate forth in lengte {COM:sic} ,
accordyng to the measure of the side of the other quadrate , which line
you maie call the grounde line , (RECORD-E1-P1,1,E4R.501)

and then haue you a right angle made on this line by an other side of
the same quadrate : (RECORD-E1-P1,1,E4R.502)

Therfore turne that into a right cornered triangle , accordyng to the
worke in the laste conclusion , by makyng of a byas line ,
(RECORD-E1-P1,1,E4R.503)

and that byas lyne will performe the worke of your desire .
(RECORD-E1-P1,1,E4R.504)

For if you take the length of that byas line with your compasse , and
then set one foote of the compas in the furthest angle of the first
quadrate <paren> whiche is the one ende of the groundline </paren> and
extend the other foote on the same line , accordyng to the measure of
the byas line , and of that line make a quadrate , enclosyng y=e= first
quadrate , then will there appere the forme of a squire about the first
quadrate , which square is equall to the second quadrate .
(RECORD-E1-P1,1,E4R.505)

<font>

Example . (RECORD-E1-P1,1,E4R.507)

</font>

{COM:figures_omitted}

The first square quadrate is A.B.C.D , (RECORD-E1-P1,1,E4R.510)

and the seconde is E . (RECORD-E1-P1,1,E4R.511)

Now would I make a squire about the quadrate A.B.C.D , whiche shall bee
equall vnto the quadrate E . (RECORD-E1-P1,1,E4R.512)

<P_1,E4V>

Therfore first I draw the line A.D , more in length , accordyng to the
measure of the side of E , as you see , from D. vnto F ,
(RECORD-E1-P1,1,E4V.514)

and so the hole line of bothe these seuerall sides is A.F ,
(RECORD-E1-P1,1,E4V.515)

the~ make I a byas line from C , to F , whiche byas line is the measure
of this woorke , wherefore I open my compas accordyng to the length of
that byas line C.F , (RECORD-E1-P1,1,E4V.516)

and set the one compas foote in A , (RECORD-E1-P1,1,E4V.517)

and extend thother foote of the compas toward F , makyng this pricke G
, from whiche I erect a plumbe line G.H , (RECORD-E1-P1,1,E4V.518)

and so make out the square quadrate A.G.H.K , whose sides are equall
eache of them to A.G . (RECORD-E1-P1,1,E4V.519)

And this square doth contain the first quadrate A.B.C.D , and also a
squire G.H.K , whiche is equall to the second quadrate E ,
(RECORD-E1-P1,1,E4V.520)

for as the last conclusion declareth , the quadrate A.G.H.K , is equall
to bothe the other quadrates proposed , (RECORD-E1-P1,1,E4V.521)

that is A.B.C.D , and E . (RECORD-E1-P1,1,E4V.522)

Then muste the squire G.H.K. needes be equall to E , consideryng that
all the rest of that great quadrate is nothyng els but the quadrate
self , A.B.C.D , (RECORD-E1-P1,1,E4V.523)

and so haue I thintent of this conclusion . (RECORD-E1-P1,1,E4V.524)

<heading>

THE .XXI. CONCLVSION . (RECORD-E1-P1,1,E4V.526)

</heading>

<heading>

TO FIND OUT THE CE~TRE OF ANY CIRCLE ASSIGNED .
(RECORD-E1-P1,1,E4V.529)

</heading>

Draw a corde or stryng line crosse the circle ,
(RECORD-E1-P1,1,E4V.531)

then deuide into .ij. equall partes , both the corde , and also the
bowe line , or arche line , that serueth to the corde ,
(RECORD-E1-P1,1,E4V.532)

and from the prickes of those diuisions , if you draw an other line
crosse the circle , it must nedes passe by the centre .
(RECORD-E1-P1,1,E4V.533)

Therfore deiude that line in the middle , (RECORD-E1-P1,1,E4V.534)

and that middle pricke is the centre of the circle proposed .
(RECORD-E1-P1,1,E4V.535)

<font>

Example . (RECORD-E1-P1,1,E4V.537)

</font>

{COM:figure_omitted}

Let the circle be A.B.C.D , whose centre I shall seke .
(RECORD-E1-P1,1,E4V.540)

First therfore I draw a corde crosse the circle ,
(RECORD-E1-P1,1,E4V.541)

that is A.C . (RECORD-E1-P1,1,E4V.542)

Then do I deuide that corde in the middle , in E ,
(RECORD-E1-P1,1,E4V.543)

and likewaies also do I deuide his arche line A.B.C , in the middle ,
in the pointe B . (RECORD-E1-P1,1,E4V.544)

Afterward I drawe a line from B. to E , and so crosse the <P_1,F1R>
circle , which line is B.D , in which line is the centre that I seeke
for . (RECORD-E1-P1,1,F1R.545)

Therefore if I parte that line B.D , in the middle in to two equall
portions , that middle pricke <paren> whiche here is F </paren> is the
verye centre of the sayde circle that I seke . (RECORD-E1-P1,1,F1R.546)

This conclusion may other waies be wrought , as the most part of
conclusions haue sondry formes of practise , (RECORD-E1-P1,1,F1R.547)

and that is , by makinge thre prickes in the circu~ference of the
circle , at liberty where you wyll , and then finding the centre to
these thre prickes , which worke bicause it serueth for sondry vses , I
thinke meet to make it a seuerall conclusion by it selfe .
(RECORD-E1-P1,1,F1R.548)

<heading>

THE XXIII. CONCLVSION . (RECORD-E1-P1,1,F1R.550)

</heading>

<heading>

TO FIND THE COMMEN CENTRE BELONGYNG TO ANYE THREE PRICKES APPOINTED ,
IF THEY BE NOT IN AN EXACTE RIGHT LINE . (RECORD-E1-P1,1,F1R.553)

</heading>

It is to be noted , that though euery small arche of a greate circle do
seeme to be a right lyne , yet in very dede it is not so ,
(RECORD-E1-P1,1,F1R.555)

for euery part of the circumference of al circles is compassed , though
in litle arches of great circles the eye $can $not {TEXT:cannot}
discerne the crokendess , (RECORD-E1-P1,1,F1R.556)

yet reason doeth alwaies declare it , (RECORD-E1-P1,1,F1R.557)

therefore iij. prickes in an exact right line can not bee brought into
the circumference of a circle . (RECORD-E1-P1,1,F1R.558)

But and if they be not in a right line how so euer they stande , thus
shall you find their com~on centre . (RECORD-E1-P1,1,F1R.559)

Ope~ your compas so wide , that it be somewhat more then the <P_1,F1V>
halfe distance of two of those prickes . (RECORD-E1-P1,1,F1V.560)

Then sette the one foote of the compas in the one pricke ,
(RECORD-E1-P1,1,F1V.561)

and with the other foot draw an arche lyne toward the other pricke ,
(RECORD-E1-P1,1,F1V.562)

then againe putte the foot of your compas in the second pricke ,
(RECORD-E1-P1,1,F1V.563)

and with the other foot make an arche line , that may crosse the firste
arch line in ij. places . (RECORD-E1-P1,1,F1V.564)

Now as you haue done with those two prickes , so do with the middle
pricke , and the thirde that remayneth . (RECORD-E1-P1,1,F1V.565)

Then draw ij. lines by the poyntes where those arche lines do crosse ,
(RECORD-E1-P1,1,F1V.566)

and where those two lines do meete , there is the centre that you seeke
for . (RECORD-E1-P1,1,F1V.567)

<font>

Example . (RECORD-E1-P1,1,F1V.569)

</font>

{COM:figure_omitted}

The iij. prickes I haue set to be A. B , and C , whiche I wold bring
into the edg of one comon circle , by finding a centre co~men to them
all , (RECORD-E1-P1,1,F1V.572)

fyrst therefore I open my co~pas , so that they occupye more then y=e=
halfe distance betwene ij. pricks <paren> as are A.B. </paren>
(RECORD-E1-P1,1,F1V.573)

and so settinge one foote in A. and extendinge the other toward B , I
make the arche line D.E . (RECORD-E1-P1,1,F1V.574)

Like wise setti~g one foot in B , and turninge the other toward A , I
draw an other arche line that crosseth the first in D. and E .
(RECORD-E1-P1,1,F1V.575)

Then from D. to E , I draw a right lyne D.H . (RECORD-E1-P1,1,F1V.576)

After this I open my co~passe to a new distance ,
(RECORD-E1-P1,1,F1V.577)

and make ij. arche lines betwene B. and C , which crosse one the other
in F. and G , by whiche two pointes I draw an other line , that is F.H
. (RECORD-E1-P1,1,F1V.578)

And bycause that the lyne D.H. and the lyne F.H. doo meete in H , I
saye that H. is the centre that serueth those iij. prickes .
(RECORD-E1-P1,1,F1V.579)

Now therfore if you set one foot of your compas in H , and extend the
other to any of the iij. prickes , you may draw a circle w=ch= shal
enclose those iij. pricks in the edg of his circu~fere~ce ,
(RECORD-E1-P1,1,F1V.580)

& thus haue you attained y=e= use of this co~clusio~
(RECORD-E1-P1,1,F1V.581)

{COM:insert_helsinki_sample_2}

